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Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time

Mean-field formulation for mean-variance asset-liability management with cash flow under an... 1IntroductionWith an explosive development of the economy in the recent 50 years, it is becoming more common that lots of private assets have been invested in the financial market. After a series of financial crises, the significance of handling private assets has been attached considerably. Mean-variance formulation is a famous tool that aims at balancing the risk and return of the investment. Owing to the seminal work of Markowitz [1], the mean-variance model has provided a fundamental basis for designing the optimal strategy balancing the contradiction between return and risk. Hundreds of applications and extensions have been developed over the past decade. For instance, Merton [2] derived the analytical expression of the mean-variance efficient frontier in a single-period setting. Li and Ng [3] developed the mean-variance model from the single period to the dynamic discrete-time version and derived the analytical solution by using the embedding method to overcome the difficulty of non-separability. Zhou and Li [4] used the same technique and further introduced the stochastic linear quadratic control as a general framework to solve the continuous-time mean-variance portfolio selection problem. Li et al. [5] developed it with the no-shorting constraint. Moreover, some recent approaches [6,7] based on enhanced index tracking are employed to deal with portfolio optimization.There is no doubt that the embedding method is indeed a classic way to solve the problems with the nonseparable property. We also need to admit that this method is liable to lead to complicated calculation and inefficiency during the derivation of the optimal portfolio selection if the problem has other constraints, such as uncertain exit time, asset-liability management, and serial correlated returns or risk control over bankruptcy. Typically, we will prefer another method called mean-field formulation to the embedding scheme, where a long list of notations should be established, and an auxiliary problem should be assumed.The mean-field formulation is a simple but powerful tool to derive the optimal strategy of a multi-period mean-variance portfolio selection problem. By using this method, which was first introduced by Cui et al. [8], we can resolve the mean-variance problem with many other additional constraints and derive the optimal strategy in a simpler and more direct manner. Yi et al. [9] used the mean-field method to study the mean-variance model under the uncertain exit time condition but did not consider the cash flow and liability. Cui et al. [10] extend it to the asset-liability management, but none of them considered the situation of random cash flow or investigated the closed-form of computational formulas for a series of coefficients.Yao et al. [11] studied the mean-variance model with a given level of expected terminal surplus. Li and Xie [12] studied the optimal investment with stochastic income under the uncertain exit time. They derived the analytical optimal strategy and explicit expression of the efficient frontier by using the Lagrange method and traditional dynamic programming with the additional conditions of endogenous liabilities if the investors exit the market randomly. Wu and Li [13] investigated the multi-period mean-variance model with different market states and stochastic cash flows. A reinforcement learning framework is employed to investigate the continuous-time mean-variance portfolio selection [14]. Ni et al. [15] derived equilibrium solutions of multi-period mean-variance and established a general theory to characterize the open-loop equilibrium control problem. However, all of the literature did not consider the correlation among cash flow, asset, and liability, which should be taken into account in the real world because the random cash flow would be affected by the return rate of companies. For example, the government will provide funding to companies in terms of their past performance. Furthermore, since the analytical solution of the mean-variance model contains the correlation coefficient, the optimal strategy will be changed due to different return rates among the asset, cash flow, and liability. Moreover, the uncorrelated case can be regarded as a special case of the correlated one, of which the correlation coefficient is zero. We extended the special case to the general case.In this paper, we employed the mean-field formulation [8,9,10] to study the general case of correlation in which the financial parameters are correlated at every period. On the basis of the aforementioned mean-field formulation, we have added some additional conditions such as random cash flow and liability to improve the accuracy of the investment strategy. During each time period, the cash flow and risky investment returns are random variables, while the risk-free investment return is deterministic. Furthermore, we have derived the analytical solutions of the mean-variance model which is lacked by using the embedding method [3,11, 12,13]. Employing the embedding method, the classical model mentioned earlier has certain limitations since they need to define a deterministic expectation of surplus, which is a single-objective optimization problem. Besides, the numerical solution needs some algorithms to compute the corresponding best auxiliary parameter or Lagrangian parameter, which will bring the inaccuracy and complexity in simulation. However, the mean-field formulation is more clear and powerful, which offers an analytical solution scheme in solving the nonseparable problems as the principle of optimality no longer applies. When both cash flow and mean-field formulation are presented in the same model, we shed light on the explicit solutions of the optimal portfolio under mean-variance criteria. In this paper, we are not only concerned about the return rate but also concerned with the volatility in the objective function in terms of a multi-objective optimization problem. We study the portfolio selection problem by adopting the mean field, and consider the cash flow, liability, etc. base on the mean variance model. Compared with the numerical solution, the analytical solution we derived in this paper is more efficient and applicable when the aforementioned additional conditions are added to our model.The rest of the paper is structured as follows. We construct a mean-variance portfolio selection problem with cash flow and define the meaning of some symbols in Section 2. In Section 3, the considered model is equivalently transferred into a linear quadratic optimal stochastic control problem in the mean-field type. Then, we identify the optimal portfolio strategy with closed-form expressions by adopting the dynamic programming approach in Section 4. Some numerical examples are provided in Section 5 to illustrate the accuracy and efficiency of the optimal strategy. Finally, the conclusion and future work are given in Section 6.2Multi period mean-variance portfolio selection modelWe assume the financial market has one liability, one risk-free asset, and nnkinds of risky assets within a time horizon TT. Let nt{n}_{t}represent the deterministic return of the risk-free asset, mt=[mt1,mt2,…,mtn]′{{\bf{m}}}_{{\bf{t}}}=\left[{m}_{t}^{1},{m}_{t}^{2},\ldots ,{m}_{t}^{n}]^{\prime} be the vector of nnkinds of risky investment return rate, and yt{y}_{t}be the rate of liability at period tt. In addition, the investor joins the financial market at the beginning of time period 0 and is proposed to quit the investment at time TT. Let w0{w}_{0}denote the wealth at the beginning, while l0{l}_{0}denotes the initial liability. Every investor can reallocate his/her portfolio selection to maximize the expected return as well as minimize the risk from the start of every time period between 0 and TT.In different time periods tt, the random variable yt{y}_{t}and the random vector mt=[mt1,…,mtn]′{{\bf{m}}}_{{\bf{t}}}=\left[{m}_{t}^{1},\ldots ,{m}_{t}^{n}]^{\prime} are assumed to be statistically independent and are defined from the probability space (Ω,ℱ,P)\left(\Omega ,{\mathcal{ {\mathcal F} }},P). The first two moments are recognized as the only information about yt{y}_{t}and mt{{\bf{m}}}_{{\bf{t}}}. We further define that the covariance matrix is positive definite, i.e., Covmtyt=Emtytmt′yt−EmtytEmt′yt≻0.{\rm{Cov}}\left(\begin{array}{c}{{\bf{m}}}_{{\bf{t}}}\\ {y}_{t}\end{array}\right)={\mathbb{E}}\left[\left(\begin{array}{c}{{\bf{m}}}_{{\bf{t}}}\\ {y}_{t}\end{array}\right)<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\left(<mml:mpadded>\begin{array}{cc}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} & {y}_{t}\end{array}</mml:mpadded>\right)</mml:mpadded>\right]-{\mathbb{E}}\left[\left(\begin{array}{c}{{\bf{m}}}_{{\bf{t}}}\\ {y}_{t}\end{array}\right)\right]{\mathbb{E}}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\left[<mml:mpadded>\left(<mml:mpadded>\begin{array}{cc}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} & {y}_{t}\end{array}</mml:mpadded>\right)</mml:mpadded>\right]</mml:mpadded>\hspace{0.33em}\succ \hspace{0.33em}0.Then, for t=0,1,…,T−1t=0,1,\ldots ,T-1, we have nt2ntE[mt′]ntE[yt]ntE[mt]E[mtmt′]E[mtyt]ntE[yt]E[ytmt′]E[yt2]≻0.\left(\begin{array}{ccc}{n}_{t}^{2}& {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {n}_{t}{\mathbb{E}}[{y}_{t}]\\ {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{y}_{t}]\\ {n}_{t}{\mathbb{E}}[{y}_{t}]& {\mathbb{E}}[{y}_{t}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {\mathbb{E}}[{y}_{t}^{2}]\end{array}\right)\hspace{0.33em}\succ \hspace{0.33em}0.Let Dt=(Dt1,…,Dtn)′=(mt1−nt,…,mtn−nt)′{D}_{t}=\left({D}_{t}^{1},\ldots ,{D}_{t}^{n})^{\prime} =\left({m}_{t}^{1}-{n}_{t},\ldots ,{m}_{t}^{n}-{n}_{t})^{\prime} represent the vector of the risky return rate minus the risk-free return rate. According to the aforementioned assumptions, we get nt2ntE[Dt′]ntE[yt]ntE[Dt]E[DtDt′]E[Dtyt]ntE[yt]E[ytDt′]E[yt2]=10′0−1I000′1nt2ntE[mt′]ntE[yt]ntE[mt]E[mtmt″]E[mtyt]ntE[yt]E[ytmt′]E[yt2]1−1′00I000′1≻0,\left(\begin{array}{ccc}{n}_{t}^{2}& {n}_{t}{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]& {n}_{t}{\mathbb{E}}[{y}_{t}]\\ {n}_{t}{\mathbb{E}}\left[{D}_{t}]& {\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]& {\mathbb{E}}\left[{D}_{t}{y}_{t}]\\ {n}_{t}{\mathbb{E}}[{y}_{t}]& {\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]& {\mathbb{E}}[{y}_{t}^{2}]\end{array}\right)=\left(\begin{array}{ccc}1& {\bf{0}}^{\prime} & 0\\ -{\bf{1}}& I& {\bf{0}}\\ 0& {\bf{0}}^{\prime} & 1\end{array}\right)\left(\begin{array}{ccc}{n}_{t}^{2}& {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {n}_{t}{\mathbb{E}}[{y}_{t}]\\ {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{{\bf{m}}}_{{\bf{t}}}^{^{\prime\prime} }]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{y}_{t}]\\ {n}_{t}{\mathbb{E}}[{y}_{t}]& {\mathbb{E}}[{y}_{t}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {\mathbb{E}}[{y}_{t}^{2}]\end{array}\right)\left(\begin{array}{ccc}1& -{\bf{1}}^{\prime} & 0\\ {\bf{0}}& I& {\bf{0}}\\ 0& {\bf{0}}^{\prime} & 1\end{array}\right)\hspace{0.33em}\succ \hspace{0.33em}0,where IIdenotes the n×nn\times nidentity matrix, and 0{\bf{0}}and 1{\bf{1}}denote the nn-dimensional all-zero and all-one vectors respectively, which signify that E[DtDt′]≻0,nt2(1−E[Dt′]E−1[DtDt′]E[Dt])>0,E[yt2]−E[ytDt′]E−1[DtDt′]E[Dtyt]>0.