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Converting Risk Preferences into Money Equivalents with Quadratic Programming

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

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Converting Risk Preferences into Money Equivalents with Quadratic Programming

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  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 manager’s 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 • E[U(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(E[x- ]) = E[U(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 people’s 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, CE’s measure the manager’s expected utility from a risky prospect • Mean-variance expected utility is a simple way to approximate CE’s

  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

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