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

Optimal solutions for optimization in practice 69

In order to determine values for risk aversion, we consider an investor who

holds a portfolio of cash and FTA only. On the basis of the constant abso-

lute risk aversion MV framework, his or her expected utility is expressed by

EU w[()] wμ ( w r)f w

λ

1 22 σ

2

, where w , μ , λ , σ 2 , r f are the optimal pro-

portion in equity, the expected rate of return, the coefficient of absolute risk

aversion, the variance of the FTA, and the riskless rate return, respectively.

Maximizing this utility yields w

rf



μ

λσ^2

or λ

μ

σ



r

w

f

2.^ Considering

current values ( μ  9%, σ  12%, r f  4.6%) reduces this to the trade-off

λ

3

w

. The proportion of equity investment for institutional investors is
indicated by peer-group benchmarks as WM to be 70%. This corresponds to
4  λ  5 as an adequate range for the average institutional investor but prob-
ably inappropriate for high net investors. On the basis of these calculations
within our mean – variance analysis, we obtain λ  2 and λ  10 to represent
high and low risk aversion. These values would fit portfolios of cash and FTA
only with equity positions of 150% and 30%, respectively.

3.5.4 Analysis and comparison

We divide our analysis into four parts, each representing a change in con-

straints, parameters, or the specification of the utility function; the main char-

acteristics are summarized.

Table 3.1 Summary of optimization results: maximum holding  100%

Mean% StdDev% Skewness Kurtosis Gain% Loss% Omega Probability

(a) G/L l  1 12.63 5.73  0.79 3.62 12.68 4.85 2.614 0.72
(b) M-V λ  2 24.40 20.65  0.92 4.41 38.65 19.05 2.029 0.66
(c) M-V λ  10 15.14 8.55  0.90 3.72 18.08 7.74 2.336 0.70

Table 3.2 Summary of optimization results: maximum holding  25%

Mean% StdDev% Skewness Kurtosis Gain% Loss% Omega Probability

(a) G/L l  1 13.66 7.73  0.94 3.86 15.67 6.82 2.298 0.70

(b) M/V λ  2 17.51 11.85  0.96 3.71 23.99 11.27 2.129 0.70

(c) M/V λ  10 15.37 9.37  0.89 3.57 19.38 8.81 2.200 0.67