154 Optimizing Optimization
estimate the covariance matrix. Table 6.2 displays the estimated volatilities and
correlations. The optimal portfolio is obtained by minimizing variance subject
to the constraint on expected return. The S & P 500 receives a weight of 37.2%,
the Nikkei 36.6%, and the FTSE 100 26.2%. The DAX index is excluded from
the optimal portfolio.
The intuition behind this outcome is straightforward: To satisfy the alpha
constraint, a portfolio must contain a long position in one of the indices with
positive alphas, i.e., DAX or Nikkei. The latter is preferred because of its low
volatility and lower correlation with S & P 500 and FTSE. Some exposure to
S & P 500 and FTSE is required by the minimum variance objective because the
two assets display the lowest volatilities.
Having characterized the optimal mean – variance portfolio, I will now focus
on the results of the mean – EVaR optimization. Using the same set of alphas
(first row of Table 6.2 ) and the same constraints, I have run the simplex opti-
mizer for a range of values of the risk tolerance parameter ω. The signal to
noise ratio parameter q was set to 10^ ^3. The resulting country allocation fron-
tier is displayed in Figure 6.4 , with the value of ω on the horizontal axis.
Setting ω 0.15 yields a portfolio that is very similar to the mean – variance
solution. As ω decreases (and therefore risk aversion increases), the optimizer
overweighs the UK index while reducing the exposure to S & P 500. This phe-
nomenon can be explained by the differences in the left tail of the distribution
of the returns to the two indices. In particular, the dynamic model that is built
into the procedure generates a weighting pattern that tends to emphasize the
recent information about downside extremes. Table 6.3 shows that during the
last 6 months of the sample period, the S & P 500 has displayed fatter tails (as
denoted by the higher kurtosis) than the FTSE 100.
As ω increases above 0.15, the portfolio manager is willing to tolerate the
high downside risk carried by the DAX in order to capture some of its upside.
As a result, the weight on the German index increases.
At ω 0.5, the risk neutrality case, the optimal portfolio is made up of just
DAX and FTSE. Even though this case is irrelevant for practical purposes, it
can help us understand the shape of the mean – EVaR frontier. It is useful to note
Table 6.2 The first row shows a vector of randomly generated alphas. The remainder
of the table displays volatilities and correlations estimated via the optimal shrinkage
method of Ledoit and Wolf (2004)spx dax nky ukx
Alpha 1.88% 2.08% 4.76% 0.15%
Ann volatility 17.73% 25.68% 22.72% 19.04%
spx 1.00 0.76 0.53 0.74
dax 0.76 1.00 0.61 0.79
nky 0.53 0.61 1.00 0.57
ukx 0.74 0.79 0.57 1.00