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

More than you ever wanted to know about conditional value at risk optimization 297

VaR with their expected value. This risk neutrality in the tail is not plausible
at all. Spectral risk measures can help here. We could, for example, suggest a
weighting function like:

φ λ

λ

() λ

()

p e

e

p









1

1 (12.11)

with λ 0, as part of the definition of the spectral risk measure:^17

MX p F ppdp

weighting

function

X

loss

quantile

φ( ) φ() ()()





1

0

1

∫

(12.12)

Higher losses (larger values of p ) get larger weights and weightings increase
even further if λ increases. Spectral risk measures allow us to include risk aver-
sion in the risk measure by allowing a (subjective) weighting on quantiles.
Interpreting λ 0 as the coefficient of absolute risk aversion from an expo-
nential utility function, we (re)introduced utility-based risk measures through
the backdoor.
We calculate Equation (12.12) via numerical integration for alternative risk
aversion coefficients and distributions. Table 12.1 provides an example. We see
that spectral risk measures explode much faster for an increase in risk aver-
sion if the underlying returns follow a “ t ” rather than a normal distribution.
In other words, the possibility of tail events has an amplifying effect on the
risk measure depending on risk aversion (weighing function). Put more bluntly,
spectral risk measures offer utility optimization in disguise. This concludes a
rather unproductive academic research cycle. After 50 years of research, we
are back to expected utility maximization. While Markowitz (1952) tried to
approximate the correct problem, many of his followers were getting further

17 This is just one example based on the exponential utility function.

Table 12.1 Spectral risk measures. We numerically integrate

Equation (12.12) to arrive at values for our spectral risk

measure with weighting function (12.11). Under a fat-tailed

t -distribution, spectral risk measures explode much faster for

an increase in risk aversion than under a normal distribution

Risk aversion λ Normal distribution t -distribution

1 0.27 3.8

5 1.08 18.26

10 1.50 34.69

25 1.95 79.69

100 2.51 274.79