\begin{array}{l}{\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]\hspace{0.33em}\succ \hspace{0.33em}0,\hspace{1.0em}{n}_{t}^{2}\left(1-{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}])\gt 0,\\ {\mathbb{E}}[{y}_{t}^{2}]-{\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}{y}_{t}]\gt 0.\end{array}Therefore, 0<E[Dt′]E−1[DtDt′]E[Dt]<10\lt {\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}]\lt 1. To express the equation more concisely, we define the following notations: (1)Zt≜E[Dt′]E−1[DtDt′]E[Dt],Z˜t≜E[ctDt′]E−1[DtDt′]E[ctDt],Z¯t≜E[ytDt′]E−1[DtDt′]E[Dt],Z^t≜E[ctDt′]E−1[DtDt′]E[Dt],Z˘t≜E[ytDt′]E−1[DtDt′]E[Dtyt],Zt¨≜E[ctDt′]E−1[DtDt′]E[Dtyt].\begin{array}{ll}{Z}_{t}\triangleq {\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],& {\widetilde{Z}}_{t}\triangleq {\mathbb{E}}\left[{c}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{c}_{t}{D}_{t}],\\ {\overline{Z}}_{t}\triangleq {\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],& {\widehat{Z}}_{t}\triangleq {\mathbb{E}}\left[{c}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],\\ {\breve{Z}}_{t}\triangleq {\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}{y}_{t}],& \ddot{{Z}_{t}}\triangleq {\mathbb{E}}\left[{c}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}{y}_{t}].\end{array}At the beginning of every period tt, investors’ wealth and liabilities are denoted by wt{w}_{t}and Qt{Q}_{t}, respectively. Therefore, the surplus is denoted by wt−Qt{w}_{t}-{Q}_{t}. If vti{v}_{t}^{i}is the money invested in the iith risky investment for i=1,2,…,ni=1,2,\ldots ,nat period tt, then wt−∑i=1nvti{w}_{t}-\mathop{\sum }\limits_{i=1}^{n}{v}_{t}^{i}is the money put into the risk-free investment. In this paper, we suppose the liability is exogenous. In other words, the investor’s strategies cannot affect the liability because of its uncontrollability. Let ℱt=σ(D0,D1,…,Dt−1,c0,c1,…,ct−1,y0,y1,…,yt−1){{\mathcal{ {\mathcal F} }}}_{t}=\sigma \left({D}_{0},{D}_{1},\ldots ,{D}_{t-1},{c}_{0},{c}_{1},\ldots ,{c}_{t-1},{y}_{0},{y}_{1},\ldots ,{y}_{t-1})represent all the information at the initial moment of ttperiod for t=1,2,…,T−1t=1,2,\ldots ,T-1, and ℱ0{{\mathcal{ {\mathcal F} }}}_{0}represent the unimportant σ\sigma -algebra over Ω\Omega . Thus, E[⋅∣ℱ0]{\mathbb{E}}\left[\cdot | {{\mathcal{ {\mathcal F} }}}_{0}]is equal to the unconditional expectation E[⋅]{\mathbb{E}}\left[\cdot ]. In this paper, all allowable portfolio selection is limited to be ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted Markov controls, i.e., vt=(vt1,vt2,…,vtn)′∈ℱt{v}_{t}=\left({v}_{t}^{1},{v}_{t}^{2},\ldots ,{v}_{t}^{n})^{\prime} \in {{\mathcal{ {\mathcal F} }}}_{t}. Therefore, Dt{D}_{t}and vt{v}_{t}are independent and ℱt=σ(wt,Qt){{\mathcal{ {\mathcal F} }}}_{t}=\sigma \left({w}_{t},{Q}_{t}).The investor plans to optimize the portfolio selection during the whole time period. However, the investment might be forced to be changed or abandoned at an uncertain time τ\tau before TTbecause of some accidents or unexpected events such as sudden resignation, serious illness, and colossal consumption. The probability mass function of the exogenous random variable κ\kappa is p˜t=Pr{κ=t}{\tilde{p}}_{t}=\hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{\kappa =t\right\}. Thus, the investor will quit the financial market eventually at time T∧κ=min{T,κ}T\wedge \kappa =\min \left\{T,\kappa \right\}. We have pt≜Pr{T∧κ=t}=p˜t,t=1,2,…,T−1,1−∑j=1T−1p˜j,t=T.{p}_{t}\triangleq \hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{T\wedge \kappa =t\right\}=\left\{\begin{array}{ll}{\tilde{p}}_{t},& t=1,2,\ldots ,T-1,\\ 1-\mathop{\displaystyle \sum }\limits_{j=1}^{T-1}{\tilde{p}}_{j},& t=T.\end{array}\right.The main investigation of this model is to find the optimal portfolio selection, vt∗=[(vt1)∗,(vt2)∗,…,(vtn)∗]′{v}_{t}^{\ast }=\left[{\left({v}_{t}^{1})}^{\ast },{\left({v}_{t}^{2})}^{\ast },\ldots ,{\left({v}_{t}^{n})}^{\ast }]^{\prime} , t=0,1,…,T−1t=0,1,\ldots ,T-1, which can be equivalent to optimizing the following optimal stochastic control problem, (2)minVar(κ)(wT∧κ−QT∧κ)−λE(κ)[wT∧κ−QT∧κ],s.t.wt+1=∑i=1nmtivti+wt−∑i=1nvtint+ct=ntwt+Dt′vt+ct,Qt+1=ytQt,fort=0,1,…,T−1,\left\{\begin{array}{ll}\min & {\text{Var}}^{\left(\kappa )}\left({w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa })-\lambda {{\mathbb{E}}}^{\left(\kappa )}\left[{w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }],\\ \hspace{0.1em}\text{s.t.}\hspace{0.1em}& \hspace{0.1em}{w}_{t+1}\hspace{0.07em}=\mathop{\displaystyle \sum }\limits_{i=1}^{n}{m}_{t}^{i}{v}_{t}^{i}+\left({w}_{t}-\mathop{\displaystyle \sum }\limits_{i=1}^{n}{v}_{t}^{i}\right){n}_{t}+{c}_{t}\\ & \hspace{1.0em}\hspace{1.1em}={n}_{t}{w}_{t}+{D}_{t}^{^{\prime} }{v}_{t}+{c}_{t},\\ & \hspace{0.1em}{Q}_{t+1}={y}_{t}{Q}_{t},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t=0,1,\ldots ,T-1,\end{array}\right.where λ>0\lambda \gt 0represents the risk aversion, and E(κ)[wT∧κ−QT∧κ]≜∑t=1TE[wT∧κ−QT∧κ∣T∧κ=t]Pr{T∧κ=t}=∑t=1TE[wt−Qt]pt,V ar(κ)(wT∧κ−QT∧κ)≜∑t=1TVar(wT∧κ−QT∧κ∣T∧κ=t)Pr{T∧κ=t}=∑t=1TVar(wt−Qt)pt.\begin{array}{rcl}{{\mathbb{E}}}^{\left(\kappa )}\left[{w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }]& \triangleq & \mathop{\displaystyle \sum }\limits_{t=1}^{T}{\mathbb{E}}\left[{w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }| T\wedge \kappa =t]\hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{T\wedge \kappa =t\right\}=\mathop{\displaystyle \sum }\limits_{t=1}^{T}{\mathbb{E}}\left[{w}_{t}-{Q}_{t}]{p}_{t},\\ {{\rm{V\; ar}}}^{\left(\kappa )}\left({w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa })& \triangleq & \mathop{\displaystyle \sum }\limits_{t=1}^{T}\hspace{0.1em}\text{Var}\hspace{0.1em}\left({w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }| T\wedge \kappa =t)\hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{T\wedge \kappa =t\right\}=\mathop{\displaystyle \sum }\limits_{t=1}^{T}{\rm{Var}}\left({w}_{t}-{Q}_{t}){p}_{t}.\end{array}Then, we can rewrite the aforementioned model as follows: (3)min∑t=1Tpt{Var(wt−Qt)−λE[wt−Qt]},s.t.wt+1=ntwt+Dt′vt+ct,Qt+1=ytQt,fort=0,1,…,T−1.\left\{\begin{array}{l}\min \hspace{0.33em}\mathop{\displaystyle \sum }\limits_{t=1}^{T}{p}_{t}\left\{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({w}_{t}-{Q}_{t})-\lambda {\mathbb{E}}\left[{w}_{t}-{Q}_{t}]\right\},\\ {\rm{s.t.}}\hspace{1em}{w}_{t+1}={n}_{t}{w}_{t}+{D}_{t}^{^{\prime} }{v}_{t}+{c}_{t},\\ \hspace{2.25em}{Q}_{t+1}={y}_{t}{Q}_{t},\hspace{1em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t=0,1,\ldots ,T-1.\end{array}\right.Since the smoothing property is no longer valid on the variance term, we cannot decompose the nonseparable problem into a stage wise backward recursion formulation, which can be tackled with traditional dynamic programming method. We solve it by employing the mean-field method.3Mean-field formulationFirst, we construct the mean-field type of model (3). According to the independence between Dt{D}_{t}and vt{v}_{t}, yt{y}_{t}and Qt{Q}_{t}, the dynamic equations of the expectation of the wealth and liability can be represented as follows: (4)E[wt+1]=ntE[wt]+E[Dt′]E[vt]+E[ct],E[w0]=w0,E[Qt+1]=E[yt]E[Qt],E[Q0]=Q0,\left\{\begin{array}{l}{\mathbb{E}}\left[{w}_{t+1}]={n}_{t}{\mathbb{E}}\left[{w}_{t}]+{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{v}_{t}]+{\mathbb{E}}\left[{c}_{t}],\\ {\mathbb{E}}\left[{w}_{0}]={w}_{0},\\ {\mathbb{E}}\left[{Q}_{t+1}]={\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{Q}_{t}],\\ {\mathbb{E}}\left[{Q}_{0}]={Q}_{0},\end{array}\right.with t=0,1,…,T−1t=0,1,\ldots ,T-1.Combining the dynamic equations in (3) and (4), we have (5)wt+1−E[wt+1]=nt(wt−E[wt])+Dt′vt−E[Dt′]E[vt]+(ct−E[ct])=nt(wt−E[wt])+Dt′(vt−E[vt])+(Dt′−E[Dt′])E[vt]+(ct−E[ct]),w0−E[w0]=0,Qt+1−E[Qt+1]=yt(Qt−E[Qt])+(yt−E[yt])E[Qt],Q0−E[Q0]=0.\left\{\begin{array}{rcl}{w}_{t+1}-{\mathbb{E}}\left[{w}_{t+1}]& =& {n}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}])+{D}_{t}^{^{\prime} }{v}_{t}-{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{v}_{t}]+\left({c}_{t}-{\mathbb{E}}\left[{c}_{t}])\\ & =& {n}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}])+{D}_{t}^{^{\prime} }\left({v}_{t}-{\mathbb{E}}\left[{v}_{t}])+\left({D}_{t}^{^{\prime} }-{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]){\mathbb{E}}\left[{v}_{t}]+\left({c}_{t}-{\mathbb{E}}\left[{c}_{t}]),\\ {w}_{0}-{\mathbb{E}}\left[{w}_{0}]& =& 0,\\ {Q}_{t+1}-{\mathbb{E}}\left[{Q}_{t+1}]& =& {y}_{t}\left({Q}_{t}-{\mathbb{E}}\left[{Q}_{t}])+({y}_{t}-{\mathbb{E}}[{y}_{t}]){\mathbb{E}}\left[{Q}_{t}],\\ {Q}_{0}-{\mathbb{E}}\left[{Q}_{0}]& =& 0.\end{array}\right.Therefore, we can equivalently reformulate problem (3) into a linear quadratic optimal problem in the mean-field type.(6)min∑t=1Tpt{E[(wT−QT−E[wT−QT])2]−λE[wT−QT]},s.t.{E[wt],E[Qt],E[vt]}satisfies dynamic equation (4),{wt−E[wt],Qt−E[Qt],vt−E[vt]}satisfies dynamic equation (5),fort=0,1,…,T−1.\left\{\begin{array}{rl}\min & \mathop{\displaystyle \sum }\limits_{t=1}^{T}{p}_{t}\left\{{\mathbb{E}}\left[{\left({w}_{T}-{Q}_{T}-{\mathbb{E}}\left[{w}_{T}-{Q}_{T}])}^{2}]-\lambda {\mathbb{E}}\left[{w}_{T}-{Q}_{T}]\right\},\\ \hspace{0.1em}\text{s.t.}\hspace{0.1em}& \left\{{\mathbb{E}}\left[{w}_{t}],{\mathbb{E}}\left[{Q}_{t}],{\mathbb{E}}\left[{v}_{t}]\right\}\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (4)}\hspace{0.1em},\\ & \left\{{w}_{t}-{\mathbb{E}}\left[{w}_{t}],{Q}_{t}-{\mathbb{E}}\left[{Q}_{t}],{v}_{t}-{\mathbb{E}}\left[{v}_{t}]\right\}\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (5),}\hspace{0.1em}\\ & \hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t=0,\hspace{0.08em}1,\ldots ,T-1.\end{array}\right.Thus, we are able to solve it by the dynamic programming method since it is separable.4The optimal strategyWith the notations given in (1), the seven parameters of the sequence {βt}\left\{{\beta }_{t}\right\}, {ηt}\left\{{\eta }_{t}\right\}, {ξt}\left\{{\xi }_{t}\right\}, {ζt}\left\{{\zeta }_{t}\right\}, {ψt}\left\{{\psi }_{t}\right\}, {δt}\left\{{\delta }_{t}\right\}, and {Δt}\left\{{\Delta }_{t}\right\}are deterministic by the following backward recursions βt=βt+1(st)2(1−Zt)+pt,ηt=ηt+1(stE[yt]−stZ¯t)+pt,ξt=ξt+1E[yt2]−ηt+12βt+1−1Bt′+pt,ζt=ζt+1st+pt,ψt=ψt+1E[yt]−2ληt+1(E[ytct]−E[yt]E[ct])+2ηt+1λE[ct]+λζt+12βt+1(E[yt]Zt−Z¯t)+Z^tZ¯t−Z^tE[yt]1−Zt+Zt¨+pt,δt=ξt+1(E[yt2]−E[yt]2)+δt+1(E[yt])2−ηt+12βt+1Z˘t−E[yt]2+(Z¯t−E[yt])21−Zt,Δt=Δt+1+βt+1E[(ct)2]−βt+1E[ct]2−λζt+1E[ct]−βt+1E[ct]+λζt+12βt+1−Z^t2Zt1−Zt−2E[ct]+λζt+12βt+1−Z^tZ^t+Z˜t−Z^t2,\hspace{-45em}\begin{array}{rcl}{\beta }_{t}& =& {\beta }_{t+1}{\left({s}_{t})}^{2}\left(1-{Z}_{t})+{p}_{t},\\ {\eta }_{t}& =& {\eta }_{t+1}\left({s}_{t}{\mathbb{E}}[{y}_{t}]-{s}_{t}{\overline{Z}}_{t})+{p}_{t},\\ {\xi }_{t}& =& {\xi }_{t+1}{\mathbb{E}}[{y}_{t}^{2}]-{\eta }_{t+1}^{2}{\beta }_{t+1}^{-1}{B}_{t}^{^{\prime} }+{p}_{t},\\ {\zeta }_{t}& =& {\zeta }_{t+1}{s}_{t}+{p}_{t},\\ {\psi }_{t}& =& {\psi }_{t+1}{\mathbb{E}}[{y}_{t}]-\frac{2}{\lambda }{\eta }_{t+1}\left({\mathbb{E}}[{y}_{t}{c}_{t}]-{\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{c}_{t}])+\frac{2{\eta }_{t+1}}{\lambda }\left(\frac{\left({\mathbb{E}}\left[{c}_{t}]+\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}\right)\left({\mathbb{E}}[{y}_{t}]{Z}_{t}-{\overline{Z}}_{t})+{\widehat{Z}}_{t}{\overline{Z}}_{t}-{\widehat{Z}}_{t}{\mathbb{E}}[{y}_{t}]}{1-{Z}_{t}}+\ddot{{Z}_{t}}\right)+{p}_{t},\\ {\delta }_{t}& =& {\xi }_{t+1}\left({\mathbb{E}}[{y}_{t}^{2}]-{\mathbb{E}}{[{y}_{t}]}^{2})+{\delta }_{t+1}{\left({\mathbb{E}}[{y}_{t}])}^{2}-\frac{{\eta }_{t+1}^{2}}{{\beta }_{t+1}}\left({\breve{Z}}_{t}-{\mathbb{E}}{[{y}_{t}]}^{2}+\frac{{\left({\overline{Z}}_{t}-{\mathbb{E}}[{y}_{t}])}^{2}}{1-{Z}_{t}}\right),\\ {\Delta }_{t}& =& {\Delta }_{t+1}+{\beta }_{t+1}{\mathbb{E}}\left[{\left({c}_{t})}^{2}]-{\beta }_{t+1}{\mathbb{E}}{\left[{c}_{t}]}^{2}-\lambda {\zeta }_{t+1}{\mathbb{E}}\left[{c}_{t}]\\ & & -{\beta }_{t+1}\left({\left({\mathbb{E}}\left[{c}_{t}]+\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}-{\widehat{Z}}_{t}\right)}^{2}\frac{{Z}_{t}}{1-{Z}_{t}}-2\left({\mathbb{E}}\left[{c}_{t}]+\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}-{\widehat{Z}}_{t}\right){\widehat{Z}}_{t}+{\widetilde{Z}}_{t}-{\widehat{Z}}_{t}^{2}\right),\end{array}with boundary conditions defined as follows: βT=pT,ηT=pT,ξT=pT,ζT=pT,ψT=pT,δT=0,ΔT=0.