LECTURE 2 : UTILITY AND RISK AVERSION

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LECTURE 2 UTILITY AND RISK AVERSION

• (Asset Pricing and Portfolio Theory)

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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

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Introduction

• Many different investment opportunities with
  different risk return characteristics
• General assumption Like returns, dislike risk
  
• Preferences of investors (like more to less)

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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
• Example ERp - rf 0.005 A s2p
• A (ERp - rf) / (0.005 s2p)

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Indifference Curves (Investors Preferences)
Indifference curve
Asset Q
ERp
Asset P
s
sp
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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
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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)
• EU(W) SpiU(Wi)

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Example Alternative Investments
Investment A Investment A Investment B Investment B Investment C Investment C
Outcome Prob Outcome Prob Oucome Prob
20 3/15 19 1/5 18 ¼
18 5/15 10 2/5 16 ¼
14 4/15 5 2/5 12 ¼
10 2/15 8 ¼
6 1/15
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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) 38.4(1/4) 33.6(1/4)
  25.6(1/4) 34.3

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Utility Function U(W) 4W (1/10)W2
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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)

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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.-

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Utility The Basics
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Utility Functions

• More is preferred to less
• U(W) ?U(W)/?W gt 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) 0.5 ( 4) 10
• If costs of gamble 10.- ? EW c 0
• How much is an individual willing to pay for
  playing the game ?

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Utility Functions (Cont.)

• Assume the following utility function
• U(W) W1/2
• Expected return from gamble
• EU(W) 0.5 U(WH) 0.5 U(WT)
• 0.5 (16)1/2 0.5(4)1/2 3

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Monetary Risk Premium
Utility
U(16) 4
U(W) W1/2
U(EW) 101/2 3.162
p
EU(W) 3 0.5(4)0.5(2)
A
U(4) 2
0
Wealth
EW10
16
4
(Wp) 9
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Degree of Risk Aversion

• An individuals 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) ?

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Absolute and Relative Risk Aversion
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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 EZE(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)

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Utility Theory (Cont.)

• Define p W Wc is the max. investor is willing
  to pay to avoid gamble. Measurement of
  investors absolute risk aversion.
• (1.) Expanding U(W Z) in a Taylor series
  expansion around W
• U(WZ) ? U(W) U(W)(WZ)-W
• (½) U(W)(WZ)-W2
• EU(WZ) EU(W) U(W)E(Z) (½)
  U(W)E(Z-0)2
• EU(WZ) U(W) (½) U(W) sz2

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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)

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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)(WZW) (U(W)/2)
  (WZ-W)2
• Taking expectations and using the assumptions
• EU(WZ) U(W) 0 (U(W)/2) W2sz2

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Relative Risk Aversion (Cont.)

• (2.)
• U(Wc) UW(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)

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Summary Attitude Towards Risk

• Risk Averse
• Definition Reject a fair gamble
• U(0) lt 0
• Risk Neutral
• Definition Indifferent to a fair game
• U(0) 0
• Risk Loving
• Definition Selects a fair game
• U(0) gt 0

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Utility Functions Graphs
Utility U(W)
Risk Neutral
Risk Averter
U(16)
U(10)
U(4)
Risk Lover
Wealth
4
10
16
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Indifference Curves in Risk Return Space
Risk Averter
Risk Lover
Expected Return
Risk Neutral
Risk, s
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Examples of Utility Functions
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Utility Function Power

• Constant Relative Risk Aversion
• U(W) W(1-g) / (1-g) g gt 0, g ? 1
• U(W) W-g
• U(W) -gW-g-1
• RA(W) g/W
• RR(W) g (a constant)

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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

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Utility Function Quadratic

• U(W) W (b/2)W2 b gt 0
• U(W) 1 bW
• U -b
• RA(W) b/(1-bW)
• RR(W) bW / (1-bW)
• Bliss point W lt 1/b

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Utility Function Negative Exponential

• Constant Absolute Risk Aversion
• U(W) a be-cW c gt 0
• RA(W) c
• RR(W) cW

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Empirical Evidence
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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 investors asset choices

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Empirical Studies

• 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)
• Data Survey data from questionnaires (brokers
  and its customers)
• Findings Investors exhibit decreasing relative
  RA and decreasing absolute RA

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Coefficient of Relative Risk Aversion (g)

• From experiments on gambles coefficient of
  relative risk aversion (g) is expected to be in
  the range of 310.
• SP500
• 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

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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 !

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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

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Avoiding a Fair Bet !
Amount of Bet () Risk Aversion g Risk Aversion g Risk Aversion g Risk Aversion g Risk Aversion g
Amount of Bet () 2 10 50 100 250
10 0.002 0.01 0.05 0.1 0.25
100 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
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Application Mean Variance Model
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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) UW0(1 Rp)

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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
• EU(Rp) U(mp) (1/2) s2p U(mp)
  E(higher-terms)
• EU(Rp) is only a function of the mean and
  variance
• Need specific utility function to know the
  functional relationship between EU(Rp) and (mp,
  sp) space

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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

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References

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

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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. 585-603
• 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. 605-620.
• Cochrane, J.H. (2001) Asset Pricing, Princeton
  University Press

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END OF LECTURE