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Issues
Estimate future value from historical returns Estimate return distribution Notion of utility Technique of maximizing expected utility Role of investment horizon
Estimate an investment’s future value
Arithmetic and geometric average 100-150-75 Arithmetic average: (1/2-1/2)= 0 Geometric average:
12[(1 0.5) (1 0.5)] 1
12[(1 0.5) (1 0.5)] 1 0.1340
Expected future value
The expected value is defined as the probability-weighted outcome.
Arithmetic average: mean Geometric average: medium Which one is bigger?
Risk (random variables)
Frequency distribution Normal distribution The central limit theorem Lognormality
n
n
Utility
(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
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 ln ln (1 )
ln150 0.5 ln50 (1 0.5)
ln ln ln (1 )
86.6
100 86.615.5 86.6
C F p U p
C F p U p
C e
e
Indifference curves
Where expected utility expected return risk aversion coefficient standard deviation of returns
2( ) ( )E U E r
( )E U ( )E r
Identifying the optimal portfolio
Portfolio composed of Stock and bond
12 2 2 2 2
2
( ) ( )
( 2 )
( )
p s s B B
P s s B B s s B B
P P
R R W R W
W W W W
E U R
Complex utility functions
Relative performance: Tracking Error TE=tracking error RF=return of fund RB=return of benchmark N=number of returns
Kinked utility functions
DR= downside risk RD= returns below target return RT= target return n= numbers of returns below target return
=aversion to standard deviation of total return
= standard deviation of total return =aversion to down side tracking error = deviations below benchmark returns
T
DTEDTE
Investment horizon
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 short-term stock returns.
Time diversification
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
The probability of losing money as a function of horizon. Say mean=10%, standard deviation=15% annually.
2(0.1 ,0.15 )
0.1 0.15
N n n
k n k n
Time diversification refuted
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
1(1 ) 100
31
(1 ) 1004
100
1 1104.7 133 75
2 21 1
ln(100) 4.60517 ln(133) ln(75)2 2
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