270 Optimizing Optimization
t 1 max[rtp,]/( D)028,28
∑ σ^ and the Calmar ratio, R p^ /| MDD p^ |.^25 To compare
diversification, we also report the maximum portfolio weight, max w i , and
the proportion of assets that are included in the optimal portfolio, No (^) p
1 || w || 0 / U 2.
According to the four performance metrics, direct maximization of the
expected utility appears superior to the mean – variance criterion for all levels of
risk aversion. What’s more, the “ No (^) p ” statistics indicate that this outperform-
ance is achieved with only 63 – 69% of the assets in the universe, which means
that including transactions costs into our analysis will only strengthen this
result. In contrast, our threshold acceptance portfolios appear less diversified
than the corresponding mean – variance portfolios owing to their greater empha-
sis on one asset. However, we can easily nullify this issue by defining upper
bounds for the weights, w b , or by imposing a minimum number of assets
to be included in the final portfolio || w || 0 d. Alternatively, we could introduce
a direct preference for diversification by considering alternative utility functions
of the form U ( w r ) λ f ( w ), where f ( · ) is a measure of diversification and λ
describes the strength of these preferences. While nonlinearities and augmented
utility functions are trivial extensions to our baseline algorithm, they lead to
many additional complications within the mean – variance framework.
Table 11.2 Mean – variance versus threshold acceptance
α^ ^1 α^ ^2 α^ ^5 α^ ^10 MV
Rp 1.7583 1.7086 2.1237 2.5064 1.7540
σ (^) p 5.2908 5.1813 4.8791 4.4295 4.7384
Skewness 0.0440 0.0613 0.0402 0.0257 0.0916
Kurtosis 3.3176 3.3019 3.2090 3.3476 3.0328
Sharpe 1.7650 1.7496 1.8451 1.8935 1.5767
Sortino 4.8249 4.6845 4.8729 5.1897 4.1756
Upside 0.8104 0.8053 0.8152 0.8214 0.8014
Calmar 0.9294 0.9036 0.9769 1.0636 0.8222
max w i 0.3574 0.3295 0.3090 0.2912 0.0919
No (^) p 0.6321 0.6436 0.6632 0.6855 1.0000
This table compares the (annualized) out-of-sample performance of the mean – variance approach
with our Johnson-based threshold acceptance algorithm. The algorithm is allowed 15 minutes
of computational power for each of the N 250 simulations, which corresponds to setting
N (^) Restarts 50, N (^) Iter 4, and N (^) Steps 3000. We set z 0.65 for convenience, and assume no
disappointment aversion, i.e., A 1. All calculations are performed using an Intel Core 2 Duo
1.86 GHz Processor and 2GB RAM.
25 We define 11 +=Rrpt∏^28 t= 1 ()+ as the cumulative portfolio return,^ σpt= ∑t^28 = 1 r^2 / 28 as
the standard deviation, and^ σpD= ∑t^28 = 1 min[ , ] /rt 0282 as the downside standard deviation.