LECTURE 2 : UTILITY AND RISK AVERSION (Asset Pricing and Portfolio Theory)

Contents Introduction to utility theory Introduction to utility theory Relative and absolute risk aversion Relative and absolute risk aversion Different forms of utility functions Different forms of utility functions Empirical evidence Empirical evidence How useful are the general findings ? How useful are the general findings ? –Equity premium puzzle –Risk free rate puzzle

Introduction Many different investment opportunities with different risk – return characteristics Many different investment opportunities with different risk – return characteristics General assumption : ‘Like returns, dislike risk’ General assumption : ‘Like returns, dislike risk’ Preferences of investors (like more to less) Preferences of investors (like more to less)

Risk Premium and Risk Aversion Risk free rate of return : Risk free rate of return : rate of return which can be earned with certainty (i.e  = 0). 3 months T-bill Risk premium : Risk premium : expected return in excess of the risk free rate (i.e. ER p – r f ) Risk aversion : Risk aversion : –measures the reluctance by investors to accept (more) risk –‘High number’ : risk averse –‘Low number’ : less risk averse Example : ER p - r f = A   p Example : ER p - r f = A   p A = (ER p - r f ) / (0.005   p )

Indifference Curves (Investor’s Preferences) ER p Asset P Indifference curve pp Asset Q

Risk and Return (US Assets : 1926 – 1998) (% p.a.) ER Standard deviation Small Company Stocks Large Company Stocks Treasury Bills Medium Term T-bonds Long Term T-bonds

Expected Utility Suppose we have a random variable, end of period wealth with ‘n’ possible outcomes W i with probabilities p i Suppose we have a random variable, end of period wealth with ‘n’ possible outcomes W i with probabilities p i Utility from any wealth outcome is denoted U(W i ) Utility from any wealth outcome is denoted U(W i ) E[U(W)] =  p i U(W i )

Example : Alternative Investments Investment A Investment B Investment C OutcomeProbOutcomeProbOucomeProb 203/15191/518¼ 185/15102/516¼ 144/1552/512¼ 102/158¼ 61/15

Example : Alternative Investments (Cont.) Assume following utility function : Assume following utility function : U(W) = 4W – (1/10) W 2 –If outcome is 20, U(W) = 80 – (1/10) 400 = 40 –… –Expected Utility Investment A : E(U A ) = … = 36.3 Investment A : E(U A ) = … = 36.3 Investment B : E(U B ) = … = Investment B : E(U B ) = … = Investment C : Investment C : E(U C ) = 39.6(1/4) (1/4) (1/4) (1/4) = 34.3 E(U C ) = 39.6(1/4) (1/4) (1/4) (1/4) = 34.3

Utility Function : U(W) = 4W – (1/10)W 2

Example : Alternative Investments (Cont.) Ranking of investments remains unchanged if Ranking of investments remains unchanged if –a constant is added to the utility function –the utility function is scaled by a constant Example : a + bU(W) gives the same ranking as U(W)

Fair Lottery A fair lottery is defined as one that has expected value of zero. A fair lottery is defined as one that has expected value of zero. Risk aversion applies that an individual would not accept a ‘fair lottery’. Risk aversion applies that an individual would not accept a ‘fair lottery’. Concave utility function over wealth Concave utility function over wealth Example : Example : –tossing a coin with $1 for WIN (heads) and -$1 for LOSS (tails). x = k 1 with probability p x = k 2 with probability 1-p E(x) = pk 1 + (1-p)k 2 = 0 k 1 /k 2 = -(1-p)/p or p = -k 2 /(k 1 -k 2 ) –Tossing a coin : p = ½ and k 1 = -k 2 = $ 1.-

Utility : The Basics

Utility Functions More is preferred to less : More is preferred to less : U’(W) = ∂ U(W)/ ∂ W > 0 Example : Example : –Tossing a (fair) coin (i.e. p = 0.5 for head) –gamble of receiving £ 16 for a ‘head’ and £ 4 for ‘tails’ –EW = 0.5 (£ 16) (£ 4) = £ 10 –If costs of ‘gamble’ = £ 10.-  EW – c = 0 –How much is an individual willing to pay for playing the game ?

Utility Functions (Cont.) Assume the following utility function Assume the following utility function U(W) = W 1/2 Expected return from gamble Expected return from gamble E[U(W)] = 0.5 U(W H ) U(W T ) = 0.5 (16) 1/ (4) 1/2 = 3

Monetary Risk Premium Wealth Utility 0 A E[U(W)] = 3 =0.5(4)+0.5(2) (W–  ) = 9 EW=  U(4) = 2 U(W)= W 1/2 U(EW) = 10 1/2 =3.162 U(16) = 4

Degree of Risk Aversion An individual’s degree of risk aversion may depend on An individual’s degree of risk aversion may depend on –Initial wealth Example : Bill Gates or You ! Example : Bill Gates or You ! –Size of the bet Risk neutral for small bets : i.e. Cost £ 10.- Risk neutral for small bets : i.e. Cost £ 10.- Gamble : Win £ 1m or £0. Would you pay £ 499,999 (being risk averse) ? Gamble : Win £ 1m or £0. Would you pay £ 499,999 (being risk averse) ?

