76 Optimizing Optimization
It follows that, mathematically, our problem is the same if we simply subtract a
portfolio variance term from a GLO. There are a number of considerations that
make Equation (3.8) more appealing than Equation (3.9) notwithstanding their
mathematical equivalence. These are the incorporation of risk model and alpha
information into the problem. We shall deal with each of these issues next.
3.6.4 Incorporation of alpha and risk model information
Incorporating risk model information
In the combined model given by Equation (3.8), risk model information can
be directly incorporated into backtesting in the usual way. In the special but
important case when we only have gain – loss, the method for incorporating risk
model information becomes much more complicated and less obvious. One
possible solution is to use the risk model’s linear factor structure combined
with the stock alphas plus Monte Carlo simulations to create artificial histories
of data, which can be used as inputs in the optimization.
Incorporating alpha information
One major difficulty that is virtually universal is trying to identify how long in
the future alphas will be effective. This can be partially addressed by informa-
tion coefficient (IC) analysis, which computes the correlation coefficient either
cross-sectionally or as a time series between alphas and future actual returns.
Alternatively, one can regress current returns on lagged alphas, this having the
advantage that the regression effectively rescales the alphas, saving one the
need to do it separately.
One could then include this forecasted return along with K historical returns
in the GLO. The difficulty with this is that the alpha information will not
enter into the overall data very much. A more ad hoc approach is to use a
weighted combination of forecasted past returns with actual past returns.
Whilst this may seem ad hoc, it should be able to be justified by some variant
of a Bayesian argument much as the one that underpins Black – Litterman, as
discussed by Scowcroft and Satchell (2000) and Meucci (2008).
Determination of the loss aversion coefficient
As mentioned above we make use of a drawdown constraint to determine the loss
aversion coefficient φ. In particular, we consider a maximum drawdown of 2.5%
per month. To find a suitable φ , we try values between 0.0 and 5.5 in our backtest
and look at the resulting drawdowns. Depending on the investor’s requirements,
φ could be chosen as the value that strictly leads to drawdowns smaller than the
defined maximum drawdown. Or, for a less constrained investor, the distribution
of drawdowns over the backtest period per trial value of φ could be examined.
One could apply a confidence level and choose φ as the value that leads to draw-
downs smaller than the defined maximum drawdown given a particular confi-
dence level. This would allow smaller values for φ than the first approach.