Two-moment decision model

Overview

In decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be known in advance, and in which choices are made based on knowledge of two moments of those random variables. The two moments are almost always the mean—that is, the expected value, which is the first moment about zero—and the variance, which is the second moment about the mean (or the standard deviation, which is the square root of the variance). The most well-known two-moment decision model is that of modern portfolio theory, which gives rise to the decision portion of the Capital Asset Pricing Model; these employ mean-variance analysis, and focus on the mean and variance of a portfolio's final value.

Two-moment models and expected utility maximization

Suppose that all relevant random variables are in the same location-scale family, meaning that the distribution of every random variable is the same as the distribution of some linear transformation of any other random variable. Then for any von Neumann–Morgenstern utility function, using a mean-variance decision framework is consistent with expected utility maximization, as illustrated in example 1: Example 1: Let there be one risky asset with random return

r

{\displaystyle r}

, and one riskfree asset with known return

r

f

{\displaystyle r_{f}}

, and let an investor's initial wealth be

w

0

{\displaystyle w_{0}}

. If the amount

q

{\displaystyle q}

, the choice variable, is to be invested in the risky asset and the amount

w

0

− q

{\displaystyle w_{0}-q}

is to be invested in the safe asset, then, contingent on

Non-expected-utility decision making

If the decision-maker is not an expected utility maximizer, decision-making can still be framed in terms of the mean and variance of a random variable if all alternative distributions for an unpredictable outcome are location-scale transformations of each other.

See also

Decision theory Intertemporal portfolio choice Microeconomics