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12.4.2 Approximation error
We have seen in Section 12.2 that CVaR optimization is essentially scenario
optimization, which models variability in returns by simulating (or bootstrap-
ping if done nonparametrically) a large numbers of scenarios. Once scenarios
are drawn, uncertainty is essentially removed and we optimize a deterministic
problem. The reliability of its solution depends on its ability to approximate
a continuous multivariate distribution from a discrete number of scenarios.
The difficulty to do this increases with the required CVaR confidence level (the
further we go into the tail) and the number of assets involved (the number of
conditional tails that need to be estimated). This is why CVaR optimization
is usually applied at an asset-allocation level rather than on a large portfolio
of individual stocks. Interestingly, this has not been widely addressed in the
finance literature, while it is well known in the stochastic optimization litera-
ture. We will engage in a simulation exercise to raise this point more clearly. To
isolate approximation error, we use the following two-step approach. First, we
estimate a variance – covariance matrix from historical data.
Second , we simulate 240, 480, 1,200, and 2,400 return draws (assuming
normality) and adjust the generated scenarios to match the estimated mean
return and variance with the original data for each asset. This step is repeated
1,000 times and the optimized portfolios (the return target is halfway between
minimum and maximum sector return, i.e., we should always arrive in the
Figure 12.4 Estimation error for alternative risk measures. Original data for the
Oil & Gas return series are bootstrapped 1,000 times and risk measures are calculated
from each resampling. The distribution of estimated risk measures is summarized in
box-plots.
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