LECTURE 2 :LECTURE 2 :

UTILITY AND RISK UTILITY AND RISK AVERSIONAVERSION

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

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 evidenceEmpirical evidence How useful are the general How useful are the general

findings ? findings ? – Equity premium puzzleEquity premium puzzle– Risk free rate puzzleRisk free rate puzzle

IntroductionIntroduction

Many different investment Many different investment opportunities with different risk – opportunities with different risk – return characteristics return characteristics

General assumption : ‘Like returns, General assumption : ‘Like returns, dislike risk’ dislike risk’

Preferences of investors (like more Preferences of investors (like more to less)to less)

Risk Premium and Risk Risk Premium and Risk AversionAversion Risk free rate of return : Risk free rate of return :

rate of return which can be earned with certainty (i.e rate of return which can be earned with certainty (i.e = 0). = 0). 3 months T-bill3 months T-bill

Risk premium : Risk premium : expected return in excess of the risk free rate (i.e. ERexpected return in excess of the risk free rate (i.e. ERpp – r – rff))

Risk aversion : Risk aversion : – measures the reluctance by investors to accept (more) riskmeasures the reluctance by investors to accept (more) risk– ‘‘High number’ : risk averse High number’ : risk averse – ‘‘Low number’ : less risk averseLow number’ : less risk averse

Example : Example : ERERpp - r - rff = 0.005 A = 0.005 A pp

A = (ERA = (ERpp - r - rff) / (0.005 ) / (0.005 pp))

Indifference Curves Indifference Curves (Investor’s (Investor’s Preferences)Preferences)

ERp Asset P

Indifference curve

p

Asset Q

Risk and Return (US Risk and Return (US Assets : 1926 – 1998) Assets : 1926 – 1998) (% p.a.)(% p.a.)

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40 45

ER

Standard deviation

Small Company Stocks

Large Company Stocks

Treasury Bills

Medium Term T-bonds

Long Term T-bonds

Expected UtilityExpected Utility

Suppose we have a random Suppose we have a random variable, end of period wealth with variable, end of period wealth with ‘n’ possible outcomes W‘n’ possible outcomes Wii with with probabilities pprobabilities pii

Utility from any wealth outcome is Utility from any wealth outcome is denoted U(Wdenoted U(Wii))

E[U(W)] = E[U(W)] = ppiiU(WU(Wii))

Example : Alternative Example : Alternative InvestmentsInvestmentsInvestment AInvestment A Investment BInvestment B Investment CInvestment C

OutcomOutcomee

ProbProb OutcomOutcomee

ProbProb OucomeOucome ProbProb

2020 3/153/15 1919 1/51/5 1818 ¼¼

1818 5/155/15 1010 2/52/5 1616 ¼¼

1414 4/154/15 55 2/52/5 1212 ¼¼

1010 2/152/15 88 ¼¼

66 1/151/15

Example : Alternative Example : Alternative Investments (Cont.)Investments (Cont.)

Assume following utility function : Assume following utility function : U(W) = 4W – (1/10) WU(W) = 4W – (1/10) W22

– If outcome is 20, U(W) = 80 – (1/10) 400 = 40If outcome is 20, U(W) = 80 – (1/10) 400 = 40– ……

– Expected Utility Expected Utility Investment A : E(UInvestment A : E(UAA) = … = 36.3) = … = 36.3 Investment B : E(UInvestment B : E(UBB) = … = 26.98) = … = 26.98 Investment C : Investment C :

E(UE(UCC) = 39.6(1/4) + 38.4(1/4) + 33.6(1/4) + 25.6(1/4) = ) = 39.6(1/4) + 38.4(1/4) + 33.6(1/4) + 25.6(1/4) = 34.334.3

Utility Function : Utility Function : U(W) = 4W – (1/10)WU(W) = 4W – (1/10)W22

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

W

U(W)

Example : Alternative Example : Alternative Investments (Cont.)Investments (Cont.) Ranking of investments remains Ranking of investments remains

unchanged if unchanged if – a constant is added to the utility function a constant is added to the utility function – the utility function is scaled by a the utility function is scaled by a

constant constant

Example : Example :

a + bU(W) gives the same ranking as a + bU(W) gives the same ranking as U(W) U(W)

Fair LotteryFair Lottery

A fair lottery is defined as one that has expected A fair lottery is defined as one that has expected value of zero.value of zero.

