The increase in expected utility from using the alternative strategy is
(50)(51)where we have made use of the concavity of ut(⋅) and the linearity of the
prospect utility term with respect to St. It is therefore enough to show that
∆(δC, δS)=0 under budget constraint (49).
Multiplying with (49) and applying Euler equations (47) and
(48), we have
(52)Summing up (52) for all tand taking expectations, we have
The budget constraint implies that δW 0 =0. By requiring feasible strategies to
use bounded units of financial securities, and with a unit of the risk-free secu-
rity priced at one, we can show that the limiting term also goes to zero if our
model parameters satisfy log ρ−γgC+gD+0.5(γ^2 σC^2 − 2 γωσCσD+σD^2 )<0, a
condition which we already noted in footnote 21.^29 So ∆(δC, δS)=0 for any
feasible alternative to (C, S).
We have thus shown that any other budget feasible strategy cannot in-
crease utility. The Euler equations are therefore necessary and sufficient
conditions of optimality.
Proof of Proposition 3. This is a special case of Proposition 1 with η=0 and
zt=1 for all t, which is in turn a special case of Proposition 2, proved above.
Computation of Average Loss Aversion. Many of the tables present the
investor’s average loss aversion over time. There is only one difficulty in
computing this quantity: when zt<1—in other words, when the investor
has prior gains—part of any subsequent loss is penalized at a rate of one,
and part of it is penalized at a rate of 2.25. Therefore, it is not obvious
what single number should be used to describe the investor’s loss aversion
∆(,) (δδCS uC W*)δ lim E[ (uC W*)].δ
T TT T
=−′(^000) →∞
E[ () ˆ(,)]
( )E[(*)].
tt t t t t t t
tt t tt t t
uC C b S R z
uC W u C W
′+
=′ − ′
- ++ +
δδ
δδ
υ 1
11 1
uC′tt++ 11 ( *)
≤≡′+
= +
∞
∆(,)E [(δδCS ∑uC C bS R ztt*)δt tt tδˆ(,)]t,
tυ 1
0E[(uCtt* Ct)(*)uCt b S R ztt tˆ(,)]t
t+− +
= + ∞∑ δδυ^1
0PROSPECT THEORY AND ASSET PRICES 267(^29) The proof of this last step is available upon request.