176 Optimizing Optimization
And , of course:
Upside Potential Ratio Upside Potential/Downside DeviationStyle analysis — determining the style blend of a fund
The costs of investing in an actively managed fund are often much higher than
owning a portfolio of index funds. So how can we determine if these extra
costs are warranted? One approach is through returns-based style analy-
sis (Sharpe, 1988) to determine a portfolio of index funds that most closely
matches the investment style of the fund.
Here , briefly, are the procedures for determining this portfolio, called the
style blend.
A small (less than 20) set of indexes is chosen to divide, without overlap, the
space of potential investments. This selection should be made so as to reduce
the problem of colinearity but still allow a good fit as measured by R 2. The
selection of the time interval of data to fit is also important. The interval must
be long enough for a good fit but short enough to capture just the current style
of the management of the fund. We call this interval the style interval.
Weights to form a passive portfolio of indexes are chosen so that the variance of
the difference between the fund’s returns and the passive portfolio’s returns over the
style interval is minimized. This choice is accomplished with a quadratic optimizer.
A style statistic
Now we have our comparison portfolio — the style blend of passive indexes.
How do we make the comparison? First, we do not even try unless the style fit,
as measured by R 2 , is high enough, say 0.75 or higher. Then, the obvious thing
is to look at the difference of annual returns over the style interval. This is not
bad, but an improvement would be to adjust for difference in risk. One way to
do this is to use a style Fishburn utility function. Here is the definition for the
statistic we use, called DTR α.
DT Risk Adjusted Return of the Fund
Risk Adjusted ReturR α
nn of the Style BlendRisk Adjusted ReturnAnnual Return λ*Downside VarianceDownside Variance is the square of the Downside Deviation and is a con-
stant we usually take to be 3.
So we will consider using a fund in our portfolio only if it has a positive
DTR α.
There is more about this selection process in the previous chapter on
optimization.