Risk aversion applies that an individual would not Risk aversion applies that an individual would not accept a ‘fair lottery’. accept a ‘fair lottery’.

Concave utility function over wealthConcave utility function over wealth Example : Example :

– tossing a coin with $1 for WIN (heads) and -$1 for LOSS tossing a coin with $1 for WIN (heads) and -$1 for LOSS (tails). (tails).

x = kx = k11 with probability p with probability p

x = kx = k22 with probability 1-p with probability 1-p

E(x) = pkE(x) = pk11 + (1-p)k + (1-p)k22 = 0 = 0

kk11/k/k22 = -(1-p)/p = -(1-p)/p or p = -kor p = -k22/(k/(k11-k-k22))– Tossing a coin : p = ½ and kTossing a coin : p = ½ and k11 = -k = -k22 = $ 1.- = $ 1.-

Utility : The BasicsUtility : The Basics

Utility Functions Utility Functions

More is preferred to less : More is preferred to less : U’(W) = U’(W) = ∂∂U(W)/U(W)/∂∂W > 0 W > 0

Example : Example : – Tossing a (fair) coin (i.e. p = 0.5 for head)Tossing a (fair) coin (i.e. p = 0.5 for head)– gamble of receiving £ 16 for a ‘head’ and £ 4 gamble of receiving £ 16 for a ‘head’ and £ 4

for ‘tails’for ‘tails’– EW = 0.5 (£ 16) + 0.5 (£ 4) = £ 10EW = 0.5 (£ 16) + 0.5 (£ 4) = £ 10– If costs of ‘gamble’ = £ 10.-If costs of ‘gamble’ = £ 10.- EW – c = 0 EW – c = 0– How much is an individual willing to pay for How much is an individual willing to pay for

playing the game ? playing the game ?

Utility Functions Utility Functions (Cont.)(Cont.) Assume the following utility function Assume the following utility function

U(W) = WU(W) = W1/21/2

Expected return from gamble Expected return from gamble E[U(W)] E[U(W)] = 0.5 U(W= 0.5 U(WHH) + 0.5 U(W) + 0.5 U(WTT) )

= 0.5 (16)= 0.5 (16)1/21/2 + 0.5(4) + 0.5(4)1/21/2 = 3 = 3

Monetary Risk Monetary Risk PremiumPremium

Wealth

Utility

0

AAE[U(W)] = 3 =0.5(4)+0.5(2)

(W–) = 9

EW=10 164

U(4) = 2

U(W)= W1/2 U(EW) = 101/2 =3.162

U(16) = 4

Degree of Risk Degree of Risk Aversion Aversion An individual’s degree of risk An individual’s degree of risk

aversion may depend on aversion may depend on – Initial wealth Initial wealth

Example : Bill Gates or You !Example : Bill Gates or You !

– Size of the bet Size of the bet Risk neutral for small bets : i.e. Cost £ Risk neutral for small bets : i.e. Cost £

10.-10.- Gamble : Win £ 1m or £0. Would you Gamble : Win £ 1m or £0. Would you

pay £ 499,999 (being risk averse) ? pay £ 499,999 (being risk averse) ?

Absolute and Relative Absolute and Relative Risk AversionRisk Aversion

Utility TheoryUtility Theory

Assumptions Assumptions – Investor has wealth W and security with Investor has wealth W and security with

outcome represented by the random variable Zoutcome represented by the random variable Z– Let Z be a fair game E(Z) = 0 and E[Z–E(Z)]Let Z be a fair game E(Z) = 0 and E[Z–E(Z)]22 = =

zz22

– Investor is indifferent between choice A and B Investor is indifferent between choice A and B

Choice A Choice A Choice B Choice B

W + ZW + Z WWcc

E[(U(W + Z)] = E[(U(W + Z)] = EU(WEU(Wcc) = U(W) = U(Wcc) )

Utility Theory (Cont.)Utility Theory (Cont.)

Define Define = W – W = W – Wcc is the max. investor is willing to is the max. investor is willing to pay to avoid gamble. Measurement of investor’s pay to avoid gamble. Measurement of investor’s absolute risk aversion. absolute risk aversion.