{\beta }_{T}={p}_{T},\hspace{1em}{\eta }_{T}={p}_{T},\hspace{1em}{\xi }_{T}={p}_{T},\hspace{1em}{\zeta }_{T}={p}_{T},\hspace{1em}{\psi }_{T}={p}_{T},\hspace{1em}{\delta }_{T}=0,\hspace{1em}{\Delta }_{T}=0.The solution scheme adopted in this paper involves two steps. The first step is to construct the cost-to-go functional and derive the backward recursion. The second step is to prove that it still holds at each period according to mathematical induction. Thus, the optimal portfolio strategy can be obtained in the following theorem.Theorem 1Assume that the return rates among asset, liability, and cash flow are correlated. Thus, we have the optimal portfolio selection of problem (6) as follows: vt−E[vt]=−st(wt−E[wt])E−1[DtDt′]E[(Dt)]+ηt+1βt+1−1(Qt−E[Qt])E−1[DtDt′]E[ytDt],\begin{array}{rcl}{v}_{t}-{\mathbb{E}}\left[{v}_{t}]& =& -{s}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}]){{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[\left({D}_{t})]+{\eta }_{t+1}{\beta }_{t+1}^{-1}\left({Q}_{t}-{\mathbb{E}}\left[{Q}_{t}]){{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}[{y}_{t}{D}_{t}],\end{array}E[vt]=−(E[DtDt′]−E[Dt]E[Dt′])−1E[ctDt]−E[ct]E[Dt]−λζt+12βt+1E[Dt]−ηt+1βt+1(E[ytDt]−E[yt]E[Dt])E[Qt].\begin{array}{rcl}{\mathbb{E}}\left[{v}_{t}]& =& -{\left({\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]-{\mathbb{E}}\left[{D}_{t}]{\mathbb{E}}\left[{D}_{t}^{^{\prime} }])}^{-1}\left({\mathbb{E}}\left[{c}_{t}{D}_{t}]-{\mathbb{E}}\left[{c}_{t}]{\mathbb{E}}\left[{D}_{t}]-\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}{\mathbb{E}}\left[{D}_{t}]-\frac{{\eta }_{t+1}}{{\beta }_{t+1}}\left({\mathbb{E}}[{y}_{t}{D}_{t}]-{\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{D}_{t}]){\mathbb{E}}\left[{Q}_{t}]\right).\end{array}The expected value of optimal wealth can be derived as follows: E[wt]=w0∏k=0t−1sk+∑j=0t−1E[cj]+λζj+12βj+1−Z^j−E[yj]+Z¯jZj1−Zj−Z^j+Z¯j+E[cj]∏l=j+1t−1sl,{\mathbb{E}}\left[{w}_{t}]={w}_{0}\mathop{\prod }\limits_{k=0}^{t-1}{s}_{k}+\mathop{\sum }\limits_{j=0}^{t-1}\left(\left({\mathbb{E}}\left[{c}_{j}]+\frac{\lambda {\zeta }_{j+1}}{2{\beta }_{j+1}}-{\widehat{Z}}_{j}-{\mathbb{E}}[{y}_{j}]+{\overline{Z}}_{j}\right)\frac{{Z}_{j}}{1-{Z}_{j}}-{\widehat{Z}}_{j}+{\overline{Z}}_{j}+{\mathbb{E}}\left[{c}_{j}]\right)\mathop{\prod }\limits_{l=j+1}^{t-1}{s}_{l},for t=1,2,…,Tt=1,2,\ldots ,T. Here, ∏∅(⋅)=1\prod _{\varnothing }\left(\cdot )=1, ∑∅(⋅)=0\sum _{\varnothing }\left(\cdot )=0.If the additional condition of liability is not considered in our case, the original model (6) would be degenerated to the one mentioned by Yao et al. [11], which will be introduced in the following corollary.Remark 1Assume that an investor participates in the initial investment under uncertain exit time without liability. Thus, the degenerated problem is equivalently reformulated as the following mean-variance model.(7)min∑t=1Tpt{E[(wt−E[wt])2]−λE[wt]},s.t.E(vt−E[vt])=0,E[wt]satisfies dynamic equation (4),wt−E[wt]satisfies dynamic equation (5).\left\{\begin{array}{rl}\min & \mathop{\displaystyle \sum }\limits_{t=1}^{T}{p}_{t}\left\{{\mathbb{E}}\left[{\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}])}^{2}]-\lambda {\mathbb{E}}\left[{w}_{t}]\right\},\\ \hspace{0.1em}\text{s.t.}\hspace{0.1em}& {\mathbb{E}}\left({v}_{t}-{\mathbb{E}}\left[{v}_{t}])={\bf{0}},\\ & {\mathbb{E}}\left[{w}_{t}]\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (4)}\hspace{0.1em},\\ & {w}_{t}-{\mathbb{E}}\left[{w}_{t}]\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (5)}\hspace{0.1em}.\end{array}\right.The optimal strategies of problem 7 are represented as follows: vt∗−E[vt∗]=−nt(wt−E[wt])E−1[DtDt′]E[Dt],E[vt∗]=wζt+12βt+1⋅11−ZtE−1[DtDt′]E[Dt].\begin{array}{rcl}{v}_{t}^{\ast }-{\mathbb{E}}\left[{v}_{t}^{\ast }]& =& -{n}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}]){{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],\\ {\mathbb{E}}\left[{v}_{t}^{\ast }]& =& \frac{w{\zeta }_{t+1}}{2{\beta }_{t+1}}\cdot \frac{1}{1-{Z}_{t}}{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}].\end{array}Thus, we get the optimal expected level of wealth E[wt]=w0∏k=0t−1sk+w2∑j=0t−1ζj+1βj+1⋅Zj1−Zj∏ℓ=j+1t−1sℓ.{\mathbb{E}}\left[{w}_{t}]={w}_{0}\mathop{\prod }\limits_{k=0}^{t-1}{s}_{k}+\frac{w}{2}\mathop{\sum }\limits_{j=0}^{t-1}\frac{{\zeta }_{j+1}}{{\beta }_{j+1}}\cdot \frac{{Z}_{j}}{1-{Z}_{j}}\mathop{\prod }\limits_{\ell =j+1}^{t-1}{s}_{\ell }.The optimal strategy of the model in Corollary 1 can be obtained according to Theorem 1, which is consistent with the results derived by Yao et al. [11]. Therefore, the accuracy of the solution derived in this paper has been verified. In comparison, Zhu et al. [16] analyzed the Lagrangian problem via the embedding method and were unable to obtain an analytical form of the optimal objective value function. Thus, they invoked a prime-dual iterative algorithm to identify the optimal Lagrangian multiplier vector. Moreover, compared with the classical embedding method, which needs a Bellman equation and the Lagrangian multiplier, the mean-field formulation has been employed in this paper, which avoids the complicated computation. In the following section, a few numerical examples from real-world applications are given to demonstrate the efficiency of the obtained optimal strategy.5Numerical exampleAccording to the data given in the study by Elton et al. [17], we investigate a portfolio selection consisting of S&P 500 (SP), the index of emerging market (EM), and small stock (MS) of the U.S. market. Moreover, we consider uncertain exit time and cash flow in the model. Table 1 presents three different assets, a liability, and a random cash flow, and it also presents the expected values, variances, and the correlation coefficients among them. The annual risk free return rate is set as 5%5 \% (nt=1.05{n}_{t}=1.05). Here, we ignore the case of uncorrelation between Dt{D}_{t}and ct{c}_{t}, i.e., the return rates and cash flow are correlated.Table 1Data for assets and cash flowSPEMMSCashflowLiabilityExpected return14%14 \% 16%16 \% 17%17 \% 110%10 \% Standard deviation18.5%18.5 \% 30%30 \% 24%24 \% 20%20 \% 20%20 \% Correlation coefficientSP10.640.79ρ1{\rho }_{1}ρ^1{\widehat{\rho }}_{1}EM0.6410.75ρ2{\rho }_{2}ρ^2{\widehat{\rho }}_{2}MS0.790.751ρ3{\rho }_{3}ρ^3{\widehat{\rho }}_{3}Cashflowρ1{\rho }_{1}ρ2{\rho }_{2}ρ3{\rho }_{3}1ρ^4{\widehat{\rho }}_{4}Liabilityρ^1{\widehat{\rho }}_{1}ρ^2{\widehat{\rho }}_{2}ρ^3{\widehat{\rho }}_{3}ρ^4{\widehat{\rho }}_{4}1Thus, for every period tt, we have the following matrices: E[Dt]=0.090.110.12,Cov(Dt)=0.03420.03550.03510.03550.09000.05400.03510.05400.0576,E[DtDt′]=0.04230.04540.04590.04540.10210.06720.04590.06720.0720.{\mathbb{E}}\left[{D}_{t}]=\left(\begin{array}{c}0.09\\ 0.11\\ 0.12\\ \end{array}\right),\hspace{1.0em}{\rm{Cov}}\left({D}_{t})=\left(\begin{array}{ccc}0.0342& 0.0355& 0.0351\\ 0.0355& 0.0900& 0.0540\\ 0.0351& 0.0540& 0.0576\end{array}\right),\hspace{1.0em}{\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]=\left(\begin{array}{ccc}0.0423& 0.0454& 0.0459\\ 0.0454& 0.1021& 0.0672\\ 0.0459& 0.0672& 0.0720\end{array}\right).The correlation coefficient between cash flow and iith asset is defined as ρ=(ρ1,ρ2,ρ3)\rho =\left({\rho }_{1},{\rho }_{2},{\rho }_{3}), while the coefficient between liability and iith asset is defined as ρ^=(ρ1^,ρ2^,ρ3^)\widehat{\rho }=\left(\widehat{{\rho }_{1}},\widehat{{\rho }_{2}},\widehat{{\rho }_{3}}), according to the definition we have ρi=Cov(ct,Dti)Var(ct)Var(Dti){\rho }_{i}=\frac{\hspace{0.1em}\text{Cov}\hspace{0.1em}\left({c}_{t},{D}_{t}^{i})}{\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})}}and ρi^=Cov(yt,Dti)Var(yt)Var(Dti).\widehat{{\rho }_{i}}=\frac{\hspace{0.1em}\text{Cov}\hspace{0.1em}({y}_{t},{D}_{t}^{i})}{\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})}}.In addition, we define the correlation of the cash flow and liability ρ4^\widehat{{\rho }_{4}}as follows: ρ4^=Cov(ct,yt)Var(ct)Var(yt).\widehat{{\rho }_{4}}=\frac{\hspace{0.1em}\text{Cov}\hspace{0.1em}\left({c}_{t},{y}_{t})}{\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})}}.Then, we have E[ctDti]=E[ct]E[Dti]+ρiVar(ct)Var(Dti),{\mathbb{E}}\left[{c}_{t}{D}_{t}^{i}]={\mathbb{E}}\left[{c}_{t}]{\mathbb{E}}\left[{D}_{t}^{i}]+{\rho }_{i}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})},E[ytDti]=E[yt]E[Dti]+ρi^Var(yt)Var(Dti),{\mathbb{E}}[{y}_{t}{D}_{t}^{i}]={\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{D}_{t}^{i}]+\widehat{{\rho }_{i}}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})},E[ctyt]=E[ct]E[yt]+ρ4^Var(ct)Var(yt),{\mathbb{E}}\left[{c}_{t}{y}_{t}]={\mathbb{E}}\left[{c}_{t}]{\mathbb{E}}[{y}_{t}]+\widehat{{\rho }_{4}}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})},E[ct2]=E[ct]2+Var(ct),{\mathbb{E}}\left[{c}_{t}^{2}]={\mathbb{E}}{\left[{c}_{t}]}^{2}+\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t}),E[yt2]=E[yt]2+Var(yt).{\mathbb{E}}[{y}_{t}^{2}]={\mathbb{E}}{[{y}_{t}]}^{2}+\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t}).Assume that ρ=(ρ1,ρ2,ρ3)=(−0.3,0.5,0.2)\rho =\left({\rho }_{1},{\rho }_{2},{\rho }_{3})=\left(-0.3,0.5,0.2), ρ^=(ρ1^,ρ2^,ρ3^)=(−0.2,0.4,0.3)\widehat{\rho }=\left(\widehat{{\rho }_{1}},\widehat{{\rho }_{2}},\widehat{{\rho }_{3}})=\left(-0.2,0.4,0.3)and ρ4^=0.1\widehat{{\rho }_{4}}=0.1. Then, CovDtct=Cov(Dt)Cov(ct,Dt)Cov(ct,Dt′)Var(ct)=0.03420.03550.0351−0.00920.03550.09000.05400.03000.03510.05400.05760.0120−0.00920.03000.01200.0400≻0.{\rm{Cov}}\left(\left(\begin{array}{c}{D}_{t}\\ {c}_{t}\end{array}\right)\right)=\left(\begin{array}{cc}{\rm{Cov}}\left({D}_{t})& {\rm{Cov}}\left({c}_{t},{D}_{t})\\ {\rm{Cov}}\left({c}_{t},{D}_{t}^{^{\prime} })& {\rm{Var}}\left({c}_{t})\end{array}\right)=\left(\begin{array}{cccc}0.0342& 0.0355& 0.0351& -0.0092\\ 0.0355& 0.0900& 0.0540& 0.0300\\ 0.0351& 0.0540& 0.0576& 0.0120\\ -0.0092& 0.0300& 0.0120& 0.0400\end{array}\right)\hspace{0.33em}\succ \hspace{0.33em}0.Substituting the data in the equations, we have E[ctDt]=(0.0898,0.1510,0.1440)′{\mathbb{E}}\left[{c}_{t}{D}_{t}]=\left(0.0898,0.1510,0.1440)^{\prime} . Moreover, we define the following notations to make the solution more concise, T1=E−1[DtDt′]E[Dt]=1.0589−0.11961.1033,T2=E−1[DtDt′]E[ctDt]=−0.34900.44931.6365,T3=E−1[DtDt′]E[ytDt]=−1.04110.17540.8209.\begin{array}{l}{T}_{1}={{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}]=\left[\begin{array}{c}1.0589\\ -0.1196\\ 1.1033\end{array}\right],\\ {T}_{2}={{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{c}_{t}{D}_{t}]=\left[\begin{array}{c}-0.3490\\ 0.4493\\ 1.6365\end{array}\right],\\ {T}_{3}={{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}[{y}_{t}{D}_{t}]=\left[\begin{array}{c}-1.0411\\ 0.1754\\ 0.8209\end{array}\right].\end{array}Zt=0.2145,Z^t=0.2144,Z˜t=0.2507,\hspace{-18em}{Z}_{t}=0.2145,\hspace{1.0em}{\widehat{Z}}_{t}=0.2144,\hspace{1.0em}{\widetilde{Z}}_{t}=0.2507,Z¯t=0.0241,Z˘t=0.0218,Zt¨=0.0488.