290 Optimizing Optimization
12.3.3 Will downside risk measures lead to “ under-diversification? ”
How can we measure diversification? The simple (and maybe na ï ve) answer
is by measuring volatility. Suppose we perform the following thought experi-
ment. Portfolio A holds 100 assets and exhibits 15% of risk, while portfolio
B holds 10 assets and also exhibits 15% of risk. Which one is riskier? If you
believe 100% in your risk model, both portfolios must look equally risky to
you. The alternative view is that investors are not sure about their risk models
and they face the risk of extreme returns in any single asset, e.g., second or
even first moments might not even exist. 8 Under this scenario, holding a large
number of small positions is preferable. 9 This is why diversification (or better
dispersion in portfolio weights) plays an important role in practical portfolio
construction and why diversification measures (weight dispersion, concentra-
tion measures, etc.) contain some information not captured in a risk measure.
In other words, in a world where extreme moves are possible, investors are
likely to underestimate the required extent of diversification. Recently, the
field of robust optimization started to put regularity constraints on weights to
enforce diversification. 10 We will now look into the dispersion optimal CVaR
portfolios weights. Are they more or less diversified than mean – variance port-
folios? To make portfolios “ comparable, ” we minimize risk subject to a return
requirement of 78.5 basis points per month, which is the average of maximum
and minimum sector return. In other words, our portfolio should be placed in
the “ middle ” of the efficient frontier. The resulting optimal portfolios can be
found in Exhibit 2 and Exhibit 3. We see that CVaR portfolios in this example
are much more concentrated than mean – variance portfolios. Consumer goods,
for example, exhibit moderate volatility risk but high tail risk. Given that we
have little reason to believe that tail risk can be reliably estimated (see the
next section on estimation risk), we feel uncomfortable with such an “ under-
diversified ” portfolio, in particular in the presence of what might turn out as
extreme tail events. One should read the above statement carefully. We can-
not generalize that CVaR optimal portfolios will always be more concentrated
than mean – variance portfolios. Counterexamples are easy enough to engineer
( Figures 12.2 and 12.3 ).
It is well known that assets with positive (co)skewness and small
(co)kurtosis are favored by CVaR optimization. The “ researcher ” can pick
his universe and time horizon to arrive at the desired result. However, in our
experience, under-diversified CVaR portfolios can be observed much more
often than not. The reason for this should be obvious as it is equally well
9 If returns exhibit “ wild randomness ” as used by Taleb (2005) , investors are better off with very
diversified portfolios even at the expense of these portfolios looking mean – variance inefficient
due to “ over-diversification. ”
10 See Uppal, DeMiguel, Garlappi, and Nogales (2008).
8 As the number of observations increases, a single observation will dominate all other observa-
tions. For a Cauchy distribution variance will become infinite as the number of observations
grows.