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An application of a cardinality-constrained multiple benchmark tracking error model on a plant enterprise selection problem

An application of a cardinality-constrained multiple benchmark tracking error model on a plant... Abstract Yield and return of plants grown in a region are generally closely related. Agricultural scientists are less likely to recommend a single-plant enterprise for a region because of risk and return concerns. From a risk/return perspective, a plant enterprise selection problem can be considered as a portfolio optimisation problem. We use a multiple benchmark tracking error (MBTE) model to select an optimal plant enterprise combination under two goals. A cardinality constraint (CC) is used to efficiently balance multiple objectives and limit over-diversification in a region. We use Chinese national and province level datasets from multiple plant enterprises over 25 years to identify the best plant enterprise combination with two objectives under consideration: return maximisation and risk minimisation. A simulated case using discrete programming is applied in order to analyse a farmer’s choice of specific plant enterprise and the transaction cost during rotation. In the continuous problem, the MBTE model is found to be efficient in choosing plant enterprises with high returns and low risk. The inclusion of a CC in the MBTE model efficiently reduces the plant enterprise number and volatility while creating smaller tracking errors than the MBTE model alone in an out-of-sample test. In the discrete problem, a CC can be used to search for the optimal number of plant enterprises to obtain high returns and low risk. The study and methods used can be helpful in choosing an optimal enterprise combination with multiple objectives when there are over-diversification concerns. 1. Introduction Several factors impact plant enterprise selection and diversification in a farm. Farmers select plant enterprises based on historical continuation, supply chain requirements, environmental concerns and risk and return reasons (Barkley and Porter, 1996; Pingali, 1997; Di Falco and Perrings, 2005; Bezabih and Sarr, 2012; Chateil et al., 2013). Bowden et al. (2001) indicate that the same level of return can be achieved by growing several different plant enterprises with complementary strengths in different fields. Although higher return and lower risk may prevail if multiple plant enterprises are chosen, farmers may still choose only a few plant enterprises because of limitations on plant enterprises production/marketing knowledge, excessive capital requirements and government policies (Omamo, 1998; Mahy et al., 2015). Given these realities, we identify and demonstrate methods that can be used to select an optimal number of plant enterprises and meet multiple objectives. Considering two objectives in a plant enterprise selection problem with diversification restrictions, we use a convex cardinality-constrained multiple benchmark tracking error model (CCMBTE). Although the core feature of the model is similar to other portfolio selection models (Barkley and Porter, 1996; Beuhler, 2006; Barkley, Hawana Peterson and Shroyer, 2010; Xu et al., 2014), it differs from others as it tracks multiple objectives by choosing a minimum number of plant enterprises. Portfolio optimisation models are widely used in health care (Sendi et al., 2003; Bridges, 2004; Prattley et al., 2007), commercial fisheries (Perruso, Weldon and Larkin, 2005), international project selection (Han et al., 2004), water usage strategies (Gaydon et al., 2012), land protection (Mallory and Ando, 2014), energy planning (Awerbuch, 2006) and climate change adaptation (Marinoni, Adkins and Hajkowicz, 2011; Mitter, Heumesser and Schmid, 2015). Unlike these papers, we are interested in the innovative combinations of different methods for finding plant enterprise combinations that can provide maximum returns and minimum risk. To achieve multiple objectives at the same time, portfolio optimisation methods with multiple benchmarks tracking error functions are used with a minimum crop diversification restriction. Roll (1992) first proposed the tracking error portfolio investment method, which is developed to minimise the tracking errors given an expected excess return. However, the performance of this model is dependent on a single benchmark, which results in large risk. El-Hassan and Kofman (2003) claim that a benchmark portfolio selection is always invalid as the selected portfolio sometimes performs even worse than Markowitz’s mean–variance (M–V) model. This will unavoidably lead to a principle-agent problem between the tracking error-minimising brokers and the total-risk-minimising investors, given a fixed expected excess return. If we use such a framework in a plant enterprise selection problem, producers can meet only one goal (maximum return or minimum risk) but not both (maximum return and minimum risk). Therefore, a single benchmark tracking error method would be an unacceptable choice in a plant enterprise selection problem. Therefore, we believe that the tool that efficiently balances multiple objectives will be attractive to agricultural producers and agricultural policy makers. Wang (1999) has taken the lead in extending the single benchmark tracking error model to a multiple benchmarks version. The objective is set as minimising the weighted sum of different benchmark tracking error values. Different asset selection strategies with different interests, such as maximising the return or minimising the return variance and so on, can be incorporated into the objective function. However, the formulation of Wang’s model compared with Xu et al. (2014)’s framework appears not so compact, and it is also not clear how to choose weights between alternative objectives. Fund and information in the financial market are often subject to certain restrictions. Similarly, transaction cost is also a problem that cannot be ignored. Since too many plant enterprises in a farmer’s portfolio increase production cost, choosing a limited number of plant enterprises are desirable in many instances. To minimise the tracking error under a limited number of trading assets, a graduated non-convex (GNC) model was proposed by Coleman and Li (2014). Also, Brito and Vicente (2014) introduced a cardinality constraint (CC) into the M–V model. The former is an approximating optimisation method that is similar to the image reconstruction idea by Blake and Zisserman (1987). GNC, essentially, approximates the discontinuous counting function by a continuous differentiable function; the latter uses a derivative-free approach to solve a multi-objective optimisation problem. Compared to these two models, a convex CC model proposed by Pilanci, Ghaoui and Chandrasekaran (2012) not only guarantees to have local optima but also provides a more concise form that is easier to compute. Considering mathematical elegance and application efficiency, we use the model by Pilanci, Ghaoui and Chandrasekaran (2012) as a sparsity penalty term on the MBTE objective function. The contributions of our article can be summarised as follows: (i) this is the first article to use an MBTE model in a plant enterprise selection problem with multiple objective functions; (ii) the target MOTAD model widely used in risk-return optimisation in agriculture is adapted with some modifications as a portfolio benchmark in the MBTE framework; (iii) a convex cardinality-constrained method is incorporated into the MBTE model for limiting the plant enterprise number in the optimal solution; and (iv) the efficiency of the CCMBTE model is compared against an MBTE model. The remaining sections of the article are organised as follows. In the next section, we introduce the framework of the MBTE model and discuss benchmark selection criteria. We also provide details on three benchmark models: modified target MOTAD, M–V model and constant rebalance method (CRM). In Section 3, we provide details on the data used in this study which come from Chinese National Crop Production data. In Section 4, we present numerical simulation results from all three benchmark models. The MBTE model results are then compared to these three benchmark models. In Section 5, we provide motivations for using CCMBTE model and simulation results. In Section 6, we estimate benchmark models, MBTE and CCMBTE models using province level data. An extension case is proposed in Section 7 to simulate the result for a farm-level problem, along with some additional constraints that are added. We conclude the paper in Section 8. 2. Modelling and analysis In this section, we introduce MBTE and other benchmark models. 2.1. MBTE model The portfolio selection optimisation methods that minimise the tracking error volatility are referred to as benchmark tracking error models. Extended from the previous work, an MBTE model simultaneously tracks more than one portfolio benchmark (Xu et al., 2014). The underlying formulation can be shown as follows: minx∈χE[ZB(r̃)−r̃Tx]2 (1) in which ZB(r̃)≔maxj={1,…,m}r̃Tpj. (2) Here the random vector r̃ denotes the return of a set of plant enterprises; x denotes the investment decision vector concerning the input applications on the interested plant enterprises. Both vectors r̃ and x have the same length, for example n ⁠. Vector pj in equation (2) denotes the jth benchmark plant enterprise selection strategy; ZB(r̃) is the term equipped to trace the highest return strategy(ies) within the m portfolios. By expanding equation (1), we obtain the following formulation which will be used in the numerical experiment1 of this paper: minxT(Σ+μμT)x−2E(ZB(r̃)r̃T)x. (3) Xu et al. (2014) indicate that the MBTE model is very sensitive to the selection of the benchmark models, and its Shapley value2 is not as high as the broker’s expectation. Still the MBTE model can be applied to a plant enterprise selection problem, as it complies with the agricultural producers’ return maximisation and risk minimisation objectives. Similar to the approach used by Wang (1999), weight selection among different tracking errors is implemented in a quadratic form.3 2.2. Benchmark selections Before using the MBTE model, we have to decide on what to use as a benchmark portfolio. There are three competitive models for the purpose: target MOTAD, M–V and CRM. The target MOTAD model developed by Tauer (1983) is widely used in plant enterprise selection problems. With an average revenue maximisation objective, its attractiveness mainly stems from its risk aversion logic of controlling the negative deviation of revenue from the target level. The M–V model developed by Markowitz (1952) was initially used to select the efficient portfolio with a minimum covariance at a given expected return level. CRM is very easy to implement even without any optimisation work, as it entails an equal allocation of capital across targeted enterprises at each decision point. CRM is widely known to be effective in minimising volatility. Some researchers even consider it to be a better performing model than the M–V model (Kirby and Ostdiek, 2012; Xu et al., 2014). 2.2.1. The modified target MOTAD model The standard Target MOTAD model (Tauer, 1983) can be formulated as maxxμTx. (4) Subject to, akTx≤bk,k={1,2,…,m} (5) T≤crTx+yr,r={1,2,…,τ} (6) pTy=λ (7) in which x ⁠, cr and μ are all n-dimensional vectors. x is the investment decision, cr is the production cost margin and μ denotes the average revenue obtained from a plant enterprise. p and y are τ-dimensional vectors which, respectively, denote the probability and negative income deviation of each risk scenario, and λ is the desired risk level measured in negative deviation from average income from each plant enterprise. Note that the use of the coefficient λ here is different from Petsakos and Rozakis (2015) and Arata et al. (2017). Since λ denotes the desired (satisfactory) level of compliance in the above model to the targeted income, it can be considered as a proxy of risk aversion coefficient rather than a calibration variable.4 The objective function maximises the expected revenue given an optimal solution x⁎ since μ averages the revenue of each plant enterprise over the estimation window. The first constraint equation (5) is the resource constraint in which k denotes the number of the resource types, and ak ⁠, bk ⁠, respectively, denote the input coefficient vector and availability of the kth resource; inequality equation (6) denotes the revenue deviation of each risk scenario r from the target revenue, which is measured by y associated with a probability weight vector p ⁠. A risk scenario denoted by r is referred to as an investment period. The occurrence of each period is given an equal probability that pr∈p is assumed to be the weight. The weighted sum is then set equal to the risk aversion coefficient λ in constraint equation (7). Considering that the income and cost data used in this study are reported in a real number, we use the following steps to modify the original target MOTAD model to comply with the portfolio optimisation problem: (i) Use the output–input ratio (OIR) as follows to reset μ on cr: OIR=aftertaxincome+totalproductioncosttotalproductioncost=1+aftertaxincomeratio The after tax income ratio is calculated by aftertaxincomeratio=grossrevenue−productioncost−taxtotalproductioncost (ii) Specify the resource constraint using production cost averaged over the estimation window. Set k=1 ⁠, then we have the constraint modified as eTx≤b. (iii) Modify T into the target income that sums each plant enterprise’s income as it is averaged over the estimation window. The maximum income through historical performance is not suitable for target T ⁠. Since, if this were the case, the portfolio would intuitively concentrate on the return-maximising strategy as we always have, with T≥crTx ⁠. If there is a plant enterprise with a very high return, to derive the pure-stand strategy is rather straight forward. The highest return target is ideal but not practical due to many constraints. (iv) The scaled-value entries of x are divided by the total production cost modified in (ii), in order to extract the portfolio vector in proportions instead of real numbers. Although the original solutions have realistic interpretations, like the proportion of each plant enterprise in the total planting area, it is not very suitable for portfolio analysis. 2.2.2. M–V model Another benchmark portfolio optimiser is the classic M–V model (Markowitz, 1952) that minimises the covariance with respect to different constraints. The formulation is mineTx=112xTΣx−ϕμTx (8) where Σ is the covariance between return of two plant enterprises, ϕ is the risk aversion parameter and e is the unit vector. All other variables are defined previously. Unlike the modified target MOTAD model, its risk controlling mechanism relies on the minimisation of the covariance. Existing studies vary ϕ by using a programming framework, such as a calibration model (Petsakos and Rozakis, 2015). 2.2.3. Constant rebalance method The CRM method assigns equal weights to all the enterprises in each period. Thus, it is also called the 1/n strategy. The model does have an attractive risk control feature. Additionally, its return performance is not as inferior as some other classic strategies, such as the M–V model (De Miguel, Garlappi and Uppal, 2009). 3. Data We use Chinese data on plant enterprise OIRs, production costs and after-tax income from 1989 to 2013. The related information regarding wheat, maize, soybean, peanut, rape seed (abbreviated as RS), cotton, flue-cured tobacco (abbreviated as TB), sugarcane, sugar beet and apple is recorded in detail in the Compilation of Cost and Income Data of Agricultural Products in China. The mean and the standard deviation of the OIR, production cost and plant enterprise income derived from the 25-year estimation window are provided in Table 1a–c, respectively. We model the whole country as a production unit and consider the correlation between plant enterprises with respect to nation-wide external conditions. Table 1. OIR, production cost and after tax income of 10 plant enterprises from 1989 to 2013 (national level) (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 a1 hectare = 15 mu. Table 1. OIR, production cost and after tax income of 10 plant enterprises from 1989 to 2013 (national level) (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 a1 hectare = 15 mu. We consider the case of a representative farmer growing crops in both national and provincial cases. Considering that homogeneity of crop/enterprise choices in a country/province may be too strong, in the latter part of the paper, we will introduce an extended farm-level case to simulate the plant enterprise selection decision. As long as the plant enterprises can be switched easily, changes to the plant enterprise mix can be welfare enhancing to farmers when commodity prices are increasing. It is worth citing an example here. During the biofuel boom corn and soybean production area increased in the USA, but cotton and other crop area decreased significantly (Chen and Önal, 2012). During that period, producers were well off. There are many policies that may affect agriculture production. For instance, the decoupled support policies are popular in industrial countries. Bhaskar and Beghin (2009) mention that these policies are always funded by taxpayers and are not related to current production, factor use or policies, and that eligibility criteria are based on historical data. Weber and Key (2012) indicate that decoupled payments have little effect on aggregate production. Unlike the decoupled policy in the United States and the European Union, Chinese agricultural policies are usually paid directly based on the production area (direct pay). Some of these policies, such as the per acre subsidy payments, were found to have little influence on production decisions, although they accounted for 7–15 per cent of the farmer’s gross income (Gale, 2013). While current Chinese agricultural policies may or may not impact aggregate production, and while it is possible to incorporate some of those policies in the modelling framework, we have limited those to only a farm-level case presented in Section 7. For the study period shown in Table 1, apples have both a high return and high variation in return (high standard deviation) compared to the other nine plant enterprises. Wheat has the lowest mean return, though its production cost is not the lowest. Wheat in many places in China is treated as a staple crop rather than a purely economic one. Note that all enterprise returns are strictly positive, since (i) the data are annually averaged and (ii) a proportion of the revenue comes from the direct government payment to farmers. These government payments are paid to all crop growers so it should not impact plant enterprise choice significantly. Using the past 10 years of data samples as an estimation window to estimate the mean return and standard deviation, the results for each year from 1999 to 2013 are shown in Figure 1.5 We find that the return as well as standard deviation of most of the plant enterprises, except rape seed and apple, shows a decreasing trend. However, rape seed shows an increase in both return and standard deviation from 2006 onward. Apples show an increasing return but decreasing standard deviation since 2001. This indicates that most of the plant enterprises show higher returns with higher uncertainty, except for apples. Fig. 1. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 based on the national level data. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 1. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 based on the national level data. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 4. Benchmark numerical experiment results In this section, we first report the in- and out-of-sample test results obtained from the modified target MOTAD and M–V models. Then, we report the performance of the CRM method as a reference. Next, a comparison among the above benchmark models along with the MBTE method is carried out. All of the numerical experiments in this paper will follow an ‘in-sample’ to ‘out-of-sample’ order for testing the effectiveness of the models. An out-of-sample test refers to estimating the model using a portion of the dataset and using the model to forecast the known values of the series that is not in the estimation window. On the contrary, the model is used to predict the values that have been already used for estimating the model (De Miguel et al., 2009). The former is more like cross-validation, which is being widely used in the optimisation and machine learning fields (Efron and Gong, 1983; Campbell et al., 1997; Rohani, Taki and Abdollahpour, 2018), and it is generally believed that the out-of-sample test can provide a measure of protection against predictability (Lo and MacKinlay, 1990; Foster, Smith and Whaley, 1997; Rapach and Wohar, 2006). 