92 Frequently Asked Questions In Quantitative Finance
outcomes, and then average the utility. This leads on to
the idea of certainty equivalent wealth.
When the wealth is random, and all outcomes can be
assigned a probability, one can ask what amount of
certain wealth has the same utility as the expected
utility of the unknown outcomes. Simply solve
U(Wc)=E[U(W)].
The quantity of wealthWcthat solves this equation is
called thecertainty equivalent wealth. One is therefore
indifferent between the average of the utilities of the
random outcomes and the guaranteed amountWc.Asan
example, consider the above dice-rolling game, suppos-
ing our utility function isU(W)=−^1 ηe−ηW.Withη= 1
0
20
40
60
80
100
120
140
160
180
200
0.001 0.01 0.1 1 10 100
ln(eta)
Certainty equivalent
Figure 2-5:Certainty equivalent as a function of the risk-aversion
parameter for example in the text.