{\overline{Z}}_{t}=0.0241,\hspace{1.0em}{\breve{Z}}_{t}=0.0218,\hspace{1.0em}\ddot{{Z}_{t}}=0.0488.Example 1An example with the terminal exit timeThe probability mass function κ\kappa is defined as follows: (α1,α2,α3,α4,α5)=(0,0,0,0,1),\left({\alpha }_{1},{\alpha }_{2},{\alpha }_{3},{\alpha }_{4},{\alpha }_{5})=\left(0,0,0,0,1),for t=1,2,3,4,5t=1,2,3,4,5. By applying the result of Theorem 1, the optimal portfolio selection of this numerical example can be obtained as follows: v0∗=−1.02(w0−3.0477)T1+1.2053T2Q0,v1∗=−1.02(w1−3.2001)T1+1.1503T2Q1,v2∗=−1.02(w2−3.3601)T1+1.0979T2Q2,v3∗=−1.02(w3−3.5281)T1+1.0478T2Q3,v4∗=−1.02(w4−3.7045)T1+1.0000T2Q4.\begin{array}{l}{v}_{0}^{\ast }=-1.02\left({w}_{0}-3.0477){T}_{1}+1.2053{T}_{2}{Q}_{0},\\ {v}_{1}^{\ast }=-1.02\left({w}_{1}-3.2001){T}_{1}+1.1503{T}_{2}{Q}_{1},\\ {v}_{2}^{\ast }=-1.02\left({w}_{2}-3.3601){T}_{1}+1.0979{T}_{2}{Q}_{2},\\ {v}_{3}^{\ast }=-1.02\left({w}_{3}-3.5281){T}_{1}+1.0478{T}_{2}{Q}_{3},\\ {v}_{4}^{\ast }=-1.02\left({w}_{4}-3.7045){T}_{1}+1.0000{T}_{2}{Q}_{4}.\end{array}Under the certain exit time, we derive the final optimal surplus as follows, E(w5−Q5)=3.3897{\mathbb{E}}\left({w}_{5}-{Q}_{5})=3.3897and Var(w5−Q5)=0.6135\hspace{0.1em}\text{Var}\hspace{0.1em}\left({w}_{5}-{Q}_{5})=0.6135, respectively.Example 2An example without liability under uncertain exit timeConsider the example as corollary. Here, we ignore the information of liability, i.e., ignore the last line and last column of Table 1 and do not fix the terminal expectation but balance the variance and expectation by the trade-off parameter.Assume that an investor plans a five-period investment with an initial wealth w0=1{w}_{0}=1and that the trade-off parameter w=1w=1, but he may exit the market at any time tt(t=1,2,3,4,5t=1,2,3,4,5).To investigate the impact of uncertain exit time on the optimal policy and efficient frontier clearly, we choose four different probability mass functions at the exit time κ\kappa , α(i)=(α1(i),α2(i),α3(i),α4(i),α5(i)),(i=1,2,3,4){\alpha }^{\left(i)}=\left({\alpha }_{1}^{\left(i)},{\alpha }_{2}^{\left(i)},{\alpha }_{3}^{\left(i)},{\alpha }_{4}^{\left(i)},{\alpha }_{5}^{\left(i)}),\left(i=1,2,3,4), as follows: α(1)=(0.1,0.15,0.2,0.25,0.3),{\alpha }^{\left(1)}=\left(0.1,0.15,0.2,0.25,0.3),α(2)=(0,0.1,0.1,0.3,0.5),\hspace{-13.15em}{\alpha }^{\left(2)}=\left(0,0.1,0.1,0.3,0.5),α(3)=(0,0,0.1,0.2,0.7),\hspace{-13.2em}{\alpha }^{\left(3)}=\left(0,0,0.1,0.2,0.7),α(4)=(0,0,0,0,1),\hspace{-13.25em}{\alpha }^{\left(4)}=\left(0,0,0,0,1),where α(4){\alpha }^{\left(4)}represents that the investor must exit the market at the terminal time.Then, the optimal expected wealth level E[w](i)=(E[w1](i),E[w2](i),E[w3](i),E[w4](i),E[w5](i)),i=1,2,3,4,{\mathbb{E}}{\left[{\bf{w}}]}^{\left(i)}=\left({\mathbb{E}}{\left[{w}_{1}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{2}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{3}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{4}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{5}]}^{\left(i)}),\hspace{1em}i=1,2,3,4,which are given by E[w](1)=(1.2675,1.5210,1.7659,2.0055,2.2423),E[w](2)=(1.3006,1.5723,1.8304,2.0756,2.3159),\begin{array}{l}{\mathbb{E}}{\left[{\bf{w}}]}^{\left(1)}=\left(1.2675,1.5210,1.7659,2.0055,2.2423),\\ {\mathbb{E}}{\left[{\bf{w}}]}^{\left(2)}=\left(1.3006,1.5723,1.8304,2.0756,2.3159),\end{array}E[w](3)=(1.3220,1.6125,1.8781,2.1304,2.3735),E[w](4)=(1.3451,1.6557,1.9392,2.2017,2.4483).\begin{array}{l}{\mathbb{E}}{\left[{\bf{w}}]}^{\left(3)}=\left(1.3220,1.6125,1.8781,2.1304,2.3735),\\ {\mathbb{E}}{\left[{\bf{w}}]}^{\left(4)}=\left(1.3451,1.6557,1.9392,2.2017,2.4483).\end{array}Therefore, the optimal strategy is specified as follows: v0(1)*=(−1.05w0+2.0635)T1,v0(2)*=(−1.05w0+2.2182)T1,v1(1)*=(−1.05w1+2.2175)T1,v1(2)*=(−1.05w1+2.3291)T1,v2(1)*=(−1.05w2+2.3841)T1,v2(2)*=(−1.05w2+2.4879)T1,v3(1)*=(−1.05w3+2.5596)T1,v3(2)*=(−1.05w3+2.6384)T1,v4(1)*=(−1.05w4+2.7423)T1,v4(2)*=(−1.05w4+2.8159)T1,\begin{array}{l}{v}_{0}^{\left(1)* }=\left(-1.05{w}_{0}+2.0635){T}_{1},\hspace{.95em}{v}_{0}^{\left(2)* }=\left(-1.05{w}_{0}+2.2182){T}_{1},\\ \hspace{0.27em}{v}_{1}^{\left(1)* }=\left(-1.05{w}_{1}+2.2175){T}_{1},\hspace{1em}{v}_{1}^{\left(2)* }=\left(-1.05{w}_{1}+2.3291){T}_{1},\\ \hspace{0.01em}{v}_{2}^{\left(1)* }=\left(-1.05{w}_{2}+2.3841){T}_{1},\hspace{1em}{v}_{2}^{\left(2)* }=\left(-1.05{w}_{2}+2.4879){T}_{1},\\ {v}_{3}^{\left(1)* }=\left(-1.05{w}_{3}+2.5596){T}_{1},\hspace{1em}{v}_{3}^{\left(2)* }=\left(-1.05{w}_{3}+2.6384){T}_{1},\\ \hspace{0.1em}{v}_{4}^{\left(1)* }=\left(-1.05{w}_{4}+2.7423){T}_{1},\hspace{1em}{v}_{4}^{\left(2)* }=\left(-1.05{w}_{4}+2.8159){T}_{1},\end{array}v0(3)*=(−1.05w0+2.3182)T1,v0(4)*=(−1.05w0+2.4256)T1,v1(3)*=(−1.05w1+2.4341)T1,v1(4)*=(−1.05w1+2.5468)T1,v2(3)*=(−1.05w2+2.5558)T1,v2(4)*=(−1.05w2+2.6742)T1,v3(3)*=(−1.05w3+2.7103)T1,v3(4)*=(−1.05w3+2.8079)T1,v4(3)*=(−1.05w4+2.8735)T1,v4(4)*=(−1.05w4+2.9483)T1,\begin{array}{l}\hspace{0.025em}{v}_{0}^{\left(3)* }=\left(-1.05{w}_{0}+2.3182){T}_{1},\hspace{1em}{v}_{0}^{\left(4)* }=\left(-1.05{w}_{0}+2.4256){T}_{1},\\ \hspace{0.015em}{v}_{1}^{\left(3)* }=\left(-1.05{w}_{1}+2.4341){T}_{1},\hspace{1em}{v}_{1}^{\left(4)* }=\left(-1.05{w}_{1}+2.5468){T}_{1},\\ \hspace{-0.05em}{v}_{2}^{\left(3)* }=\left(-1.05{w}_{2}+2.5558){T}_{1},\hspace{1em}{v}_{2}^{\left(4)* }=\left(-1.05{w}_{2}+2.6742){T}_{1},\\ \hspace{0.01em}{v}_{3}^{\left(3)* }=\left(-1.05{w}_{3}+2.7103){T}_{1},\hspace{1em}{v}_{3}^{\left(4)* }=\left(-1.05{w}_{3}+2.8079){T}_{1},\\ \hspace{0.05em}{v}_{4}^{\left(3)* }=\left(-1.05{w}_{4}+2.8735){T}_{1},\hspace{1em}{v}_{4}^{\left(4)* }=\left(-1.05{w}_{4}+2.9483){T}_{1},\end{array}where T1{T}_{1}is the same as Example 1. The optimal variances under the best strategy can be derived as follows: Var(w)(i)=(Var[w1](i),Var[w2](i),Var[w3](i),Var[w4](i),Var[w5](i)),i=1,2,3,4\hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(i)}=\left(\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{1}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{2}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{3}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{4}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{5}]}^{\left(i)}),\hspace{1em}i=1,2,3,4which are given as follows: Var(w)(1)=(0.1731,0.2824,0.3489,0.3860,0.4026),Var(w)(2)=(0.2299,0.3555,0.4260,0.4554,0.4626),Var(w)(3)=(0.2710,0.4190,0.4882,0.5146,0.5140),Var(w)(4)=(0.3188,0.4930,0.5744,0.5978,0.5860).\begin{array}{l}\hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(1)}=\left(0.1731,0.2824,0.3489,0.3860,0.4026),\\ \hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(2)}=\left(0.2299,0.3555,0.4260,0.4554,0.4626),\\ \hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(3)}=\left(0.2710,0.4190,0.4882,0.5146,0.5140),\\ \hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(4)}=\left(0.3188,0.4930,0.5744,0.5978,0.5860).\end{array}Thus,we have E(κ)[w5∧κ](1)=1.8821,Var(κ)(w5∧κ)(1)=0.3467,E(κ)[w5∧κ](2)=2.1209,Var(κ)(w5∧κ)(2)=0.4461,E(κ)[w5∧κ](3)=2.2753,Var(κ)(w5∧κ)(3)=0.5115,E(κ)[w5∧κ](4)=2.4483,Var(κ)(w5∧κ)(4)=0.5860.\hspace{-22.9em}\begin{array}{l}\hspace{0.1em}{{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(1)}=1.8821,\hspace{1em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(1)}=0.3467,\\ \hspace{0.1em}{{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(2)}=2.1209,\hspace{0.9em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(2)}=0.4461,\\ \hspace{0.05em}{{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(3)}=2.2753,\hspace{0.97em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(3)}=0.5115,\\ {{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(4)}=2.4483,\hspace{0.75em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(4)}=0.5860.\\ \end{array}Figure 1 depicts the efficient frontier with different probability mass function of the exit time. We can see that the one exits at the terminal time gets the most expected wealth return at the same risk level compared with others. It is also indicated that if the investment is more stable, the investors can obtain higher expected returns at the same level of the risk, which is consistent with the real life.Figure 1Efficient frontiers with different probability mass functions of exit time.Example 3An example under uncertain exit time with liabilityThe probability mass function of an exit time κ\kappa is defined as follows: (D1,D2,D3,D4,D5)=(0.10,0.15,0.2,0.25,0.3).\left({D}_{1},{D}_{2},{D}_{3},{D}_{4},{D}_{5})=\left(0.10,0.15,0.2,0.25,0.3).Thus, the optimal expected value of assets in different time periods is given by E[w]=(4.3710,5.7834,7.2450,8.7621,10.3403).{\mathbb{E}}\left[{\bf{w}}]=\left(4.3710,5.7834,7.2450,8.7621,10.3403).Suppose the initial wealth of the investor w0=3{w}_{0}=3, initial liability Q0=1{Q}_{0}=1, and risk aversion parameter λ=1\lambda =1, we can derive the optimal portfolio selection after substituting the number to the aforementioned equations in Theorem 1: v0∗=−1.05(w0−4.9032)T1−T2+0.1595T3Q0,v1∗=−1.05(w1−6.1660)T1−T2+0.2377T3Q1,v2∗=−1.05(w2−7.4853)T1−T2+0.3458T3Q2,v3∗=−1.05(w3−8.8694)T1−T2+0.5373T3Q3,v4∗=−1.05(w4−10.3209)T1−T2+1.0000T3Q4.\begin{array}{l}{v}_{0}^{\ast }=-1.05\left({w}_{0}-4.9032){T}_{1}-{T}_{2}+0.1595{T}_{3}{Q}_{0},\\ {v}_{1}^{\ast }=-1.05\left({w}_{1}-6.1660){T}_{1}-{T}_{2}+0.2377{T}_{3}{Q}_{1},\\ {v}_{2}^{\ast }=-1.05\left({w}_{2}-7.4853){T}_{1}-{T}_{2}+0.3458{T}_{3}{Q}_{2},\\ {v}_{3}^{\ast }=-1.05\left({w}_{3}-8.8694){T}_{1}-{T}_{2}+0.5373{T}_{3}{Q}_{3},\\ {v}_{4}^{\ast }=-1.05\left({w}_{4}-10.3209){T}_{1}-{T}_{2}+1.0000{T}_{3}{Q}_{4}.\end{array}Furthermore, the final value of mean and variance under the optimal strategy are E(κ)(w5∧κ)=8.0462{{\mathbb{E}}}^{\left(\kappa )}\left({w}_{5\wedge \kappa })=8.0462and Var(κ)(w5∧κ)=0.3901{\text{Var}}^{\left(\kappa )}\left({w}_{5\wedge \kappa })=0.3901, respectively.Following Example 2, we choose four different probability mass functions at the exit time κ\kappa , α(i)=(α1(i),α2(i),α3(i),α4(i),α5(i)){\alpha }^{\left(i)}=\left({\alpha }_{1}^{\left(i)},{\alpha }_{2}^{\left(i)},{\alpha }_{3}^{\left(i)},{\alpha }_{4}^{\left(i)},{\alpha }_{5}^{\left(i)}), (i=1,2,3,4)\left(i=1,2,3,4), as follows: α(1)=(0.1,0.15,0.2,0.25,0.3),α(2)=(0,0.1,0.1,0.3,0.5),α(3)=(0,0,0.1,0.2,0.7),α(4)=(0,0,0,0,1).\begin{array}{rcl}{\alpha }^{\left(1)}& =& \left(0.1,0.15,0.2,0.25,0.3),\\ {\alpha }^{\left(2)}& =& \left(0,0.1,0.1,0.3,0.5),\\ {\alpha }^{\left(3)}& =& \left(0,0,0.1,0.2,0.7),\\ {\alpha }^{\left(4)}& =& \left(0,0,0,0,1).\end{array}Figure 2 is the efficient frontier of M–V model with liability and random cashflow under uncertain exit time. It can be seen that as the expectation go up, the more stable the investment, the less risk it takes, which has the same conclusion as Figure 1. Actually, Example 2 is a special case of Example 3, where we degenerate the term of liabilities to zero.Figure 2Efficient frontiers with different exit time.6ConclusionThe focus of the paper is placed on investigating the optimal strategy of multi-period mean-variance model with cash flow, and liability under uncertain exit time. It is a nonseparable dynamic programming problem that cannot be solved by the traditional method. In this paper, we transform the original model into a mean-field type and apply a dynamic programming approach and matrix theory to derive the optimal strategy explicitly. Our methods are shown to be much more efficient and accurate compared with other methods in the literature. For further research, we will try to employ the mean-field method to derive the mean-variance model with various additional conditions such as regime switching, bankruptcy constraints, and time inconsistency. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time

Open Mathematics , Volume 20 (1): 14 – Jan 1, 2022

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de Gruyter
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© 2022 Wei Liu et al., published by De Gruyter
