Chapter 2: FAQs 91
have certain commonsense properties. In the following′
denotes differentiation with respect toW.
- The functionU(W) can vary among investors, each
will have a different attitude to risk for example. - U′(W)≥0: more is preferred to less. If it is a strict
inequality then satiation is not possible, the investor
will always prefer more than he has. This slope
measures the marginal improvement in utility with
changes in wealth. - UsuallyU′′(W)<0: the utility function is strictly
concave. Since this is the rate of change of the
marginal ‘happiness,’ it gets harder and harder to
increase happiness as wealth increases. An investor
with a concave utility function is said to berisk
averse. This property is often referred to as the law
of diminishing returns.
The final point in the above leads to definitions for mea-
surement ofrisk aversion.Theabsolute risk aversion
functionis defined as
A(W)=−U′′(W)
U′(W).Therelative risk aversion functionis defined as
R(W)=−WU′′(W)
U′(W)=WA(W).Utility functions are often used to analyze random
events. Suppose a monetary amount is associated with
the number of spots on a rolled dice. You could
calculate the expected winnings as the average of all
of the six amounts. But what if the amounts were
$1, $2, $3, $4, $5 and $6,000,000? Would the average,
$1,000,002.5, be meaningful? Would you be willing to pay
$1,000,000 to enter this as a bet? After all, you expect
to make a profit. A more sensible way of valuing this
game might be to look at the utility of each of the six