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

Computing optimal mean/downside risk frontiers: the role of ellipticity 195

Step 5: Repeat Step 2 to Step 4, M times.

Step 6: From the M sets of risk – return measures, rank the returns in
ascending order and allocate the returns and allocate each
pair to B buckets equally spaced from the smallest to the
largest return. Within each bucket, determine the portfolio
with the minimum (or maximum as required) risk measure.
Across the B buckets, these portfolios define the approximate
risk – return frontier.

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Mean return (% per day)

Standard deviation and value at risk (% per day)

Value at risk (5%) Standard deviation

Figure 8.1 Illustration of mean/standard deviation and mean value at risk surfaces
using 8,000 portfolios.

We illustrate the feasibility and accuracy of this algorithm using data from
the Australian Stock Exchange. Daily returns were obtained for the trading days
from 1 June 1995 to 30 August 2002 on the eight largest capitalization stocks
giving a history of 1,836 returns. Summary statistics for the daily percentage
return on these stocks are reported in Table 8.1. They display typical proper-
ties of stock returns, in particular, with all stocks displaying significant excess
kurtosis. 128,000 random portfolios were generated and for each portfolio
the average return, the sample standard deviation, the sample semi – standard
deviation, the 5% value at risk, and the sample expected loss below zero were
computed. Figure 8.1 illustrates the surface of the mean/standard deviation and
mean/value at risk pairs obtained from 8,000 random portfolios. The fron-
tiers for these risk measures are readily identified. From the 128,000 random
portfolios, the approximated risk frontiers are presented in Figure 8.2 for