(1.) Expanding U(W + Z) in a Taylor series expansion around W(1.) Expanding U(W + Z) in a Taylor series expansion around W

U(W+Z) U(W+Z) U(W) U(W) + U’(W)[(W+Z)-W] + U’(W)[(W+Z)-W]

+ (½) U’’(W)[(W+Z)-W]+ (½) U’’(W)[(W+Z)-W]22 + … + …

E[U(W+Z)] = E[U(W)] + U’(W)E(Z) + (½) U’’(W)E(Z-E[U(W+Z)] = E[U(W)] + U’(W)E(Z) + (½) U’’(W)E(Z-0)0)22

E[U(W+Z)] = U(W) + (½) U’’(W) E[U(W+Z)] = U(W) + (½) U’’(W) zz22

Utility Theory (Cont.)Utility Theory (Cont.)

(2.) Expanding U(W - (2.) Expanding U(W - ) in a Taylor series expansion around W) in a Taylor series expansion around W

U(WU(Wcc) = U(W – ) = U(W – ) ) U(W) + U’(W)[(W- U(W) + U’(W)[(W-)-W] + )-W] + ……

U(WU(Wcc) = U(W) + U’(W)(-) = U(W) + U’(W)(-))

Rem. : E(U(W + Z)) = U(WRem. : E(U(W + Z)) = U(Wcc) ) U(W) + (½) U’’(W) U(W) + (½) U’’(W) zz

22 = U(W) + U’(W)(- = U(W) + U’(W)(-) )

Rearranging Rearranging = -½ = -½ zz22 [U’’(W)] / [U’(W)] [U’’(W)] / [U’(W)]

Hence : A(W) = -U’’(W) / U’(W)Hence : A(W) = -U’’(W) / U’(W)

Relative Risk AversionRelative Risk Aversion

Percentage ‘insurance premium’ is Percentage ‘insurance premium’ is = (W-W = (W-Wcc)/W Or W)/W Or Wcc = W(1- = W(1-) )

Z is now outcome per Dollar invested WZZ is now outcome per Dollar invested WZ Let E(Z) = 1 and E(Z – E(Z))Let E(Z) = 1 and E(Z – E(Z))22 = = zz

22

Choice A Choice A Choice B Choice B WZ WZ = = WWcc

Applying a Taylor series expansion : Applying a Taylor series expansion :

(1.) (1.) U(WZ) = U(W) + U’(W)(WZ–W) + (U’’(W)/2) (WZ-W)U(WZ) = U(W) + U’(W)(WZ–W) + (U’’(W)/2) (WZ-W)22 + … + …

Taking expectations and using the assumptionsTaking expectations and using the assumptions

EU(WZ) = U(W) + 0 + (U’’(W)/2) [WEU(WZ) = U(W) + 0 + (U’’(W)/2) [W22zz22]]

Relative Risk Aversion Relative Risk Aversion (Cont.)(Cont.)

(2.) (2.)

U(WU(Wcc) ) = U[W(1 – = U[W(1 – )])]

= U(W) + U’(W)[W(1 – = U(W) + U’(W)[W(1 – ) – W] + … ) – W] + …

U(WU(Wcc) = U(W) + U’(W) (-) = U(W) + U’(W) (-W) W)

U(W) + ½ U’’(W) U(W) + ½ U’’(W) zz22 W W22 = U(W) – = U(W) – WU’(W) WU’(W)

= -(= -(zz2 2 /2) [WU’’(W) / U’(W)] /2) [WU’’(W) / U’(W)]

Hence : R(W) = -WU’’(W) / U’(W) Hence : R(W) = -WU’’(W) / U’(W)

Summary : Attitude Summary : Attitude Towards Risk Towards Risk Risk AverseRisk Averse

Definition : Reject a fair gambleDefinition : Reject a fair gambleU’’(0) < 0 U’’(0) < 0

Risk NeutralRisk NeutralDefinition : Indifferent to a fair game Definition : Indifferent to a fair game U’’(0) = 0 U’’(0) = 0

Risk LovingRisk LovingDefinition : Selects a fair gameDefinition : Selects a fair gameU’’(0) > 0 U’’(0) > 0

Utility Functions : Utility Functions : GraphsGraphs

Wealth

Utility U(W) Risk Neutral

Risk Lover

U(16)

