244 Optimizing Optimization
Select 201 values of^ π^ centered on bcˆ/ˆ, i.e., choose an interval bcˆ/ˆ^3 (bcˆ/ˆ) with
100 points above^ bcˆ/ˆ and 100 below^ bcˆ/ˆ.
Now for each value of π ,,,
^1 201, we can use
σπjjjkk 12 ()(( ) /^2 k 3 j) ( / 1 Tcˆ)to generate 5,000 values of σ (^) j.
4. From the 5,000 values of σ (^) j , for each π we can calculate the mean and the 2.5 and
97.5 percentile, i.e., σσ σ ,,Luand respectively.
- Now plot the pairs of points (, )( , )σπ σπ ,,L and (, )σπ u for^ ^1 ,,...^201 giving
the average mean – variance frontier and the 95% confidence limits.
From the discussion in Section 5, it is clear that a similar algorithm to that above
can be used to generate the frontier in a situation with general linear restrictions.
However, a simple algorithm for inequality constraints based on our approach
does not seem easily attainable. We do not present results on computational effi-
ciency gains but it is clear that for large N and T , they should be considerable.
10.10 Section 7: Conclusion
This chapter has explored the link between Monte Carlo averages and popula-
tion moments to assess the merits of Monte Carlo – based optimization meth-
ods that compute average frontiers. To this end, we have collected together a
number of results on exact properties of portfolio measures, some of which
already exist in the literature. We extend these results to include relative and
absolute return utility and relative and absolute mean – variance frontiers.
We compute biases for the optimal portfolios, alpha, volatility, and the infor-
mation ratio. We detect significant biases for the case when the number of assets
increases with the sample size, a case of great practical relevance. We further
show that when the problem is constrained, these biases are reduced. This sheds
some light on the practitioner approach to mean – variance optimization of impos-
ing large numbers of constraints. Not only does this control the optimization,
but, if the constraints are valid, it reduces the bias as well. Finally, averaging simu-
lated optimization can be seen as a satisfactory procedure if N is small relative to
T , or if N is large and K is large relative to T , or if the average optimal portfolio
or its moments are bias-corrected. We have not investigated the impact of N , K ,
and T on the width of the simulated confidence intervals for the key parameters
nor have we considered how we might extend our analysis to the Kuhn – Tucker
problems discussed in Section 5; these remain topics for future research.
Acknowledgment
We wish to thank Harry Markowitz for his comments and encourage-
ment. We also thank Phelim Boyle, participants of the 8th Annual Financial