Download Notes 2

Risk and Utility
2003,3,6
Purpose, Goal
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Quickly-risky
Gradually-comfort
Absolute goal or benchmark
Investment horizon
Issues
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Estimate future value from historical returns
Estimate return distribution
Notion of utility
Technique of maximizing expected utility
Role of investment horizon
1
0.5)  (1  0.5)] 2  1
Estimate an investment’s future value
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Arithmetic and geometric average
100-150-75
Arithmetic average: (1/2-1/2)= 0
Geometric average: [(1  0.5)  (1  0.5)] 12  1  0.1340
Expected future value
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The expected value is defined as the
probability-weighted outcome.
Arithmetic average: mean
Geometric average: medium
Which one is bigger?
  n
Risk (random variables)
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Frequency distribution
Normal distribution
The central limit theorem
Lognormality
  n
Continuous return
r n
r
lim(1  )  e
n 
n
Utility
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(1738) Bernoulli: the determination of the value of an
item must not be based on its price, but rather on the
utility it yield.
Diminishing marginal utility
is increasing but
is decreasing
Examples: Natural log, power functions
Risk aversion
U ( x)
U ( x )

ln x, x
Certainty equivalent
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The value of a certain prospect that yields the same
utility as the expected utility of an uncertain prospect
is called a certainty equivalent.
Risk premium: 100---150 or 50
ln C  ln F  p  ln U  (1  p )
C e
ln C  ln F  p  ln U (1 p )
86.6  eln1500.5 ln 50(1 0.5)
15.5  100  86.6
86.6
Risk preference
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Risk averse: curvature of the utility
Risk averse, risk neutral, risk seeking
Indifference curves
E (U )  E (r )     2
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Where E (U )  expected utility
E ( r ) expected return
 risk aversion coefficient
  standard deviation of returns
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0.03  (0.08  5  0.1 )
2
0.028  (0.1  5  0.12 )
2
0.05  (0.08  3  0.1 )
2
0.057  (0.1  3  0.12 )
2
E (r )  E (U )  4   , E (U )  u
2
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E(r)
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The optimal portfolio
Identifying the optimal portfolio
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Portfolio composed of Stock and bond
Rp  ( Rs  Ws )  ( RB  WB )
 P  ( s  Ws    W  2     s  Ws   B  WB )
2
2
E (U )  RP  
2
B
2
P
2
B
1
2
E (U )
 Rs   (2   s2Ws  2     s   B  WB )
Ws
E (U )
 RB   (2   B2WB  2     B   s  Ws )
WB
Complex utility functions
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Relative performance: Tracking Error
TE=tracking error
RF=return of fund
RB=return of benchmark
N=number of returns
Complex utility functions
n
TE 
 (R
i 1
F
 RB )
n
2
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Expected return, absolute risk and relative
risk
E (U )  E (r )    TE
2
2
Kinked utility functions
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DR= downside risk
RD= returns below target return
RT= target return
n= numbers of returns below target return
n
DR 
 (R
i 1
D
 RT )
n
2
T
=aversion to standard deviation of total
return
  = standard deviation of total return
 DTE =aversion to down side tracking error
 DTE= deviations below benchmark returns
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E (U )  E (r )  T   DTE DTE
2
2
Investment horizon
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Investor with longer horizon should allocate a
larger fraction of their saving to risky assets
than investor with shorter horizon.
Over a long horizon, favorable short-term
stock returns are likely to offset poor shortterm stock returns.
Time diversification
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The above-average returns tend to offset
below average returns over long horizon.
If returns are independent from on year to
next, the standard deviation of annualized
returns diminishes with time.
Regression to the mean
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The probability of losing money as a function
of horizon. Say mean=10%, standard
deviation=15% annually.
N (0.1 n, 0.15  n)
2
  k  0.1 n  k  0.15  n
Time diversification refuted
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The relative metric is terminal wealth not
annualized returns
As the investment horizon increases, the
dispersion of terminal wealth diverges from
the expected wealth.
Time does not diversify risk
100
1
(1 )100
3
1
(1 )100
4
1
1
104.7  133   75 
2
2
1
1
ln(100)  4.60517  ln(133)   ln(75) 
2
2
100
1
(1 )100
3
1
(1 )100
4
(11/ 3)2 100
(11/ 3)(11/ 4 ) 100