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Unformatted text preview: n risk aversion RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/11/2012. © P. KOLM. 24 Generalization: Joint Asset Uncertainty Consider the joint uncertainty set ì
ïm | (m - m)¢ S-1 (m - m) £ e2 ü
U e (m ) = í
þ Solve the M-V problem for the worst-case realization of the uncertain expected returns max min w ¢m - lw ¢Sw
m ÎUe ( m ) s.t. w Î C This problem is equivalent to the second-order cone program (SOCP): ˆ
max w ¢m - lw ¢Sw - e w ¢Smw
w s.t. w Î C
where w ¢Smw : estimation risk, e : aversion to estimation risk RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/11/2012. © P. KOLM. 25 Observations from Practice Robust efficient portfolios… Have better worst-case behavior and are only slightly inefficient with “average” inputs Remain relevant for longer periods (better for buy-and-hold approaches than standard M-V) Have lower turnover (transaction costs) But… Worst-case/conservative models are not for everyone Robust methods are in some cases equivalent to shrinkage estimators and leave the efficient set unchanged but come at the expense of having to solve
a SOCP RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/11/2012. © P. KOLM. 26 References
Aharon Ben-Tal and Arkadi S. Nemirovski (1998), "Robust Convex Optimization," Mathematics of Operations Research
Aharon Ben-Tal and Arkadi S. Nemirovski (1999), "Robust Solutions to Uncertain Linear Progra...
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