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2391-5455
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2391-5455
DOI
10.1515/math-2022-0007
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Abstract

1IntroductionWith an explosive development of the economy in the recent 50 years, it is becoming more common that lots of private assets have been invested in the financial market. After a series of financial crises, the significance of handling private assets has been attached considerably. Mean-variance formulation is a famous tool that aims at balancing the risk and return of the investment. Owing to the seminal work of Markowitz [1], the mean-variance model has provided a fundamental basis for designing the optimal strategy balancing the contradiction between return and risk. Hundreds of applications and extensions have been developed over the past decade. For instance, Merton [2] derived the analytical expression of the mean-variance efficient frontier in a single-period setting. Li and Ng [3] developed the mean-variance model from the single period to the dynamic discrete-time version and derived the analytical solution by using the embedding method to overcome the difficulty of non-separability. Zhou and Li [4] used the same technique and further introduced the stochastic linear quadratic control as a general framework to solve the continuous-time mean-variance portfolio selection problem. Li et al. [5] developed it with the no-shorting constraint. Moreover, some recent approaches [6,7] based on enhanced index tracking are employed to deal with portfolio optimization.There is no doubt that the embedding method is indeed a classic way to solve the problems with the nonseparable property. We also need to admit that this method is liable to lead to complicated calculation and inefficiency during the derivation of the optimal portfolio selection if the problem has other constraints, such as uncertain exit time, asset-liability management, and serial correlated returns or risk control over bankruptcy. Typically, we will prefer another method called mean-field formulation to the embedding scheme, where a long list of notations should be established, and an auxiliary problem should be assumed.The mean-field formulation is a simple but powerful tool to derive the optimal strategy of a multi-period mean-variance portfolio selection problem. By using this method, which was first introduced by Cui et al. [8], we can resolve the mean-variance problem with many other additional constraints and derive the optimal strategy in a simpler and more direct manner. Yi et al. [9] used the mean-field method to study the mean-variance model under the uncertain exit time condition but did not consider the cash flow and liability. Cui et al. [10] extend it to the asset-liability management, but none of them considered the situation of random cash flow or investigated the closed-form of computational formulas for a series of coefficients.Yao et al. [11] studied the mean-variance model with a given level of expected terminal surplus. Li and Xie [12] studied the optimal investment with stochastic income under the uncertain exit time. They derived the analytical optimal strategy and explicit expression of the efficient frontier by using the Lagrange method and traditional dynamic programming with the additional conditions of endogenous liabilities if the investors exit the market randomly. Wu and Li [13] investigated the multi-period mean-variance model with different market states and stochastic cash flows. A reinforcement learning framework is employed to investigate the continuous-time mean-variance portfolio selection [14]. Ni et al. [15] derived equilibrium solutions of multi-period mean-variance and established a general theory to characterize the open-loop equilibrium control problem. However, all of the literature did not consider the correlation among cash flow, asset, and liability, which should be taken into account in the real world because the random cash flow would be affected by the return rate of companies. For example, the government will provide funding to companies in terms of their past performance. Furthermore, since the analytical solution of the mean-variance model contains the correlation coefficient, the optimal strategy will be changed due to different return rates among the asset, cash flow, and liability. Moreover, the uncorrelated case can be regarded as a special case of the correlated one, of which the correlation coefficient is zero. We extended the special case to the general case.In this paper, we employed the mean-field formulation [8,9,10] to study the general case of correlation in which the financial parameters are correlated at every period. On the basis of the aforementioned mean-field formulation, we have added some additional conditions such as random cash flow and liability to improve the accuracy of the investment strategy. During each time period, the cash flow and risky investment returns are random variables, while the risk-free investment return is deterministic. Furthermore, we have derived the analytical solutions of the mean-variance model which is lacked by using the embedding method [3,11, 12,13]. Employing the embedding method, the classical model mentioned earlier has certain limitations since they need to define a deterministic expectation of surplus, which is a single-objective optimization problem. Besides, the numerical solution needs some algorithms to compute the corresponding best auxiliary parameter or Lagrangian parameter, which will bring the inaccuracy and complexity in simulation. However, the mean-field formulation is more clear and powerful, which offers an analytical solution scheme in solving the nonseparable problems as the principle of optimality no longer applies. When both cash flow and mean-field formulation are presented in the same model, we shed light on the explicit solutions of the optimal portfolio under mean-variance criteria. In this paper, we are not only concerned about the return rate but also concerned with the volatility in the objective function in terms of a multi-objective optimization problem. We study the portfolio selection problem by adopting the mean field, and consider the cash flow, liability, etc. base on the mean variance model. Compared with the numerical solution, the analytical solution we derived in this paper is more efficient and applicable when the aforementioned additional conditions are added to our model.The rest of the paper is structured as follows. We construct a mean-variance portfolio selection problem with cash flow and define the meaning of some symbols in Section 2. In Section 3, the considered model is equivalently transferred into a linear quadratic optimal stochastic control problem in the mean-field type. Then, we identify the optimal portfolio strategy with closed-form expressions by adopting the dynamic programming approach in Section 4. Some numerical examples are provided in Section 5 to illustrate the accuracy and efficiency of the optimal strategy. Finally, the conclusion and future work are given in Section 6.2Multi period mean-variance portfolio selection modelWe assume the financial market has one liability, one risk-free asset, and nnkinds of risky assets within a time horizon TT. Let nt{n}_{t}represent the deterministic return of the risk-free asset, mt=[mt1,mt2,…,mtn]′{{\bf{m}}}_{{\bf{t}}}=\left[{m}_{t}^{1},{m}_{t}^{2},\ldots ,{m}_{t}^{n}]^{\prime} be the vector of nnkinds of risky investment return rate, and yt{y}_{t}be the rate of liability at period tt. In addition, the investor joins the financial market at the beginning of time period 0 and is proposed to quit the investment at time TT. Let w0{w}_{0}denote the wealth at the beginning, while l0{l}_{0}denotes the initial liability. Every investor can reallocate his/her portfolio selection to maximize the expected return as well as minimize the risk from the start of every time period between 0 and TT.In different time periods tt, the random variable yt{y}_{t}and the random vector mt=[mt1,…,mtn]′{{\bf{m}}}_{{\bf{t}}}=\left[{m}_{t}^{1},\ldots ,{m}_{t}^{n}]^{\prime} are assumed to be statistically independent and are defined from the probability space (Ω,ℱ,P)\left(\Omega ,{\mathcal{ {\mathcal F} }},P). The first two moments are recognized as the only information about yt{y}_{t}and mt{{\bf{m}}}_{{\bf{t}}}. We further define that the covariance matrix is positive definite, i.e., Covmtyt=Emtytmt′yt−EmtytEmt′yt≻0.{\rm{Cov}}\left(\begin{array}{c}{{\bf{m}}}_{{\bf{t}}}\\ {y}_{t}\end{array}\right)={\mathbb{E}}\left[\left(\begin{array}{c}{{\bf{m}}}_{{\bf{t}}}\\ {y}_{t}\end{array}\right)<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\left(<mml:mpadded>\begin{array}{cc}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} & {y}_{t}\end{array}</mml:mpadded>\right)</mml:mpadded>\right]-{\mathbb{E}}\left[\left(\begin{array}{c}{{\bf{m}}}_{{\bf{t}}}\\ {y}_{t}\end{array}\right)\right]{\mathbb{E}}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\left[<mml:mpadded>\left(<mml:mpadded>\begin{array}{cc}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} & {y}_{t}\end{array}</mml:mpadded>\right)</mml:mpadded>\right]</mml:mpadded>\hspace{0.33em}\succ \hspace{0.33em}0.Then, for t=0,1,…,T−1t=0,1,\ldots ,T-1, we have nt2ntE[mt′]ntE[yt]ntE[mt]E[mtmt′]E[mtyt]ntE[yt]E[ytmt′]E[yt2]≻0.\left(\begin{array}{ccc}{n}_{t}^{2}& {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {n}_{t}{\mathbb{E}}[{y}_{t}]\\ {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{y}_{t}]\\ {n}_{t}{\mathbb{E}}[{y}_{t}]& {\mathbb{E}}[{y}_{t}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {\mathbb{E}}[{y}_{t}^{2}]\end{array}\right)\hspace{0.33em}\succ \hspace{0.33em}0.Let Dt=(Dt1,…,Dtn)′=(mt1−nt,…,mtn−nt)′{D}_{t}=\left({D}_{t}^{1},\ldots ,{D}_{t}^{n})^{\prime} =\left({m}_{t}^{1}-{n}_{t},\ldots ,{m}_{t}^{n}-{n}_{t})^{\prime} represent the vector of the risky return rate minus the risk-free return rate. According to the aforementioned assumptions, we get nt2ntE[Dt′]ntE[yt]ntE[Dt]E[DtDt′]E[Dtyt]ntE[yt]E[ytDt′]E[yt2]=10′0−1I000′1nt2ntE[mt′]ntE[yt]ntE[mt]E[mtmt″]E[mtyt]ntE[yt]E[ytmt′]E[yt2]1−1′00I000′1≻0,\left(\begin{array}{ccc}{n}_{t}^{2}& {n}_{t}{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]& {n}_{t}{\mathbb{E}}[{y}_{t}]\\ {n}_{t}{\mathbb{E}}\left[{D}_{t}]& {\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]& {\mathbb{E}}\left[{D}_{t}{y}_{t}]\\ {n}_{t}{\mathbb{E}}[{y}_{t}]& {\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]& {\mathbb{E}}[{y}_{t}^{2}]\end{array}\right)=\left(\begin{array}{ccc}1& {\bf{0}}^{\prime} & 0\\ -{\bf{1}}& I& {\bf{0}}\\ 0& {\bf{0}}^{\prime} & 1\end{array}\right)\left(\begin{array}{ccc}{n}_{t}^{2}& {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {n}_{t}{\mathbb{E}}[{y}_{t}]\\ {n}_{t}{\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{{\bf{m}}}_{{\bf{t}}}^{^{\prime\prime} }]& {\mathbb{E}}\left[{{\bf{m}}}_{{\bf{t}}}{y}_{t}]\\ {n}_{t}{\mathbb{E}}[{y}_{t}]& {\mathbb{E}}[{y}_{t}{{\bf{m}}}_{{\bf{t}}}\hspace{-0.4em}^{\prime} ]& {\mathbb{E}}[{y}_{t}^{2}]\end{array}\right)\left(\begin{array}{ccc}1& -{\bf{1}}^{\prime} & 0\\ {\bf{0}}& I& {\bf{0}}\\ 0& {\bf{0}}^{\prime} & 1\end{array}\right)\hspace{0.33em}\succ \hspace{0.33em}0,where IIdenotes the n×nn\times nidentity matrix, and 0{\bf{0}}and 1{\bf{1}}denote the nn-dimensional all-zero and all-one vectors respectively, which signify that E[DtDt′]≻0,nt2(1−E[Dt′]E−1[DtDt′]E[Dt])>0,E[yt2]−E[ytDt′]E−1[DtDt′]E[Dtyt]>0.