4.1. Numerical experiment of in-sample case–modified target MOTAD and M–V models 4.1.1. The modified target MOTAD model In the in-sample test, we identify an optimal plant enterprise mix for various λ value using the entire 25-year estimation window. The result converges to a specific portfolio as we increase λ ⁠, say from 0 to 55 at a step length of 1. Corresponding selection combinations are shown in Figure 2a, which shows that the model always chooses the most high-average return enterprises.6 It complies with the return orientation of the modified target MOTAD model. The details of the input weight of each plant enterprise are reported in Table 2a, with the corresponding return and covariance averaged over the past 25 years. Based on the 56 solutions on the 56 λs, we take the mean and variance of the after-tax income and report them against the investment horizon in Figure 2b and c. Figure 2b denotes the return averaged on λ ⁠; the corresponding variance is in Figure 2c, which measures the risk controllability of a coefficient-averaged level, or, say, the solution’s sensitivity to different risk aversion degrees. To practically interpret the above results, we claim that by the modified target MOTAD method a producer can obtain a favourable income in the first and last several years, but s/he must handle a relatively large uncertainty. Fig. 2. View largeDownload slide (a)–(c) In-sample test on the modified target MOTAD model using the test sample of national level: (a) plant enterprise selection strategy, (b) pareto frontier, (c) return averaged on λ and (d)–(f) in-sample test on the M–V model using the test sample of national level: (d) plant enterprise selection strategy, (e) pareto frontier and (f) return averaged on λ ⁠. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 2. View largeDownload slide (a)–(c) In-sample test on the modified target MOTAD model using the test sample of national level: (a) plant enterprise selection strategy, (b) pareto frontier, (c) return averaged on λ and (d)–(f) in-sample test on the M–V model using the test sample of national level: (d) plant enterprise selection strategy, (e) pareto frontier and (f) return averaged on λ ⁠. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Table 2. In-sample test for the modified target MOTAD and M–V models using the national level data (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 Table 2. In-sample test for the modified target MOTAD and M–V models using the national level data (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 4.1.2. The M–V model The optimised results of the in-sample experiment of the M–V method on the 25-year estimation window are numerically shown in Table 2b and graphically depicted in Figure 2d–f. Specifically, the investment weight, average return and covariance are derived by increasing ϕ from 0 to 0.5 at a step length of 0.01. The corresponding plant enterprise selection strategies are depicted in Figure 2d against ϕ ⁠. The investment weights with different values of ϕ are found to converge from six-enterprise diversified strategies, including wheat, peanuts, rape seed, cotton, TB and sugar beet, to a single-plant enterprise selection: apples. More specifically, apples attract more investment as ϕ increases due to its high-return rate (1.9647), but wheat rapidly decreases with the increment of ϕ since this enterprise has the lowest estimated average return (1.2240) among all the plant enterprises; peanuts have the second highest return (1.6355) and a relatively low variance (0.0555); their production at first increases but then declines by giving more investment weight to apples; soybeans are not as popular as peanuts, although the return is ranked third due to its larger variance compared to peanuts. Note that since the M–V solutions are more diversified than those of the modified target MOTAD solutions, they lead to a smaller mean return and smaller covariance. Also, we find both models solve to high-income plant enterprises as λ gets larger. Both in the modified target MOTAD and M–V models, food staples such as soybeans and wheat are ignored when λ and ϕ values are high. An exception is rape seed, which is a cash crop but loses investment interest due to its low returns. The annual return and the annual variance both averaged on ϕ are shown in Figure 2e and f, respectively. Results show that the returns until 1995 stay at a high level but sharply decrease after that year. From 2001, returns increase but fall again after 2012. Accordingly, the variance follows a similar dynamic except it falls earlier than the return during 1993–1995, but again in 2004. After experiencing a relatively long downturn period from 1995 to 2003, it rises again. We find (i) the optimal plant enterprise selection strategies drawn in Figure 2 under either the modified target MOTAD or M–V models draw some similar dynamics between the return and the variance. For practical interpretation, the agricultural producers may face similar income and yield risk irrespective of the two models chosen and (ii) the M–V model performs worse in controlling the return volatility than the modified target MOTAD model. The t-test on the variances indicates that the M–V model is more sensitive to the risk coefficient in statistical meaning. The same problem is observed by Xu et al. (2014), when applying the M–V model to a financial portfolio optimisation problem. 4.2. Numerical experiment of out-of-sample tests–modified target MOTAD and M–V models In the out-of-sample test, we use a 1-year rolling process in which the immediate past 10 years are used to estimate the sample mean, variance and covariance of the plant enterprise return, and the 11th data sample is used to evaluate the performance. The whole process moves 1-year forward, updating the estimates of the sample mean, variance and covariance each time. For instance, the dataset for 1999 is verified using the portfolio derived from the estimation window of 1989–1998; the data sample from 2000 is verified using information from 1990 to 1999, and so forth. Taking the modified target MOTAD as an example to identify all possible solution portfolios, we increase λ from 0 to 400 with a step length of 1 and report the result through the planning horizon, i.e. from 1999 to 2013. For each rolling cycle, there is a feasible region for λ ⁠, which is reported in Table 3a. The lower bound (e.g. 119 in 1999) denotes the first appearance of the modified target MOTAD solution, while the upper bound indicates a threshold beyond which the solution converges to one plant enterprise. In the simulation process, we find that as we move the estimation window forward, the feasible value of λ decreases in general. This is because, as the expected return gets closer to the target value with each estimation window, a producer may face smaller return deviation from such a target; thus, the risk tolerance level could be reduced. Table 3. One-year rolling out-of-sample test for the modified target MOTAD and M–V models using the national level data (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 Table 3. One-year rolling out-of-sample test for the modified target MOTAD and M–V models using the national level data (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 For the optimised plant enterprise selection strategies for the modified target MOTAD, we show the patterns in Figure 3a by averaging the input weight of each plant enterprise on λ and report the results from 1999 to 2013. Results show that selection mostly concentrates on three plant enterprises, viz., soybeans, peanuts and apples. As λ increases, only apples are selected. Although soybeans are a staple food in Northeast China, apples outweigh soybeans in profitability. Figure 3b shows the variance of return averaged on year against λ ⁠, which shows the periodical variability of the return of the modified target MOTAD model. Figure 3c and d reports the annual return and the annual variance both averaged on λ ⁠. The results show that both annual return and the annual variance increase sharply in the last several years. This is mainly due to the increased income of the plant enterprise. These values are obtained by using the following process: first we fix the lower bound of the feasible λ (from the modified target MOTAD model) subject to the largest lower bound of a specific solution among them all, while fixing the upper bound using the smallest upper bound from another specific solution set (it is possible that these two solution sets could be the same but it will not affect our analyses); then we report the annual variances of the return averaged against the above range of λ ⁠. For instance, the solution set from the estimation window of 1990 to 1999 is feasible in λ in [150, 298], of which the lower bound of 150 is the largest among all the other feasible λ ranges of all the solution sets. Meanwhile, the solution for the estimation window between 1991 and 2000 is feasible in λ [122,313], of which the upper bound of 313 is the largest among all the upper bounds. Therefore, we present the return variance averaged on years against the λ range of [150, 313] in the x-axis. For the solutions which have already converged with λ<313 ⁠, say λ=200 ⁠, we continue to use the converged solutions in λ [200, 313]. This process helps to find coefficient patterns from the benchmark models. Fig. 3. View largeDownload slide One-year rolling out-of-sample test on the modified target MOTAD and M–V models over 10-year estimation window based on the national level data. (a)–(d) Modified target MOTAD model: (a) plant enterprise selection strategy, (b) volatility measured by variance averaged on year, (c) return averaged on λs and (d) variance averaged on λs. (e)–(h) M–V Model: (e) plant enterprise selection strategy, (f) volatility of the M–V model, (g) return averaged on ϕs and (h) variance averaged on ϕs. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 3. View largeDownload slide One-year rolling out-of-sample test on the modified target MOTAD and M–V models over 10-year estimation window based on the national level data. (a)–(d) Modified target MOTAD model: (a) plant enterprise selection strategy, (b) volatility measured by variance averaged on year, (c) return averaged on λs and (d) variance averaged on λs. (e)–(h) M–V Model: (e) plant enterprise selection strategy, (f) volatility of the M–V model, (g) return averaged on ϕs and (h) variance averaged on ϕs. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Each solution set of the M–V model during the rolling out-of-sample experiment is derived from a particular feasible region of ϕ ⁠. Unlike the modified target MOTAD model, the existence of the solutions is always guaranteed. The annual return and annual variance both averaged on ϕ are reported in Table 3b, by choosing 0.5 as the upper bound of ϕ and a step length of 0.02. The annual plant enterprise selection strategies averaged on ϕ are shown in Figure 3e, where soybeans attract more investment than other plant enterprises until 2006. From 2006 onward, apples take priority within the portfolio after experiencing a down turning trend. The third high-performing plant enterprise is peanuts, which as a typical economic and oil crop has always obtained a certain amount of investment except in 2008, 2009 and 2010. The annual average variance of return against ϕ is shown in Figure 3f. Based on return/risk in these two models (M–V and modified target MOTAD model), we cannot clearly tell the dominance of one model compared to the other. The annual return and the annual variance of both averaged on ϕ are exhibited in Figure 3g and h. Comparing them with the counter-part result of the M–V model, as depicted in Figure 4a and b, both show an upward trend. Fig. 4. View largeDownload slide Comparison between the modified target MOTAD and M–V models in the out-of-sample test using the test sample of national level: (a) return and (b) variance. Fig. 4. View largeDownload slide Comparison between the modified target MOTAD and M–V models in the out-of-sample test using the test sample of national level: (a) return and (b) variance. 4.3. Numerical experiment–CRM We report results from this method in Figure 5. The mean return and the variance are 1.4825 and 0.1660, respectively, which are significantly smaller (on t-test) than the result of either the modified target MOTAD or M–V models. This is attributed to the high diversity of the CRM’s plant enterprise selection strategies. Fig. 5. View largeDownload slide Annual performances of CRM from 1989 to 2013 of national level: (a) return and (b) variance. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 5. View largeDownload slide Annual performances of CRM from 1989 to 2013 of national level: (a) return and (b) variance. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 4.4. Comparison of MBTE, modified target MOTAD, M–V and CRM We use modified target MOTAD, M–V and CRM on the MBTE model as the benchmark plant enterprise selection methods. To simulate the result, we gradually change the value of ϕ based on a continuum of λ. This is essentially an ergodic process for searching all the possible solutions through the risk coefficient combinations of the modified target MOTAD and M–V models. Here, we have P×L MBTE solutions where L is the number of λ ⁠, which is also the number of the modified target MOTAD solutions, while P is the number of ϕ ⁠, which is also the number of the M–V solutions. Using these portfolios along with the CRM portfolios, we seek the MBTE solutions and compare them with the performances of all the benchmark portfolios. 4.4.1. Numerical experiments First, we provide results from a 25-year estimation window. This is then followed by the results from the 1-year rolling out-of-sample case with a 10-year estimation window. In-sample forecasting results for different geographic regions are presented in Figure 6a–d. Figure 6a and b indicates that the MBTE return is as good as the return from a modified target MOTAD model but is less sensitive to the risk coefficient than the M–V model. More specifically, in Figure 6a, the MBTE return denoted by the red line has an upper bound very close to that of the M–V model. The MBTE’s lower bound shows a return similar to the modified target MOTAD return. The MBTE method is always tracking the return of a good performing plant enterprise mix but never touches the lower values derived from either the M–V (green dot) or the CRM methods (black dash). As ϕ varies, we find the upper bound of the MBTE variance is always smaller than that of the M–V model. The lower bound is larger than the M–V method since the MBTE model on these data points is meant to track the higher return, either from the modified target MOTAD or M–V model, by sacrificing some risk control. Figure 6c and d shows that the MBTE covariance always performs better than that of either the modified target MOTAD or the M–V method. As we can see from Figure 6c, when the MBTE result does not track other models, its value is smaller than that of the modified target MOTAD model. As the MBTE covariance tracks the M–V result periodically, the covariance of the modified target MOTAD method is almost covered by the red curve. Since M–V covariance has a relatively high value here, it can be only observed in Figure 6d as we zoom out the area. We do not elaborate on the results from the CRM model as its return is clearly inferior to other models. Fig. 6. View largeDownload slide In-sample experiment of the MBTE model equipped with the benchmarks of the modified target MOTAD, M–V and CRM model, using the test sample of national level. (a)–(d) Comparison between the modified target MOTAD, M–V and CRM models; (e) optimal crop numbers in the MBTE, modified target MOTAD and M–V models. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 6. View largeDownload slide In-sample experiment of the MBTE model equipped with the benchmarks of the modified target MOTAD, M–V and CRM model, using the test sample of national level. (a)–(d) Comparison between the modified target MOTAD, M–V and CRM models; (e) optimal crop numbers in the MBTE, modified target MOTAD and M–V models. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. For the number of the plant enterprise selections (see Figure 6e), we find that if the MBTE model (denoted by the red curve) tracks the modified target MOTAD model with low-risk measures within λ[0,15], it will find more diversified strategies than that of the M–V model. Specifically, for the optimal plant enterprise choice without any restrictions, the solutions always include soybeans, peanuts, cotton, sugarcane, sugar beets and apples. When the number of plant enterprise choices in the optimal solution is reduced to five, the optimal plant enterprise mix includes soybeans, peanuts, cotton, sugarcane, sugar beets and apples. The optimal plant enterprise choice with four enterprise selection restrictions includes soybeans, peanuts, sugarcane and apples; the optimal solution with three enterprise restrictions includes soybeans, peanuts and apples and the optimal solution with two enterprise restriction include soybeans and apples. Table 4, collectively, reports the above results. This in-sample result indicates that using the MBTE model has more practical contribution to the peoples’ livelihood issues since it also concerns itself with growing staples instead of only tracking the economic enterprises. Table 4. MBTE portfolios with different CCs (national level) Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple Table 4. MBTE portfolios with different CCs (national level) Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple To streamline the analysis and convey practical significance, we only focus on the 1-year rolling method in the out-of-sample test. Since the test sample size duration is 15 years, i.e. 1999–2013, the computation scale is correspondingly expanded to ∑t=1τλt×ϕt ⁠, where τ is the rolling length. For instance, we use a 10-year estimation window from 1989 to 1998 to find the MBTE solutions tested on the data sample of 1999. This entails running 167×25=4,175 optimisation models for each time period. Based on Table 3 for 1999, we have λ=167 and ϕ=125 ⁠. Next, we test samples using the estimation window from 1990 to 1999; the number of optimisation models needed is 150×25=3,750 for t=2 ⁠, and so on. Since we have fixed the available range of ϕ ⁠, its subscript t can be removed. As an example, we report the result for the year 1999 in Figure 7a–d. The comparison of returns and variances among the modified target MOTAD, M–V and CRM is shown in Figure 7a. Figure 7b highlights the difference between the MBTE and modified target MOTAD models. The result indicates that although the MBTE model (red curve) uses the modified target MOTAD (blue curve) and M–V (green curve) models as a benchmark, it depends more on the modified target MOTAD model by considering its high-return level. The comparison of variances among the four models is shown in Figure 7c, from which we find that the MBTE model generally has a lower variance than the modified target MOTAD variance (red vs. blue curves). Although here the CRM variance is too low to be observed, its return and some other performances are not comparable to the other models. When the modified target MOTAD model is solved using a large λ (=156), the MBTE will no longer closely follow it but chooses to balance the risk by gradually approaching the M–V solutions. For the solutions through 1999–2013, we compare the volatilities of the MBTE, modified target MOTAD and M–V models averaged on a specific range of feasible λ ⁠, and report the result in Figure 7d. It shows that the MBTE model clearly performs better than the other two models based on the risk and return criteria. Fig. 7. View largeDownload slide Comparison of return, variance and volatility on the MBTE, modified target MOTAD, M–V as well as CRM models in 1-year rolling out-of-sample experiment. (a) return, (b) return and (c) variance: using the test sample (year = 1999, national level); (d) volatility: using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 7. View largeDownload slide Comparison of return, variance and volatility on the MBTE, modified target MOTAD, M–V as well as CRM models in 1-year rolling out-of-sample experiment. (a) return, (b) return and (c) variance: using the test sample (year = 1999, national level); (d) volatility: using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. The tracking error averaged on (λ,ϕ) pair is reported in Figure 8a, which shows that the value becomes larger in the last several years. In the asset investment area, tracking error is defined as the deviation of the portfolio return from the benchmark return (Roll, 1992; Jorion, 2003). This value is widely used to measure the performance of stock brokers. However, unlike stock brokers, agricultural producers should be more concerned about plant enterprise return than the tracking error. Results indicate that it would be increasingly difficult for the decision-maker to ‘track a desired economic target’ over time. Figure 8b reports the plant enterprise number based on average, maximum, and minimum levels of (λ,ϕ) over the 15-year test data samples. The result shows that although the average MBTE plant enterprise numbers are always lower than five, the maximum level could be much higher, e.g. as high as 10. It is clear that the MBTE model provides for fairly diversified strategies that may result in higher transaction cost. This motivates us to introduce a CCMBTE model which restricts over-diversification strategies. This method, as we show later, performs well for sparsity control in continuous programming. Although this advantage does not appear in the discrete environment of our extended cases in Section 7, it still provides a better solution for return and risk control compared to the other models by adjusting the cardinality of the decision vector. Fig. 8. View largeDownload slide Tracking error and plant enterprise number in the MBTE model (a) tracking error of the MBTE model averaged on λ−ϕ pair in 1-year rolling out-of-sample experiment and (b) plant enterprise number of λ−ϕ pair average, maximum and minimum levels (national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 8. View largeDownload slide Tracking error and plant enterprise number in the MBTE model (a) tracking error of the MBTE model averaged on λ−ϕ pair in 1-year rolling out-of-sample experiment and (b) plant enterprise number of λ−ϕ pair average, maximum and minimum levels (national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 5. Motivation for using the CCMBTE model Comparing the out-of-sample results of return and variance from the M–V and modified target MOTAD model, we found in Figure 4 that neither of them dominates the other all the time (figure legend MOTAD). The same problem exists in the comparison of the volatilities of both models, although the average volatility level of the M–V model is better than that of the modified target MOTAD model. The MBTE model is useful here as it is always able to track the good features of both of these models. The development of the CCMBTE model, however, should be more useful since it not only tracks the MBTE’s desirable features but also considers controlling too much enterprise diversity. Moreover, it is easy to implement in practice. We adjust the cardinality coefficient under an already constructed MBTE framework. Rădulescu, Rădulescu and Zbăganu (2014) mention that farmers have to decide ‘how much diversification is enough’ to capture the most potential gains from expanding their enterprise mix. We can rephrase the plant enterprise selection problem as ‘How many plant enterprises should a farmer choose to approach the best performing diversified strategies?’ The potential users of such models include the policy makers of a country or a province, the agri-business investors, producers of an agricultural production base or even individual farmers. To mitigate the disadvantages of limiting plant enterprises within a single economic area, different plant enterprises can be geographically spread out in practice, which is called location diversification. For instance, soybeans are the major staple crop of the northern part of China, whereas wheat is a staple crop in the southern part. Yang et al. (2014,,2015) use data from a relatively large area, for example, northern China, and Werners et al. (2007) use data from the Guadiana River Basin, Portugal. In these studies, spatially/geographically diversified crop selection strategies are assumed to be available for the underlying regions. Another implicit assumption is that through policies the government could successfully guide the farmers in the targeted regions to have the most profitable and risk reducing combination of plant enterprises. In Section 7, the CCMBTE approach is applied in a discretized situation in order to address a farm-level problem. We also present its advantage compared to the MBTE and benchmark models. 5.1. Cardinality constraint To adjust the plant enterprise number to reduce over diversification, a CC developed by Pilanci, Ghaoui and Chandrasekaran (2012) is introduced as follows: mineTx=1,x≥0f(x)+θcard(x) (9) where card(x) denotes the number of non-negative entries of vector x ⁠, i.e. the invested plant enterprise number; θ is a given trade-off parameter for adjusting the desired cardinality level. The above formulation runs into difficulty in finding a sparse solution since ‖x‖1=∑i∈N|xi|≤=1 is a constant wherein we cannot fix the number zero. To address this concern, Pilanci et al. use the l∞ norm as the maximum relaxation, as shown in equation (10). This helps us to find a lower bound for the original problem ‖x‖1=∑i∈N|xi|≤card(x)maxi|xi|≤card(x)‖x‖∞. (10) If we denote the original problem as P⁎ ⁠, then we have P⁎≥P∞⁎=mineTx=1{f(x)+θ1maxixi} (11) which according to Proposition 2.1 of the original work by Pilanci et al. can be reformulated as a convex optimisation as follows: (PSP⁎)=mini{mineTx=1,x≥0f(x)+z:z≥θxi}. (12) When solving PSP⁎ ⁠, we first solve the sub-problem inside the curly brackets given the plant enterprise indexed i ⁠. The inner problem will generate a solution that minimises the portfolio objective f(x) summed by z ⁠, an extra variable for limiting the cardinality. z reaches the minimum value if the condition z≥θxi is active. Thus, we should test each xi,i={1,2,…,m} to find the minimum objective of the inner problem. Since m denotes the length of the portfolio vector, i.e. the number of plant enterprises, it also denotes the computation times for the inner sub-problem. The outside programme will then extract a minimum value among the m values, which will compare the inner objective value m times. If we incorporate the above described conditions in equation (12), the following objective function is obtained: mini{mineTx=1,x≥0E[ZB(r̃)−r̃Tx]2+z:z≥θxi}. (13) We call this objective function, along with the constraints presented earlier, the CCMBTE model. Since we are solving a two-layer optimisation problem in equation (13), the computation time should again be increased to N #×∑t=1τλtϕt ⁠, where N # denotes the size of the candidate plant enterprises. 5.2. Numerical experiment–CCMBTE model We report the result in Figure 9 by using the 2008 dataset. Arbitrarily choosing θ at three values, say, 0.15, 0.1 and 0.05 (green, red and yellow), we find the CCMBTE return is always close to or even higher than the modified target MOTAD and MBTE returns (see Figure 9a). Correspondingly, its variance reported in Figure 9b increases but moves closer to the MBTE solutions when we decrease the cardinality coefficient. Fig. 9. View largeDownload slide Comparison of (a) returns, (b) variances, (c) plant enterprise numbers and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 2008, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 9. View largeDownload slide Comparison of (a) returns, (b) variances, (c) plant enterprise numbers and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 2008, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. As shown in Figure 9c, we find the CCMBTE chooses a smaller number of plant enterprises than the MBTE model. It is also interesting to note that the CCMBTE tracking error (depicted in Figure 9d) is always lower than that generated by the MBTE model. This is because the data sample always provides a higher return from a set of small plant enterprise selection strategies, and the CCMBTE model at the same time is motivated to solve the problem by just tracking these plant enterprises. If the plant enterprise number is increased, the tracking error will accordingly increase. It is quite intuitive, since different models use different selection strategies and these strategies derived from different models, that these should be less likely to be equal to each other. Even so, this increasing number will always be lower than that of the MBTE method. Simulation with a larger sample size (as in the example from 1999) gives the average CCMBTE plant enterprise number (in Figure 10a) for an arbitrarily chosen coefficient, say θ=0.1 ⁠, significantly smaller than that of the MBTE model (with the F-value of 407.63 and a p-value of 0.000). Also, we find the return, variance and tracking error of the CCMBTE model (in Figure 10b–d, respectively) are also close to those of the MBTE model, in the region of λ around 19.5. For the specific selection strategies of the CCMBTE model, not only do the plant enterprise numbers reduce but also the usually invested number of plant enterprises is different from the other models. For instance, in a two-enterprise pattern, wheat becomes more popular than soybeans, peanuts, or any other eight plant enterprises. It is interesting that the application of the CCMBTE model can keep the food staples in a sparser solution environment instead of diversifying the strategies to more high-return plant enterprises. Fig. 10. View largeDownload slide Comparison of (a) plant enterprise numbers, (b) returns, (c) variances and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 10. View largeDownload slide Comparison of (a) plant enterprise numbers, (b) returns, (c) variances and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. To compare the averaged volatilities of the MBTE and CCMBTE models through the 15-year test sample, we follow the same process as that of the MBTE model. The feasible λ is still from 150 to 313. The result is reported on the left panel of Figure 11, which shows that the CCMBTE model under relatively small λ’s (e.g. less than about 160) performs better than the M–V model within the upper bound of the volatility. Besides, the right subfigure also shows that the CCMBTE has successfully controlled the plant enterprise number. As we select the threshold of sparsity measure of between 0.0001 and 0.00005, the result shows that the CCMBTE plant enterprise number is smaller than that of the MBTE model except in the last four years. As we relax the requirement on the solution precision by setting the threshold as 0.001, the CCMBTE plant enterprise number is never larger than that of the MBTE model. Relatively, the plant enterprise number of the MBTE is less sensitive to the sparsity coefficient. Fig. 11. View largeDownload slide Comparison of volatilities and plant enterprise number of average level of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 11. View largeDownload slide Comparison of volatilities and plant enterprise number of average level of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 6. Extension–province level We believe that a differently scaled dataset can be used to test model robustness. Accordingly, we apply the optimisation models to a smaller scale dataset from Hebei province, China. This dataset was also collected from the Compilation of Cost and Income Data of Agricultural Products in China. There are only six candidate plant enterprises (soybeans, maize, peanuts, japonica rice (JR), cotton and apples) included in this case. The related descriptive statistics are listed in Table 5, which shows that apples still have the highest average return over the 25-year estimation window. Table 5. Mean and standard deviation of six plant enterprises in Hebei province from 1989 to 2013 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 Table 5. Mean and standard deviation of six plant enterprises in Hebei province from 1989 to 2013 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 6.1. The modified target MOTAD model Figure 12 shows the return and standard deviation over a 10-year rolling estimation window from 1999 to 2013. This figure shows a generally downward trend for all the plant enterprises in both return and standard deviation. Fig. 12. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 12. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Table 6 lists the portfolios derived at 15 estimation windows in the 1-year rolling experiments from 1999 to 2013. We find that portfolios are usually more diversified than in the national case, i.e. four to five plant enterprises are always selected among six potential plant enterprises. We find that compared with the national case in which a single-plant enterprise such as apples always generates the highest return, the provincial case is different, as there are multiple enterprises with similar types of returns. The solution estimated over 1989–1998 contains the most diversified strategy, with all enterprises except soybeans in the solution. Only the solution of 2013 generates a single-plant enterprise strategy where only maize is selected. This is contrary to other optimal solutions in which both food staples and cash crops were in the solution. Table 6. Modified target MOTAD solutions of the out-of-sample rolling test (province level) 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N Table 6. Modified target MOTAD solutions of the out-of-sample rolling test (province level) 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N In Figure 13a and b, the red curve indicates that the producers may face a larger risk when desiring a higher return, especially during the last several years. The return and the variance both averaged by years are listed in Figure 13c and d. The former shows an increasing return with λ ⁠, while the latter exhibits a consistent trend. Fig. 13. View largeDownload slide Return and variance of the modified target MOTAD model averaged on λ (shown in (a) and (b)) and year (shown in (c) and (d)) in 1-year rolling out-of-sample experiment (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 13. View largeDownload slide Return and variance of the modified target MOTAD model averaged on λ (shown in (a) and (b)) and year (shown in (c) and (d)) in 1-year rolling out-of-sample experiment (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 6.2. M–V model Figure 14 reports the performance of the M–V model using the provincial dataset. The mean values of return and variance averaged either on ϕ or year are more consistent with each other than when both of them are derived from the modified target MOTAD model (see Figure 14a and b). The variance of M–V return averaged on year (see Figure 14d) is generally larger than that of the modified target MOTAD model, although the former performed relatively well in the in-sample case (see Figure 14f). For plant enterprise selection strategies as when ϕ=0 ⁠, it includes all the six plant enterprises. On the other hand, when ϕ exceeds 2, the solution converges to two plant enterprises, i.e. maize and peanuts. Compared to the nation-wide case, the M–V’s plant enterprise selection strategy is to select only a few enterprises rather than large diversification. Fig. 14. View largeDownload slide (a), (b), (c) and (d) Return and variance of the M–V model averaged on λ and year; (e) and (f) comparison of returns and variances of the M–V and modified target MOTAD models averaged on ϕ and λ (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 14. View largeDownload slide (a), (b), (c) and (d) Return and variance of the M–V model averaged on λ and year; (e) and (f) comparison of returns and variances of the M–V and modified target MOTAD models averaged on ϕ and λ (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 6.3. Comparison among M–V, modified target MOTAD and MBTE models We run the model by choosing a specific range of risk coefficients (⁠ ϕ=[0,4]) with a step length of 0.02 for the M–V benchmark model. First, we use the sample year 1999 as an example and compare the results between the modified target MOTAD and M–V models (see Figure 15a and b). The result shows that the MBTE returns always follow the M–V’s target as ϕ increases, but the variance reaches a high value at the M–V benchmark. In the independent t-test for the MBTE and M–V return, the M–V return is significantly higher than the returns obtained from the MBTE model. The variance of the M–V model is significantly larger than that of the MBTE method. Fig. 15. View largeDownload slide Return and variance of the MBTE, M–V and modified target MOTAD models in 1-year rolling out-of-sample experiment: (a) and (b)–using test sample (year = 1999); (c) and (d)–using the test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 15. View largeDownload slide Return and variance of the MBTE, M–V and modified target MOTAD models in 1-year rolling out-of-sample experiment: (a) and (b)–using test sample (year = 1999); (c) and (d)–using the test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Figure 15c and d shows the annually averaged performances of return and variance of three models (MBTE, modified target MOTAD and M–V) when λ ≥ 485 for the years 1999–2013. Figure 15c shows that the MBTE return is lower than the modified target MOTAD and M–V returns. We find the MBTE variance still performs very well compared with those of the other two models (see Figure 15d). 6.4. MBTE and CCMBTE models Within the same data sample (year 1999), Figure 16 compares return, variance, tracking error and plant enterprise selection (in Figure 6a–d, respectively) between MBTE and CCMBTE models. The results show that the CCMBTE model with an arbitrarily chosen cardinality coefficient (θ = 0.005) gives a similar return level, variance, and tracking error as the MBTE method. For the plant enterprise number, the CCMBTE solution is significantly smaller than the MBTE strategies, say $3.1439<3.2813 under 0.00 statistical significance level. For the specific patterns, the CCMBTE model always provides maize, peanuts, apples and sometimes includes cotton in the optimal solution. Both food staples and economic crops are included in the optimal solution. Fig. 16. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 16. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Based on the dataset between 1999 and 2013, Figure 17 shows that (a) the CCMBTE return is always larger than that of the MBTE model; (b) the CCMBTE variance averaged on year is not as low as in the MBTE model, but variance is lower than in the M–V model; (c) CCMBTE tracking error is mostly smaller than that of the MBTE model; and (d) the coefficient-averaged plant enterprise number of the CCMBTE is almost always smaller than that of the MBTE model and shows greater robustness along the investment horizon. Fig. 17. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE the CCMBTE models test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 17. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE the CCMBTE models test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. In summary, the MBTE model in the country- and province-wide data sample is able to efficiently track high return with low risk. The CCMBTE model, however, is able to effectively track the MBTE model given the over diversification control power. 7. Simulated case for a farm-level In this section, we extend the application of our models to a farm-level. We follow these steps: (i) simulate based on the statistical information of the provincial/national data; (ii) transform the continuous optimisation models into discrete problems; and (iii) introduce the policy impact into the simulation. For the first step, we use a scaled down value of standard deviation but use the same mean return as at the national or state level. We use a scaled down variance but the same mean return because cropping conditions in a smaller area are more homogenous than those in a larger area. In the second step, we transform the continuous decision variable x∈{0,1}n into a discrete (binary) term x∈{0,1} ⁠. Discretisation assumes farmers can plant only one crop per land parcel in a given year. We believe a discretised variable is more appropriate for describing a farmer’s enterprise selection decision than a fractional variable. Transformation of a continuous problem to a discrete problem also makes it easier to analyse rotations. The corresponding recast of the original models to a discrete space is listed below: Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Let 0<v≤n,v∈Z+ be the constraint on the number of plant enterprise choices. Assuming n (number of plant enterprises under consideration) = 6 in the farm-level scenario, a farmer cannot choose more than n crops to plant on these v land parcels. The combination of constraints can be rewritten as eTx=v,x∈{0,1} ⁠. Then, the modified target MOTAD is transformed as an integer problem and solved using integer programming. Similarly, the M–V and MBTE models can be reformulated as a mixed integer quadratic model and solved using a mixed integer quadratic programming. We re-cast zxi≥θ,xi∈[0,1] from the original CCMBTE model to a mixed integer quadratic constraint. Each of these transformed problems can be efficiently solved using the CPLEX solver. 7.1. Simulation of farm-level data sample We introduce six plant enterprises, including wheat, soybeans, maize, peanuts, japonica rice and cotton (a total of six plant enterprises), in our simulated farm-level case. These are the plant enterprises found in Northern China. We keep the mean return vector [1.4757, 1.2079, 2.0387, 1.7724, 1.7364, 1.3732] of the six plant enterprises from the original dataset that come from both national and provincial cases, but scale down the standard deviation vector [0.2875, 0.3607, 0.6559, 0.5400, 0.5390, 0.3718] arbitrarily by 0.15 for each entry to regenerate a set of data samples using a multivariate normal distribution. We chose one of these vectors with the lowest averaged sum of all the entries in the covariance matrix.7 We show the results in Table 7. Table 7. Covariance of simulated data sample for six plant enterprises Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 Table 7. Covariance of simulated data sample for six plant enterprises Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 7.2. Experiment on the farm-level case We arbitrarily set the maximum plant enterprise number at v=3 and derive the results from the modified target MOTAD and M–V models, and then compare the results to those derived from the MBTE and CCMBTE models. Detailed results from the benchmark models can be informative when viewing the MBTE/CCMBTE results. All the results, including the experiments in discrete choice cases, are derived from an out-of-the sample forecasting. 