of the stock one year ago, which the investor still remembers. The differ-
ence St−Zt=$10 represents the cushion, or reserve of prior gains that the
investor has built up. Suppose finally that the risk-free rate is zero.
Imagine that over the next year, the value of the stock falls from
St=$100 down to StRt+ 1 =$80. In the case of zt=1, where the investor has
no prior gains or losses, equations (3) and (4) show that we measure the
pain of this loss as
(80−100)(λ)=− 40for a λof 2.
When the investor has some prior gains, this calculation probably over-
states actual discomfort. We propose a more realistic measure of the pain
caused: since the first $10 drop, from St=$100 down to Zt=$90, is com-
pletely cushioned by the $10 reserve of prior gains, we penalize it at a rate
of only 1, rather than λ. The second part of the loss, from Zt=$90 down
to StRt+ 1 =$80 will be morepainful since all prior gains have already been
depleted, and we penalize it at the higher rate of λ. Using a λof 2 again, the
overall disutility of the $20 loss is
(90−100)(1)+(80−90)(λ)=(90−100)(1)+(80−90)(2)=−30,or in general terms
(Zt−St)(1)+(StRt+ 1 −Zt)(λ)=St(zt−1)(1)+St(Rt+ 1 −zt)(λ).232 BARBERIS, HUANG, SANTOS
Figure 7.1. Utility of gains and losses. The top line represents the case where the in-
vestor has prior gains, the dotted line the case of prior losses, and the middle line
the case where he has neither prior gains nor losses.