Absolute and Relative Risk Aversion

Utility Theory Assumptions Assumptions –Investor has wealth W and security with outcome represented by the random variable Z –Let Z be a fair game E(Z) = 0 and E[Z–E(Z)] 2 =  z 2 –Investor is indifferent between choice A and B Choice A Choice B W + ZW c E[(U(W + Z)] = EU(W c ) = U(W c )

Utility Theory (Cont.) Define  = W – W c is the max. investor is willing to pay to avoid gamble. Measurement of investor’s absolute risk aversion. Define  = W – W c is the max. investor is willing to pay to avoid gamble. Measurement of investor’s absolute risk aversion. (1.) Expanding U(W + Z) in a Taylor series expansion around W U(W+Z)  U(W) + U’(W)[(W+Z)-W] + (½) U’’(W)[(W+Z)-W] 2 + … E[U(W+Z)] = E[U(W)] + U’(W)E(Z) + (½) U’’(W)E(Z-0) 2 E[U(W+Z)] = U(W) + (½) U’’(W)  z 2

Utility Theory (Cont.) (2.) Expanding U(W -  ) in a Taylor series expansion around W U(W c ) = U(W –  )  U(W) + U’(W)[(W-  )-W] + … U(W c ) = U(W) + U’(W)(-  ) Rem. : E(U(W + Z)) = U(W c )  U(W) + (½) U’’(W)  z 2 = U(W) + U’(W)(-  )  U(W) + (½) U’’(W)  z 2 = U(W) + U’(W)(-  ) Rearranging  = -½  z 2 [U’’(W)] / [U’(W)] Hence : A(W) = -U’’(W) / U’(W)

Relative Risk Aversion Percentage ‘insurance premium’ is  = (W-W c )/W Or W c = W(1-  ) Percentage ‘insurance premium’ is  = (W-W c )/W Or W c = W(1-  ) Z is now outcome per Dollar invested WZ Z is now outcome per Dollar invested WZ Let E(Z) = 1 and E(Z – E(Z)) 2 =  z 2 Let E(Z) = 1 and E(Z – E(Z)) 2 =  z 2 Choice A Choice B WZ = W c Applying a Taylor series expansion : (1.) U(WZ) = U(W) + U’(W)(WZ–W) + (U’’(W)/2) (WZ-W) 2 + … Taking expectations and using the assumptions EU(WZ) = U(W) (U’’(W)/2) [W 2  z 2 ]

Relative Risk Aversion (Cont.) (2.) U(W c ) = U[W(1 –  )] = U(W) + U’(W)[W(1 –  ) – W] + … U(W c ) = U(W) + U’(W) (-  W) U(W) + ½ U’’(W)  z 2 W 2 = U(W) –  WU’(W)  = -(  z 2 /2) [WU’’(W) / U’(W)] Hence : R(W) = -WU’’(W) / U’(W)

Summary : Attitude Towards Risk Risk Averse Risk Averse Definition : Reject a fair gamble U’’(0) < 0 Risk Neutral Risk Neutral Definition : Indifferent to a fair game U’’(0) = 0 Risk Loving Risk Loving Definition : Selects a fair game U’’(0) > 0

Utility Functions : Graphs Wealth Utility U(W) Risk Neutral Risk Lover U(16) U(4) U(10) 4 Risk Averter 10 16

Indifference Curves in Risk – Return Space Risk,  Expected Return Risk Neutral Risk Lover Risk Averter

Examples of Utility Functions

Utility Function : Power Constant Relative Risk Aversion Constant Relative Risk Aversion U(W) = W (1-  ) / (1-  )  > 0,  ≠ 1 U(W) = W (1-  ) / (1-  )  > 0,  ≠ 1 U’(W) = W -  U’(W) = W -  U’’(W) = -  W -  -1 U’’(W) = -  W -  -1 R A (W) =  /W R A (W) =  /W R R (W) =  (a constant) R R (W) =  (a constant)

Utility Function : Logarithmic As   1, logarithmic utility is a limiting case of power utility As   1, logarithmic utility is a limiting case of power utility U(W) = ln(W) U(W) = ln(W) U’(W) = 1/W U’(W) = 1/W U’’(W) = -1/W 2 U’’(W) = -1/W 2 R A (W) = 1/W R A (W) = 1/W R R (W) = 1 R R (W) = 1

Utility Function : Quadratic U(W) = W – (b/2)W 2 b > 0 U(W) = W – (b/2)W 2 b > 0 U’(W) = 1 – bW U’(W) = 1 – bW U’’ = -b U’’ = -b R A (W) = b/(1-bW) R A (W) = b/(1-bW) R R (W) = bW / (1-bW) R R (W) = bW / (1-bW) Bliss point : W < 1/b Bliss point : W < 1/b