7.3. Comparison of the modified target MOTAD and M–V models in the farm-level case We compare the results of both the modified target MOTAD and M–V models using the feasible sets of risk coefficients. Generally speaking, the solution number in the discrete space is smaller than the continuous space as many different risk coefficients will lead to the same discretised solution set. We report risk and return values obtained from the modified target MOTAD and M–V models averaged by the feasible risk coefficient for each year in Figure 18a–d. We find that both the return and the variance of the modified target MOTAD model are always higher than those obtained from the M–V model. For the plant enterprise selection strategies, the modified target MOTAD model shows that its convergent solutions choose soybeans, maize and peanuts regardless of the estimation windows. The model always tracks these three highest return generating plant enterprise combinations irrespective of the estimation windows. In contrast, the convergent M–V selection strategies are more sensitive to the estimation window. For example, the solution combination includes maize, peanuts, and japonica rice under the estimation windows from 1989–1998 to 2001–2010, but solves to wheat, maize and japonica rice under the years of 2002–2011, and to maize, japonica rice and cotton under 2003–2012. To find the shadow price within the application of the models, we relax the plant enterprise number constraint to an inequality form, i.e. eTx≤v ⁠. None of the plant enterprises gets selected when we use a small ϕ in the M–V model. The result implies that under such circumstance a farmer prefers to avoid the risk rather than attempt to obtain some return. On the contrary, the modified target MOTAD model always uses v land parcels no matter whether the plant enterprise number constraint is relaxed or not. Since the modified target MOTAD model’s objective is to maximise return subject to some randomly selected λ ⁠, this result is understandable. Fig. 18. View largeDownload slide Out-of-sample test for the M–V and modified target MOTAD model in the farm-level case: (a) and (b) return and variance of the modified target MOTAD model averaged on λ ⁠; (c) and (d) return and variance of the M–V model. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 18. View largeDownload slide Out-of-sample test for the M–V and modified target MOTAD model in the farm-level case: (a) and (b) return and variance of the modified target MOTAD model averaged on λ ⁠; (c) and (d) return and variance of the M–V model. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 7.4. Comparison of the MBTE and CCMBTE models in the farm-level case In this section, we compare the performance of the MBTE and CCMBTE models against two benchmark models. We do not consider the CRM method further as it generates nothing particularly of interest for either return or variance. The optimisation result of the CCMBTE model provides similar returns, variance and crop enterprise choice as the MBTE within a range of CC, namely when θ=0.01~1,000 ⁠. Table 8 shows the MBTE model results, in which we consider λ=1,2,3 ⁠, ϕ={11,12,…,16} ⁠, and restrict eTx≤3 ⁠. It shows that both the CCMBTE and MBTE models choose maize, peanuts and cotton as an optimal crop combination. These results reflect the advantage of using both the CCMBTE and MBTE models, since the return is (i) larger than the M–V solution and close to the modified target MOTAD return, while the variance is smaller than the modified target MOTAD variance or (ii) no worse than any of the benchmark results. It is also possible to obtain a higher return and smaller variance simultaneously by slightly adjusting the plant enterprise numbers in the CCMBTE result. Table 9 shows one example with λ=1 and ϕ=2 ⁠, in which we find the CCMBTE results provide four plant enterprises under an estimation window subset including 1989–1998, 1990–1999, 1998–2007, 1999–2008 and 2000–2009, 2001–2010. Additionally, results from the CCMBTE are always better than in the MBTE in both return and risk. The results from these two models within the other estimation windows are similar as their selection strategies are always identical. In summary, irrespective of estimation window chosen, the CCMBTE model, with just one more plant enterprise, performs no worse than the MBTE model in terms of return and variance. On the other hand, the MBTE model with plant enterprise selection constraint always undertakes a lower return with higher risk when selecting one less plant enterprise than the CCMBTE model does. The difference in the return and risk can be seen as the shadow price. One disadvantage of considering many plant enterprises is the increase in the transaction cost as more resources, including arable land, fertilisers and production technologies, are required. Table 8. Comparison between the MBTE and CCMBTE models, θ=0.01 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 Table 8. Comparison between the MBTE and CCMBTE models, θ=0.01 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 Table 9. Comparison between the MBTE and CCMBTE models, θ=100,000 CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C Note: W, S, M, P, J and C represent wheat, soybean, maize, peanuts, japonica rice and cotton, respectively. Bolded values indicate CCMBTE has advantage over MBTE. Table 9. Comparison between the MBTE and CCMBTE models, θ=100,000 CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C Note: W, S, M, P, J and C represent wheat, soybean, maize, peanuts, japonica rice and cotton, respectively. Bolded values indicate CCMBTE has advantage over MBTE. 7.5. Policy impact on the CCMBTE result in the farm-level case We would like to assess the effects of a subsidy on wheat on plant enterprise selection strategy, given that wheat is one of the three major cereal crops in China (Li et al., 2016)8. Based on the prevailing subsidy paid for wheat grown in the northern Chinese provinces, such as Shandong, Hubei and Henan, we use a direct government payment of 115.74 RMB per mu (Gale, 2013) in calculation. We also refer to this paper for production cost and per acre yield, which are 364 RMB and 392.4 kg per mu, respectively.9–11 The optimisation result shows that wheat still does not appear in the optimal solution as it did not without considering the subsidy impact. In some instances, within the estimation window of 1998–2007, the model chooses wheat. As we compare results from other estimation windows, e.g. 1991–2000, 1997–2006 and 2001–2010, etc., we find the CCMBTE model chooses wheat only when either one of the two benchmark models choose it. This illustrates that some agriculture policy may have little impact on our optimal result, if it has no significant influence on return. It also demonstrates the robustness of our model against some changes within the data sample. 7.6. Transaction cost in the plant enterprise rotation We would like to compare the transaction cost obtained from the modified target MOTAD, M–V and MBTE models with eTx≤(=)v and the crop rotation scenario to the results obtained from the CCMBTE model. First, we assume that the land parcel for growing a specific plant enterprise has the same area through different decision periods.12 For instance, the rotation pattern from 2011 to 2012 in one of the M–V results is [maize, peanuts, japonica rice]→[wheat, soybeans, japonica rice]. The transaction cost is 2, as only two plant enterprises differ from the preceding year’s plant enterprise portfolio. If the rotation pattern from 2000 to 2001 in one of the CCMBTE results is [wheat, soybeans, japonica rice, cotton]→[maize, peanuts, japonica rice], the transaction cost is 5 as five plant enterprises in total differ between these 2 years. In Figure 19, we summarise the underlying results for all five models used in this paper. The number is averaged by the risk coefficients λ and ϕ (or λ−ϕ pair). It shows that although the CCMBTE model has a relatively higher transaction cost than the others, the value is much smaller than the M–V result during some specific year periods. Also, this value is very close to the MBTE model. Fig. 19. View largeDownload slide Transaction cost for the modified target MOTAD, M–V, MBTE and CCMBTE models based on the 1-year rolling out-of-sample forecasting. Numbers inside parentheses denote an average value for each method. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 19. View largeDownload slide Transaction cost for the modified target MOTAD, M–V, MBTE and CCMBTE models based on the 1-year rolling out-of-sample forecasting. Numbers inside parentheses denote an average value for each method. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 8. Conclusions We applied an MBTE model to the plant enterprise selection problem with multiple objectives. The portfolio solution in the MBTE model was found to track the highest return benchmark portfolio of the benchmark set while considering other benchmark portfolio characteristics such as risk control. Given that the MBTE is very sensitive to the selection of the benchmark, we used return maximisation and risk minimisation objectives in the MBTE framework. The constant rebalanced method, which is found to perform well in financial investment returns, was included as a reference benchmark portfolio. We modified the target MOTAD to follow the portfolio analysis. This modification, along with its experiment result, is a byproduct of this paper. We believe this modified target MOTAD model can also be applied in an asset investment problem. Considering the trade-off between the diversified and single-plant-enterprise strategies, we further added a convex CC onto the MBTE objective and adjusted the sparsity coefficient to limit the plant enterprise number. Chinese national data with 10 plant enterprises were used as a study case. This was followed by an extended case of provincial datasets with six plant enterprises to test the performance of alternative models. During the in-sample numerical experiment over a 25-year estimation window with the nation-wide data, we compared the annual return and the annual variance averaged on λ and ϕ of the benchmark models as modified target MOTAD and M–V, respectively. The result showed that neither of these models can dominate the other in all aspects (i.e. return and risk). In the MBTE framework, we chose a range of ϕ against each specific λ to generate benchmark solutions. Tracking these solutions in the in-sample case, the MBTE model produced a higher return level on average than that of the M–V and CRM models and a lower variance than those of the modified target MOTAD and M–V. For the plant enterprise selections in the nation-wide case, the modified target MOTAD model provided high returns but neglected diversification. In both in-sample and out-of-sample cases, it gave three or fewer plant enterprises, and these choices could not be further diversified by choosing any different risk coefficients. The M–V model, on the contrary, had more plant enterprises in a choice set within its low-risk portfolios but did not perform as well as the modified target MOTAD model regarding return. The MBTE model, however, generated even more diversified solutions and simultaneously searched for highest return. Additionally, it tracked all the benchmark portfolios well. The CCMBTE model under different cardinality coefficients always obtained a higher return than the MBTE model, but produced higher variance as well. For reducing the plant enterprise number, the CCMBTE model is found to perform better than the MBTE model if there was not a very rigid requirement on the computation precision. A province-wide dataset was used to assess the robustness of models under different settings. When considering only a 1-year rolling test, we found that the modified target MOTAD model provides more diversified plant enterprise selection than it did in nation-wide solutions. The estimation error of the M–V model on variance was larger than the nation-wide case. The modified target MOTAD return was not significantly larger than the M–V return in the province-wide solution. The MBTE model, however, based on its return-tracking characteristics, performed as well as the M–V model in return. Besides, it could control the variance even when the variance from the M–V model increases. The statistical significance between the two variances demonstrated the advantage of using MBTE’s to control variance. The CCMBTE performed as well as the MBTE in returns, variance and tracking error under an arbitrarily selected ϕ ⁠. It provided an optimal solution comprised of maize, peanuts, apples and sometimes cotton. In order to understand plant enterprise selection, rotation and policy effects on a farm level, we generated a new dataset using the same mean but reduced covariance of the original data. By additionally considering the plant enterprise number constraint, we transformed the decision variables from continuous to binary in all models. The numerical experiments compare the performance between the discretised MBTE and CCMBTE models and discretised benchmark models. After imposing the CC on the plant enterprise number, the CCMBTE model in the farm-level case performs no worse than the MBTE model in terms of return and variance within all estimation windows. Further, the MBTE and CCMBTE models in the discrete space perform no worse than the benchmark models. In order to simulate the impact of the direct subsidy policy for grain on the optimisation result, the return vector of wheat is modified. The result shows that the increase in return in wheat does not substantially change the selection strategies of the CCMBTE model. The normalised transaction cost analysis shows the CCMBTE results are no worse than those of the MBTE and two benchmark models. To summarise, the CCMBTE model tracks the MBTE model by adding a constraint on the number of plant enterprises to avoid over diversification. Therefore, CCMBTE provides a better modelling approach when a bi-criteria objective function is being considered along with a restriction on the number of plant enterprise choices. Within an agricultural production system, a decision-maker has to decide what plant enterprises to grow considering various policies, bio-geophysical constraints and return and risk factors. For the macro-policy makers, our method would be constructive in creating strategic policies concerning the agricultural production over some particular geographical areas or even the whole country. For the micro-level decision makers, it is valuable to provide entrepreneurs/farmers with some suggestions on their farm-level plant enterprise selection so that they can maximise return, minimise risk and reduce the transaction cost associated with growing too many plant enterprises. In future studies, it would be possible to introduce other non-linear benchmark models. However, it should also be noted that a non-linear benchmark model requires more computing resources. The CCMBTE model developed in our research can be an efficient method to meet the diversification requirement set by the EU’s common agricultural policy (CAP) while maximising the return and minimising the risk in plant enterprise selection. We can optimise our result by tracking the optimal results from the models with CAP constraints (as stated by Sckokai and Moro, 2006, 2009), the models under the consideration of subsidy scheme impact (as stated by Lien and Hardaker, 2001) or some calibration method like positive mathematical programming (as stated by Petsakos and Rozakis, 2015) as long as we have the required data. Acknowledgment The authors would like to sincerely thank Teo Chung Piaw, Zhichao Zheng, and Yunchao Xu for many stimulating discussions. Wen's time in this paper was supported by the key project of the National Science Foundation of China: Research of Food Risk Identification and Early Warning In the Production and Supply Process (grant No: 71633002). Paudel's time in this paper was supported by United States Department of Agriculture Hatch/Multistate project number LAB 94358. Ma was an assistant professor at South China Agricultural University when this paper was completed. Footnotes 1 Numerical experiment indicates both optimisation and simulation. 2 Shapley value was initially developed for efficiently allocating the risk among different assets. See Aumann and Shapley (1974). 3 Note that introducing price variabilities into the quadratic form will generate a polynomial term of degree four (non-convex). Solving such a model would require a prohibitive amount of computational resources. 4 Here, we should note that using the average crop revenue per unit of land may result in loss of some information about plant enterprise rotation; e.g. yield of apples in land previously cultivated by oilseeds will give the same return and variability as in land cultivated with apples, etc. Barkley et al. (2010) state that farmers often select crop varieties according to the published average yields. Although it is possible to incorporate additional constraints (such as for rotation) in this model, a simplified parameter structure can facilitate the analysis better, especially when the model is already complex. 5 These values are calculated based on the information from more than 60,000 households collected in the Compilation of Cost and Income Data of Agricultural Products in China. 6 Note that these solutions are fairly straight forward in a linear programming problem since we can choose those favourable plant enterprises by estimating the information on the existing data. However, as we expand our research to non-linear MBTE and CCMBTE models, the results cannot be easily derived. 7 Considering a new standard deviation is used in the regeneration process, we assume the new data sample is independent to the original set. 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Reducing agricultural carbon footprint through diversified crop rotation systems in the North China Plain . Journal of Cleaner Production 76 : 131 – 139 . Google Scholar Crossref Search ADS Author notes Review coordinated by Giannis Karagiannis © Oxford University Press and Foundation for the European Review of Agricultural Economics 2018; all rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png European Review of Agricultural Economics Oxford University Press

An application of a cardinality-constrained multiple benchmark tracking error model on a plant enterprise selection problem

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Abstract

Abstract Yield and return of plants grown in a region are generally closely related. Agricultural scientists are less likely to recommend a single-plant enterprise for a region because of risk and return concerns. From a risk/return perspective, a plant enterprise selection problem can be considered as a portfolio optimisation problem. We use a multiple benchmark tracking error (MBTE) model to select an optimal plant enterprise combination under two goals. A cardinality constraint (CC) is used to efficiently balance multiple objectives and limit over-diversification in a region. We use Chinese national and province level datasets from multiple plant enterprises over 25 years to identify the best plant enterprise combination with two objectives under consideration: return maximisation and risk minimisation. A simulated case using discrete programming is applied in order to analyse a farmer’s choice of specific plant enterprise and the transaction cost during rotation. In the continuous problem, the MBTE model is found to be efficient in choosing plant enterprises with high returns and low risk. The inclusion of a CC in the MBTE model efficiently reduces the plant enterprise number and volatility while creating smaller tracking errors than the MBTE model alone in an out-of-sample test. In the discrete problem, a CC can be used to search for the optimal number of plant enterprises to obtain high returns and low risk. The study and methods used can be helpful in choosing an optimal enterprise combination with multiple objectives when there are over-diversification concerns. 