Converting Risk Preferences into Money Equivalents with Quadratic Programming

1
Converting Risk Preferences into Money
Equivalents with Quadratic Programming

• AEC 851 Agribusiness Operations Management
• Spring, 2006

2
Expected Utility Model

• A numerical utility value can be linked to any
  risky prospect if a managers preferences meet
  these conditions
• Can be ordered
• Are transitive
• Are continuous
• Are independent of irrelevant alternatives

3
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
• EU(x) U(xCE)

4
Utility function showing risk aversity,
certainty equivalent and risk premium
Source Boisvert McCarl (1990)
5

• Risk premium (?) is amount a risk-averse manager
  would be willing to pay to avoid a risky
  prospect
• U(Ex- ?) EU(x)

6
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 peoples preferences are
  accurately described by CARA or CRRA

7
Money measures of EU

• Certainty equivalents are money values that can
  be derived from expected utility functions
• In money units, CEs measure the managers
  expected utility from a risky prospect
• Mean-variance expected utility is a simple way to
  approximate CEs

8
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)?x2
• 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

9
Mean-Variance (EV) indifference curve and
feasible set
Source Robison Barry (1987)
10
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)

11
Other risk programming models

• Extensions of sensitivity analysis
• Breakeven values (parametric programming)
• Catastrophic risk modeling
• Safety-first programming
• Chance-constrained programming