INVESTOR PREFERENCES

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INVESTOR PREFERENCES

• Pricing of derivatives relies primarily on
  arbitrage arguments
• To get very far in portfolio management, however,
  we need to make some assumptions about investor
  utility
• What are investors attitudes toward risk?
• How can we allow for different attitudes?

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ST. PETERSBURG PARADOX

• Do investors maximize expected wealth?
• Toss a fair coin repeatedly
• Receive 1 if heads comes up for the first time
  on first toss
• Receive 2 if heads comes up for first time on
  second toss
• Receive 2n-1 if heads comes up for first time on
  nth toss

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COIN FLIPPING PAYOFF
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RISK AVERSION

• Conclusion People are risk averse
• A sure amount is worth more than the same
  expected (but uncertain) amount
• Willingness to pay to avoid a gamble
• Utility function concave

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ATTITUDE TOWARD A GAMBLE

• Gamble pays 1000 (prob. .6) or
  0 (prob. .4)
• E(W) .6(1000).4(0) 600
• Linear function of probabilities
• U(1000) 100 U(0) 0
• EU(W) .6(100).4(0) 60
• Less than utility of 600 for sure

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CONCAVE UTILITY FUNCTION
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MEASURING RISK AVERSION

• Markowitz Risk Premium
• E(W) - CEQ
• Amount investor would pay to avoid a gamble with
  expected value E(W)
• Coefficient of Absolute Risk Aversion
• ARA -U(W)/U(W)
• Coefficient of Relative Risk Aversion
• RRA -WU(W)/U(W)

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EXAMPLE POWER UTILITY
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EXAMPLE LOG UTILITY
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PROBLEM 1 LOG UTILITY
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MEAN-VARIANCE UTILITY

• E(Rp) expected portfolio return
• s2(Rp) variance of portfolio return
• A coefficient of risk aversion (constant
  relative risk aversion)

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MEAN-VARIANCE DEFINITIONS

• Mean return
• Variance
• Std. Dev.
• Covariance
• Correlation

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PORTFOLIO MEAN AND VARIANCE
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PORTFOLIO MEAN AND VARIANCEn securities
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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

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PORTFOLIO PROBLEM 1ONE RISKY SECURITY AND A
RISK-FREE SECURITY
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SOLUTION TO PP1

• Proportion of asset x held in portfolio relative
  to risk-free asset depends
• Positively on risk premium on x
• Negatively on xs risk
• Negatively on investor risk aversion

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PORTFOLIO PROBLEM 2TWO RISKY SECURITIES
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SOLUTION TO PORT. PROB. 2