Risk and Utility

2003,3,6

Purpose, Goal

Quickly-risky Gradually-comfort Absolute goal or benchmark Investment horizon

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

Continuous return

lim(1 )n r

n

re

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

Risk preference

Risk averse: curvature of the utility Risk averse, risk neutral, risk seeking

Indifference curves

Where expected utility expected return risk aversion coefficient standard deviation of returns

2( ) ( )E U E r

( )E U ( )E r

2

2

0.03 (0.08 5 0.1 )

0.028 (0.1 5 0.12 )

2

2

0.05 (0.08 3 0.1 )

0.057 (0.1 3 0.12 )

E(r)

2( ) ( ) 4 , ( )E r E U E U u

The optimal portfolio

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

2

2

( )(2 2 )

( )(2 2 )

s s s s B Bs

B B B B s sB

E UR W W

W

E UR W W

W

Complex utility functions

Relative performance: Tracking Error TE=tracking error RF=return of fund RB=return of benchmark N=number of returns

Complex utility functions

2

1

( )n

F Bi

R RTE

n

Expected return, absolute risk and relative risk

2 2( ) ( )E U E r TE

Kinked utility functions

DR= downside risk RD= returns below target return RT= target return n= numbers of returns below target return

2

1

( )n

D Ti

R RDR

n

=aversion to standard deviation of total return

= standard deviation of total return =aversion to down side tracking error = deviations below benchmark returns

T

DTEDTE

2 2( ) ( ) T DTEE U E r DTE

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

2(1 1/ 3) 100(1 1/ 3)(1 1/ 4 ) 100

1(1 ) 100

31

(1 ) 1004

100