156 Optimizing Optimization
Table 6.4 further compares the mean – variance and the mean – EVaR portfo-
lios. By using the covariance matrix estimated ex ante ( Table 6.2 ), I computed
ex ante estimates of the volatility of each optimal portfolio. The minimum is
reached by the mean – variance solution, by construction, at 17.1%. However,
for low risk tolerance levels, the EVaR portfolios yield very similar levels of ex
ante volatility (ranging from 17.1% to 17.4% for ω between 0.05 and 0.2). In
other words, the portfolios obtained by minimizing EVaR at low levels of risk
tolerance are also low risk according to a volatility criterion. Similarly, the pre-
dicted 5% EVaR of the optimal mean – variance portfolio is similar to the one
obtained by optimizing on EVaR directly.
The illustration so far has focused on one particular vector of simulated
alphas. I carried out a Monte Carlo analysis by simulating 100 vectors with
an information coefficient of 0.5. The three-step simulation procedure follows
Grinold and Kahn (2000) and Ledoit and Wolf (2004). The benchmark weights
are taken to reflect the capitalization of each market, in US dollars, at the
end of November 2008: The US receives a weight of 63.9%, Germany 5.7%,
Japan 18.7%, and the UK 11.7%. The goal in each optimization exercise was
to obtain a portfolio alpha greater than or equal to the expected benchmark
return plus 3% (annualized).^6
Table 6.5 shows summary statistics for the simulated alphas. The first row
displays the actual return of each asset for the month of December 2008. The
means of the simulated values mirror the pattern found in the actual returns,
while the accuracy of each prediction is negatively related to the volatility of
the corresponding asset ( Table 6.2 ).
The Monte Carlo results are displayed in Figure 6.5. Each plot refers to an
individual asset and the horizontal axis (drawn on the same scale for all plots)
measures the difference, in percentage points, between the asset’s weight in
the mean – variance portfolio and the weight in the mean – 5% EVaR portfolio.
If the distribution is concentrated to the right of zero, then the mean – EVaR
Table 6.4 Annualized ex ante volatility for alternative portfolios, based on the
optimal shrinkage estimate of the covariance matrix displayed in Table 6.3Annualized ex ante vol
Mean – variance solution 17.1%
Mean, 5% EVaR 17.3%
Mean, 10% EVaR 17.1%
Mean, 20% EVaR 17.4%
Mean, 30% EVaR 19.7%
Mean, 40% EVaR 21.3%
50% EVaR (risk neutral) 21.3%6 In the event that for a particular set of alphas the constraints are inconsistent, the simulated
alpha vector is discarded.