 Twomoment decision models

 Meanvariance analysis redirects here. For meanvariance portfolio theory, see Modern portfolio theory or Mutual fund separation theorem.
In decision theory, economics, and finance, a twomoment decision model is a model that describes or prescribes the process of making decisions in a context in which the decisionmaker 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).
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Twomoment models and expected utility maximization
Suppose that all relevant random variables are in the same locationscale 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 NeumannMorgenstern utility function, using a meanvariance decision framework is consistent with expected utility maximization,^{[1]}^{[2]} as illustrated in example 1:
Example 1:^{[3]}^{[4]} Let there be one risky asset with random return r, and one riskfree asset with known return r_{f}, and let an investor's initial wealth be w_{0}. If the amount q, the choice variable, is to be invested in the risky asset and the amount w_{0} – q is to be invested in the safe asset, then contingent on q the investor's random final wealth will be w=(w_{0} – q)r_{f} + qr. Then for any choice of q, w is distributed as a locationscale transformation of r. If we define random variable x as equal in distribution to then w is equal in distribution to (Ew + σ_{w}x), where E 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 w :
where u is the von NeumannMorgenstern utility function, f is the density function of x, and v is the derived meanstandard deviation choice function, which depends in form on the density function f. The von NeumannMorgenstern utility function is assumed to be increasing, implying that more wealth is preferred to less, and it is assumed to be concave, which is the same as assuming that the individual is risk averse.
It can be shown that the partial derivative of v with respect to Ew is positive, and the partial derivative of v with respect to σ_{w} is negative; thus more expected wealth is always liked, and more risk (as measured by the standard deviation of wealth) is always disliked. A meanstandard deviation indifference curve is defined as the locus of points (σ_{w} , Ew) with σ_{w} plotted horizontally, such that Eu(w) has the same value at all points on the locus. Then the derivatives of v imply that every indifference curve is upward sloped: that is, along any indifference curve dEw / dσ_{w} > 0. Moreover, it can be shown^{[3]} that all such indifference curves are convex: along any indifference curve, d^{2}Ew / d(σ_{w})^{2} > 0.
Example 2: The portfolio analysis in example 1 can be generalized. If there are n risky assets instead of just one, and if their returns are jointly elliptically distributed, then^{[5]}^{[6]} all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return—and all possible portfolios have return distributions that are locationscalerelated to each other. Thus portfolio optimization can be implemented using a twomoment decision model.
Example 3: Suppose that a pricetaking, riskaverse firm must commit to producing a quantity of output q before observing the market realization p of the product's price.^{[7]} Its decision problem is to choose q so as to maximize the expected utility of profit:
 Maximize Eu(pq – c(q) – g),
where E is the expected value operator, u is the firm's utility function, c is its variable cost function, and g is its fixed cost. All possible distributions of the firm's random revenue pq, based on all possible choices of q, are locationscale related; so the decision problem can be framed in terms of the expected value and variance of revenue.
Nonexpectedutility decision making
If the decisionmaker is not an expected utility maximizer, decisionmaking can still be framed in terms of the mean and variance of a random variable if all alternative distributions for an unpredictable outcome are locationscale transformations of each other.^{[8]}
See also
References
 ^ Mayshar, J., "A note on Feldstein's criticism of meanvariance analysis," Review of Economic Studies 45, 1978, 197199.
 ^ Sinn, H.W., Economic Decisions under Uncertainty, second English edition, 1983, NorthHolland.
 ^ ^{a} ^{b} Meyer, Jack. "Twomoment decision models and expected utility maximization," American Economic Review 77, June 1987, 421430.
 ^ Tobin, J., "Liquidity preference as behavior towards risk," Review of Economic Studies 25(1), February 1958, 65–86. Also in: (1) M. G. Mueller, ed., Readings in Macroeconomics, Holt, Rinehart & Winston, Inc., 1966, pp. 6586; (2) Richard S. Thorn, ed., Monetary Theory and Policy, Random House, 1966, pp. 172–191; (3) H. R. Williams and J. D. Huffnagle, eds., Macroeconomic Theory, AppletonCenturyCrofts, 1969, pp. 299–324; (4) Essays in Economics: Macroeconomics, Vol. 1, chapter 15; (5) J. Tobin and D. Hester, eds., Risk Aversion and Portfolio Choice, Cowles Monograph No. 19, John Wiley & Sons, 1967; (6) David Laidler, ed., The Foundations of Monetary Economics, Vol. 1, Edward Elgar Publishing Ltd., 1999.
 ^ Chamberlain, G., "A characterization of the distributions that imply meanvariance utility functions", Journal of Economic Theory 29, 1983, 185201.
 ^ Owen, J., and Rabinovitch, R. "On the class of elliptical distributions and their applications to the theory of portfolio choice", Journal of Finance 38, 1983, 745752.
 ^ Sandmo, Agnar. "On the theory of the competitive firm under price uncertainty," American Economic Review 61, March 1971, 6573.
 ^ BarShira, Z., and Finkelshtain, I., "Twomoments decision models and utilityrepresentable preferences," Journal of Economic Behavior and Organization 38, 1999, 237244. See also Mitchell, Douglas W., and Gelles, Gregory M., "Twomoments decision models and utilityrepresentable preferences: A comment on BarShira and Finkelshtain, vol. 49, 2002, 423427.
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