Converting Risk Preferences into Money Equivalents with

Quadratic Programming

AEC 851 – Agribusiness Operations Management

Spring, 2006

Expected Utility Model

• A numerical “utility” value can be linked to any risky prospect if a manager’s preferences meet these conditions:– Can be ordered– Are transitive– Are continuous– Are independent of irrelevant alternatives

Key results from EUM

• A manager is risk averse if he or she prefers the expected outcome of a risky prospect to the risky prospect itself– So utility function is concave

• Certainty equivalent (xCE) is value that would leave manager indifferent between that and expected outcome – E[U(x)] = U(xCE)

Utility function showing risk aversity, certainty equivalent and risk premium

Source: Boisvert & McCarl (1990)

• Risk premium () is amount a risk-averse manager would be willing to pay to avoid a risky prospect:– U(E[x- ]) = E[U(x)]

EU functions

• Risk aversion is shown by the degree of curvature of the utility function

• Math functions exist that characterize– Constant absolute risk aversion (constant rate of

curvature of utility function) (CARA)– Constant relative risk aversion (constant rate of risk

aversion relative to total wealth) (CRRA)

• However, these functions have limitations:– 1) Complicated forms for certainty equivalent– 2) Not clear how many people’s preferences are

accurately described by CARA or CRRA

Money measures of EU

• Certainty equivalents are money values that can be derived from expected utility functions– In money units, CE’s measure the

manager’s expected utility from a risky prospect

• Mean-variance expected utility is a simple way to approximate CE’s

Mean-variance (E-V) to express Expected Utility

• Expected utility can be expressed as a function of mean and variance, i.e.,– UEV(x) = xCE = E(x) – (/2)x

2

– What is the risk premium () in this equation?– (/2) weights the variance

• Alternative assumptions:– Manager has CARA utility and outcomes (x) follow

normal distribution: x ~ N(x, x2)

– Want local approximation to a generic expected utility function, using a Taylor series approximation

Source: Robison & Barry (1987)

Mean-Variance (EV) indifference curve and feasible set

E-V risk programming models

• Quadratic programming (QP)– Max E(x) subject to max Var(x)– Min Var(x) subject to min E(x)– Max E(x) – (/2)Var(x)

• Minimization of Total Absolute Deviations (MOTAD) is analogous to QP but is linear (so uses LP algorithm)

Other risk programming models

• Extensions of sensitivity analysis– Breakeven values (parametric

programming)

• Catastrophic risk modeling– Safety-first programming– Chance-constrained programming