Mean-variance analysis

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.

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 ${\displaystyle r}$, and one riskfree asset with known return ${\displaystyle r_{f}}$, and let an investor's initial wealth be ${\displaystyle w_{0}}$. If the amount ${\displaystyle q}$, the choice variable, is to be invested in the risky asset and the amount ${\displaystyle w_{0}-q}$ is to be invested in the safe asset, then, contingent on ${\displaystyle q}$, the investor's random final wealth will be ${\displaystyle w=(w_{0}-q)r_{f}+qr}$. Then for any choice of ${\displaystyle q}$, ${\displaystyle w}$ is distributed as a location-scale transformation of ${\displaystyle r}$. If we define random variable ${\displaystyle x}$ as equal in distribution to ${\displaystyle {\tfrac {w-\mu _{w}}{\sigma _{w}}},}$ then ${\displaystyle w}$ is equal in distribution to ${\displaystyle \mu _{w}+\sigma _{w}x}$, where μ represents an expected value and σ represents a random variable's standard deviation (the square root of its second moment). Thus we can write expected utility in terms of two moments of ${\displaystyle w}$:

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