Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Staying ahead on downside risk 153

In particular, this class of utility functions introduces loss aversion. Suppose

ω  0.1. When we are below the target level μ , the decrease in utility caused by

a 1% decrease in return is equal to 9. When we subtract 1% from the portfolio

return above the target , utility drops by 1. Figure 6.3 illustrates the different

utility functions corresponding to alternative levels of risk aversion.

An example will help illustrate the proposed methodology. Assume that the

only four available assets are the S & P 500, DAX, Nikkei and FTSE 100 indi-

ces. Domestic returns are used, i.e., all currency risk is assumed to be hedged.

The data consists of 400 weekly returns, obtained from Bloomberg, covering

the period April 2001 – November 2008. The goal is to identify the optimal

country allocation at the end of November 2008.

To abstract from the problem of estimating alphas, following Grinold and

Kahn (2000) , I generated a vector of predicted returns for December 2008

assuming a high information coefficient (0.5). The vector is displayed in the

first row of Table 6.2.

The equality constraint used in the optimization forces the portfolio to be

fully invested, i.e., Σ (^) i λ (^) i  1. The inequality constraints ensure that the portfo-
lio is long-only ( λ (^) i  0 ∀ i ) and that the expected monthly return based on the
alphas in Table 6.2 is greater than or equal to 1% ( λ  α  0.01).
I will first present the results obtained by using a mean – variance optimizer. I
adopted the optimal shrinkage estimator devised by Ledoit and Wolf (2004) to
U (X)
x =μ
X
ω= 0.5
ω= 0.25
ω= 0.1
Figure 6.3 Shape of the loss aversion utility function for alternative values of ω.

6.4 Empirical illustration