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

68 Optimizing Optimization

The expected shortfall values of the optimal portfolio generated by the

mean – ES framework are reduced relative to those of the traditional mean –

variance and mean – VaR approaches. Furthermore, when portfolio returns

are assumed to be normally distributed, the optimal portfolios selected within

the mean – VaR and mean – ES frameworks are identical to those generated by

the traditional mean – variance approach. Our results are consistent with a

growing risk management literature which suggests that, when dealing with

fat-tailed distributions, there is a need to focus on the tail region, and further,

that expected shortfall is a better framework to base decisions upon than

Value-at-Risk. ”

Of course, the computation of Ω corresponding to our maximized expected
utility remains possible.

3.5.3 Choice of inputs

In order to present the behavior of this GLO, it seems appropriate to compare
its outcomes with mean – variance optimization (expected return minus lambda
times variance) for various parameterizations. The asset classes chosen are sub-
ject to the constraint of long-only investments. We fix the target rate or thresh-
old R that decides if a return is treated as a gain or a loss to 4.8%, which is
comparable to the current annual interest rate.^2
In order to nominate a value for l for the GLO, we confront a virtual inves-
tor with a gamble comparable to the Basic Reference Lottery. 3 This earns
him or her either a utility rate U (probability  ½) or he or she loses U (also
probability  ½), so that his or her total utility equals:

1

2

1

2

UlU() 1

The insurance D he or she would pay to avoid the gamble (i.e., the difference
between the gamble and a certainty equivalent) reveals his or her loss aversion:

   

 



() ()1

1

2

1

2

1

2

2

lD U lU

l

D

UD

Giving up 3% of one’s wealth to avoid the uncertainty of a 10% gain or loss
with equal probability would, hence, imply l  1.5. Evaluating this as a fairly
high degree of loss aversion, we choose a more conservative value of l  1.0.
Another procedure to capture loss aversion, which might be more suitable for
an institutional investor, is described in section 6.

3 Gains/Losses utility is linear in wealth and, hence, independent of an investor’s wealth level. This
allows calculations in terms of rates.

2 At the time of the analysis (early 2008).