More General Conditions Under Which Mean-Variance Decision Making Is Unjustified

Risk Analysis, Vol. 30, No. 3, 2010 DOI: 10.1111/j.1539-6924.2010.01358.x Response More General Conditions Under Which Mean-Variance Decision Making Is Unjustified Bill Huber has presented a new, simpler, and more general proof of the fact that mean-variance decision making violates the principle that a rational decisionmaker should prefer higher to lower proba- bilities of receiving a fixed gain, all else being equal. The new version holds even for preferences that cannot be represented by well-behaved (e.g., continu- ous, downward-sloping) indifference curves, showing that the result is quite fundamental. It is perhaps somewhat startling that mean-variance methods con- tinue to enjoy widespread practical application in risk management, as this practice risks failing to se- lect dominant choices unless restrictive assumptions (such as normally distributed consequences) can be justified. I thank Bill Huber for improving upon and gen- eralizing my note on the inconsistency of mean- variance preferences with the dominance principle (“first-order stochastic dominance” (FSD), in the ter- minology of the field), that rational decisionmakers should, whenever possible, choose prospects that make preferred outcomes more probable. My orig- inal note mentioned that it did not cover the case of vertical indifference curves. Bill Huber’s proof takes care of that case, and several others in which the con- venient assumptions of smooth, downward-sloping indifference curves do not hold. As Bill pointed out in private correspondence, the new proof es- tablishes the result with great generality: not only is the framework of expected utility theory dis- pensed with, but even the assumption that preferences are well-defined and that prospects can be well-ordered is relaxed. Thus, the inconsistency of mean-variance decision making with FSD seems to be quite fundamental. Interestingly, this observation, which builds on several decades of theoretical research, has not prevented continuing enthusiasm for mean-variance models among theorists and practitioners, even when the special circumstances that would prevent FSD violations (such as normal distributions of consequences) cannot be assumed. It is an interesting empirical question to what extent practitioners who apply mean-variance risk models in different fields are led thereby to make dominated decisions. Tony Cox 329 0272-4332/10/0100-0329$22.00/1 C 2010 Society for Risk Analysis

Upload
tony-cox
Category

Documents
view
218
download
0

Embed Size (px):

Transcript of More General Conditions Under Which Mean-Variance Decision Making Is Unjustified

Page 1: More General Conditions Under Which Mean-Variance Decision Making Is Unjustified

Risk Analysis, Vol. 30, No. 3, 2010 DOI: 10.1111/j.1539-6924.2010.01358.x

Response

More General Conditions Under Which Mean-VarianceDecision Making Is Unjustified

Bill Huber has presented a new, simpler, andmore general proof of the fact that mean-variancedecision making violates the principle that a rationaldecisionmaker should prefer higher to lower proba-bilities of receiving a fixed gain, all else being equal.The new version holds even for preferences that can-not be represented by well-behaved (e.g., continu-ous, downward-sloping) indifference curves, showingthat the result is quite fundamental. It is perhapssomewhat startling that mean-variance methods con-tinue to enjoy widespread practical application inrisk management, as this practice risks failing to se-lect dominant choices unless restrictive assumptions(such as normally distributed consequences) can bejustified.

I thank Bill Huber for improving upon and gen-eralizing my note on the inconsistency of mean-variance preferences with the dominance principle(“first-order stochastic dominance” (FSD), in the ter-minology of the field), that rational decisionmakersshould, whenever possible, choose prospects thatmake preferred outcomes more probable. My orig-inal note mentioned that it did not cover the case ofvertical indifference curves. Bill Huber’s proof takes

care of that case, and several others in which the con-venient assumptions of smooth, downward-slopingindifference curves do not hold. As Bill pointedout in private correspondence, the new proof es-tablishes the result with great generality: not onlyis the framework of expected utility theory dis-pensed with, but even the assumption that prefer-ences are well-defined and that prospects can bewell-ordered is relaxed. Thus, the inconsistency ofmean-variance decision making with FSD seems tobe quite fundamental.

Interestingly, this observation, which builds onseveral decades of theoretical research, has notprevented continuing enthusiasm for mean-variancemodels among theorists and practitioners, even whenthe special circumstances that would prevent FSDviolations (such as normal distributions of conse-quences) cannot be assumed. It is an interestingempirical question to what extent practitioners whoapply mean-variance risk models in different fieldsare led thereby to make dominated decisions.

Tony Cox