\begin{array}{l}{\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]\hspace{0.33em}\succ \hspace{0.33em}0,\hspace{1.0em}{n}_{t}^{2}\left(1-{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}])\gt 0,\\ {\mathbb{E}}[{y}_{t}^{2}]-{\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}{y}_{t}]\gt 0.\end{array}Therefore, 0<E[Dt′]E−1[DtDt′]E[Dt]<10\lt {\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}]\lt 1. To express the equation more concisely, we define the following notations: (1)Zt≜E[Dt′]E−1[DtDt′]E[Dt],Z˜t≜E[ctDt′]E−1[DtDt′]E[ctDt],Z¯t≜E[ytDt′]E−1[DtDt′]E[Dt],Z^t≜E[ctDt′]E−1[DtDt′]E[Dt],Z˘t≜E[ytDt′]E−1[DtDt′]E[Dtyt],Zt¨≜E[ctDt′]E−1[DtDt′]E[Dtyt].\begin{array}{ll}{Z}_{t}\triangleq {\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],& {\widetilde{Z}}_{t}\triangleq {\mathbb{E}}\left[{c}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{c}_{t}{D}_{t}],\\ {\overline{Z}}_{t}\triangleq {\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],& {\widehat{Z}}_{t}\triangleq {\mathbb{E}}\left[{c}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],\\ {\breve{Z}}_{t}\triangleq {\mathbb{E}}[{y}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}{y}_{t}],& \ddot{{Z}_{t}}\triangleq {\mathbb{E}}\left[{c}_{t}{D}_{t}^{^{\prime} }]{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}{y}_{t}].\end{array}At the beginning of every period tt, investors’ wealth and liabilities are denoted by wt{w}_{t}and Qt{Q}_{t}, respectively. Therefore, the surplus is denoted by wt−Qt{w}_{t}-{Q}_{t}. If vti{v}_{t}^{i}is the money invested in the iith risky investment for i=1,2,…,ni=1,2,\ldots ,nat period tt, then wt−∑i=1nvti{w}_{t}-\mathop{\sum }\limits_{i=1}^{n}{v}_{t}^{i}is the money put into the risk-free investment. In this paper, we suppose the liability is exogenous. In other words, the investor’s strategies cannot affect the liability because of its uncontrollability. Let ℱt=σ(D0,D1,…,Dt−1,c0,c1,…,ct−1,y0,y1,…,yt−1){{\mathcal{ {\mathcal F} }}}_{t}=\sigma \left({D}_{0},{D}_{1},\ldots ,{D}_{t-1},{c}_{0},{c}_{1},\ldots ,{c}_{t-1},{y}_{0},{y}_{1},\ldots ,{y}_{t-1})represent all the information at the initial moment of ttperiod for t=1,2,…,T−1t=1,2,\ldots ,T-1, and ℱ0{{\mathcal{ {\mathcal F} }}}_{0}represent the unimportant σ\sigma -algebra over Ω\Omega . Thus, E[⋅∣ℱ0]{\mathbb{E}}\left[\cdot | {{\mathcal{ {\mathcal F} }}}_{0}]is equal to the unconditional expectation E[⋅]{\mathbb{E}}\left[\cdot ]. In this paper, all allowable portfolio selection is limited to be ℱt{{\mathcal{ {\mathcal F} }}}_{t}-adapted Markov controls, i.e., vt=(vt1,vt2,…,vtn)′∈ℱt{v}_{t}=\left({v}_{t}^{1},{v}_{t}^{2},\ldots ,{v}_{t}^{n})^{\prime} \in {{\mathcal{ {\mathcal F} }}}_{t}. Therefore, Dt{D}_{t}and vt{v}_{t}are independent and ℱt=σ(wt,Qt){{\mathcal{ {\mathcal F} }}}_{t}=\sigma \left({w}_{t},{Q}_{t}).The investor plans to optimize the portfolio selection during the whole time period. However, the investment might be forced to be changed or abandoned at an uncertain time τ\tau before TTbecause of some accidents or unexpected events such as sudden resignation, serious illness, and colossal consumption. The probability mass function of the exogenous random variable κ\kappa is p˜t=Pr{κ=t}{\tilde{p}}_{t}=\hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{\kappa =t\right\}. Thus, the investor will quit the financial market eventually at time T∧κ=min{T,κ}T\wedge \kappa =\min \left\{T,\kappa \right\}. We have pt≜Pr{T∧κ=t}=p˜t,t=1,2,…,T−1,1−∑j=1T−1p˜j,t=T.{p}_{t}\triangleq \hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{T\wedge \kappa =t\right\}=\left\{\begin{array}{ll}{\tilde{p}}_{t},& t=1,2,\ldots ,T-1,\\ 1-\mathop{\displaystyle \sum }\limits_{j=1}^{T-1}{\tilde{p}}_{j},& t=T.\end{array}\right.The main investigation of this model is to find the optimal portfolio selection, vt∗=[(vt1)∗,(vt2)∗,…,(vtn)∗]′{v}_{t}^{\ast }=\left[{\left({v}_{t}^{1})}^{\ast },{\left({v}_{t}^{2})}^{\ast },\ldots ,{\left({v}_{t}^{n})}^{\ast }]^{\prime} , t=0,1,…,T−1t=0,1,\ldots ,T-1, which can be equivalent to optimizing the following optimal stochastic control problem, (2)minVar(κ)(wT∧κ−QT∧κ)−λE(κ)[wT∧κ−QT∧κ],s.t.wt+1=∑i=1nmtivti+wt−∑i=1nvtint+ct=ntwt+Dt′vt+ct,Qt+1=ytQt,fort=0,1,…,T−1,\left\{\begin{array}{ll}\min & {\text{Var}}^{\left(\kappa )}\left({w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa })-\lambda {{\mathbb{E}}}^{\left(\kappa )}\left[{w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }],\\ \hspace{0.1em}\text{s.t.}\hspace{0.1em}& \hspace{0.1em}{w}_{t+1}\hspace{0.07em}=\mathop{\displaystyle \sum }\limits_{i=1}^{n}{m}_{t}^{i}{v}_{t}^{i}+\left({w}_{t}-\mathop{\displaystyle \sum }\limits_{i=1}^{n}{v}_{t}^{i}\right){n}_{t}+{c}_{t}\\ & \hspace{1.0em}\hspace{1.1em}={n}_{t}{w}_{t}+{D}_{t}^{^{\prime} }{v}_{t}+{c}_{t},\\ & \hspace{0.1em}{Q}_{t+1}={y}_{t}{Q}_{t},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t=0,1,\ldots ,T-1,\end{array}\right.where λ>0\lambda \gt 0represents the risk aversion, and E(κ)[wT∧κ−QT∧κ]≜∑t=1TE[wT∧κ−QT∧κ∣T∧κ=t]Pr{T∧κ=t}=∑t=1TE[wt−Qt]pt,V ar(κ)(wT∧κ−QT∧κ)≜∑t=1TVar(wT∧κ−QT∧κ∣T∧κ=t)Pr{T∧κ=t}=∑t=1TVar(wt−Qt)pt.\begin{array}{rcl}{{\mathbb{E}}}^{\left(\kappa )}\left[{w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }]& \triangleq & \mathop{\displaystyle \sum }\limits_{t=1}^{T}{\mathbb{E}}\left[{w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }| T\wedge \kappa =t]\hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{T\wedge \kappa =t\right\}=\mathop{\displaystyle \sum }\limits_{t=1}^{T}{\mathbb{E}}\left[{w}_{t}-{Q}_{t}]{p}_{t},\\ {{\rm{V\; ar}}}^{\left(\kappa )}\left({w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa })& \triangleq & \mathop{\displaystyle \sum }\limits_{t=1}^{T}\hspace{0.1em}\text{Var}\hspace{0.1em}\left({w}_{T\wedge \kappa }-{Q}_{T\wedge \kappa }| T\wedge \kappa =t)\hspace{0.1em}\text{Pr}\hspace{0.1em}\left\{T\wedge \kappa =t\right\}=\mathop{\displaystyle \sum }\limits_{t=1}^{T}{\rm{Var}}\left({w}_{t}-{Q}_{t}){p}_{t}.\end{array}Then, we can rewrite the aforementioned model as follows: (3)min∑t=1Tpt{Var(wt−Qt)−λE[wt−Qt]},s.t.wt+1=ntwt+Dt′vt+ct,Qt+1=ytQt,fort=0,1,…,T−1.\left\{\begin{array}{l}\min \hspace{0.33em}\mathop{\displaystyle \sum }\limits_{t=1}^{T}{p}_{t}\left\{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({w}_{t}-{Q}_{t})-\lambda {\mathbb{E}}\left[{w}_{t}-{Q}_{t}]\right\},\\ {\rm{s.t.}}\hspace{1em}{w}_{t+1}={n}_{t}{w}_{t}+{D}_{t}^{^{\prime} }{v}_{t}+{c}_{t},\\ \hspace{2.25em}{Q}_{t+1}={y}_{t}{Q}_{t},\hspace{1em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t=0,1,\ldots ,T-1.\end{array}\right.Since the smoothing property is no longer valid on the variance term, we cannot decompose the nonseparable problem into a stage wise backward recursion formulation, which can be tackled with traditional dynamic programming method. We solve it by employing the mean-field method.3Mean-field formulationFirst, we construct the mean-field type of model (3). According to the independence between Dt{D}_{t}and vt{v}_{t}, yt{y}_{t}and Qt{Q}_{t}, the dynamic equations of the expectation of the wealth and liability can be represented as follows: (4)E[wt+1]=ntE[wt]+E[Dt′]E[vt]+E[ct],E[w0]=w0,E[Qt+1]=E[yt]E[Qt],E[Q0]=Q0,\left\{\begin{array}{l}{\mathbb{E}}\left[{w}_{t+1}]={n}_{t}{\mathbb{E}}\left[{w}_{t}]+{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{v}_{t}]+{\mathbb{E}}\left[{c}_{t}],\\ {\mathbb{E}}\left[{w}_{0}]={w}_{0},\\ {\mathbb{E}}\left[{Q}_{t+1}]={\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{Q}_{t}],\\ {\mathbb{E}}\left[{Q}_{0}]={Q}_{0},\end{array}\right.with t=0,1,…,T−1t=0,1,\ldots ,T-1.Combining the dynamic equations in (3) and (4), we have (5)wt+1−E[wt+1]=nt(wt−E[wt])+Dt′vt−E[Dt′]E[vt]+(ct−E[ct])=nt(wt−E[wt])+Dt′(vt−E[vt])+(Dt′−E[Dt′])E[vt]+(ct−E[ct]),w0−E[w0]=0,Qt+1−E[Qt+1]=yt(Qt−E[Qt])+(yt−E[yt])E[Qt],Q0−E[Q0]=0.\left\{\begin{array}{rcl}{w}_{t+1}-{\mathbb{E}}\left[{w}_{t+1}]& =& {n}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}])+{D}_{t}^{^{\prime} }{v}_{t}-{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{v}_{t}]+\left({c}_{t}-{\mathbb{E}}\left[{c}_{t}])\\ & =& {n}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}])+{D}_{t}^{^{\prime} }\left({v}_{t}-{\mathbb{E}}\left[{v}_{t}])+\left({D}_{t}^{^{\prime} }-{\mathbb{E}}\left[{D}_{t}^{^{\prime} }]){\mathbb{E}}\left[{v}_{t}]+\left({c}_{t}-{\mathbb{E}}\left[{c}_{t}]),\\ {w}_{0}-{\mathbb{E}}\left[{w}_{0}]& =& 0,\\ {Q}_{t+1}-{\mathbb{E}}\left[{Q}_{t+1}]& =& {y}_{t}\left({Q}_{t}-{\mathbb{E}}\left[{Q}_{t}])+({y}_{t}-{\mathbb{E}}[{y}_{t}]){\mathbb{E}}\left[{Q}_{t}],\\ {Q}_{0}-{\mathbb{E}}\left[{Q}_{0}]& =& 0.\end{array}\right.Therefore, we can equivalently reformulate problem (3) into a linear quadratic optimal problem in the mean-field type.(6)min∑t=1Tpt{E[(wT−QT−E[wT−QT])2]−λE[wT−QT]},s.t.{E[wt],E[Qt],E[vt]}satisfies dynamic equation (4),{wt−E[wt],Qt−E[Qt],vt−E[vt]}satisfies dynamic equation (5),fort=0,1,…,T−1.\left\{\begin{array}{rl}\min & \mathop{\displaystyle \sum }\limits_{t=1}^{T}{p}_{t}\left\{{\mathbb{E}}\left[{\left({w}_{T}-{Q}_{T}-{\mathbb{E}}\left[{w}_{T}-{Q}_{T}])}^{2}]-\lambda {\mathbb{E}}\left[{w}_{T}-{Q}_{T}]\right\},\\ \hspace{0.1em}\text{s.t.}\hspace{0.1em}& \left\{{\mathbb{E}}\left[{w}_{t}],{\mathbb{E}}\left[{Q}_{t}],{\mathbb{E}}\left[{v}_{t}]\right\}\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (4)}\hspace{0.1em},\\ & \left\{{w}_{t}-{\mathbb{E}}\left[{w}_{t}],{Q}_{t}-{\mathbb{E}}\left[{Q}_{t}],{v}_{t}-{\mathbb{E}}\left[{v}_{t}]\right\}\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (5),}\hspace{0.1em}\\ & \hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t=0,\hspace{0.08em}1,\ldots ,T-1.\end{array}\right.Thus, we are able to solve it by the dynamic programming method since it is separable.4The optimal strategyWith the notations given in (1), the seven parameters of the sequence {βt}\left\{{\beta }_{t}\right\}, {ηt}\left\{{\eta }_{t}\right\}, {ξt}\left\{{\xi }_{t}\right\}, {ζt}\left\{{\zeta }_{t}\right\}, {ψt}\left\{{\psi }_{t}\right\}, {δt}\left\{{\delta }_{t}\right\}, and {Δt}\left\{{\Delta }_{t}\right\}are deterministic by the following backward recursions βt=βt+1(st)2(1−Zt)+pt,ηt=ηt+1(stE[yt]−stZ¯t)+pt,ξt=ξt+1E[yt2]−ηt+12βt+1−1Bt′+pt,ζt=ζt+1st+pt,ψt=ψt+1E[yt]−2ληt+1(E[ytct]−E[yt]E[ct])+2ηt+1λE[ct]+λζt+12βt+1(E[yt]Zt−Z¯t)+Z^tZ¯t−Z^tE[yt]1−Zt+Zt¨+pt,δt=ξt+1(E[yt2]−E[yt]2)+δt+1(E[yt])2−ηt+12βt+1Z˘t−E[yt]2+(Z¯t−E[yt])21−Zt,Δt=Δt+1+βt+1E[(ct)2]−βt+1E[ct]2−λζt+1E[ct]−βt+1E[ct]+λζt+12βt+1−Z^t2Zt1−Zt−2E[ct]+λζt+12βt+1−Z^tZ^t+Z˜t−Z^t2,\hspace{-45em}\begin{array}{rcl}{\beta }_{t}& =& {\beta }_{t+1}{\left({s}_{t})}^{2}\left(1-{Z}_{t})+{p}_{t},\\ {\eta }_{t}& =& {\eta }_{t+1}\left({s}_{t}{\mathbb{E}}[{y}_{t}]-{s}_{t}{\overline{Z}}_{t})+{p}_{t},\\ {\xi }_{t}& =& {\xi }_{t+1}{\mathbb{E}}[{y}_{t}^{2}]-{\eta }_{t+1}^{2}{\beta }_{t+1}^{-1}{B}_{t}^{^{\prime} }+{p}_{t},\\ {\zeta }_{t}& =& {\zeta }_{t+1}{s}_{t}+{p}_{t},\\ {\psi }_{t}& =& {\psi }_{t+1}{\mathbb{E}}[{y}_{t}]-\frac{2}{\lambda }{\eta }_{t+1}\left({\mathbb{E}}[{y}_{t}{c}_{t}]-{\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{c}_{t}])+\frac{2{\eta }_{t+1}}{\lambda }\left(\frac{\left({\mathbb{E}}\left[{c}_{t}]+\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}\right)\left({\mathbb{E}}[{y}_{t}]{Z}_{t}-{\overline{Z}}_{t})+{\widehat{Z}}_{t}{\overline{Z}}_{t}-{\widehat{Z}}_{t}{\mathbb{E}}[{y}_{t}]}{1-{Z}_{t}}+\ddot{{Z}_{t}}\right)+{p}_{t},\\ {\delta }_{t}& =& {\xi }_{t+1}\left({\mathbb{E}}[{y}_{t}^{2}]-{\mathbb{E}}{[{y}_{t}]}^{2})+{\delta }_{t+1}{\left({\mathbb{E}}[{y}_{t}])}^{2}-\frac{{\eta }_{t+1}^{2}}{{\beta }_{t+1}}\left({\breve{Z}}_{t}-{\mathbb{E}}{[{y}_{t}]}^{2}+\frac{{\left({\overline{Z}}_{t}-{\mathbb{E}}[{y}_{t}])}^{2}}{1-{Z}_{t}}\right),\\ {\Delta }_{t}& =& {\Delta }_{t+1}+{\beta }_{t+1}{\mathbb{E}}\left[{\left({c}_{t})}^{2}]-{\beta }_{t+1}{\mathbb{E}}{\left[{c}_{t}]}^{2}-\lambda {\zeta }_{t+1}{\mathbb{E}}\left[{c}_{t}]\\ & & -{\beta }_{t+1}\left({\left({\mathbb{E}}\left[{c}_{t}]+\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}-{\widehat{Z}}_{t}\right)}^{2}\frac{{Z}_{t}}{1-{Z}_{t}}-2\left({\mathbb{E}}\left[{c}_{t}]+\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}-{\widehat{Z}}_{t}\right){\widehat{Z}}_{t}+{\widetilde{Z}}_{t}-{\widehat{Z}}_{t}^{2}\right),\end{array}with boundary conditions defined as follows: βT=pT,ηT=pT,ξT=pT,ζT=pT,ψT=pT,δT=0,ΔT=0.