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 Arithmetic average: (1/2-1/2)= 0 Geometric average:

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

Continuous return

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

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: or 50

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

E(r)

The optimal portfolio

Identifying the optimal portfolio Portfolio composed of Stock and bond

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

Expected return, absolute risk and relative risk

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

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.

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