292 Optimizing Optimization
known that estimation error in conjunction with long-only constraints causes
extreme, i.e., concentrated mean – variance portfolios. Given that CVaR is
much more sensitive to estimation error, we conjecture without proof that
CVaR optimal portfolios will also tend to be more concentrated for very much
the same reasons. There are not too many assets with positive skewness and
small kurtosis.
12.4 Scenario generation I: The impact of estimation and
approximation error
12.4.1 Estimation error
Estimation error is a serious concern in portfolio construction. If we have little
confidence that our inputs used to describe the randomness of future returns are
accurate, we will also have little confidence in the normative nature of optimal
portfolios. Any portfolio optimization process will spot high return, low risk,
low common correlation opportunities and try to leverage on them. These are
precisely the estimates that will be the most error laden, as high returns with
little risk and low correlation are not an equilibrium proposition but “ free
lunches. ” This is the economic basis of the “ error maximization ” argument and
it is one of the most serious objections to portfolio optimization of practitioners
and academics alike. We will not attempt to review the vast literature on how to
best deal with estimation error ranging from Bayesian statistics to robust statis-
tics and robust portfolio optimization. However, we need to make the point that
not all risk measures are equally sensitive to estimation error. 11 How does the
estimation error in CVaR compare to alternative risk measures ( Figure 12.4 )?
We employ a simple bootstrapping exercise, where we take the returns for
an arbitrary sector (here oil) sample downside and dispersion risk measures
1,000 times and plot the distribution of percentage deviations from the sample
risk measure (which serves as the true risk measure). Exhibit 4 summarizes our
results. Downside risk measures like value at risk ( VaR ), CVaR , and semivari-
ance ( SV ) are many times more sensitive to estimation error than dispersion
measure like mean – absolute deviation ( MAD ) or volatility ( VOL ). This should
not be surprising given that dispersion measures are symmetric risk measures
that use all available return information. Downside risk measures on the other
hand use at best half of the information in the case of semivariance and only
extreme returns for value at risk and CVaR. CVaR in particular is very sensi-
tive to a few outliers in the tail of the distribution.^12
12 Note that we can estimate the minimum variance portfolio without having to rely on return
expectations. Whatever return expectations investors have, they will not change the minimum
variance portfolio. This does not apply to the minimum CVaR portfolio, the higher the return,
the lower the conditional value at risk.
11 After all, this is the foundation of robust statistics, i.e. not all statistics have the same sensitivity
to outliers.