specific stock price at salient moments in the past, such as the end of a
year. Whichever way the benchmark level is formed, the difference St−Zt,
when positive, is the investor’s personal measure of how much “he is up”
on his investment at time tand conversely, when negative, how much “he
is down.”
Introducing Ztis helpful in modeling the influence of prior outcomes
on the way subsequent gains and losses are experienced. When St>Zt, the
investor has had prior gains, making subsequent losses less painful and
lowering the investor’s risk aversion. Conversely, when St<Zt, the in-
vestor has endured prior losses. Subsequent losses are more painful, and
the investor is more risk averse than usual.
Since Stand Ztsummarize how the investor perceives his past perfor-
mance, a simple way of capturing the effect of prior outcomes would be to
write the utility of financial wealth fluctuations as v(Xt+ 1 ,St,Zt). For model-
ing purposes, we find it more convenient to write it as v(Xt+ 1 ,St,zt), where
zt=Zt/St.
C. Utility from Gains and LossesIn defining v(Xt+ 1 ,St,zt), we consider three separate cases: zt=1, where the
investor has neither prior gains nor prior losses on his investments; zt<1,
the case of prior gains; and zt>1, the case of prior losses.
We start with the case of zt=1. We want to model the idea that investors
are much more sensitive to reductions in financial wealth than to increases,
a feature sometimes known as loss aversion. We capture this by defining
(4)with λ>1. This is a piecewise linear function, shown as the line marked
“zt=1” in figure 7.1. It is kinked at the origin, where the gain equals zero.
We now turn to zt<1, where the investor has accumulated prior gains in
the stock market. The upper-most line in figure 7.1 shows the form of
v(Xt+ 1 ,St,zt) in this case. It differs from v(Xt+ 1 ,St,1) in the way losses are
penalized. Small losses are not penalized very heavily, but once the loss ex-
ceeds a certain amount, it starts being penalized at a more severe rate. The
intuition is that if the investor has built up a cushion of prior gains, these
gains may soften the blow of small subsequent losses, although they may
not be enough to protect him against larger losses.
To understand how we formalize this intuition, an example may be
helpful. Suppose that the current stock value is St=$100, but that the in-
vestor has recently accumulated some gains on his investments. A reason-
able historical benchmark level is Zt=$90, since the stock must have gone
up in value recently. As discussed above, we can think of $90 as the value
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λ 0PROSPECT THEORY AND ASSET PRICES 231