1. Introduction Several factors impact plant enterprise selection and diversification in a farm. Farmers select plant enterprises based on historical continuation, supply chain requirements, environmental concerns and risk and return reasons (Barkley and Porter, 1996; Pingali, 1997; Di Falco and Perrings, 2005; Bezabih and Sarr, 2012; Chateil et al., 2013). Bowden et al. (2001) indicate that the same level of return can be achieved by growing several different plant enterprises with complementary strengths in different fields. Although higher return and lower risk may prevail if multiple plant enterprises are chosen, farmers may still choose only a few plant enterprises because of limitations on plant enterprises production/marketing knowledge, excessive capital requirements and government policies (Omamo, 1998; Mahy et al., 2015). Given these realities, we identify and demonstrate methods that can be used to select an optimal number of plant enterprises and meet multiple objectives. Considering two objectives in a plant enterprise selection problem with diversification restrictions, we use a convex cardinality-constrained multiple benchmark tracking error model (CCMBTE). Although the core feature of the model is similar to other portfolio selection models (Barkley and Porter, 1996; Beuhler, 2006; Barkley, Hawana Peterson and Shroyer, 2010; Xu et al., 2014), it differs from others as it tracks multiple objectives by choosing a minimum number of plant enterprises. Portfolio optimisation models are widely used in health care (Sendi et al., 2003; Bridges, 2004; Prattley et al., 2007), commercial fisheries (Perruso, Weldon and Larkin, 2005), international project selection (Han et al., 2004), water usage strategies (Gaydon et al., 2012), land protection (Mallory and Ando, 2014), energy planning (Awerbuch, 2006) and climate change adaptation (Marinoni, Adkins and Hajkowicz, 2011; Mitter, Heumesser and Schmid, 2015). Unlike these papers, we are interested in the innovative combinations of different methods for finding plant enterprise combinations that can provide maximum returns and minimum risk. To achieve multiple objectives at the same time, portfolio optimisation methods with multiple benchmarks tracking error functions are used with a minimum crop diversification restriction. Roll (1992) first proposed the tracking error portfolio investment method, which is developed to minimise the tracking errors given an expected excess return. However, the performance of this model is dependent on a single benchmark, which results in large risk. El-Hassan and Kofman (2003) claim that a benchmark portfolio selection is always invalid as the selected portfolio sometimes performs even worse than Markowitz’s mean–variance (M–V) model. This will unavoidably lead to a principle-agent problem between the tracking error-minimising brokers and the total-risk-minimising investors, given a fixed expected excess return. If we use such a framework in a plant enterprise selection problem, producers can meet only one goal (maximum return or minimum risk) but not both (maximum return and minimum risk). Therefore, a single benchmark tracking error method would be an unacceptable choice in a plant enterprise selection problem. Therefore, we believe that the tool that efficiently balances multiple objectives will be attractive to agricultural producers and agricultural policy makers. Wang (1999) has taken the lead in extending the single benchmark tracking error model to a multiple benchmarks version. The objective is set as minimising the weighted sum of different benchmark tracking error values. Different asset selection strategies with different interests, such as maximising the return or minimising the return variance and so on, can be incorporated into the objective function. However, the formulation of Wang’s model compared with Xu et al. (2014)’s framework appears not so compact, and it is also not clear how to choose weights between alternative objectives. Fund and information in the financial market are often subject to certain restrictions. Similarly, transaction cost is also a problem that cannot be ignored. Since too many plant enterprises in a farmer’s portfolio increase production cost, choosing a limited number of plant enterprises are desirable in many instances. To minimise the tracking error under a limited number of trading assets, a graduated non-convex (GNC) model was proposed by Coleman and Li (2014). Also, Brito and Vicente (2014) introduced a cardinality constraint (CC) into the M–V model. The former is an approximating optimisation method that is similar to the image reconstruction idea by Blake and Zisserman (1987). GNC, essentially, approximates the discontinuous counting function by a continuous differentiable function; the latter uses a derivative-free approach to solve a multi-objective optimisation problem. Compared to these two models, a convex CC model proposed by Pilanci, Ghaoui and Chandrasekaran (2012) not only guarantees to have local optima but also provides a more concise form that is easier to compute. Considering mathematical elegance and application efficiency, we use the model by Pilanci, Ghaoui and Chandrasekaran (2012) as a sparsity penalty term on the MBTE objective function. The contributions of our article can be summarised as follows: (i) this is the first article to use an MBTE model in a plant enterprise selection problem with multiple objective functions; (ii) the target MOTAD model widely used in risk-return optimisation in agriculture is adapted with some modifications as a portfolio benchmark in the MBTE framework; (iii) a convex cardinality-constrained method is incorporated into the MBTE model for limiting the plant enterprise number in the optimal solution; and (iv) the efficiency of the CCMBTE model is compared against an MBTE model. The remaining sections of the article are organised as follows. In the next section, we introduce the framework of the MBTE model and discuss benchmark selection criteria. We also provide details on three benchmark models: modified target MOTAD, M–V model and constant rebalance method (CRM). In Section 3, we provide details on the data used in this study which come from Chinese National Crop Production data. In Section 4, we present numerical simulation results from all three benchmark models. The MBTE model results are then compared to these three benchmark models. In Section 5, we provide motivations for using CCMBTE model and simulation results. In Section 6, we estimate benchmark models, MBTE and CCMBTE models using province level data. An extension case is proposed in Section 7 to simulate the result for a farm-level problem, along with some additional constraints that are added. We conclude the paper in Section 8. 2. Modelling and analysis In this section, we introduce MBTE and other benchmark models. 2.1. MBTE model The portfolio selection optimisation methods that minimise the tracking error volatility are referred to as benchmark tracking error models. Extended from the previous work, an MBTE model simultaneously tracks more than one portfolio benchmark (Xu et al., 2014). The underlying formulation can be shown as follows: minx∈χE[ZB(r̃)−r̃Tx]2 (1) in which ZB(r̃)≔maxj={1,…,m}r̃Tpj. (2) Here the random vector r̃ denotes the return of a set of plant enterprises; x denotes the investment decision vector concerning the input applications on the interested plant enterprises. Both vectors r̃ and x have the same length, for example n ⁠. Vector pj in equation (2) denotes the jth benchmark plant enterprise selection strategy; ZB(r̃) is the term equipped to trace the highest return strategy(ies) within the m portfolios. By expanding equation (1), we obtain the following formulation which will be used in the numerical experiment1 of this paper: minxT(Σ+μμT)x−2E(ZB(r̃)r̃T)x. (3) Xu et al. (2014) indicate that the MBTE model is very sensitive to the selection of the benchmark models, and its Shapley value2 is not as high as the broker’s expectation. Still the MBTE model can be applied to a plant enterprise selection problem, as it complies with the agricultural producers’ return maximisation and risk minimisation objectives. Similar to the approach used by Wang (1999), weight selection among different tracking errors is implemented in a quadratic form.3 2.2. Benchmark selections Before using the MBTE model, we have to decide on what to use as a benchmark portfolio. There are three competitive models for the purpose: target MOTAD, M–V and CRM. The target MOTAD model developed by Tauer (1983) is widely used in plant enterprise selection problems. With an average revenue maximisation objective, its attractiveness mainly stems from its risk aversion logic of controlling the negative deviation of revenue from the target level. The M–V model developed by Markowitz (1952) was initially used to select the efficient portfolio with a minimum covariance at a given expected return level. CRM is very easy to implement even without any optimisation work, as it entails an equal allocation of capital across targeted enterprises at each decision point. CRM is widely known to be effective in minimising volatility. Some researchers even consider it to be a better performing model than the M–V model (Kirby and Ostdiek, 2012; Xu et al., 2014). 2.2.1. The modified target MOTAD model The standard Target MOTAD model (Tauer, 1983) can be formulated as maxxμTx. (4) Subject to, akTx≤bk,k={1,2,…,m} (5) T≤crTx+yr,r={1,2,…,τ} (6) pTy=λ (7) in which x ⁠, cr and μ are all n-dimensional vectors. x is the investment decision, cr is the production cost margin and μ denotes the average revenue obtained from a plant enterprise. p and y are τ-dimensional vectors which, respectively, denote the probability and negative income deviation of each risk scenario, and λ is the desired risk level measured in negative deviation from average income from each plant enterprise. Note that the use of the coefficient λ here is different from Petsakos and Rozakis (2015) and Arata et al. (2017). Since λ denotes the desired (satisfactory) level of compliance in the above model to the targeted income, it can be considered as a proxy of risk aversion coefficient rather than a calibration variable.4 The objective function maximises the expected revenue given an optimal solution x⁎ since μ averages the revenue of each plant enterprise over the estimation window. The first constraint equation (5) is the resource constraint in which k denotes the number of the resource types, and ak ⁠, bk ⁠, respectively, denote the input coefficient vector and availability of the kth resource; inequality equation (6) denotes the revenue deviation of each risk scenario r from the target revenue, which is measured by y associated with a probability weight vector p ⁠. A risk scenario denoted by r is referred to as an investment period. The occurrence of each period is given an equal probability that pr∈p is assumed to be the weight. The weighted sum is then set equal to the risk aversion coefficient λ in constraint equation (7). Considering that the income and cost data used in this study are reported in a real number, we use the following steps to modify the original target MOTAD model to comply with the portfolio optimisation problem: (i) Use the output–input ratio (OIR) as follows to reset μ on cr: OIR=aftertaxincome+totalproductioncosttotalproductioncost=1+aftertaxincomeratio The after tax income ratio is calculated by aftertaxincomeratio=grossrevenue−productioncost−taxtotalproductioncost (ii) Specify the resource constraint using production cost averaged over the estimation window. Set k=1 ⁠, then we have the constraint modified as eTx≤b. (iii) Modify T into the target income that sums each plant enterprise’s income as it is averaged over the estimation window. The maximum income through historical performance is not suitable for target T ⁠. Since, if this were the case, the portfolio would intuitively concentrate on the return-maximising strategy as we always have, with T≥crTx ⁠. If there is a plant enterprise with a very high return, to derive the pure-stand strategy is rather straight forward. The highest return target is ideal but not practical due to many constraints. (iv) The scaled-value entries of x are divided by the total production cost modified in (ii), in order to extract the portfolio vector in proportions instead of real numbers. Although the original solutions have realistic interpretations, like the proportion of each plant enterprise in the total planting area, it is not very suitable for portfolio analysis. 2.2.2. M–V model Another benchmark portfolio optimiser is the classic M–V model (Markowitz, 1952) that minimises the covariance with respect to different constraints. The formulation is mineTx=112xTΣx−ϕμTx (8) where Σ is the covariance between return of two plant enterprises, ϕ is the risk aversion parameter and e is the unit vector. All other variables are defined previously. Unlike the modified target MOTAD model, its risk controlling mechanism relies on the minimisation of the covariance. Existing studies vary ϕ by using a programming framework, such as a calibration model (Petsakos and Rozakis, 2015). 2.2.3. Constant rebalance method The CRM method assigns equal weights to all the enterprises in each period. Thus, it is also called the 1/n strategy. The model does have an attractive risk control feature. Additionally, its return performance is not as inferior as some other classic strategies, such as the M–V model (De Miguel, Garlappi and Uppal, 2009). 3. Data We use Chinese data on plant enterprise OIRs, production costs and after-tax income from 1989 to 2013. The related information regarding wheat, maize, soybean, peanut, rape seed (abbreviated as RS), cotton, flue-cured tobacco (abbreviated as TB), sugarcane, sugar beet and apple is recorded in detail in the Compilation of Cost and Income Data of Agricultural Products in China. The mean and the standard deviation of the OIR, production cost and plant enterprise income derived from the 25-year estimation window are provided in Table 1a–c, respectively. We model the whole country as a production unit and consider the correlation between plant enterprises with respect to nation-wide external conditions. Table 1. OIR, production cost and after tax income of 10 plant enterprises from 1989 to 2013 (national level) (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 a1 hectare = 15 mu. Table 1. OIR, production cost and after tax income of 10 plant enterprises from 1989 to 2013 (national level) (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (a) Mean and standard deviation of OIR Crop Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Total Mean 1.2240 1.4260 1.5885 1.6355 1.2738 1.4640 1.2815 1.4847 1.4819 1.9647 14.8245 SD 0.1796 0.2400 0.3056 0.2356 0.3655 0.3367 0.2700 0.3353 0.2308 0.4670 2.9660 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 (b) Mean and standard deviation of production cost (unit: RMB per mu)a Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 304.0325 326.1504 188.8664 420.8680 274.5828 722.8200 SD 168.9349 181.0069 89.2111 244.1846 159.5941 427.7958 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 Crop TB Sugarcane Sugar beet Apple Total Mean 1,080.1016 819.7624 445.5984 1,598.6004 6,201.4456 SD 785.0631 422.6807 252.9626 1,290.1464 4,021.5352 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 (c) Mean and standard deviation of gross revenue (unit: RMB per mu) Crop Wheat Maize Soybeans Peanuts RS Cotton Mean 387.8552 442.8311 280.2453 668.6952 333.3880 970.8264 SD 181.6362 203.9288 93.6498 352.6116 169.6292 418.1268 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 Crop TB Sugarcane Sugar beet Apple Total Mean 1,261.1952 1,140.6189 636.7277 3,055.7487 9,178.1319 SD 750.6582 471.4040 334.5629 2,509.9714 5,486.1790 a1 hectare = 15 mu. We consider the case of a representative farmer growing crops in both national and provincial cases. Considering that homogeneity of crop/enterprise choices in a country/province may be too strong, in the latter part of the paper, we will introduce an extended farm-level case to simulate the plant enterprise selection decision. As long as the plant enterprises can be switched easily, changes to the plant enterprise mix can be welfare enhancing to farmers when commodity prices are increasing. It is worth citing an example here. During the biofuel boom corn and soybean production area increased in the USA, but cotton and other crop area decreased significantly (Chen and Önal, 2012). During that period, producers were well off. There are many policies that may affect agriculture production. For instance, the decoupled support policies are popular in industrial countries. Bhaskar and Beghin (2009) mention that these policies are always funded by taxpayers and are not related to current production, factor use or policies, and that eligibility criteria are based on historical data. Weber and Key (2012) indicate that decoupled payments have little effect on aggregate production. Unlike the decoupled policy in the United States and the European Union, Chinese agricultural policies are usually paid directly based on the production area (direct pay). Some of these policies, such as the per acre subsidy payments, were found to have little influence on production decisions, although they accounted for 7–15 per cent of the farmer’s gross income (Gale, 2013). While current Chinese agricultural policies may or may not impact aggregate production, and while it is possible to incorporate some of those policies in the modelling framework, we have limited those to only a farm-level case presented in Section 7. For the study period shown in Table 1, apples have both a high return and high variation in return (high standard deviation) compared to the other nine plant enterprises. Wheat has the lowest mean return, though its production cost is not the lowest. Wheat in many places in China is treated as a staple crop rather than a purely economic one. Note that all enterprise returns are strictly positive, since (i) the data are annually averaged and (ii) a proportion of the revenue comes from the direct government payment to farmers. These government payments are paid to all crop growers so it should not impact plant enterprise choice significantly. Using the past 10 years of data samples as an estimation window to estimate the mean return and standard deviation, the results for each year from 1999 to 2013 are shown in Figure 1.5 We find that the return as well as standard deviation of most of the plant enterprises, except rape seed and apple, shows a decreasing trend. However, rape seed shows an increase in both return and standard deviation from 2006 onward. Apples show an increasing return but decreasing standard deviation since 2001. This indicates that most of the plant enterprises show higher returns with higher uncertainty, except for apples. Fig. 1. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 based on the national level data. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 1. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 based on the national level data. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 4. Benchmark numerical experiment results In this section, we first report the in- and out-of-sample test results obtained from the modified target MOTAD and M–V models. Then, we report the performance of the CRM method as a reference. Next, a comparison among the above benchmark models along with the MBTE method is carried out. All of the numerical experiments in this paper will follow an ‘in-sample’ to ‘out-of-sample’ order for testing the effectiveness of the models. An out-of-sample test refers to estimating the model using a portion of the dataset and using the model to forecast the known values of the series that is not in the estimation window. On the contrary, the model is used to predict the values that have been already used for estimating the model (De Miguel et al., 2009). The former is more like cross-validation, which is being widely used in the optimisation and machine learning fields (Efron and Gong, 1983; Campbell et al., 1997; Rohani, Taki and Abdollahpour, 2018), and it is generally believed that the out-of-sample test can provide a measure of protection against predictability (Lo and MacKinlay, 1990; Foster, Smith and Whaley, 1997; Rapach and Wohar, 2006). 4.1. Numerical experiment of in-sample case–modified target MOTAD and M–V models 4.1.1. The modified target MOTAD model In the in-sample test, we identify an optimal plant enterprise mix for various λ value using the entire 25-year estimation window. The result converges to a specific portfolio as we increase λ ⁠, say from 0 to 55 at a step length of 1. Corresponding selection combinations are shown in Figure 2a, which shows that the model always chooses the most high-average return enterprises.6 It complies with the return orientation of the modified target MOTAD model. The details of the input weight of each plant enterprise are reported in Table 2a, with the corresponding return and covariance averaged over the past 25 years. Based on the 56 solutions on the 56 λs, we take the mean and variance of the after-tax income and report them against the investment horizon in Figure 2b and c. Figure 2b denotes the return averaged on λ ⁠; the corresponding variance is in Figure 2c, which measures the risk controllability of a coefficient-averaged level, or, say, the solution’s sensitivity to different risk aversion degrees. To practically interpret the above results, we claim that by the modified target MOTAD method a producer can obtain a favourable income in the first and last several years, but s/he must handle a relatively large uncertainty. Fig. 2. View largeDownload slide (a)–(c) In-sample test on the modified target MOTAD model using the test sample of national level: (a) plant enterprise selection strategy, (b) pareto frontier, (c) return averaged on λ and (d)–(f) in-sample test on the M–V model using the test sample of national level: (d) plant enterprise selection strategy, (e) pareto frontier and (f) return averaged on λ ⁠. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 2. View largeDownload slide (a)–(c) In-sample test on the modified target MOTAD model using the test sample of national level: (a) plant enterprise selection strategy, (b) pareto frontier, (c) return averaged on λ and (d)–(f) in-sample test on the M–V model using the test sample of national level: (d) plant enterprise selection strategy, (e) pareto frontier and (f) return averaged on λ ⁠. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Table 2. In-sample test for the modified target MOTAD and M–V models using the national level data (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 Table 2. In-sample test for the modified target MOTAD and M–V models using the national level data (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (a) Modified target MOTAD model on 25-year estimation window λ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 1 0.0000 0.0000 0.4094 0.2523 0.0000 0.0000 0.0000 0.0000 0.0000 0.3383 1.7346 0.0816 5 0.0000 0.0000 0.3196 0.2485 0.0000 0.0000 0.0000 0.0000 0.0000 0.4319 1.7627 0.0900 10 0.0000 0.0000 0.2299 0.2446 0.0000 0.0000 0.0000 0.0000 0.0000 0.5256 1.7977 0.1033 15 0.0000 0.0000 0.1401 0.2407 0.0000 0.0000 0.0000 0.0000 0.0000 0.6192 1.8328 0.1198 20 0.0000 0.0000 0.0504 0.2368 0.0000 0.0000 0.0000 0.0000 0.0000 0.7129 1.8678 0.1393 25 0.0000 0.0000 0.0492 0.1681 0.0000 0.0000 0.0000 0.0000 0.0000 0.7828 1.8909 0.1563 30 0.0000 0.0000 0.0736 0.0808 0.0000 0.0000 0.0000 0.0000 0.0000 0.8456 1.9104 0.1733 35 0.0000 0.0000 0.0941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9050 1.9293 0.1909 40 0.0000 0.0000 0.0673 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9327 0.1983 1.9394 45 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9594 0.2060 1.9494 50 0.0000 0.0000 0.0138 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9862 0.2139 1.9595 55 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.2180 1.9647 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 (b) M–V model on 25-year estimation window ϕ Wheat Maize Soybeans Peanuts RS Cotton TB Sugarcane Sugar beet Apple Return Covariance 0.0000 0.5916 0.0000 0.0000 0.1483 0.0101 0.0157 0.0905 0.0000 0.1438 0.0000 1.3316 0.0286 0.0100 0.4495 0.0000 0.0000 0.2655 0.0000 0.0000 0.0599 0.0000 0.2251 0.0000 1.3947 0.0292 0.0200 0.3078 0.0000 0.0000 0.3703 0.0000 0.0000 0.0266 0.0000 0.2953 0.0000 1.4541 0.0310 0.0400 0.0126 0.0000 0.0000 0.5681 0.0000 0.0000 0.0000 0.0000 0.4192 0.0000 1.5659 0.0377 0.0800 0.0000 0.0000 0.0315 0.6378 0.0000 0.0000 0.0000 0.0000 0.2724 0.0583 1.6114 0.0436 0.1000 0.0000 0.0000 0.0544 0.6553 0.0000 0.0000 0.0000 0.0000 0.1563 0.1339 1.6530 0.0829 0.2000 0.0000 0.0000 0.0000 0.5977 0.0000 0.0000 0.0000 0.0000 0.0000 0.4023 1.7679 0.0511 0.3000 0.0000 0.0000 0.0000 0.3891 0.0000 0.0000 0.0000 0.0000 0.0000 0.6109 1.8336 0.1173 0.4000 0.0000 0.0000 0.0000 0.1805 0.0000 0.0000 0.0000 0.0000 0.0000 0.8195 1.9053 0.1654 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.9647 0.2180 4.1.2. The M–V model The optimised results of the in-sample experiment of the M–V method on the 25-year estimation window are numerically shown in Table 2b and graphically depicted in Figure 2d–f. Specifically, the investment weight, average return and covariance are derived by increasing ϕ from 0 to 0.5 at a step length of 0.01. The corresponding plant enterprise selection strategies are depicted in Figure 2d against ϕ ⁠. The investment weights with different values of ϕ are found to converge from six-enterprise diversified strategies, including wheat, peanuts, rape seed, cotton, TB and sugar beet, to a single-plant enterprise selection: apples. More specifically, apples attract more investment as ϕ increases due to its high-return rate (1.9647), but wheat rapidly decreases with the increment of ϕ since this enterprise has the lowest estimated average return (1.2240) among all the plant enterprises; peanuts have the second highest return (1.6355) and a relatively low variance (0.0555); their production at first increases but then declines by giving more investment weight to apples; soybeans are not as popular as peanuts, although the return is ranked third due to its larger variance compared to peanuts. Note that since the M–V solutions are more diversified than those of the modified target MOTAD solutions, they lead to a smaller mean return and smaller covariance. Also, we find both models solve to high-income plant enterprises as λ gets larger. Both in the modified target MOTAD and M–V models, food staples such as soybeans and wheat are ignored when λ and ϕ values are high. An exception is rape seed, which is a cash crop but loses investment interest due to its low returns. The annual return and the annual variance both averaged on ϕ are shown in Figure 2e and f, respectively. Results show that the returns until 1995 stay at a high level but sharply decrease after that year. From 2001, returns increase but fall again after 2012. Accordingly, the variance follows a similar dynamic except it falls earlier than the return during 1993–1995, but again in 2004. After experiencing a relatively long downturn period from 1995 to 2003, it rises again. We find (i) the optimal plant enterprise selection strategies drawn in Figure 2 under either the modified target MOTAD or M–V models draw some similar dynamics between the return and the variance. For practical interpretation, the agricultural producers may face similar income and yield risk irrespective of the two models chosen and (ii) the M–V model performs worse in controlling the return volatility than the modified target MOTAD model. The t-test on the variances indicates that the M–V model is more sensitive to the risk coefficient in statistical meaning. The same problem is observed by Xu et al. (2014), when applying the M–V model to a financial portfolio optimisation problem. 4.2. Numerical experiment of out-of-sample tests–modified target MOTAD and M–V models In the out-of-sample test, we use a 1-year rolling process in which the immediate past 10 years are used to estimate the sample mean, variance and covariance of the plant enterprise return, and the 11th data sample is used to evaluate the performance. The whole process moves 1-year forward, updating the estimates of the sample mean, variance and covariance each time. For instance, the dataset for 1999 is verified using the portfolio derived from the estimation window of 1989–1998; the data sample from 2000 is verified using information from 1990 to 1999, and so forth. Taking the modified target MOTAD as an example to identify all possible solution portfolios, we increase λ from 0 to 400 with a step length of 1 and report the result through the planning horizon, i.e. from 1999 to 2013. For each rolling cycle, there is a feasible region for λ ⁠, which is reported in Table 3a. The lower bound (e.g. 119 in 1999) denotes the first appearance of the modified target MOTAD solution, while the upper bound indicates a threshold beyond which the solution converges to one plant enterprise. In the simulation process, we find that as we move the estimation window forward, the feasible value of λ decreases in general. This is because, as the expected return gets closer to the target value with each estimation window, a producer may face smaller return deviation from such a target; thus, the risk tolerance level could be reduced. Table 3. One-year rolling out-of-sample test for the modified target MOTAD and M–V models using the national level data (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 Table 3. One-year rolling out-of-sample test for the modified target MOTAD and M–V models using the national level data (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (a) Rolling out-of-sample test for the modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Feasible λ 119–285 150–299 122–313 101–256 66–192 49–54 9–21 0–53 Mean return 1.4217 1.3981 1.4037 1.6100 1.7442 1.5268 1.5235 1.9033 Mean variance 0.1288 0.1284 0.1322 0.1718 0.1926 0.1742 0.1308 0.2956 Year 2007 2008 2009 2010 2011 2012 2013 Feasible λ 0–54 6–34 12–40 29–69 49–49 0–0 0–0 0–0 0–0 Mean return 2.0150 1.7919 1.7021 2.0083 2.1085 1.8486 1.6633 Mean variance 0.2925 0.2327 0.2055 0.2805 0.4446 0.3417 0.2767 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 (b) Rolling out-of-sample test for the M–V and modified target MOTAD on the 10-year estimation window Year 1999 2000 2001 2002 2003 2004 2005 2006 Mean return 1.3614 1.4126 1.3383 1.5599 1.7838 1.5268 1.3580 1.5910 Mean variance 1.5828 1.7112 1.5623 2.1680 2.8146 2.0515 1.6097 2.1394 Year 2007 2008 2009 2010 2011 2012 2013 Mean return 1.8134 1.7046 1.6531 2.0734 1.9462 1.7675 1.5955 Mean variance 2.7990 2.5212 2.4030 3.8602 3.3984 2.7874 2.2861 For the optimised plant enterprise selection strategies for the modified target MOTAD, we show the patterns in Figure 3a by averaging the input weight of each plant enterprise on λ and report the results from 1999 to 2013. Results show that selection mostly concentrates on three plant enterprises, viz., soybeans, peanuts and apples. As λ increases, only apples are selected. Although soybeans are a staple food in Northeast China, apples outweigh soybeans in profitability. Figure 3b shows the variance of return averaged on year against λ ⁠, which shows the periodical variability of the return of the modified target MOTAD model. Figure 3c and d reports the annual return and the annual variance both averaged on λ ⁠. The results show that both annual return and the annual variance increase sharply in the last several years. This is mainly due to the increased income of the plant enterprise. These values are obtained by using the following process: first we fix the lower bound of the feasible λ (from the modified target MOTAD model) subject to the largest lower bound of a specific solution among them all, while fixing the upper bound using the smallest upper bound from another specific solution set (it is possible that these two solution sets could be the same but it will not affect our analyses); then we report the annual variances of the return averaged against the above range of λ ⁠. For instance, the solution set from the estimation window of 1990 to 1999 is feasible in λ in [150, 298], of which the lower bound of 150 is the largest among all the other feasible λ ranges of all the solution sets. Meanwhile, the solution for the estimation window between 1991 and 2000 is feasible in λ [122,313], of which the upper bound of 313 is the largest among all the upper bounds. Therefore, we present the return variance averaged on years against the λ range of [150, 313] in the x-axis. For the solutions which have already converged with λ<313 ⁠, say λ=200 ⁠, we continue to use the converged solutions in λ [200, 313]. This process helps to find coefficient patterns from the benchmark models. Fig. 3. View largeDownload slide One-year rolling out-of-sample test on the modified target MOTAD and M–V models over 10-year estimation window based on the national level data. (a)–(d) Modified target MOTAD model: (a) plant enterprise selection strategy, (b) volatility measured by variance averaged on year, (c) return averaged on λs and (d) variance averaged on λs. (e)–(h) M–V Model: (e) plant enterprise selection strategy, (f) volatility of the M–V model, (g) return averaged on ϕs and (h) variance averaged on ϕs. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 3. View largeDownload slide One-year rolling out-of-sample test on the modified target MOTAD and M–V models over 10-year estimation window based on the national level data. (a)–(d) Modified target MOTAD model: (a) plant enterprise selection strategy, (b) volatility measured by variance averaged on year, (c) return averaged on λs and (d) variance averaged on λs. (e)–(h) M–V Model: (e) plant enterprise selection strategy, (f) volatility of the M–V model, (g) return averaged on ϕs and (h) variance averaged on ϕs. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Each solution set of the M–V model during the rolling out-of-sample experiment is derived from a particular feasible region of ϕ ⁠. Unlike the modified target MOTAD model, the existence of the solutions is always guaranteed. The annual return and annual variance both averaged on ϕ are reported in Table 3b, by choosing 0.5 as the upper bound of ϕ and a step length of 0.02. The annual plant enterprise selection strategies averaged on ϕ are shown in Figure 3e, where soybeans attract more investment than other plant enterprises until 2006. From 2006 onward, apples take priority within the portfolio after experiencing a down turning trend. The third high-performing plant enterprise is peanuts, which as a typical economic and oil crop has always obtained a certain amount of investment except in 2008, 2009 and 2010. The annual average variance of return against ϕ is shown in Figure 3f. Based on return/risk in these two models (M–V and modified target MOTAD model), we cannot clearly tell the dominance of one model compared to the other. The annual return and the annual variance of both averaged on ϕ are exhibited in Figure 3g and h. Comparing them with the counter-part result of the M–V model, as depicted in Figure 4a and b, both show an upward trend. Fig. 4. View largeDownload slide Comparison between the modified target MOTAD and M–V models in the out-of-sample test using the test sample of national level: (a) return and (b) variance. Fig. 4. View largeDownload slide Comparison between the modified target MOTAD and M–V models in the out-of-sample test using the test sample of national level: (a) return and (b) variance. 4.3. Numerical experiment–CRM We report results from this method in Figure 5. The mean return and the variance are 1.4825 and 0.1660, respectively, which are significantly smaller (on t-test) than the result of either the modified target MOTAD or M–V models. This is attributed to the high diversity of the CRM’s plant enterprise selection strategies. Fig. 5. View largeDownload slide Annual performances of CRM from 1989 to 2013 of national level: (a) return and (b) variance. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 5. View largeDownload slide Annual performances of CRM from 1989 to 2013 of national level: (a) return and (b) variance. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 4.4. Comparison of MBTE, modified target MOTAD, M–V and CRM We use modified target MOTAD, M–V and CRM on the MBTE model as the benchmark plant enterprise selection methods. To simulate the result, we gradually change the value of ϕ based on a continuum of λ. This is essentially an ergodic process for searching all the possible solutions through the risk coefficient combinations of the modified target MOTAD and M–V models. Here, we have P×L MBTE solutions where L is the number of λ ⁠, which is also the number of the modified target MOTAD solutions, while P is the number of ϕ ⁠, which is also the number of the M–V solutions. Using these portfolios along with the CRM portfolios, we seek the MBTE solutions and compare them with the performances of all the benchmark portfolios. 4.4.1. Numerical experiments First, we provide results from a 25-year estimation window. This is then followed by the results from the 1-year rolling out-of-sample case with a 10-year estimation window. In-sample forecasting results for different geographic regions are presented in Figure 6a–d. Figure 6a and b indicates that the MBTE return is as good as the return from a modified target MOTAD model but is less sensitive to the risk coefficient than the M–V model. More specifically, in Figure 6a, the MBTE return denoted by the red line has an upper bound very close to that of the M–V model. The MBTE’s lower bound shows a return similar to the modified target MOTAD return. The MBTE method is always tracking the return of a good performing plant enterprise mix but never touches the lower values derived from either the M–V (green dot) or the CRM methods (black dash). As ϕ varies, we find the upper bound of the MBTE variance is always smaller than that of the M–V model. The lower bound is larger than the M–V method since the MBTE model on these data points is meant to track the higher return, either from the modified target MOTAD or M–V model, by sacrificing some risk control. Figure 6c and d shows that the MBTE covariance always performs better than that of either the modified target MOTAD or the M–V method. As we can see from Figure 6c, when the MBTE result does not track other models, its value is smaller than that of the modified target MOTAD model. As the MBTE covariance tracks the M–V result periodically, the covariance of the modified target MOTAD method is almost covered by the red curve. Since M–V covariance has a relatively high value here, it can be only observed in Figure 6d as we zoom out the area. We do not elaborate on the results from the CRM model as its return is clearly inferior to other models. Fig. 6. View largeDownload slide In-sample experiment of the MBTE model equipped with the benchmarks of the modified target MOTAD, M–V and CRM model, using the test sample of national level. (a)–(d) Comparison between the modified target MOTAD, M–V and CRM models; (e) optimal crop numbers in the MBTE, modified target MOTAD and M–V models. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 6. View largeDownload slide In-sample experiment of the MBTE model equipped with the benchmarks of the modified target MOTAD, M–V and CRM model, using the test sample of national level. (a)–(d) Comparison between the modified target MOTAD, M–V and CRM models; (e) optimal crop numbers in the MBTE, modified target MOTAD and M–V models. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. For the number of the plant enterprise selections (see Figure 6e), we find that if the MBTE model (denoted by the red curve) tracks the modified target MOTAD model with low-risk measures within λ[0,15], it will find more diversified strategies than that of the M–V model. Specifically, for the optimal plant enterprise choice without any restrictions, the solutions always include soybeans, peanuts, cotton, sugarcane, sugar beets and apples. When the number of plant enterprise choices in the optimal solution is reduced to five, the optimal plant enterprise mix includes soybeans, peanuts, cotton, sugarcane, sugar beets and apples. The optimal plant enterprise choice with four enterprise selection restrictions includes soybeans, peanuts, sugarcane and apples; the optimal solution with three enterprise restrictions includes soybeans, peanuts and apples and the optimal solution with two enterprise restriction include soybeans and apples. Table 4, collectively, reports the above results. This in-sample result indicates that using the MBTE model has more practical contribution to the peoples’ livelihood issues since it also concerns itself with growing staples instead of only tracking the economic enterprises. Table 4. MBTE portfolios with different CCs (national level) Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple Table 4. MBTE portfolios with different CCs (national level) Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple Cardinality 6 5 4 Crops Soybeans, peanuts, cotton, sugarcane, sugar beet, apple Soybeans, peanuts, cotton, sugarcane, apple Soybeans, peanuts, sugarcane, apple Cardinality 3 2 1 Crops Soybeans, peanut, apple Soybeans, apple Apple To streamline the analysis and convey practical significance, we only focus on the 1-year rolling method in the out-of-sample test. Since the test sample size duration is 15 years, i.e. 1999–2013, the computation scale is correspondingly expanded to ∑t=1τλt×ϕt ⁠, where τ is the rolling length. For instance, we use a 10-year estimation window from 1989 to 1998 to find the MBTE solutions tested on the data sample of 1999. This entails running 167×25=4,175 optimisation models for each time period. Based on Table 3 for 1999, we have λ=167 and ϕ=125 ⁠. Next, we test samples using the estimation window from 1990 to 1999; the number of optimisation models needed is 150×25=3,750 for t=2 ⁠, and so on. Since we have fixed the available range of ϕ ⁠, its subscript t can be removed. As an example, we report the result for the year 1999 in Figure 7a–d. The comparison of returns and variances among the modified target MOTAD, M–V and CRM is shown in Figure 7a. Figure 7b highlights the difference between the MBTE and modified target MOTAD models. The result indicates that although the MBTE model (red curve) uses the modified target MOTAD (blue curve) and M–V (green curve) models as a benchmark, it depends more on the modified target MOTAD model by considering its high-return level. The comparison of variances among the four models is shown in Figure 7c, from which we find that the MBTE model generally has a lower variance than the modified target MOTAD variance (red vs. blue curves). Although here the CRM variance is too low to be observed, its return and some other performances are not comparable to the other models. When the modified target MOTAD model is solved using a large λ (=156), the MBTE will no longer closely follow it but chooses to balance the risk by gradually approaching the M–V solutions. For the solutions through 1999–2013, we compare the volatilities of the MBTE, modified target MOTAD and M–V models averaged on a specific range of feasible λ ⁠, and report the result in Figure 7d. It shows that the MBTE model clearly performs better than the other two models based on the risk and return criteria. Fig. 7. View largeDownload slide Comparison of return, variance and volatility on the MBTE, modified target MOTAD, M–V as well as CRM models in 1-year rolling out-of-sample experiment. (a) return, (b) return and (c) variance: using the test sample (year = 1999, national level); (d) volatility: using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 7. View largeDownload slide Comparison of return, variance and volatility on the MBTE, modified target MOTAD, M–V as well as CRM models in 1-year rolling out-of-sample experiment. (a) return, (b) return and (c) variance: using the test sample (year = 1999, national level); (d) volatility: using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. The tracking error averaged on (λ,ϕ) pair is reported in Figure 8a, which shows that the value becomes larger in the last several years. In the asset investment area, tracking error is defined as the deviation of the portfolio return from the benchmark return (Roll, 1992; Jorion, 2003). This value is widely used to measure the performance of stock brokers. However, unlike stock brokers, agricultural producers should be more concerned about plant enterprise return than the tracking error. Results indicate that it would be increasingly difficult for the decision-maker to ‘track a desired economic target’ over time. Figure 8b reports the plant enterprise number based on average, maximum, and minimum levels of (λ,ϕ) over the 15-year test data samples. The result shows that although the average MBTE plant enterprise numbers are always lower than five, the maximum level could be much higher, e.g. as high as 10. It is clear that the MBTE model provides for fairly diversified strategies that may result in higher transaction cost. This motivates us to introduce a CCMBTE model which restricts over-diversification strategies. This method, as we show later, performs well for sparsity control in continuous programming. Although this advantage does not appear in the discrete environment of our extended cases in Section 7, it still provides a better solution for return and risk control compared to the other models by adjusting the cardinality of the decision vector. Fig. 8. View largeDownload slide Tracking error and plant enterprise number in the MBTE model (a) tracking error of the MBTE model averaged on λ−ϕ pair in 1-year rolling out-of-sample experiment and (b) plant enterprise number of λ−ϕ pair average, maximum and minimum levels (national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 8. View largeDownload slide Tracking error and plant enterprise number in the MBTE model (a) tracking error of the MBTE model averaged on λ−ϕ pair in 1-year rolling out-of-sample experiment and (b) plant enterprise number of λ−ϕ pair average, maximum and minimum levels (national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 5. Motivation for using the CCMBTE model Comparing the out-of-sample results of return and variance from the M–V and modified target MOTAD model, we found in Figure 4 that neither of them dominates the other all the time (figure legend MOTAD). The same problem exists in the comparison of the volatilities of both models, although the average volatility level of the M–V model is better than that of the modified target MOTAD model. The MBTE model is useful here as it is always able to track the good features of both of these models. The development of the CCMBTE model, however, should be more useful since it not only tracks the MBTE’s desirable features but also considers controlling too much enterprise diversity. Moreover, it is easy to implement in practice. We adjust the cardinality coefficient under an already constructed MBTE framework. Rădulescu, Rădulescu and Zbăganu (2014) mention that farmers have to decide ‘how much diversification is enough’ to capture the most potential gains from expanding their enterprise mix. We can rephrase the plant enterprise selection problem as ‘How many plant enterprises should a farmer choose to approach the best performing diversified strategies?’ The potential users of such models include the policy makers of a country or a province, the agri-business investors, producers of an agricultural production base or even individual farmers. To mitigate the disadvantages of limiting plant enterprises within a single economic area, different plant enterprises can be geographically spread out in practice, which is called location diversification. For instance, soybeans are the major staple crop of the northern part of China, whereas wheat is a staple crop in the southern part. Yang et al. (2014,,2015) use data from a relatively large area, for example, northern China, and Werners et al. (2007) use data from the Guadiana River Basin, Portugal. In these studies, spatially/geographically diversified crop selection strategies are assumed to be available for the underlying regions. Another implicit assumption is that through policies the government could successfully guide the farmers in the targeted regions to have the most profitable and risk reducing combination of plant enterprises. In Section 7, the CCMBTE approach is applied in a discretized situation in order to address a farm-level problem. We also present its advantage compared to the MBTE and benchmark models. 5.1. Cardinality constraint To adjust the plant enterprise number to reduce over diversification, a CC developed by Pilanci, Ghaoui and Chandrasekaran (2012) is introduced as follows: mineTx=1,x≥0f(x)+θcard(x) (9) where card(x) denotes the number of non-negative entries of vector x ⁠, i.e. the invested plant enterprise number; θ is a given trade-off parameter for adjusting the desired cardinality level. The above formulation runs into difficulty in finding a sparse solution since ‖x‖1=∑i∈N|xi|≤=1 is a constant wherein we cannot fix the number zero. To address this concern, Pilanci et al. use the l∞ norm as the maximum relaxation, as shown in equation (10). This helps us to find a lower bound for the original problem ‖x‖1=∑i∈N|xi|≤card(x)maxi|xi|≤card(x)‖x‖∞. (10) If we denote the original problem as P⁎ ⁠, then we have P⁎≥P∞⁎=mineTx=1{f(x)+θ1maxixi} (11) which according to Proposition 2.1 of the original work by Pilanci et al. can be reformulated as a convex optimisation as follows: (PSP⁎)=mini{mineTx=1,x≥0f(x)+z:z≥θxi}. (12) When solving PSP⁎ ⁠, we first solve the sub-problem inside the curly brackets given the plant enterprise indexed i ⁠. The inner problem will generate a solution that minimises the portfolio objective f(x) summed by z ⁠, an extra variable for limiting the cardinality. z reaches the minimum value if the condition z≥θxi is active. Thus, we should test each xi,i={1,2,…,m} to find the minimum objective of the inner problem. Since m denotes the length of the portfolio vector, i.e. the number of plant enterprises, it also denotes the computation times for the inner sub-problem. The outside programme will then extract a minimum value among the m values, which will compare the inner objective value m times. If we incorporate the above described conditions in equation (12), the following objective function is obtained: mini{mineTx=1,x≥0E[ZB(r̃)−r̃Tx]2+z:z≥θxi}. (13) We call this objective function, along with the constraints presented earlier, the CCMBTE model. Since we are solving a two-layer optimisation problem in equation (13), the computation time should again be increased to N #×∑t=1τλtϕt ⁠, where N # denotes the size of the candidate plant enterprises. 5.2. Numerical experiment–CCMBTE model We report the result in Figure 9 by using the 2008 dataset. Arbitrarily choosing θ at three values, say, 0.15, 0.1 and 0.05 (green, red and yellow), we find the CCMBTE return is always close to or even higher than the modified target MOTAD and MBTE returns (see Figure 9a). Correspondingly, its variance reported in Figure 9b increases but moves closer to the MBTE solutions when we decrease the cardinality coefficient. Fig. 9. View largeDownload slide Comparison of (a) returns, (b) variances, (c) plant enterprise numbers and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 2008, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 9. View largeDownload slide Comparison of (a) returns, (b) variances, (c) plant enterprise numbers and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 2008, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. As shown in Figure 9c, we find the CCMBTE chooses a smaller number of plant enterprises than the MBTE model. It is also interesting to note that the CCMBTE tracking error (depicted in Figure 9d) is always lower than that generated by the MBTE model. This is because the data sample always provides a higher return from a set of small plant enterprise selection strategies, and the CCMBTE model at the same time is motivated to solve the problem by just tracking these plant enterprises. If the plant enterprise number is increased, the tracking error will accordingly increase. It is quite intuitive, since different models use different selection strategies and these strategies derived from different models, that these should be less likely to be equal to each other. Even so, this increasing number will always be lower than that of the MBTE method. Simulation with a larger sample size (as in the example from 1999) gives the average CCMBTE plant enterprise number (in Figure 10a) for an arbitrarily chosen coefficient, say θ=0.1 ⁠, significantly smaller than that of the MBTE model (with the F-value of 407.63 and a p-value of 0.000). Also, we find the return, variance and tracking error of the CCMBTE model (in Figure 10b–d, respectively) are also close to those of the MBTE model, in the region of λ around 19.5. For the specific selection strategies of the CCMBTE model, not only do the plant enterprise numbers reduce but also the usually invested number of plant enterprises is different from the other models. For instance, in a two-enterprise pattern, wheat becomes more popular than soybeans, peanuts, or any other eight plant enterprises. It is interesting that the application of the CCMBTE model can keep the food staples in a sparser solution environment instead of diversifying the strategies to more high-return plant enterprises. Fig. 10. View largeDownload slide Comparison of (a) plant enterprise numbers, (b) returns, (c) variances and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 10. View largeDownload slide Comparison of (a) plant enterprise numbers, (b) returns, (c) variances and (d) tracking errors of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. To compare the averaged volatilities of the MBTE and CCMBTE models through the 15-year test sample, we follow the same process as that of the MBTE model. The feasible λ is still from 150 to 313. The result is reported on the left panel of Figure 11, which shows that the CCMBTE model under relatively small λ’s (e.g. less than about 160) performs better than the M–V model within the upper bound of the volatility. Besides, the right subfigure also shows that the CCMBTE has successfully controlled the plant enterprise number. As we select the threshold of sparsity measure of between 0.0001 and 0.00005, the result shows that the CCMBTE plant enterprise number is smaller than that of the MBTE model except in the last four years. As we relax the requirement on the solution precision by setting the threshold as 0.001, the CCMBTE plant enterprise number is never larger than that of the MBTE model. Relatively, the plant enterprise number of the MBTE is less sensitive to the sparsity coefficient. Fig. 11. View largeDownload slide Comparison of volatilities and plant enterprise number of average level of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 11. View largeDownload slide Comparison of volatilities and plant enterprise number of average level of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999–2013, national level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 6. Extension–province level We believe that a differently scaled dataset can be used to test model robustness. Accordingly, we apply the optimisation models to a smaller scale dataset from Hebei province, China. This dataset was also collected from the Compilation of Cost and Income Data of Agricultural Products in China. There are only six candidate plant enterprises (soybeans, maize, peanuts, japonica rice (JR), cotton and apples) included in this case. The related descriptive statistics are listed in Table 5, which shows that apples still have the highest average return over the 25-year estimation window. Table 5. Mean and standard deviation of six plant enterprises in Hebei province from 1989 to 2013 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 Table 5. Mean and standard deviation of six plant enterprises in Hebei province from 1989 to 2013 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 Crop Soybeans Maize Peanuts JR Cotton Apple (a) 25-year estimation window for OIR  Mean 1.29 2.02 1.82 1.70 1.57 1.56  SD 0.30 0.65 0.51 0.34 0.41 0.58 (b) 25-year estimation window for revenue (unit: RMB per mu)  Mean 287.32 547.32 800.58 1,112.22 1,005.60 1,544.58  SD 187.23 297.37 481.98 645.29 574.62 946.49 (c) 25-year estimation window for cost (unit: RMB per mu)  Mean 238.82 310.22 488.27 698.50 727.10 1,184.96  SD 174.73 216.54 360.47 479.89 553.29 978.86 6.1. The modified target MOTAD model Figure 12 shows the return and standard deviation over a 10-year rolling estimation window from 1999 to 2013. This figure shows a generally downward trend for all the plant enterprises in both return and standard deviation. Fig. 12. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 12. View largeDownload slide Return (a) and standard deviation (b) over 10-year rolling estimation window from 1999 to 2013 (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Table 6 lists the portfolios derived at 15 estimation windows in the 1-year rolling experiments from 1999 to 2013. We find that portfolios are usually more diversified than in the national case, i.e. four to five plant enterprises are always selected among six potential plant enterprises. We find that compared with the national case in which a single-plant enterprise such as apples always generates the highest return, the provincial case is different, as there are multiple enterprises with similar types of returns. The solution estimated over 1989–1998 contains the most diversified strategy, with all enterprises except soybeans in the solution. Only the solution of 2013 generates a single-plant enterprise strategy where only maize is selected. This is contrary to other optimal solutions in which both food staples and cash crops were in the solution. Table 6. Modified target MOTAD solutions of the out-of-sample rolling test (province level) 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N Table 6. Modified target MOTAD solutions of the out-of-sample rolling test (province level) 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Soybeans N N N N N N N N N N N N N N N Maize Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Peanuts Y Y Y Y Y Y Y Y Y Y N N N N N JR Y Y Y Y Y Y Y Y Y Y N N N N N Cotton Y Y N Y N N N N N N N N N N N Apple N Y Y N Y Y Y Y Y Y Y Y Y Y N In Figure 13a and b, the red curve indicates that the producers may face a larger risk when desiring a higher return, especially during the last several years. The return and the variance both averaged by years are listed in Figure 13c and d. The former shows an increasing return with λ ⁠, while the latter exhibits a consistent trend. Fig. 13. View largeDownload slide Return and variance of the modified target MOTAD model averaged on λ (shown in (a) and (b)) and year (shown in (c) and (d)) in 1-year rolling out-of-sample experiment (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 13. View largeDownload slide Return and variance of the modified target MOTAD model averaged on λ (shown in (a) and (b)) and year (shown in (c) and (d)) in 1-year rolling out-of-sample experiment (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 6.2. M–V model Figure 14 reports the performance of the M–V model using the provincial dataset. The mean values of return and variance averaged either on ϕ or year are more consistent with each other than when both of them are derived from the modified target MOTAD model (see Figure 14a and b). The variance of M–V return averaged on year (see Figure 14d) is generally larger than that of the modified target MOTAD model, although the former performed relatively well in the in-sample case (see Figure 14f). For plant enterprise selection strategies as when ϕ=0 ⁠, it includes all the six plant enterprises. On the other hand, when ϕ exceeds 2, the solution converges to two plant enterprises, i.e. maize and peanuts. Compared to the nation-wide case, the M–V’s plant enterprise selection strategy is to select only a few enterprises rather than large diversification. Fig. 14. View largeDownload slide (a), (b), (c) and (d) Return and variance of the M–V model averaged on λ and year; (e) and (f) comparison of returns and variances of the M–V and modified target MOTAD models averaged on ϕ and λ (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 14. View largeDownload slide (a), (b), (c) and (d) Return and variance of the M–V model averaged on λ and year; (e) and (f) comparison of returns and variances of the M–V and modified target MOTAD models averaged on ϕ and λ (province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 6.3. Comparison among M–V, modified target MOTAD and MBTE models We run the model by choosing a specific range of risk coefficients (⁠ ϕ=[0,4]) with a step length of 0.02 for the M–V benchmark model. First, we use the sample year 1999 as an example and compare the results between the modified target MOTAD and M–V models (see Figure 15a and b). The result shows that the MBTE returns always follow the M–V’s target as ϕ increases, but the variance reaches a high value at the M–V benchmark. In the independent t-test for the MBTE and M–V return, the M–V return is significantly higher than the returns obtained from the MBTE model. The variance of the M–V model is significantly larger than that of the MBTE method. Fig. 15. View largeDownload slide Return and variance of the MBTE, M–V and modified target MOTAD models in 1-year rolling out-of-sample experiment: (a) and (b)–using test sample (year = 1999); (c) and (d)–using the test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 15. View largeDownload slide Return and variance of the MBTE, M–V and modified target MOTAD models in 1-year rolling out-of-sample experiment: (a) and (b)–using test sample (year = 1999); (c) and (d)–using the test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Figure 15c and d shows the annually averaged performances of return and variance of three models (MBTE, modified target MOTAD and M–V) when λ ≥ 485 for the years 1999–2013. Figure 15c shows that the MBTE return is lower than the modified target MOTAD and M–V returns. We find the MBTE variance still performs very well compared with those of the other two models (see Figure 15d). 6.4. MBTE and CCMBTE models Within the same data sample (year 1999), Figure 16 compares return, variance, tracking error and plant enterprise selection (in Figure 6a–d, respectively) between MBTE and CCMBTE models. The results show that the CCMBTE model with an arbitrarily chosen cardinality coefficient (θ = 0.005) gives a similar return level, variance, and tracking error as the MBTE method. For the plant enterprise number, the CCMBTE solution is significantly smaller than the MBTE strategies, say $3.1439<3.2813 under 0.00 statistical significance level. For the specific patterns, the CCMBTE model always provides maize, peanuts, apples and sometimes includes cotton in the optimal solution. Both food staples and economic crops are included in the optimal solution. Fig. 16. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 16. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE and CCMBTE models in 1-year rolling out-of-sample experiment using the test sample (year = 1999, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Based on the dataset between 1999 and 2013, Figure 17 shows that (a) the CCMBTE return is always larger than that of the MBTE model; (b) the CCMBTE variance averaged on year is not as low as in the MBTE model, but variance is lower than in the M–V model; (c) CCMBTE tracking error is mostly smaller than that of the MBTE model; and (d) the coefficient-averaged plant enterprise number of the CCMBTE is almost always smaller than that of the MBTE model and shows greater robustness along the investment horizon. Fig. 17. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE the CCMBTE models test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 17. View largeDownload slide Comparison of (a) return, (b) variance, (c) tracking error and (d) plant enterprise number of the MBTE the CCMBTE models test sample (year = 1999–2013, province level). This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. In summary, the MBTE model in the country- and province-wide data sample is able to efficiently track high return with low risk. The CCMBTE model, however, is able to effectively track the MBTE model given the over diversification control power. 7. Simulated case for a farm-level In this section, we extend the application of our models to a farm-level. We follow these steps: (i) simulate based on the statistical information of the provincial/national data; (ii) transform the continuous optimisation models into discrete problems; and (iii) introduce the policy impact into the simulation. For the first step, we use a scaled down value of standard deviation but use the same mean return as at the national or state level. We use a scaled down variance but the same mean return because cropping conditions in a smaller area are more homogenous than those in a larger area. In the second step, we transform the continuous decision variable x∈{0,1}n into a discrete (binary) term x∈{0,1} ⁠. Discretisation assumes farmers can plant only one crop per land parcel in a given year. We believe a discretised variable is more appropriate for describing a farmer’s enterprise selection decision than a fractional variable. Transformation of a continuous problem to a discrete problem also makes it easier to analyse rotations. The corresponding recast of the original models to a discrete space is listed below: Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Modelling approach Original problem (continuous case) Transformed problem (discrete case) Modified target MOTAD Linear programming (LP) Integer programming (IP) M–V Quadratic programming (QP) Mixed integer quadratic programming (MIQP) MBTE QP MIQP CCMBTE Quadratic constrained programming (QCP) Mixed integer quadratic constrained programming (MIQCP) Let 0<v≤n,v∈Z+ be the constraint on the number of plant enterprise choices. Assuming n (number of plant enterprises under consideration) = 6 in the farm-level scenario, a farmer cannot choose more than n crops to plant on these v land parcels. The combination of constraints can be rewritten as eTx=v,x∈{0,1} ⁠. Then, the modified target MOTAD is transformed as an integer problem and solved using integer programming. Similarly, the M–V and MBTE models can be reformulated as a mixed integer quadratic model and solved using a mixed integer quadratic programming. We re-cast zxi≥θ,xi∈[0,1] from the original CCMBTE model to a mixed integer quadratic constraint. Each of these transformed problems can be efficiently solved using the CPLEX solver. 