Theory of portfolio and risk based
on incremental entropy
Investment Director, Qingdao Development and Investment Company,
Qingdao, PR China
Purpose – To develop a new theory of portfolio and risk based on incremental entropy and
Design/methodology/approach – Replacing arithmetic, the mean return adopted by M.H.
Markowitz, with geometric mean return as a criterion for assessing a portfolio, one gets
incremental entropy: one of the generalized entropies. It indicates that the incremental speed of capital
is a more objective and testable criterion.
Findings – The difference between the new theory based on incremental entropy and Markowitz’s
theory is that the new theory emphasizes that there is an objectively optimal portfolio for given
probability of returns.
Originality/value – This paper provides some formulas for optimizing portfolio allocations. Based
on the new portfolio theory, this paper also presents a new measure of information value, analyzes the
differences and similarities between this measure and K.J. Arrow’s measure of information value, and
discusses how to optimize forecasts with the new measure.
Keywords Portfolio theory, Risk management, Optimization techniques
Paper type Technical paper
The author found that incremental entropy, one of the generalized entropies, could be
used to optimize portfolios. The new portfolio theory based on incremental entropy
carries on some aspects of Markowitz’s (1959, 1991) theory, but it emphasizes that the
incremental speed of capital is a more objective criterion for assessing portfolios. Given
probability forecasts of returns, we can obtain the optimal investment ratio. Combining
the new portfolio theory and the general theory of information, we can approach a
meaning-explicit measure, which represents the increment of capital-increasing speed
after information is provided.
Assume there is an investment whose return is determined by tossing a coin. If A-side
appears, you will win 200 percent of your wager. If B-side appears, you will lose 100
percent of your wager. You have only 100 dollars and have no way to borrow. What is
the optimal investment ratio so that your capital could increase fastest, or say, you
could become a millionaire fastest after many times of gambling?
Staking all of your money every time is obviously infeasible because once B-side
appears you will lose all of money and lose the opportunity of becoming rich forever.
Staking 10 percent of your money is not too bad since you might not lose forever. But
the capital-increasing speed might be too low. How can we ﬁnd the optimal investment
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portfolio and risk
Journal of Risk Finance
Vol. 6 No. 1, 2005
q Emerald Group Publishing Limited