level of the stock market? If the return on the stock market is particularly
good, investors should feel as though they have increased their reserve of prior
gains. Mathematically, this means that the benchmark level Ztshould move
up less than the stock price itself, so that the cushion at time t+1, namely
St+ 1 −Zt+ 1 , be larger than the cushion at time t, St−Zt. Conversely, if the re-
turn on market is particularly poor, the investor should feel as though his re-
serves of prior gains are depleted. For this to happen, Ztmust fall less than St.
A simple way of modeling the sluggishness of the benchmark level Ztis
to write the dynamics of ztas
(9)where is a fixed parameter. This equation then says that if the return on
the risky asset is particularly good, so that Rt+ 1 > , the state variable
z=Z/Sfalls in value. This is consistent with the benchmark level Ztbehav-
ing sluggishly, rising less than the stock price itself. Conversely, if the return
is poor and Rt+ 1 < , then zgoes up. This is consistent with the benchmark
level falling less than the stock price.^10
is not a free parameter in our model, but is determined endogenously
by imposing the reasonable requirement that in equilibrium, the median
value of ztbe equal to one. In other words, half the time the investor has
prior gains, and the rest of the time he has prior losses. It turns out that is
typically of similar magnitude to the average stock return.
We can generalize (9) slightly to allow for varying degrees of sluggishness in
the dynamics of the historical benchmark level. One way to do this is to write
(10)When η=1, this reduces to (9), which represents a sluggish benchmark
level. When η=0, it reduces to zt+ 1 =1, which means that the benchmark
level Zttracks the stock value St one-for-one throughout—a very fast-
moving benchmark level.
The parameter ηcan be given an interpretation in terms of the investor’s
memory: it measures how far back the investor’s mind stretches when re-
calling past gains and losses. When ηis near zero, the benchmark level Ztis
always close to the value of the stock St: prior gains and losses are quickly
swallowed up and are not allowed to affect the investor for long. In effect,
the investor has a short-term memory, recalling only the most recent prior
outcomes. When ηis closer to one, though, the benchmark level moves
zz
R
tt+ Rt+
=1 +−
1ηη()(). 11RRRRRzzR
tt+ Rt+
1 =
1,PROSPECT THEORY AND ASSET PRICES 235(^10) The benchmark level dynamics in (9) are one simple way of capturing sluggishness. More
generally, we can assume dynamics of the form zt+ 1 =g(zt,Rt+ 1 ), where g(zt,Rt+ 1 ) is strictly in-
creasing in ztand strictly decreasing in Rt+ 1.