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

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

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

4. Risk Premium and Risk Aversion
Risk free rate of return :
rate of return which can be earned with certainty (i.e s = 0).
3 months T-bill
Risk premium :
expected return in excess of the risk free rate (i.e. ERp – rf)
Risk aversion :
measures the reluctance by investors to accept (more) risk
‘High number’ : risk averse
‘Low number’ : less risk averse

5. Indifference Curves (Investor’s Preferences)
Asset Q
ERp
Asset P sp s

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

7. Risk and Return (US Assets : 1926 – 1998) (% p.a.)
Access:
French data library;
St Louis Federal Reserve
Shiller dataset
and verify the assumption

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

9. Example : Alternative Investments
Investment A
Investment B
Investment C
Outcome
Prob
Oucome 20
3/15 19
1/5 18
5/15 10
2/5 16 14
4/15 5 12
2/15 8 6
1/15

10. Example : Alternative Investments (Cont.)
Assume following utility function :
U(W) = 4W – (1/10) W2
If outcome is 20, U(W) = 80 – (1/10) 400 = 40 …
Expected Utility
Investment A : E(UA) = … = 36.3
Investment B : E(UB) = … = 26.98
Investment C :
E(UC) = 39.6(1/4) (1/4) (1/4) (1/4) = 34.3

11. Utility Function : U(W) = 4W – (1/10)W2

12. Example : Alternative Investments (Cont.)
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)

13. Fair Lottery
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’.
Concave utility function over wealth
Example :
tossing a coin with $1 for WIN (heads) and -$1 for LOSS (tails).
x = k1 with probability p
x = k2 with probability 1-p
E(x) = pk1 + (1-p)k2 = 0
k1/k2 = -(1-p)/p or p = -k2/(k1-k2)
Tossing a coin : p = ½ and k1 = -k2 = $ 1.-

14. Utility : The Basics

15. Utility Functions More is preferred to less : U’(W) = ∂U(W)/∂W > 0
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’ = £  EW – c = 0
How much is an individual willing to pay for playing the game ?

16. Utility Functions (Cont.)
Assume the following utility function
U(W) = W1/2
Expected return from gamble
E[U(W)] = 0.5 U(WH) U(WT)
= 0.5 (16)1/ (4)1/2 = 3

17. Monetary Risk Premium Utility Wealth U(16) = 4 U(W)= W1/2
U(EW) = 101/2 =3.162 p
E[U(W)] = 3
=0.5(4)+0.5(2) A
U(4) = 2
EW=10
Wealth 4 16
(W–p) = 9

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

19. Absolute and Relative Risk Aversion

20. Utility Theory 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 = sz2
Investor is indifferent between choice A and B
Choice A Choice B
W + Z Wc
E[(U(W + Z)] = EU(Wc) = U(Wc)

21. Utility Theory (Cont.)
Define p = W – Wc 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) sz2

22. Hence : A(W) = -U’’(W) / U’(W)
Utility Theory (Cont.)
(2.) Expanding U(W - p) in a Taylor series expansion around W
U(Wc) = U(W – p)  U(W) + U’(W)[(W-p)-W] + …
U(Wc) = U(W) + U’(W)(-p)
Rem. : E(U(W + Z)) = U(Wc)
 U(W) + (½) U’’(W) sz2 = U(W) + U’(W)(-p)
Rearranging p = -½ sz2 [U’’(W)] / [U’(W)]
Hence : A(W) = -U’’(W) / U’(W)

23. Relative Risk Aversion
Percentage ‘insurance premium’ is p = (W-Wc)/W Or Wc = W(1-p)
Z is now outcome per Dollar invested WZ
Let E(Z) = 1 and E(Z – E(Z))2 = sz2
Choice A Choice B
WZ = Wc
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) [W2sz2]

24. Relative Risk Aversion (Cont.)
(2.)
U(Wc) = U[W(1 – p)]
= U(W) + U’(W)[W(1 – p) – W] + …
U(Wc) = U(W) + U’(W) (-pW)
U(W) + ½ U’’(W) sz2 W2 = U(W) – pWU’(W)
p = -(sz2 /2) [WU’’(W) / U’(W)]
Hence : R(W) = -WU’’(W) / U’(W)

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

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

27. Indifference Curves in Risk – Return Space
Risk Averter
Risk Lover
Expected Return
Risk Neutral
Risk, s

28. Examples of Utility Functions

29. Utility Function : Power
Constant Relative Risk Aversion
U(W) = W(1-g) / (1-g) g > 0, g ≠ 1
U’(W) = W-g
U’’(W) = -gW-g-1
RA(W) = g/W
RR(W) = g (a constant)

30. Utility Function : Logarithmic
As g  1, logarithmic utility is a limiting case of power utility
U(W) = ln(W)
U’(W) = 1/W
U’’(W) = -1/W2
RA(W) = 1/W
RR(W) = 1

31. Utility Function : Quadratic
U(W) = W – (b/2)W2 b > 0
U’(W) = 1 – bW
U’’ = -b
RA(W) = b/(1-bW)
RR(W) = bW / (1-bW)
Bliss point : W < 1/b

32. Utility Function : Negative Exponential
Constant Absolute Risk Aversion
U(W) = a – be-cW c > 0
RA(W) = c
RR(W) = cW

33. Empirical Evidence

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

35. Empirical Studies Blume and Friend (1975) Cohn et al (1978)
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)
Data : Survey data from questionnaires (brokers and its customers)
Findings : Investors exhibit decreasing relative RA and decreasing absolute RA

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

37. Is g = 50 Acceptable ? Based on the C-CAPM
For g = 50, risk free rate must be 49%
Cochrane (2001) presents a nice example
Annual earnings $ 50,000
Annual expenditure on holidays (5%) is $ 2,500
Rft ≈ (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% !

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

39. Avoiding a Fair Bet ! Risk Aversion g 2 10 50 100 250 1,000 10,000
Amount of Bet ($)
Risk Aversion g 2 10 50
100
250
0.002
0.01
0.05
0.1
0.25
0.2 1 5
9.9 24
1,000 20 99
435
655
863
10,000
2,000
6,920
9,430
9,718
9,888
20,000
8,000
17,600
19,573
19,789
19,916

40. Application : Mean Variance Model

41. Mean-Variance Model and Utility Functions
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
Either utility is quadratic, or
Portfolio returns are normally distributed (and utility is concave)
W = W0(1 + Rp)
U(W) = U[W0(1 + Rp)]

42. Mean-Variance Model and Utility Functions (Cont.)
Expanding U(Rp) in Taylor series around mean of Rp (=mp)
U(Rp) = U(mp) + (Rp – mp) U’(mp)
+ (1/2)(Rp – mp)2 U’’(mp)
+ higher order terms
Taking expectations
E[U(Rp)] = U(mp) + (1/2) s2p U’’(mp) +E(higher-terms)
E[U(Rp)] is only a function of the mean and variance
Need specific utility function to know the functional relationship between E[U(Rp)] and (mp, sp) space

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

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

45. 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
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

46. END OF LECTURE