288 Optimizing Optimization
complete and arbitrage possibilities might exist). Virtually, all implementation
problems relate to the scenario matrix and its generation.
12.3 Downside risk measures
12.3.1 Do we need downside risk measures?
Mean – variance optimization and mean – CVaR optimization yield identi-
cal results if returns are normally distributed. The efficient set (i.e., the set of
optimal portfolios) is identical as the return distribution would be completely
determined by the first two moments. In a normally distributed world, CVaR
optimization has nothing to exploit to its advantage. Downside risk measures
need distributional asymmetry (positive or negative skewness) for their justifi-
cation. While we know the world is not normally distributed most of the time,
we need to ask ourselves the following question to assess the merit of deviating
from mean – variance optimization: Are the deviations from symmetry statisti-
cally significant and do these deviations display an exploitable pattern, i.e., are
they stable?
Throughout this chapter, we will use monthly total return data for 10 US
sector indices (Oil, Basic Materials, Industrials, Health Care, Consumer Goods,
Consumer Services, Telecom, Utilities, Financials, Technology) for the last 20
years (from March 1989 to March 2009) to illustrate all our results. In order
to assess the stability of risk estimates, we split the sample into two 10-year
periods and estimate the excess skewness for each series and time period. We
then run a regression of the form:
skewtt^101 ββ εskew t^1 (12.10)If deviations from normality were stable, we would expect β 0 0 and β 1 1
together with a high R 2. For the data set above, we estimate β 0 0.00003,
β 1 2.187 with an R 2 of 68%. The slope coefficient β 1 is highly significant
with a t -value of 4.2. Data and regression fit are displayed in Figure 12.1.
Deviations from symmetry seem to be reasonably stable and given both sub-
periods have shown a negative skew, we would expect CVaR optimization to
arrive at different solutions than mean – variance investing. Of course, these
results cannot be generalized and a different investment universe might exhibit
different properties.
12.3.2 How much momentum investing is in a downside risk measure?
Let us first distinguish between dispersion measures (volatility, mean – absolute
deviation) and downside risk measures ( CVaR , minimum regret, etc.). 6 While
dispersion measures look at the whole distribution (they measure risk as dispersion
6 See Scherer (2007) for a review of these measures.