U(4)

U(10)

4

Risk Averter

10 16

Indifference Curves in Indifference Curves in Risk – Return SpaceRisk – Return Space

Risk,

Exp

ecte

d R

etu

rn

Risk Neutral

Risk LoverRisk Averter

Examples of Utility Examples of Utility FunctionsFunctions

Utility Function : Utility Function : PowerPower Constant Relative Risk AversionConstant Relative Risk Aversion

U(W) = WU(W) = W(1-(1-)) / (1- / (1-) ) > 0, > 0, ≠ 1 ≠ 1 U’(W) = WU’(W) = W--

U’’(W) = -U’’(W) = -WW---1-1

RRAA(W) = (W) = /W/W

RRRR(W) = (W) = (a constant) (a constant)

Utility Function : Utility Function : LogarithmicLogarithmic As As 1, logarithmic utility is a 1, logarithmic utility is a

limiting case of power utility limiting case of power utility

U(W) = ln(W) U(W) = ln(W) U’(W) = 1/WU’(W) = 1/W U’’(W) = -1/WU’’(W) = -1/W22

RRAA(W) = 1/W(W) = 1/W RRRR(W) = 1(W) = 1

Utility Function : Utility Function : QuadraticQuadratic U(W) = W – (b/2)WU(W) = W – (b/2)W22 b > 0 b > 0 U’(W) = 1 – bWU’(W) = 1 – bW U’’ = -bU’’ = -b

RRAA(W) = b/(1-bW) (W) = b/(1-bW)

RRRR(W) = bW / (1-bW)(W) = bW / (1-bW)

Bliss point : W < 1/bBliss point : W < 1/b

Utility Function : Utility Function : Negative ExponentialNegative Exponential Constant Absolute Risk Aversion Constant Absolute Risk Aversion

U(W) = a – beU(W) = a – be-cW-cW c > 0 c > 0

RRAA(W) = c (W) = c

RRRR(W) = cW(W) = cW

Empirical EvidenceEmpirical Evidence

How does it Work in How does it Work in the ‘Real World’ ? the ‘Real World’ ? To investigate whether (specific) To investigate whether (specific)

utility functions represent utility functions represent behaviour of economic agents : behaviour of economic agents : – Experimental evidence from simple Experimental evidence from simple

choice situations choice situations – Survey data on investor’s asset Survey data on investor’s asset

choiceschoices

Empirical Studies Empirical Studies

Blume and Friend (1975) Blume and Friend (1975) – Data : Federal Reserve Board survey of Data : Federal Reserve Board survey of

financial characteristics of consumersfinancial characteristics of consumers– Findings : Percentage invested in risky Findings : Percentage invested in risky

asset unchanged for investors with asset unchanged for investors with different wealthdifferent wealth

Cohn et al (1978) Cohn et al (1978) – Data : Survey data from questionnaires Data : Survey data from questionnaires

(brokers and its customers)(brokers and its customers)– Findings : Investors exhibit decreasing Findings : Investors exhibit decreasing

relative RA and decreasing absolute RArelative RA and decreasing absolute RA

Coefficient of Relative Coefficient of Relative Risk Aversion (Risk Aversion ()) From experiments on gambles coefficient From experiments on gambles coefficient

of relative risk aversion (of relative risk aversion () is expected to ) is expected to be in the range of 3–10. be in the range of 3–10.

S&P500 S&P500 Average real return (since WW II) 9% p.a. with Average real return (since WW II) 9% p.a. with SD 16% p.a. SD 16% p.a.

C-CAPM suggests coefficient of relative risk C-CAPM suggests coefficient of relative risk aversion (aversion () of 50. ) of 50.

Equity Premium PuzzleEquity Premium Puzzle

Is Is = 50 Acceptable ? = 50 Acceptable ?

Based on the C-CAPMBased on the C-CAPM For For = 50, risk free rate must be 49% = 50, risk free rate must be 49% Cochrane (2001) presents a nice Cochrane (2001) presents a nice

example example – Annual earnings $ 50,000Annual earnings $ 50,000– Annual expenditure on holidays (5%) is $ Annual expenditure on holidays (5%) is $

2,5002,500– RRftft ≈ (52,500/47,500) ≈ (52,500/47,500)5050 – 1 = 14,800% p.a. – 1 = 14,800% p.a. – Interpretation : Would skip holidays this Interpretation : Would skip holidays this

year only if the risk free rate is 14,800% ! year only if the risk free rate is 14,800% !