Utility Function : Negative Exponential Constant Absolute Risk Aversion Constant Absolute Risk Aversion U(W) = a – be -cW c > 0 U(W) = a – be -cW c > 0 R A (W) = c R A (W) = c R R (W) = cW R R (W) = cW

Empirical Evidence

How does it Work in the ‘Real World’ ? To investigate whether (specific) utility functions represent behaviour of economic agents : To investigate whether (specific) utility functions represent behaviour of economic agents : –Experimental evidence from simple choice situations –Survey data on investor’s asset choices

Empirical Studies Blume and Friend (1975) Blume and Friend (1975) –Data : Federal Reserve Board survey of financial characteristics of consumers –Findings : Percentage invested in risky asset unchanged for investors with different wealth Cohn et al (1978) Cohn et al (1978) –Data : Survey data from questionnaires (brokers and its customers) –Findings : Investors exhibit decreasing relative RA and decreasing absolute RA

Coefficient of Relative Risk Aversion (  ) From experiments on gambles coefficient of relative risk aversion (  ) is expected to be in the range of 3–10. From experiments on gambles coefficient of relative risk aversion (  ) is expected to be in the range of 3–10. S&P500 S&P500 Average real return (since WW II) 9% p.a. with SD 16% p.a. C-CAPM suggests coefficient of relative risk aversion (  ) of 50.  Equity Premium Puzzle

Is  = 50 Acceptable ? Based on the C-CAPM Based on the C-CAPM For  = 50, risk free rate must be 49% For  = 50, risk free rate must be 49% Cochrane (2001) presents a nice example Cochrane (2001) presents a nice example –Annual earnings $ 50,000 –Annual expenditure on holidays (5%) is $ 2,500 –R ft ≈ (52,500/47,500) 50 – 1 = 14,800% p.a. –Interpretation : Would skip holidays this year only if the risk free rate is 14,800% !

How Risk Averse are You ? Investigate the plausibility of different values of  to examine the certainty equivalent amount for various bets. Investigate the plausibility of different values of  to examine the certainty equivalent amount for various bets. Avoiding a fair bet (i.e. win or lose $x) Avoiding a fair bet (i.e. win or lose $x) –Power utility –Initial consumption : $ 50,000

Avoiding a Fair Bet ! Amount of Bet ($) Risk Aversion  , ,0002,0006,9209,4309,7189,888 20,0008,00017,60019,57319,78919,916

Application : Mean Variance Model

Mean-Variance Model and Utility Functions Investors maximise expected utility of end-of-period wealth Investors maximise expected utility of end-of-period wealth Can be shown that above implies maximise a function of expected portfolio returns and portfolio variance providing Can be shown that above implies maximise a function of expected portfolio returns and portfolio variance providing –Either utility is quadratic, or –Portfolio returns are normally distributed (and utility is concave) W = W 0 (1 + R p ) W = W 0 (1 + R p ) U(W) = U[W 0 (1 + R p )] U(W) = U[W 0 (1 + R p )]

Mean-Variance Model and Utility Functions (Cont.) Expanding U(R p ) in Taylor series around mean of R p (=  p ) U(R p ) = U(  p ) + (R p –  p ) U’(  p ) U(R p ) = U(  p ) + (R p –  p ) U’(  p ) + (1/2)(R p –  p ) 2 U’’(  p ) + higher order terms Taking expectations Taking expectations E[U(R p )] = U(  p ) + (1/2)  2 p U’’(  p ) +E(higher-terms) –E[U(R p )] is only a function of the mean and variance –Need specific utility function to know the functional relationship between E[U(R p )] and (  p,  p ) space

Summary Utility functions, expected utility Utility functions, expected utility Different measures of risk aversion : absolute, relative Different measures of risk aversion : absolute, relative Attitude towards risk, indifference curves Attitude towards risk, indifference curves Empirical evidence and an application of utility analysis Empirical evidence and an application of utility analysis

References Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapter 1 Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapter 1

References Blume, M. and Friend, I. (1975) ‘The Asset Structure of Individual Portfolios and Some Implications for Utility Functions’, Journal of Finance, Vol. 10(2), pp Blume, M. and Friend, I. (1975) ‘The Asset Structure of Individual Portfolios and Some Implications for Utility Functions’, Journal of Finance, Vol. 10(2), pp Cohn, R., Lewellen, W., Lease, R. and Schlarbaum, G. (1975) ‘Individual Investor Risk Aversion and Investment Portfolio Composition’, Journal of Finance, Vol. 10(2), pp Cohn, R., Lewellen, W., Lease, R. and Schlarbaum, G. (1975) ‘Individual Investor Risk Aversion and Investment Portfolio Composition’, Journal of Finance, Vol. 10(2), pp Cochrane, J.H. (2001) Asset Pricing, Princeton University Press Cochrane, J.H. (2001) Asset Pricing, Princeton University Press

END OF LECTURE