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

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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
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