INVESTOR PREFERENCES èPricing of derivatives relies primarily on arbitrage arguments èTo get very far in portfolio management, however, we need to make some assumptions about investor utility èWhat are investors’ attitudes toward risk? èHow can we allow for different attitudes?

ST. PETERSBURG PARADOX èDo investors maximize expected wealth? èToss a fair coin repeatedly èReceive $1 if heads comes up for the first time on first toss èReceive $2 if heads comes up for first time on second toss èReceive $2 n-1 if heads comes up for first time on n th toss

COIN FLIPPING PAYOFF

RISK AVERSION èConclusion: People are risk averse èA sure amount is worth more than the same expected (but uncertain) amount èWillingness to pay to avoid a gamble èUtility function concave

ATTITUDE TOWARD A GAMBLE èGamble pays $1000 (prob. =.6) or $0 (prob. =.4) èE(W) =.6(1000)+.4(0) = $600 èLinear function of probabilities èU(1000) = 100U(0) = 0 èE[U(W)] =.6(100)+.4(0) = 60 èLess than utility of $600 for sure

CONCAVE UTILITY FUNCTION

MEASURING RISK AVERSION èMarkowitz Risk Premium èE(W) - CEQ èAmount investor would pay to avoid a gamble with expected value E(W) èCoefficient of Absolute Risk Aversion èARA = -U’’(W)/U’(W) èCoefficient of Relative Risk Aversion èRRA = -WU’’(W)/U’(W)

EXAMPLE: POWER UTILITY

EXAMPLE: LOG UTILITY

PROBLEM 1: LOG UTILITY

MEAN-VARIANCE UTILITY E(R p ) = expected portfolio return    R p ) = variance of portfolio return A = coefficient of risk aversion (constant relative risk aversion)

MEAN-VARIANCE DEFINITIONS Mean return Variance Std. Dev. Covariance Correlation

PORTFOLIO MEAN AND VARIANCE

PORTFOLIO MEAN AND VARIANCE: n securities

LINEARLY RELATED SECURITIES a, b are constants Useful when each security has a return component linearly related to the market return plus an unrelated return component

PORTFOLIO PROBLEM #1 ONE RISKY SECURITY AND A RISK-FREE SECURITY

SOLUTION TO PP#1 Proportion of asset x held in portfolio relative to risk-free asset depends èPositively on risk premium on x èNegatively on x’s risk èNegatively on investor risk aversion

PORTFOLIO PROBLEM #2 TWO RISKY SECURITIES

SOLUTION TO PORT. PROB. #2