294 Optimizing Optimization
“ middle ” of the efficient frontier) are visualized in below box-plots provided
in Figure 12.5.^13 Given that we employ a 10-asset problem only, it becomes
clear that even for small problems we cannot rely on historical data. Note that
even for 2,400 monthly returns (which equals 200 years of data) in the bottom
panel, the estimation error is still far from negligible. Not only do we not have
the luxury of such data, but even if we did we would find these data unlikely to
be usable due to nonstationarity. Given the above simulation results, the mini-
mum advice that needs to be given is to limit the application of CVaR opti-
mization to problems with a small set of assets. In any case, investors should
check the accuracy of their results with alternative samples for the scenario
matrix to estimate the impact of approximation error. They should ask: How
many scenarios do we need to generate to get the approximation error down
to a predefined level? A CVaR optimization based on a nonparametric historic
scenario matrix (in other words, a download of historical return series from
a data provider) is likely to disappoint due to its large in-built approximation
error, even if there was no estimation error at all.
0.00.20.40.60.00.20.40.60.00.20.40.60.00.20.40.6Oil and Gas Basic Mats Industrials Cons Gds Health Care Cons Svs Telecom Utilities Financials TechnologyOil and Gas Basic Mats Industrials Cons Gds Health Care Cons Svs Telecom Utilities Financials TechnologyOil and Gas Basic Mats Industrials Cons Gds Health Care Cons Svs Telecom Utilities Financials TechnologyOil and Gas Basic Mats Industrials Cons Gds Health Care Cons Svs Telecom Utilities Financials TechnologyFigure 12.5 Approximation error for alternative size of scenario matrices. The
variability of optimal weights is described in the above box-plots. The top panel uses
1,000 optimizations with 240 generated scenarios (to approximate a 10-asset problem),
while the last panel uses 2,400 generated scenarios. The intermediate panels use 480
and 1,200 scenarios respectively. Approximation error (measured as the indicated
dispersion shown in the above box-plots) falls with an increasing number of employed
scenarios.
13 Note that our setup (normal scenarios) offers no advantage over mean – variance investing; we
just build an extremely slow and imprecise mean – variance optimizer.