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

196 Optimizing Optimization

the mean/standard deviation, the mean/semi – standard deviation, the mean/
value at risk (5%), and the mean/expected loss. The black lines represent the
frontiers if it is assumed that the returns are elliptically distributed. In this case,
a quadratic programming algorithm has solved for the optimal mean/standard
deviation portfolio with no short selling for a number of target returns. The
portfolio weights were then used to compute a history of returns to obtain the

–3.00 –2.50 –2.00 –1.50 –1.00 –0.50 0.00 0.50 1.00 1.50

Value at risk (5%) Expected loss

Risk measure (% per day)

Semi–standard

deviation

Standard

deviation

2.00

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Expected return (% per day)

Figure 8.2 Risk frontiers obtained using quadratic programming and 128,000 simulated
portfolios. Black line: frontiers using weights from solving minimum standard deviation
frontiers with no short selling. Grey line: frontiers obtained using the simulated portfolios
where portfolio weights are derived by minimizing standard deviation.

Table 8.1 Summary statistics for eight stocks

Rank Stock Average St Dev Min Max Skewness Kurtosis

1 NAB 0.058 1.407  13.871 4.999  0.823 9.688
2 CBA 0.066 1.254  7.131 7.435  0.188 5.022
3 BHP 0.008 1.675  7.617 7.843 0.107 4.205
4 ANZ 0.073 1.503  7.064 9.195  0.122 4.617
5 WBC 0.059 1.357  6.397 5.123  0.181 3.985
6 NCP 0.013 2.432  14.891 24.573 0.564 11.321
7 RIO 0.035 1.659  12.002 7.663 0.069 5.360
8 WOW 0.078 1.482  8.392 11.483 0.025 6.846

Note : Stocks are ranked by market capitalization on 30 August 2002. The statistics are based on
1,836 daily returns, and expressed as percentage per day. Average is the sample mean; St Dev is the
sample standard deviation; Min is the minimum observed return; Max is the maximum observed
return; Skewness is the sample skewnees; and Kurtosis is the sample kurtosis.