198 Optimizing Optimization
measured in daily percentages. For the mean/standard deviation frontier, we
measure the error as the distance of the approximation from the quadratic
programming solution, and across 100 points on the frontier, the average error
is 1.05 basis points with a standard deviation of 0.49 basis points and the larg-
est error is 2.42 basis points. For the mean/semi – standard deviation frontier,
the average error is 0.72 basis points with a standard deviation of 0.36 basis
points. The expected loss frontiers almost coincide, with the average error
0.01 basis points with a standard deviation of 0.08 basis points. The value
at risk frontiers show the largest discrepancies with an average error of 1.75
basis points, a standard deviation of 2.84 basis points and a range in values
from – 5.08 up to 9.55 basis points. Panels B, C, and D report similar mea-
sures when the optimal portfolios in each bucket are chosen by minimizing
semi – standard deviation, maximizing value at risk (5%), and maximizing the
associated expected loss. The results are qualitatively the same as in Panel A,
but it is notable that the portfolios chosen using value at risk result in the larg-
est deviations from the quadratic programming solutions.
This example does illustrate that it is feasible to approximate the frontiers
for a variety of risk measures using this intuitive simulation methodology, but
that in this case there is little to be gained over the frontiers identified from
assuming that the returns are elliptically distributed.
8.6 Conclusion
In this chapter, we have presented analytical results that allow us to under-
stand better what optimal mean – risk frontiers look like. For elliptical returns,
these simplify to explicit formulae and we present closed form expressions for
mean/value at risk frontiers under ellipticity and mean/expected loss and mean/
semivariance frontiers under normality. For nonelliptical distributions, a simu-
lation methodology is presented that can be applied easily to historical data.
We do not consider the case of a riskless asset since this only has relevance
when index-linked bonds are available. However, our results could be extended
to this case.
References
Alexander , G. J. , & Bapista , A. M. ( 2001 ). Economic implications of using a mean – VaR
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Bensalah, Y. Asset allocation using extreme value theory. Bank of Canada Working
Paper 2002-2.
Bradley , B. O. , & Jaggia , M. S. ( 2002 ). Financial risk and heavy tails. In F. Comroy &
S. T. Rader (Eds.) , Heavy-tailed distributions in finance. North Holland.
Brown , J. T. , & Manson , C. W. M. ( 1959 ). The elements of analytical geometry. New
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