The Value of Judgment: A Mathematical Theory

Abstract

Superior investment returns are often attributed to superior information processing capabilities of investors. Inferior investment returns are often attributed to inferior cognitive capabilities of investors. While investment return can be quantified, there does not exist a quantitative measure of judgment. In this paper, we develop a mathematical theory of judgment. It provides a quantitative measure on the value of judgment. This theory is generalized from the entropy theory of information when subjective assessment of the probability distribution of a random event differs from the objective probability distribution. The formula of the value of judgment defined in this theory bridges the chasm between the concept of information and cognitive bias. To evaluate the validity of the new theory, we will compare the value of a judgment calculated from this theory with the expected rates of return of the portfolio constructed from the same judgment. Investment decisions are made according to investors' judgment about stocks. For example, if an investor believes one stock will significantly outperform the general market, he will put a significant portion of his investment fund into that stock. To quantify the relation between judgment and investment decision, we will consider a simple portfolio with only two assets: a risk free asset and a risky asset. Based on the subjective assessment of the return distribution of the risky asset, an investor can determine the optimal portion of the risky asset in the portfolio and calculate the expected rate of return of this portfolio. We prove that the first order approximation of the expected rate of return of the portfolios constructed from a judgment is equal to the value of the same judgment. This indicates the value of judgment defined in this theory is built on a solid foundation. Since the value of judgment provides a good approximation to the rate of return on investment, it can be conveniently used to understand the relation between human judgment and patterns in investment returns and stock market.

Keywords: judgment, information, entropy, bias, investment return, behavioral finance

JEL Classification: G14