{\beta }_{T}={p}_{T},\hspace{1em}{\eta }_{T}={p}_{T},\hspace{1em}{\xi }_{T}={p}_{T},\hspace{1em}{\zeta }_{T}={p}_{T},\hspace{1em}{\psi }_{T}={p}_{T},\hspace{1em}{\delta }_{T}=0,\hspace{1em}{\Delta }_{T}=0.The solution scheme adopted in this paper involves two steps. The first step is to construct the cost-to-go functional and derive the backward recursion. The second step is to prove that it still holds at each period according to mathematical induction. Thus, the optimal portfolio strategy can be obtained in the following theorem.Theorem 1Assume that the return rates among asset, liability, and cash flow are correlated. Thus, we have the optimal portfolio selection of problem (6) as follows: vt−E[vt]=−st(wt−E[wt])E−1[DtDt′]E[(Dt)]+ηt+1βt+1−1(Qt−E[Qt])E−1[DtDt′]E[ytDt],\begin{array}{rcl}{v}_{t}-{\mathbb{E}}\left[{v}_{t}]& =& -{s}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}]){{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[\left({D}_{t})]+{\eta }_{t+1}{\beta }_{t+1}^{-1}\left({Q}_{t}-{\mathbb{E}}\left[{Q}_{t}]){{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}[{y}_{t}{D}_{t}],\end{array}E[vt]=−(E[DtDt′]−E[Dt]E[Dt′])−1E[ctDt]−E[ct]E[Dt]−λζt+12βt+1E[Dt]−ηt+1βt+1(E[ytDt]−E[yt]E[Dt])E[Qt].\begin{array}{rcl}{\mathbb{E}}\left[{v}_{t}]& =& -{\left({\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]-{\mathbb{E}}\left[{D}_{t}]{\mathbb{E}}\left[{D}_{t}^{^{\prime} }])}^{-1}\left({\mathbb{E}}\left[{c}_{t}{D}_{t}]-{\mathbb{E}}\left[{c}_{t}]{\mathbb{E}}\left[{D}_{t}]-\frac{\lambda {\zeta }_{t+1}}{2{\beta }_{t+1}}{\mathbb{E}}\left[{D}_{t}]-\frac{{\eta }_{t+1}}{{\beta }_{t+1}}\left({\mathbb{E}}[{y}_{t}{D}_{t}]-{\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{D}_{t}]){\mathbb{E}}\left[{Q}_{t}]\right).\end{array}The expected value of optimal wealth can be derived as follows: E[wt]=w0∏k=0t−1sk+∑j=0t−1E[cj]+λζj+12βj+1−Z^j−E[yj]+Z¯jZj1−Zj−Z^j+Z¯j+E[cj]∏l=j+1t−1sl,{\mathbb{E}}\left[{w}_{t}]={w}_{0}\mathop{\prod }\limits_{k=0}^{t-1}{s}_{k}+\mathop{\sum }\limits_{j=0}^{t-1}\left(\left({\mathbb{E}}\left[{c}_{j}]+\frac{\lambda {\zeta }_{j+1}}{2{\beta }_{j+1}}-{\widehat{Z}}_{j}-{\mathbb{E}}[{y}_{j}]+{\overline{Z}}_{j}\right)\frac{{Z}_{j}}{1-{Z}_{j}}-{\widehat{Z}}_{j}+{\overline{Z}}_{j}+{\mathbb{E}}\left[{c}_{j}]\right)\mathop{\prod }\limits_{l=j+1}^{t-1}{s}_{l},for t=1,2,…,Tt=1,2,\ldots ,T. Here, ∏∅(⋅)=1\prod _{\varnothing }\left(\cdot )=1, ∑∅(⋅)=0\sum _{\varnothing }\left(\cdot )=0.If the additional condition of liability is not considered in our case, the original model (6) would be degenerated to the one mentioned by Yao et al. [11], which will be introduced in the following corollary.Remark 1Assume that an investor participates in the initial investment under uncertain exit time without liability. Thus, the degenerated problem is equivalently reformulated as the following mean-variance model.(7)min∑t=1Tpt{E[(wt−E[wt])2]−λE[wt]},s.t.E(vt−E[vt])=0,E[wt]satisfies dynamic equation (4),wt−E[wt]satisfies dynamic equation (5).\left\{\begin{array}{rl}\min & \mathop{\displaystyle \sum }\limits_{t=1}^{T}{p}_{t}\left\{{\mathbb{E}}\left[{\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}])}^{2}]-\lambda {\mathbb{E}}\left[{w}_{t}]\right\},\\ \hspace{0.1em}\text{s.t.}\hspace{0.1em}& {\mathbb{E}}\left({v}_{t}-{\mathbb{E}}\left[{v}_{t}])={\bf{0}},\\ & {\mathbb{E}}\left[{w}_{t}]\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (4)}\hspace{0.1em},\\ & {w}_{t}-{\mathbb{E}}\left[{w}_{t}]\hspace{0.33em}\hspace{0.1em}\text{satisfies dynamic equation (5)}\hspace{0.1em}.\end{array}\right.The optimal strategies of problem 7 are represented as follows: vt∗−E[vt∗]=−nt(wt−E[wt])E−1[DtDt′]E[Dt],E[vt∗]=wζt+12βt+1⋅11−ZtE−1[DtDt′]E[Dt].\begin{array}{rcl}{v}_{t}^{\ast }-{\mathbb{E}}\left[{v}_{t}^{\ast }]& =& -{n}_{t}\left({w}_{t}-{\mathbb{E}}\left[{w}_{t}]){{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}],\\ {\mathbb{E}}\left[{v}_{t}^{\ast }]& =& \frac{w{\zeta }_{t+1}}{2{\beta }_{t+1}}\cdot \frac{1}{1-{Z}_{t}}{{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}].\end{array}Thus, we get the optimal expected level of wealth E[wt]=w0∏k=0t−1sk+w2∑j=0t−1ζj+1βj+1⋅Zj1−Zj∏ℓ=j+1t−1sℓ.{\mathbb{E}}\left[{w}_{t}]={w}_{0}\mathop{\prod }\limits_{k=0}^{t-1}{s}_{k}+\frac{w}{2}\mathop{\sum }\limits_{j=0}^{t-1}\frac{{\zeta }_{j+1}}{{\beta }_{j+1}}\cdot \frac{{Z}_{j}}{1-{Z}_{j}}\mathop{\prod }\limits_{\ell =j+1}^{t-1}{s}_{\ell }.The optimal strategy of the model in Corollary 1 can be obtained according to Theorem 1, which is consistent with the results derived by Yao et al. [11]. Therefore, the accuracy of the solution derived in this paper has been verified. In comparison, Zhu et al. [16] analyzed the Lagrangian problem via the embedding method and were unable to obtain an analytical form of the optimal objective value function. Thus, they invoked a prime-dual iterative algorithm to identify the optimal Lagrangian multiplier vector. Moreover, compared with the classical embedding method, which needs a Bellman equation and the Lagrangian multiplier, the mean-field formulation has been employed in this paper, which avoids the complicated computation. In the following section, a few numerical examples from real-world applications are given to demonstrate the efficiency of the obtained optimal strategy.5Numerical exampleAccording to the data given in the study by Elton et al. [17], we investigate a portfolio selection consisting of S&P 500 (SP), the index of emerging market (EM), and small stock (MS) of the U.S. market. Moreover, we consider uncertain exit time and cash flow in the model. Table 1 presents three different assets, a liability, and a random cash flow, and it also presents the expected values, variances, and the correlation coefficients among them. The annual risk free return rate is set as 5%5 \% (nt=1.05{n}_{t}=1.05). Here, we ignore the case of uncorrelation between Dt{D}_{t}and ct{c}_{t}, i.e., the return rates and cash flow are correlated.Table 1Data for assets and cash flowSPEMMSCashflowLiabilityExpected return14%14 \% 16%16 \% 17%17 \% 110%10 \% Standard deviation18.5%18.5 \% 30%30 \% 24%24 \% 20%20 \% 20%20 \% Correlation coefficientSP10.640.79ρ1{\rho }_{1}ρ^1{\widehat{\rho }}_{1}EM0.6410.75ρ2{\rho }_{2}ρ^2{\widehat{\rho }}_{2}MS0.790.751ρ3{\rho }_{3}ρ^3{\widehat{\rho }}_{3}Cashflowρ1{\rho }_{1}ρ2{\rho }_{2}ρ3{\rho }_{3}1ρ^4{\widehat{\rho }}_{4}Liabilityρ^1{\widehat{\rho }}_{1}ρ^2{\widehat{\rho }}_{2}ρ^3{\widehat{\rho }}_{3}ρ^4{\widehat{\rho }}_{4}1Thus, for every period tt, we have the following matrices: E[Dt]=0.090.110.12,Cov(Dt)=0.03420.03550.03510.03550.09000.05400.03510.05400.0576,E[DtDt′]=0.04230.04540.04590.04540.10210.06720.04590.06720.0720.{\mathbb{E}}\left[{D}_{t}]=\left(\begin{array}{c}0.09\\ 0.11\\ 0.12\\ \end{array}\right),\hspace{1.0em}{\rm{Cov}}\left({D}_{t})=\left(\begin{array}{ccc}0.0342& 0.0355& 0.0351\\ 0.0355& 0.0900& 0.0540\\ 0.0351& 0.0540& 0.0576\end{array}\right),\hspace{1.0em}{\mathbb{E}}\left[{D}_{t}{D}_{t}^{^{\prime} }]=\left(\begin{array}{ccc}0.0423& 0.0454& 0.0459\\ 0.0454& 0.1021& 0.0672\\ 0.0459& 0.0672& 0.0720\end{array}\right).The correlation coefficient between cash flow and iith asset is defined as ρ=(ρ1,ρ2,ρ3)\rho =\left({\rho }_{1},{\rho }_{2},{\rho }_{3}), while the coefficient between liability and iith asset is defined as ρ^=(ρ1^,ρ2^,ρ3^)\widehat{\rho }=\left(\widehat{{\rho }_{1}},\widehat{{\rho }_{2}},\widehat{{\rho }_{3}}), according to the definition we have ρi=Cov(ct,Dti)Var(ct)Var(Dti){\rho }_{i}=\frac{\hspace{0.1em}\text{Cov}\hspace{0.1em}\left({c}_{t},{D}_{t}^{i})}{\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})}}and ρi^=Cov(yt,Dti)Var(yt)Var(Dti).\widehat{{\rho }_{i}}=\frac{\hspace{0.1em}\text{Cov}\hspace{0.1em}({y}_{t},{D}_{t}^{i})}{\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})}}.In addition, we define the correlation of the cash flow and liability ρ4^\widehat{{\rho }_{4}}as follows: ρ4^=Cov(ct,yt)Var(ct)Var(yt).\widehat{{\rho }_{4}}=\frac{\hspace{0.1em}\text{Cov}\hspace{0.1em}\left({c}_{t},{y}_{t})}{\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})}}.Then, we have E[ctDti]=E[ct]E[Dti]+ρiVar(ct)Var(Dti),{\mathbb{E}}\left[{c}_{t}{D}_{t}^{i}]={\mathbb{E}}\left[{c}_{t}]{\mathbb{E}}\left[{D}_{t}^{i}]+{\rho }_{i}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})},E[ytDti]=E[yt]E[Dti]+ρi^Var(yt)Var(Dti),{\mathbb{E}}[{y}_{t}{D}_{t}^{i}]={\mathbb{E}}[{y}_{t}]{\mathbb{E}}\left[{D}_{t}^{i}]+\widehat{{\rho }_{i}}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({D}_{t}^{i})},E[ctyt]=E[ct]E[yt]+ρ4^Var(ct)Var(yt),{\mathbb{E}}\left[{c}_{t}{y}_{t}]={\mathbb{E}}\left[{c}_{t}]{\mathbb{E}}[{y}_{t}]+\widehat{{\rho }_{4}}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t})}\sqrt{\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t})},E[ct2]=E[ct]2+Var(ct),{\mathbb{E}}\left[{c}_{t}^{2}]={\mathbb{E}}{\left[{c}_{t}]}^{2}+\hspace{0.1em}\text{Var}\hspace{0.1em}\left({c}_{t}),E[yt2]=E[yt]2+Var(yt).{\mathbb{E}}[{y}_{t}^{2}]={\mathbb{E}}{[{y}_{t}]}^{2}+\hspace{0.1em}\text{Var}\hspace{0.1em}({y}_{t}).Assume that ρ=(ρ1,ρ2,ρ3)=(−0.3,0.5,0.2)\rho =\left({\rho }_{1},{\rho }_{2},{\rho }_{3})=\left(-0.3,0.5,0.2), ρ^=(ρ1^,ρ2^,ρ3^)=(−0.2,0.4,0.3)\widehat{\rho }=\left(\widehat{{\rho }_{1}},\widehat{{\rho }_{2}},\widehat{{\rho }_{3}})=\left(-0.2,0.4,0.3)and ρ4^=0.1\widehat{{\rho }_{4}}=0.1. Then, CovDtct=Cov(Dt)Cov(ct,Dt)Cov(ct,Dt′)Var(ct)=0.03420.03550.0351−0.00920.03550.09000.05400.03000.03510.05400.05760.0120−0.00920.03000.01200.0400≻0.{\rm{Cov}}\left(\left(\begin{array}{c}{D}_{t}\\ {c}_{t}\end{array}\right)\right)=\left(\begin{array}{cc}{\rm{Cov}}\left({D}_{t})& {\rm{Cov}}\left({c}_{t},{D}_{t})\\ {\rm{Cov}}\left({c}_{t},{D}_{t}^{^{\prime} })& {\rm{Var}}\left({c}_{t})\end{array}\right)=\left(\begin{array}{cccc}0.0342& 0.0355& 0.0351& -0.0092\\ 0.0355& 0.0900& 0.0540& 0.0300\\ 0.0351& 0.0540& 0.0576& 0.0120\\ -0.0092& 0.0300& 0.0120& 0.0400\end{array}\right)\hspace{0.33em}\succ \hspace{0.33em}0.Substituting the data in the equations, we have E[ctDt]=(0.0898,0.1510,0.1440)′{\mathbb{E}}\left[{c}_{t}{D}_{t}]=\left(0.0898,0.1510,0.1440)^{\prime} . Moreover, we define the following notations to make the solution more concise, T1=E−1[DtDt′]E[Dt]=1.0589−0.11961.1033,T2=E−1[DtDt′]E[ctDt]=−0.34900.44931.6365,T3=E−1[DtDt′]E[ytDt]=−1.04110.17540.8209.\begin{array}{l}{T}_{1}={{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{D}_{t}]=\left[\begin{array}{c}1.0589\\ -0.1196\\ 1.1033\end{array}\right],\\ {T}_{2}={{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}\left[{c}_{t}{D}_{t}]=\left[\begin{array}{c}-0.3490\\ 0.4493\\ 1.6365\end{array}\right],\\ {T}_{3}={{\mathbb{E}}}^{-1}\left[{D}_{t}{D}_{t}^{^{\prime} }]{\mathbb{E}}[{y}_{t}{D}_{t}]=\left[\begin{array}{c}-1.0411\\ 0.1754\\ 0.8209\end{array}\right].\end{array}Zt=0.2145,Z^t=0.2144,Z˜t=0.2507,\hspace{-18em}{Z}_{t}=0.2145,\hspace{1.0em}{\widehat{Z}}_{t}=0.2144,\hspace{1.0em}{\widetilde{Z}}_{t}=0.2507,Z¯t=0.0241,Z˘t=0.0218,Zt¨=0.0488.