7.1. Simulation of farm-level data sample We introduce six plant enterprises, including wheat, soybeans, maize, peanuts, japonica rice and cotton (a total of six plant enterprises), in our simulated farm-level case. These are the plant enterprises found in Northern China. We keep the mean return vector [1.4757, 1.2079, 2.0387, 1.7724, 1.7364, 1.3732] of the six plant enterprises from the original dataset that come from both national and provincial cases, but scale down the standard deviation vector [0.2875, 0.3607, 0.6559, 0.5400, 0.5390, 0.3718] arbitrarily by 0.15 for each entry to regenerate a set of data samples using a multivariate normal distribution. We chose one of these vectors with the lowest averaged sum of all the entries in the covariance matrix.7 We show the results in Table 7. Table 7. Covariance of simulated data sample for six plant enterprises Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 Table 7. Covariance of simulated data sample for six plant enterprises Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 Wheat Soybeans Maize Peanuts JR Cotton Wheat 0.1914 −0.0541 0.0545 0.0578 0.0288 −0.0250 Soybeans −0.0541 0.3022 −0.0848 0.0112 −0.1265 −0.0159 Maize 0.0545 −0.0848 0.6495 −0.1891 −0.0755 −0.0541 Peanuts 0.0578 0.0112 −0.1891 0.4762 −0.0716 −0.0274 JR 0.0288 −0.1265 −0.0755 −0.0716 0.4747 0.0024 Cotton −0.0250 −0.0159 −0.0541 −0.0274 0.0024 0.2722 7.2. Experiment on the farm-level case We arbitrarily set the maximum plant enterprise number at v=3 and derive the results from the modified target MOTAD and M–V models, and then compare the results to those derived from the MBTE and CCMBTE models. Detailed results from the benchmark models can be informative when viewing the MBTE/CCMBTE results. All the results, including the experiments in discrete choice cases, are derived from an out-of-the sample forecasting. 7.3. Comparison of the modified target MOTAD and M–V models in the farm-level case We compare the results of both the modified target MOTAD and M–V models using the feasible sets of risk coefficients. Generally speaking, the solution number in the discrete space is smaller than the continuous space as many different risk coefficients will lead to the same discretised solution set. We report risk and return values obtained from the modified target MOTAD and M–V models averaged by the feasible risk coefficient for each year in Figure 18a–d. We find that both the return and the variance of the modified target MOTAD model are always higher than those obtained from the M–V model. For the plant enterprise selection strategies, the modified target MOTAD model shows that its convergent solutions choose soybeans, maize and peanuts regardless of the estimation windows. The model always tracks these three highest return generating plant enterprise combinations irrespective of the estimation windows. In contrast, the convergent M–V selection strategies are more sensitive to the estimation window. For example, the solution combination includes maize, peanuts, and japonica rice under the estimation windows from 1989–1998 to 2001–2010, but solves to wheat, maize and japonica rice under the years of 2002–2011, and to maize, japonica rice and cotton under 2003–2012. To find the shadow price within the application of the models, we relax the plant enterprise number constraint to an inequality form, i.e. eTx≤v ⁠. None of the plant enterprises gets selected when we use a small ϕ in the M–V model. The result implies that under such circumstance a farmer prefers to avoid the risk rather than attempt to obtain some return. On the contrary, the modified target MOTAD model always uses v land parcels no matter whether the plant enterprise number constraint is relaxed or not. Since the modified target MOTAD model’s objective is to maximise return subject to some randomly selected λ ⁠, this result is understandable. Fig. 18. View largeDownload slide Out-of-sample test for the M–V and modified target MOTAD model in the farm-level case: (a) and (b) return and variance of the modified target MOTAD model averaged on λ ⁠; (c) and (d) return and variance of the M–V model. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 18. View largeDownload slide Out-of-sample test for the M–V and modified target MOTAD model in the farm-level case: (a) and (b) return and variance of the modified target MOTAD model averaged on λ ⁠; (c) and (d) return and variance of the M–V model. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 7.4. Comparison of the MBTE and CCMBTE models in the farm-level case In this section, we compare the performance of the MBTE and CCMBTE models against two benchmark models. We do not consider the CRM method further as it generates nothing particularly of interest for either return or variance. The optimisation result of the CCMBTE model provides similar returns, variance and crop enterprise choice as the MBTE within a range of CC, namely when θ=0.01~1,000 ⁠. Table 8 shows the MBTE model results, in which we consider λ=1,2,3 ⁠, ϕ={11,12,…,16} ⁠, and restrict eTx≤3 ⁠. It shows that both the CCMBTE and MBTE models choose maize, peanuts and cotton as an optimal crop combination. These results reflect the advantage of using both the CCMBTE and MBTE models, since the return is (i) larger than the M–V solution and close to the modified target MOTAD return, while the variance is smaller than the modified target MOTAD variance or (ii) no worse than any of the benchmark results. It is also possible to obtain a higher return and smaller variance simultaneously by slightly adjusting the plant enterprise numbers in the CCMBTE result. Table 9 shows one example with λ=1 and ϕ=2 ⁠, in which we find the CCMBTE results provide four plant enterprises under an estimation window subset including 1989–1998, 1990–1999, 1998–2007, 1999–2008 and 2000–2009, 2001–2010. Additionally, results from the CCMBTE are always better than in the MBTE in both return and risk. The results from these two models within the other estimation windows are similar as their selection strategies are always identical. In summary, irrespective of estimation window chosen, the CCMBTE model, with just one more plant enterprise, performs no worse than the MBTE model in terms of return and variance. On the other hand, the MBTE model with plant enterprise selection constraint always undertakes a lower return with higher risk when selecting one less plant enterprise than the CCMBTE model does. The difference in the return and risk can be seen as the shadow price. One disadvantage of considering many plant enterprises is the increase in the transaction cost as more resources, including arable land, fertilisers and production technologies, are required. Table 8. Comparison between the MBTE and CCMBTE models, θ=0.01 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 Table 8. Comparison between the MBTE and CCMBTE models, θ=0.01 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 MOTAD M–V λ ϕ Return Var Return Var Return Var Wheat Soybeans Maize Peanuts JR Cotton 1,2,3 11,12,.,16 5.8325 1.20572 3.3226 0.428062 CCMBTE 4.8662 1.00852 0 0 1 1 1 0 MBTE 4.8662 1.00852 0 0 1 1 1 0 Table 9. Comparison between the MBTE and CCMBTE models, θ=100,000 CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C Note: W, S, M, P, J and C represent wheat, soybean, maize, peanuts, japonica rice and cotton, respectively. Bolded values indicate CCMBTE has advantage over MBTE. Table 9. Comparison between the MBTE and CCMBTE models, θ=100,000 CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C CCMBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.3524 0.541732 4.3524 0.541732 6.5799 2.03415 3.4678 0.413346 6.1938 1.81424 W, S, J, C W, S, J, C P, J, C P, J, C P, J, C MBTE λ ϕ 1999 2001 2002 2003 2004 1 2 return var return var return var return var return var 4.1462 0.68165 3.4535 0.804526 6.5799 2.03415 3.4678 0.413346 4.4548 0.757335 M, P, C P, J, C P, J, C P, J, C M, P, C CCMBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 6.6832 0.995552 6.3887 0.822958 P, J, C M, P, J P, J, C S, P, J, C S, P, J, C MBTE λ ϕ 2005 2006 2007 2008 2009 1 2 return var return var return var return var return var 4.7723 0.932865 2.9917 0.439933 4.1882 0.831906 5.8325 1.20572 5.0598 0.976893 P, J, C M, P, J P, J, C P, J, C P, J, C CCMBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 5.6559 0.694961 5.1311 0.586378 3.3971 0.559011 6.1336 1.61252 S, P, J, C S, M, P, C W, M, J M, J, C MBTE λ ϕ 2010 2011 2012 2013 1 2 return var return var return var return var 4.9834 0.843007 5.4067 1.01692 3.3971 0.559011 6.1336 1.61252 P, J, C M, P, J W, M, J M, J, C Note: W, S, M, P, J and C represent wheat, soybean, maize, peanuts, japonica rice and cotton, respectively. Bolded values indicate CCMBTE has advantage over MBTE. 7.5. Policy impact on the CCMBTE result in the farm-level case We would like to assess the effects of a subsidy on wheat on plant enterprise selection strategy, given that wheat is one of the three major cereal crops in China (Li et al., 2016)8. Based on the prevailing subsidy paid for wheat grown in the northern Chinese provinces, such as Shandong, Hubei and Henan, we use a direct government payment of 115.74 RMB per mu (Gale, 2013) in calculation. We also refer to this paper for production cost and per acre yield, which are 364 RMB and 392.4 kg per mu, respectively.9–11 The optimisation result shows that wheat still does not appear in the optimal solution as it did not without considering the subsidy impact. In some instances, within the estimation window of 1998–2007, the model chooses wheat. As we compare results from other estimation windows, e.g. 1991–2000, 1997–2006 and 2001–2010, etc., we find the CCMBTE model chooses wheat only when either one of the two benchmark models choose it. This illustrates that some agriculture policy may have little impact on our optimal result, if it has no significant influence on return. It also demonstrates the robustness of our model against some changes within the data sample. 7.6. Transaction cost in the plant enterprise rotation We would like to compare the transaction cost obtained from the modified target MOTAD, M–V and MBTE models with eTx≤(=)v and the crop rotation scenario to the results obtained from the CCMBTE model. First, we assume that the land parcel for growing a specific plant enterprise has the same area through different decision periods.12 For instance, the rotation pattern from 2011 to 2012 in one of the M–V results is [maize, peanuts, japonica rice]→[wheat, soybeans, japonica rice]. The transaction cost is 2, as only two plant enterprises differ from the preceding year’s plant enterprise portfolio. If the rotation pattern from 2000 to 2001 in one of the CCMBTE results is [wheat, soybeans, japonica rice, cotton]→[maize, peanuts, japonica rice], the transaction cost is 5 as five plant enterprises in total differ between these 2 years. In Figure 19, we summarise the underlying results for all five models used in this paper. The number is averaged by the risk coefficients λ and ϕ (or λ−ϕ pair). It shows that although the CCMBTE model has a relatively higher transaction cost than the others, the value is much smaller than the M–V result during some specific year periods. Also, this value is very close to the MBTE model. Fig. 19. View largeDownload slide Transaction cost for the modified target MOTAD, M–V, MBTE and CCMBTE models based on the 1-year rolling out-of-sample forecasting. Numbers inside parentheses denote an average value for each method. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. Fig. 19. View largeDownload slide Transaction cost for the modified target MOTAD, M–V, MBTE and CCMBTE models based on the 1-year rolling out-of-sample forecasting. Numbers inside parentheses denote an average value for each method. This figure is available in black and white in print and in colour at European Review of Agricultural Economics online. 8. Conclusions We applied an MBTE model to the plant enterprise selection problem with multiple objectives. The portfolio solution in the MBTE model was found to track the highest return benchmark portfolio of the benchmark set while considering other benchmark portfolio characteristics such as risk control. Given that the MBTE is very sensitive to the selection of the benchmark, we used return maximisation and risk minimisation objectives in the MBTE framework. The constant rebalanced method, which is found to perform well in financial investment returns, was included as a reference benchmark portfolio. We modified the target MOTAD to follow the portfolio analysis. This modification, along with its experiment result, is a byproduct of this paper. We believe this modified target MOTAD model can also be applied in an asset investment problem. Considering the trade-off between the diversified and single-plant-enterprise strategies, we further added a convex CC onto the MBTE objective and adjusted the sparsity coefficient to limit the plant enterprise number. Chinese national data with 10 plant enterprises were used as a study case. This was followed by an extended case of provincial datasets with six plant enterprises to test the performance of alternative models. During the in-sample numerical experiment over a 25-year estimation window with the nation-wide data, we compared the annual return and the annual variance averaged on λ and ϕ of the benchmark models as modified target MOTAD and M–V, respectively. The result showed that neither of these models can dominate the other in all aspects (i.e. return and risk). In the MBTE framework, we chose a range of ϕ against each specific λ to generate benchmark solutions. Tracking these solutions in the in-sample case, the MBTE model produced a higher return level on average than that of the M–V and CRM models and a lower variance than those of the modified target MOTAD and M–V. For the plant enterprise selections in the nation-wide case, the modified target MOTAD model provided high returns but neglected diversification. In both in-sample and out-of-sample cases, it gave three or fewer plant enterprises, and these choices could not be further diversified by choosing any different risk coefficients. The M–V model, on the contrary, had more plant enterprises in a choice set within its low-risk portfolios but did not perform as well as the modified target MOTAD model regarding return. The MBTE model, however, generated even more diversified solutions and simultaneously searched for highest return. Additionally, it tracked all the benchmark portfolios well. The CCMBTE model under different cardinality coefficients always obtained a higher return than the MBTE model, but produced higher variance as well. For reducing the plant enterprise number, the CCMBTE model is found to perform better than the MBTE model if there was not a very rigid requirement on the computation precision. A province-wide dataset was used to assess the robustness of models under different settings. When considering only a 1-year rolling test, we found that the modified target MOTAD model provides more diversified plant enterprise selection than it did in nation-wide solutions. The estimation error of the M–V model on variance was larger than the nation-wide case. The modified target MOTAD return was not significantly larger than the M–V return in the province-wide solution. The MBTE model, however, based on its return-tracking characteristics, performed as well as the M–V model in return. Besides, it could control the variance even when the variance from the M–V model increases. The statistical significance between the two variances demonstrated the advantage of using MBTE’s to control variance. The CCMBTE performed as well as the MBTE in returns, variance and tracking error under an arbitrarily selected ϕ ⁠. It provided an optimal solution comprised of maize, peanuts, apples and sometimes cotton. In order to understand plant enterprise selection, rotation and policy effects on a farm level, we generated a new dataset using the same mean but reduced covariance of the original data. By additionally considering the plant enterprise number constraint, we transformed the decision variables from continuous to binary in all models. The numerical experiments compare the performance between the discretised MBTE and CCMBTE models and discretised benchmark models. After imposing the CC on the plant enterprise number, the CCMBTE model in the farm-level case performs no worse than the MBTE model in terms of return and variance within all estimation windows. Further, the MBTE and CCMBTE models in the discrete space perform no worse than the benchmark models. In order to simulate the impact of the direct subsidy policy for grain on the optimisation result, the return vector of wheat is modified. The result shows that the increase in return in wheat does not substantially change the selection strategies of the CCMBTE model. The normalised transaction cost analysis shows the CCMBTE results are no worse than those of the MBTE and two benchmark models. To summarise, the CCMBTE model tracks the MBTE model by adding a constraint on the number of plant enterprises to avoid over diversification. Therefore, CCMBTE provides a better modelling approach when a bi-criteria objective function is being considered along with a restriction on the number of plant enterprise choices. Within an agricultural production system, a decision-maker has to decide what plant enterprises to grow considering various policies, bio-geophysical constraints and return and risk factors. For the macro-policy makers, our method would be constructive in creating strategic policies concerning the agricultural production over some particular geographical areas or even the whole country. For the micro-level decision makers, it is valuable to provide entrepreneurs/farmers with some suggestions on their farm-level plant enterprise selection so that they can maximise return, minimise risk and reduce the transaction cost associated with growing too many plant enterprises. In future studies, it would be possible to introduce other non-linear benchmark models. However, it should also be noted that a non-linear benchmark model requires more computing resources. The CCMBTE model developed in our research can be an efficient method to meet the diversification requirement set by the EU’s common agricultural policy (CAP) while maximising the return and minimising the risk in plant enterprise selection. We can optimise our result by tracking the optimal results from the models with CAP constraints (as stated by Sckokai and Moro, 2006, 2009), the models under the consideration of subsidy scheme impact (as stated by Lien and Hardaker, 2001) or some calibration method like positive mathematical programming (as stated by Petsakos and Rozakis, 2015) as long as we have the required data. Acknowledgment The authors would like to sincerely thank Teo Chung Piaw, Zhichao Zheng, and Yunchao Xu for many stimulating discussions. Wen's time in this paper was supported by the key project of the National Science Foundation of China: Research of Food Risk Identification and Early Warning In the Production and Supply Process (grant No: 71633002). Paudel's time in this paper was supported by United States Department of Agriculture Hatch/Multistate project number LAB 94358. Ma was an assistant professor at South China Agricultural University when this paper was completed. Footnotes 1 Numerical experiment indicates both optimisation and simulation. 2 Shapley value was initially developed for efficiently allocating the risk among different assets. See Aumann and Shapley (1974). 3 Note that introducing price variabilities into the quadratic form will generate a polynomial term of degree four (non-convex). Solving such a model would require a prohibitive amount of computational resources. 4 Here, we should note that using the average crop revenue per unit of land may result in loss of some information about plant enterprise rotation; e.g. yield of apples in land previously cultivated by oilseeds will give the same return and variability as in land cultivated with apples, etc. Barkley et al. (2010) state that farmers often select crop varieties according to the published average yields. Although it is possible to incorporate additional constraints (such as for rotation) in this model, a simplified parameter structure can facilitate the analysis better, especially when the model is already complex. 5 These values are calculated based on the information from more than 60,000 households collected in the Compilation of Cost and Income Data of Agricultural Products in China. 6 Note that these solutions are fairly straight forward in a linear programming problem since we can choose those favourable plant enterprises by estimating the information on the existing data. However, as we expand our research to non-linear MBTE and CCMBTE models, the results cannot be easily derived. 7 Considering a new standard deviation is used in the regeneration process, we assume the new data sample is independent to the original set. 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European Review of Agricultural Economics – Oxford University Press

Published: Dec 1, 2018

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