How Risk Averse are How Risk Averse are You ? You ? Investigate the plausibility of different Investigate the plausibility of different

values of values of to examine the certainty to examine the certainty equivalent amount for various bets. 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 Power utility – Initial consumption : $ 50,000Initial consumption : $ 50,000

Avoiding a Fair Bet !Avoiding a Fair Bet !

Amount Amount of Bet of Bet

($)($)

Risk Aversion Risk Aversion 22 1010 5050 100100 250250

1010 0.0020.002 0.010.01 0.050.05 0.10.1 0.250.25

100100 0.20.2 11 55 9.99.9 2424

1,0001,000 2020 9999 435435 655655 863863

10,0010,0000

2,0002,000 6,9206,920 9,4309,430 9,7189,718 9,8889,888

20,0020,0000

8,0008,000 17,60017,600 19,57319,573 19,78919,789 19,91619,916

Application : Mean Application : Mean Variance ModelVariance Model

Mean-Variance Model Mean-Variance Model and Utility Functionsand Utility Functions Investors maximise expected utility of end-of-Investors maximise expected utility of end-of-

period wealthperiod wealth Can be shown that above implies maximise a Can be shown that above implies maximise a

function of expected portfolio returns and function of expected portfolio returns and portfolio variance providing portfolio variance providing – Either utility is quadratic, or Either utility is quadratic, or – Portfolio returns are normally distributed (and utility is Portfolio returns are normally distributed (and utility is

concave)concave)

W = WW = W00(1 + R(1 + Rpp) ) U(W) = U[WU(W) = U[W00(1 + R(1 + Rpp)] )]

Mean-Variance Model Mean-Variance Model and Utility Functions and Utility Functions (Cont.)(Cont.)Expanding U(RExpanding U(Rpp) in Taylor series around mean of R) in Taylor series around mean of Rpp (= (=pp) ) U(RU(Rpp) = U() = U(pp) + (R) + (Rpp – – pp) U’() U’(pp) )

+ (1/2)(R+ (1/2)(Rpp – – pp))22 U’’( U’’(pp) ) + higher order terms+ higher order terms

Taking expectations Taking expectations E[U(RE[U(Rpp)] = U()] = U(pp) + (1/2) ) + (1/2) 22

pp U’’( U’’(pp) +E(higher-terms)) +E(higher-terms)

– E[U(RE[U(Rpp)] is only a function of the mean and variance)] is only a function of the mean and variance– Need specific utility function to know the functional Need specific utility function to know the functional

relationship between E[U(Rrelationship between E[U(Rpp)] and ()] and (pp, , pp) space) space

SummarySummary

Utility functions, expected utilityUtility functions, expected utility Different measures of risk Different measures of risk

aversion : absolute, relativeaversion : absolute, relative Attitude towards risk, indifference Attitude towards risk, indifference

curves curves Empirical evidence and an Empirical evidence and an

application of utility analysisapplication of utility analysis

References References

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

ReferencesReferences

Blume, M. and Friend, I. (1975) ‘The Asset Blume, M. and Friend, I. (1975) ‘The Asset Structure of Individual Portfolios and Some Structure of Individual Portfolios and Some Implications for Utility Functions’, Implications for Utility Functions’, Journal of Journal of FinanceFinance, Vol. 10(2), pp. 585-603, Vol. 10(2), pp. 585-603

Cohn, R., Lewellen, W., Lease, R. and Cohn, R., Lewellen, W., Lease, R. and Schlarbaum, G. (1975) ‘Individual Investor Risk Schlarbaum, G. (1975) ‘Individual Investor Risk Aversion and Investment Portfolio Aversion and Investment Portfolio Composition’, Composition’, Journal of FinanceJournal of Finance, Vol. 10(2), pp. , Vol. 10(2), pp. 605-620. 605-620.

Cochrane, J.H. (2001) Asset Pricing, Princeton Cochrane, J.H. (2001) Asset Pricing, Princeton University PressUniversity Press

END OF LECTUREEND OF LECTURE