Risk-Return Analysis, Volume 2 by Harry M. Markowitz

Author:Harry M. Markowitz [Harry M. Markowitz]
Language: eng
Format: epub
Publisher: McGraw-Hill Education
Published: 2016-04-05T00:00:00+00:00

The next-to-last section of Markowitz (1959) tries to carry out this general advice in various portfolio contexts using what we have called implicit approximate EU maximization. We now turn to explicit approximate EU maximization.

THE MARKOWITZ AND VAN DIJK METHODOLOGY

In Chapter 8 we discussed dynamic programming’s (DP’s) “curse of dimensionality.” DP calculations with one or two state-variables in well-confined ranges proceed with ease. Those with three or four state-variables typically become computationally intense. Beyond that, the DP procedure quickly becomes out of the question.

Markowitz and van Dijk (2003) (MvD) and Kritzman, Myrgren, and Page (2009) (KMP) present two versions of a heuristic that has proven quite effective in dealing with DP’s curse of dimensionality. Specifically, the MvD experiment applies the method to a problem for which the DP optimum can be computed, and finds that the expected utility provided by the MvD heuristic is almost indistinguishable from that of the optimum strategy, as well as being substantially superior to standard heuristics. The KMP experiment applied the MvD heuristic to both small and large problems—namely ones for which the optimal solution can, versus those for which it cannot, be computed economically. They found that the MvD heuristic gives near-optimum results when the latter can be computed, and substantially outperforms standard heuristics in both small and large problems. The MvD heuristic has been used in fact for large rebalancing applications.1

The basic idea behind the MvD heuristic is this: The Markowitz (1959) Chapter 13 proposals, summarized earlier, in effect assume that the derived utility function Ut(st) is approximately a quadratic function of the state-variables that comprise st. Therefore, the investor can approximately maximize E(Ut+1(st+1)) by choosing a “good” combination of E(R), V(R), and the covariance of R with other state-variables. Markowitz (1959) uses this argument to justify displaying these statistics to the investor so that the investor can implicitly maximize EU.

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