Optimal solutions for optimization in practice 75
framework. The theory also includes that an investor is actually concerned about
changes in wealth (and not about the absolute level of wealth) and that he or she
experiences a higher sensitivity to losses than to gains. The latter is expressed by a
factor ( 1), which addresses the loss aversion within a prospect utility function.
Hence , it appears that GLO is more amenable to modeling individual utility
and is valid for arbitrary distributions; however, it is not set up for using the
great deal of valuable information that active managers have accumulated over
years of running successful funds. This information is usually in the form of a
history of stock or asset alphas and a history of risk model information. Whilst
it is possible to use this to improve GLO, it is obvious that gain and loss is
a different reward and risk paradigm from the more traditional mean and
variance.
3.6.3 The model
We thus propose a utility function V , where:
Vg θμ λσ θ()()(())pp^211 p φlp
(3.8)where
μ (^) p mean of portfolio p ;
(^) σp^2 variance of portfolio p ;
g p gain of portfolio p relative to some target t ;
l p loss of portfolio p relative to some target t ;
λ risk aversion coefficient in the MV framework;^4
φ loss aversion coefficient in the GL framework.
The parameter θ is a weight parameter that takes values between 0 and 1. If
θ 1, we have MV and if θ 0, we have GLO. This can also be set to resolve
scale issues that arise from using alphas in the mean – variance that are not exactly
expected annualized rates of return. To determine φ , a number of possible
approaches can be considered. One we investigate in this chapter is based on the
idea that the maximum drawdown on the resulting optimal strategy should be less
than 2.5% per month. This approach is described in the final part of this section.
We demonstrate that the above model is equivalent to subtracting variance
from a GLO problem. Using the fact that μ (^) p t g p l p where t is the target
rate, it follows that Equation (3.8) becomes:
Vg tppθθ(( ))11φθλσlp^2
(3.9)
4 We understand that the loss aversion coefficient and the risk aversion coefficient will probably
change if they are combined in a mutual utility function. However, this matter will be excluded
from this chapter.