Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

258 Optimizing Optimization

lems.^14 For practical purposes, however, we prefer the more parsimonious dis-
appointment aversion (DA) framework described by Gul (1991). In essence,
DA preferences are a one-parameter extension of the expected utility frame-
work and have the characteristic that favorable outcomes, i.e., outcomes above
a certain target level, are given a lower weight relative to unfavorable out-
comes. Since DA preferences are derived by relaxing the much lambasted inde-
pendence axiom from standard decision theory, they allow us to retain many
of the insights offered by expected utility theory and also to make comparisons
between existing empirical works that use standard preference settings.
We therefore consider an investor who chooses a set of portfolio weights,
w *, in order to maximize the expected value of his or her disappointment-
averse CRRA utility function:

UW

K

t UW dFW A UW dFW

WTA

TtW

TA

() 11   ( )() 11 ( )(Tt

1

∫∫∞

⎛

⎝

⎜⎜

⎜ +

∞

11 )

⎞

⎠

⎟⎟

⎟⎟⎟

(11.5)

where U ̃ (.) is the CRRA utility function,

UW

W

W

t

t

t

()

ln( )









 



1

1

1

1

1

1

1

α

α

α

α

for

for

⎧

⎨

⎪⎪

⎪⎪

⎩

⎪⎪

⎪⎪

(11.6)

W t (^)  1 is (normalized) end of period wealth, W TA is the target level of wealth,
0  A  1 is the coefficient of disappointment aversion, α is the coefficient of
relative risk aversion,^15 and K is a scalar given by:
KPWWAPWW ()()tT^11 A tTA (11.7)
Although this is a nonexpected utility function, CRRA preferences are a special
case for A  1. When 0  A 1, individuals are averse to disappointment,
i.e., outcomes below the target are weighted more heavily than the outcomes
above the target. The relative reweighting of expected utility is illustrated in
Figure 11.2 for A  0.5 and A  1.
15 Larger values of this parameter signify increasingly risk-averse behavior. A value of α  1 cor-
responds to the maximization of logarithmic utility, which was popularized by Edward Thorp
in the early 1960s for the game of Blackjack and subsequently applied to portfolio construction
later that decade ( Thorp, 1960 ; Thorp & Kassouf, 1967 ).
14 See Benartzi and Thaler (1995) , A ï t-Sahalia and Brandt (2001) , Berkelaar, Kouwenberg, and
Post (2004) , and Gomes (2005) for recent applications of this approach.