{\overline{Z}}_{t}=0.0241,\hspace{1.0em}{\breve{Z}}_{t}=0.0218,\hspace{1.0em}\ddot{{Z}_{t}}=0.0488.Example 1An example with the terminal exit timeThe probability mass function κ\kappa is defined as follows: (α1,α2,α3,α4,α5)=(0,0,0,0,1),\left({\alpha }_{1},{\alpha }_{2},{\alpha }_{3},{\alpha }_{4},{\alpha }_{5})=\left(0,0,0,0,1),for t=1,2,3,4,5t=1,2,3,4,5. By applying the result of Theorem 1, the optimal portfolio selection of this numerical example can be obtained as follows: v0∗=−1.02(w0−3.0477)T1+1.2053T2Q0,v1∗=−1.02(w1−3.2001)T1+1.1503T2Q1,v2∗=−1.02(w2−3.3601)T1+1.0979T2Q2,v3∗=−1.02(w3−3.5281)T1+1.0478T2Q3,v4∗=−1.02(w4−3.7045)T1+1.0000T2Q4.\begin{array}{l}{v}_{0}^{\ast }=-1.02\left({w}_{0}-3.0477){T}_{1}+1.2053{T}_{2}{Q}_{0},\\ {v}_{1}^{\ast }=-1.02\left({w}_{1}-3.2001){T}_{1}+1.1503{T}_{2}{Q}_{1},\\ {v}_{2}^{\ast }=-1.02\left({w}_{2}-3.3601){T}_{1}+1.0979{T}_{2}{Q}_{2},\\ {v}_{3}^{\ast }=-1.02\left({w}_{3}-3.5281){T}_{1}+1.0478{T}_{2}{Q}_{3},\\ {v}_{4}^{\ast }=-1.02\left({w}_{4}-3.7045){T}_{1}+1.0000{T}_{2}{Q}_{4}.\end{array}Under the certain exit time, we derive the final optimal surplus as follows, E(w5−Q5)=3.3897{\mathbb{E}}\left({w}_{5}-{Q}_{5})=3.3897and Var(w5−Q5)=0.6135\hspace{0.1em}\text{Var}\hspace{0.1em}\left({w}_{5}-{Q}_{5})=0.6135, respectively.Example 2An example without liability under uncertain exit timeConsider the example as corollary. Here, we ignore the information of liability, i.e., ignore the last line and last column of Table 1 and do not fix the terminal expectation but balance the variance and expectation by the trade-off parameter.Assume that an investor plans a five-period investment with an initial wealth w0=1{w}_{0}=1and that the trade-off parameter w=1w=1, but he may exit the market at any time tt(t=1,2,3,4,5t=1,2,3,4,5).To investigate the impact of uncertain exit time on the optimal policy and efficient frontier clearly, we choose four different probability mass functions at the exit time κ\kappa , α(i)=(α1(i),α2(i),α3(i),α4(i),α5(i)),(i=1,2,3,4){\alpha }^{\left(i)}=\left({\alpha }_{1}^{\left(i)},{\alpha }_{2}^{\left(i)},{\alpha }_{3}^{\left(i)},{\alpha }_{4}^{\left(i)},{\alpha }_{5}^{\left(i)}),\left(i=1,2,3,4), as follows: α(1)=(0.1,0.15,0.2,0.25,0.3),{\alpha }^{\left(1)}=\left(0.1,0.15,0.2,0.25,0.3),α(2)=(0,0.1,0.1,0.3,0.5),\hspace{-13.15em}{\alpha }^{\left(2)}=\left(0,0.1,0.1,0.3,0.5),α(3)=(0,0,0.1,0.2,0.7),\hspace{-13.2em}{\alpha }^{\left(3)}=\left(0,0,0.1,0.2,0.7),α(4)=(0,0,0,0,1),\hspace{-13.25em}{\alpha }^{\left(4)}=\left(0,0,0,0,1),where α(4){\alpha }^{\left(4)}represents that the investor must exit the market at the terminal time.Then, the optimal expected wealth level E[w](i)=(E[w1](i),E[w2](i),E[w3](i),E[w4](i),E[w5](i)),i=1,2,3,4,{\mathbb{E}}{\left[{\bf{w}}]}^{\left(i)}=\left({\mathbb{E}}{\left[{w}_{1}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{2}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{3}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{4}]}^{\left(i)},{\mathbb{E}}{\left[{w}_{5}]}^{\left(i)}),\hspace{1em}i=1,2,3,4,which are given by E[w](1)=(1.2675,1.5210,1.7659,2.0055,2.2423),E[w](2)=(1.3006,1.5723,1.8304,2.0756,2.3159),\begin{array}{l}{\mathbb{E}}{\left[{\bf{w}}]}^{\left(1)}=\left(1.2675,1.5210,1.7659,2.0055,2.2423),\\ {\mathbb{E}}{\left[{\bf{w}}]}^{\left(2)}=\left(1.3006,1.5723,1.8304,2.0756,2.3159),\end{array}E[w](3)=(1.3220,1.6125,1.8781,2.1304,2.3735),E[w](4)=(1.3451,1.6557,1.9392,2.2017,2.4483).\begin{array}{l}{\mathbb{E}}{\left[{\bf{w}}]}^{\left(3)}=\left(1.3220,1.6125,1.8781,2.1304,2.3735),\\ {\mathbb{E}}{\left[{\bf{w}}]}^{\left(4)}=\left(1.3451,1.6557,1.9392,2.2017,2.4483).\end{array}Therefore, the optimal strategy is specified as follows: v0(1)*=(−1.05w0+2.0635)T1,v0(2)*=(−1.05w0+2.2182)T1,v1(1)*=(−1.05w1+2.2175)T1,v1(2)*=(−1.05w1+2.3291)T1,v2(1)*=(−1.05w2+2.3841)T1,v2(2)*=(−1.05w2+2.4879)T1,v3(1)*=(−1.05w3+2.5596)T1,v3(2)*=(−1.05w3+2.6384)T1,v4(1)*=(−1.05w4+2.7423)T1,v4(2)*=(−1.05w4+2.8159)T1,\begin{array}{l}{v}_{0}^{\left(1)* }=\left(-1.05{w}_{0}+2.0635){T}_{1},\hspace{.95em}{v}_{0}^{\left(2)* }=\left(-1.05{w}_{0}+2.2182){T}_{1},\\ \hspace{0.27em}{v}_{1}^{\left(1)* }=\left(-1.05{w}_{1}+2.2175){T}_{1},\hspace{1em}{v}_{1}^{\left(2)* }=\left(-1.05{w}_{1}+2.3291){T}_{1},\\ \hspace{0.01em}{v}_{2}^{\left(1)* }=\left(-1.05{w}_{2}+2.3841){T}_{1},\hspace{1em}{v}_{2}^{\left(2)* }=\left(-1.05{w}_{2}+2.4879){T}_{1},\\ {v}_{3}^{\left(1)* }=\left(-1.05{w}_{3}+2.5596){T}_{1},\hspace{1em}{v}_{3}^{\left(2)* }=\left(-1.05{w}_{3}+2.6384){T}_{1},\\ \hspace{0.1em}{v}_{4}^{\left(1)* }=\left(-1.05{w}_{4}+2.7423){T}_{1},\hspace{1em}{v}_{4}^{\left(2)* }=\left(-1.05{w}_{4}+2.8159){T}_{1},\end{array}v0(3)*=(−1.05w0+2.3182)T1,v0(4)*=(−1.05w0+2.4256)T1,v1(3)*=(−1.05w1+2.4341)T1,v1(4)*=(−1.05w1+2.5468)T1,v2(3)*=(−1.05w2+2.5558)T1,v2(4)*=(−1.05w2+2.6742)T1,v3(3)*=(−1.05w3+2.7103)T1,v3(4)*=(−1.05w3+2.8079)T1,v4(3)*=(−1.05w4+2.8735)T1,v4(4)*=(−1.05w4+2.9483)T1,\begin{array}{l}\hspace{0.025em}{v}_{0}^{\left(3)* }=\left(-1.05{w}_{0}+2.3182){T}_{1},\hspace{1em}{v}_{0}^{\left(4)* }=\left(-1.05{w}_{0}+2.4256){T}_{1},\\ \hspace{0.015em}{v}_{1}^{\left(3)* }=\left(-1.05{w}_{1}+2.4341){T}_{1},\hspace{1em}{v}_{1}^{\left(4)* }=\left(-1.05{w}_{1}+2.5468){T}_{1},\\ \hspace{-0.05em}{v}_{2}^{\left(3)* }=\left(-1.05{w}_{2}+2.5558){T}_{1},\hspace{1em}{v}_{2}^{\left(4)* }=\left(-1.05{w}_{2}+2.6742){T}_{1},\\ \hspace{0.01em}{v}_{3}^{\left(3)* }=\left(-1.05{w}_{3}+2.7103){T}_{1},\hspace{1em}{v}_{3}^{\left(4)* }=\left(-1.05{w}_{3}+2.8079){T}_{1},\\ \hspace{0.05em}{v}_{4}^{\left(3)* }=\left(-1.05{w}_{4}+2.8735){T}_{1},\hspace{1em}{v}_{4}^{\left(4)* }=\left(-1.05{w}_{4}+2.9483){T}_{1},\end{array}where T1{T}_{1}is the same as Example 1. The optimal variances under the best strategy can be derived as follows: Var(w)(i)=(Var[w1](i),Var[w2](i),Var[w3](i),Var[w4](i),Var[w5](i)),i=1,2,3,4\hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(i)}=\left(\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{1}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{2}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{3}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{4}]}^{\left(i)},\hspace{0.1em}\text{Var}\hspace{0.1em}{\left[{w}_{5}]}^{\left(i)}),\hspace{1em}i=1,2,3,4which are given as follows: Var(w)(1)=(0.1731,0.2824,0.3489,0.3860,0.4026),Var(w)(2)=(0.2299,0.3555,0.4260,0.4554,0.4626),Var(w)(3)=(0.2710,0.4190,0.4882,0.5146,0.5140),Var(w)(4)=(0.3188,0.4930,0.5744,0.5978,0.5860).\begin{array}{l}\hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(1)}=\left(0.1731,0.2824,0.3489,0.3860,0.4026),\\ \hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(2)}=\left(0.2299,0.3555,0.4260,0.4554,0.4626),\\ \hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(3)}=\left(0.2710,0.4190,0.4882,0.5146,0.5140),\\ \hspace{0.1em}\text{Var}\hspace{0.1em}{\left({\bf{w}})}^{\left(4)}=\left(0.3188,0.4930,0.5744,0.5978,0.5860).\end{array}Thus,we have E(κ)[w5∧κ](1)=1.8821,Var(κ)(w5∧κ)(1)=0.3467,E(κ)[w5∧κ](2)=2.1209,Var(κ)(w5∧κ)(2)=0.4461,E(κ)[w5∧κ](3)=2.2753,Var(κ)(w5∧κ)(3)=0.5115,E(κ)[w5∧κ](4)=2.4483,Var(κ)(w5∧κ)(4)=0.5860.\hspace{-22.9em}\begin{array}{l}\hspace{0.1em}{{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(1)}=1.8821,\hspace{1em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(1)}=0.3467,\\ \hspace{0.1em}{{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(2)}=2.1209,\hspace{0.9em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(2)}=0.4461,\\ \hspace{0.05em}{{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(3)}=2.2753,\hspace{0.97em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(3)}=0.5115,\\ {{\mathbb{E}}}^{\left(\kappa )}{\left[{w}_{5\wedge \kappa }]}^{\left(4)}=2.4483,\hspace{0.75em}{\text{Var}}^{\left(\kappa )}{\left({w}_{5\wedge \kappa })}^{\left(4)}=0.5860.\\ \end{array}Figure 1 depicts the efficient frontier with different probability mass function of the exit time. We can see that the one exits at the terminal time gets the most expected wealth return at the same risk level compared with others. It is also indicated that if the investment is more stable, the investors can obtain higher expected returns at the same level of the risk, which is consistent with the real life.Figure 1Efficient frontiers with different probability mass functions of exit time.Example 3An example under uncertain exit time with liabilityThe probability mass function of an exit time κ\kappa is defined as follows: (D1,D2,D3,D4,D5)=(0.10,0.15,0.2,0.25,0.3).\left({D}_{1},{D}_{2},{D}_{3},{D}_{4},{D}_{5})=\left(0.10,0.15,0.2,0.25,0.3).Thus, the optimal expected value of assets in different time periods is given by E[w]=(4.3710,5.7834,7.2450,8.7621,10.3403).{\mathbb{E}}\left[{\bf{w}}]=\left(4.3710,5.7834,7.2450,8.7621,10.3403).Suppose the initial wealth of the investor w0=3{w}_{0}=3, initial liability Q0=1{Q}_{0}=1, and risk aversion parameter λ=1\lambda =1, we can derive the optimal portfolio selection after substituting the number to the aforementioned equations in Theorem 1: v0∗=−1.05(w0−4.9032)T1−T2+0.1595T3Q0,v1∗=−1.05(w1−6.1660)T1−T2+0.2377T3Q1,v2∗=−1.05(w2−7.4853)T1−T2+0.3458T3Q2,v3∗=−1.05(w3−8.8694)T1−T2+0.5373T3Q3,v4∗=−1.05(w4−10.3209)T1−T2+1.0000T3Q4.\begin{array}{l}{v}_{0}^{\ast }=-1.05\left({w}_{0}-4.9032){T}_{1}-{T}_{2}+0.1595{T}_{3}{Q}_{0},\\ {v}_{1}^{\ast }=-1.05\left({w}_{1}-6.1660){T}_{1}-{T}_{2}+0.2377{T}_{3}{Q}_{1},\\ {v}_{2}^{\ast }=-1.05\left({w}_{2}-7.4853){T}_{1}-{T}_{2}+0.3458{T}_{3}{Q}_{2},\\ {v}_{3}^{\ast }=-1.05\left({w}_{3}-8.8694){T}_{1}-{T}_{2}+0.5373{T}_{3}{Q}_{3},\\ {v}_{4}^{\ast }=-1.05\left({w}_{4}-10.3209){T}_{1}-{T}_{2}+1.0000{T}_{3}{Q}_{4}.\end{array}Furthermore, the final value of mean and variance under the optimal strategy are E(κ)(w5∧κ)=8.0462{{\mathbb{E}}}^{\left(\kappa )}\left({w}_{5\wedge \kappa })=8.0462and Var(κ)(w5∧κ)=0.3901{\text{Var}}^{\left(\kappa )}\left({w}_{5\wedge \kappa })=0.3901, respectively.Following Example 2, we choose four different probability mass functions at the exit time κ\kappa , α(i)=(α1(i),α2(i),α3(i),α4(i),α5(i)){\alpha }^{\left(i)}=\left({\alpha }_{1}^{\left(i)},{\alpha }_{2}^{\left(i)},{\alpha }_{3}^{\left(i)},{\alpha }_{4}^{\left(i)},{\alpha }_{5}^{\left(i)}), (i=1,2,3,4)\left(i=1,2,3,4), as follows: α(1)=(0.1,0.15,0.2,0.25,0.3),α(2)=(0,0.1,0.1,0.3,0.5),α(3)=(0,0,0.1,0.2,0.7),α(4)=(0,0,0,0,1).\begin{array}{rcl}{\alpha }^{\left(1)}& =& \left(0.1,0.15,0.2,0.25,0.3),\\ {\alpha }^{\left(2)}& =& \left(0,0.1,0.1,0.3,0.5),\\ {\alpha }^{\left(3)}& =& \left(0,0,0.1,0.2,0.7),\\ {\alpha }^{\left(4)}& =& \left(0,0,0,0,1).\end{array}Figure 2 is the efficient frontier of M–V model with liability and random cashflow under uncertain exit time. It can be seen that as the expectation go up, the more stable the investment, the less risk it takes, which has the same conclusion as Figure 1. Actually, Example 2 is a special case of Example 3, where we degenerate the term of liabilities to zero.Figure 2Efficient frontiers with different exit time.6ConclusionThe focus of the paper is placed on investigating the optimal strategy of multi-period mean-variance model with cash flow, and liability under uncertain exit time. It is a nonseparable dynamic programming problem that cannot be solved by the traditional method. In this paper, we transform the original model into a mean-field type and apply a dynamic programming approach and matrix theory to derive the optimal strategy explicitly. Our methods are shown to be much more efficient and accurate compared with other methods in the literature. For further research, we will try to employ the mean-field method to derive the mean-variance model with various additional conditions such as regime switching, bankruptcy constraints, and time inconsistency.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: mean-field formulation; optimal strategy; closed-form expressions; 49L20; 91G10; 93E20

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