has no prior gains or losses, equations (3) and (4) show that we measure
the pain of this loss as
(90−100)(λ)=(90−100)(2)=−20.In our example, though, there has been a prior loss and zt=1.1. This
means that the pain will now be
(90−100)(λ+3(0.1))=(90−100)(2+3(0.1))=−23,capturing the idea that losses are more painful after prior losses.
D. Dynamics of the Benchmark LevelTo complete our description of the model, we need to discuss how the in-
vestor’s cushion of prior gains changes over time. In formal terms, we have
to specify how ztmoves over time, or equivalently how the historical
benchmark level Ztreacts to changes in stock value St. There are two ways
that the value of the investor’s stock holdings can change. First, it can
change at time tbecause of an action taken by the investor: he may take out
the dividend and consume it, or he may buy or sell some shares. For this
type of change, we assume that Ztchanges in proportion to St, so that ztre-
mains constant. For example, suppose that the initial value of the investor’s
stock holdings is St=$100 and that Zt=$80, implying that he has accumu-
lated $20 of prior gains. If he sells $10 of stock for consumption purposes,
bringing Stdown to $90, we assume that Ztfalls to $72, so that ztremains
constant at 0.8. In other words, when the investor sells stock for consump-
tion, we assume that he uses up some of his prior gains.
The assumption that the investor’s actions do not affect the evolution of
ztis reasonable for transactions of moderate size, or more precisely, for
moderate deviations from a strategy in which the investor holds a fixed
number of shares and consumes the dividend each period. However, larger
deviations—a complete exit from the stock market, for example—might
plausibly affect the way ztevolves. In supposing that they do not, we make
a strong assumption, but one that is very helpful in keeping our analysis
tractable. We discuss the economic interpretation of this assumption fur-
ther in section 4 when we compute equilibrium prices.
The second way stock value can change is simply through its return be-
tween time tand time t+1. In this case, the only requirement we impose on
Ztis that it respond sluggishlyto changes in the value of the risky asset. By
this we mean that when the stock price moves up by a lot, the benchmark
level also moves up, but by less. Conversely, if the stock price falls sharply,
the benchmark level does not adjust downwards by as much.
Sluggishness turns out to be a very intuitive requirement to impose. To see
this, recall that the difference St−Ztis the investor’s measure of his reserve of
prior gains. How should this quantity change as a result of a change in the
234 BARBERIS, HUANG, SANTOS