Asset Allocation without Unobservable Parameters

# 4 a simple proof follows set w 1 in equation 2 which

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4. A simple proof follows. Set W 0 = 1 in Equation 2, which produces the equivalent criterion function of . When returns are independently distributed, we can move the expectation operator inside the product to yield . We immediately see that the portfolio weights that maximize this function will be the weights that minimize . Moreover, because the logarithmic function is monotonically increas- ing, the same weight vectors will also minimize . Because this prob- lem is additively separable, the first-order necessary condition for any particular p t will not include any of the other p t ’s. As a result, p t can be found simply by minimizing . If we make use of the assumption that portfolio returns are independently distributed across the H equal-length periods, we can remove the subscript t from p t in this problem, producing . Multiplying by –1 converts it into the one-period maximization in the article (Problem 3). 5. In practice, rebalancing may also be a useful strategy for maximizing other criterion functions, even when a more elaborate dynamic strategy could, in theory, do better (e.g., when portfolio returns are not IID). Buetow, Sellers, Trotter, Hunt, and Whipple (2002) documented some of the general benefits of rebalancing. 6. Later, I describe how the analysis should be modified to address realistic applications with multiple assets, more realistic return processes, and a situation in which no asset earns a constant real rate of return. 7. Matching the average real stock return requires the solution of one equation in the two unknowns, u and d ; matching the standard deviation requires the solution of another equa- tion in the two unknowns. The assumed values of u and d are the only ones satisfying both equations. They are reported to only four decimal places in the text, but more accurate floating point values were used in the computa- tions. Because of this discrepancy, readers may be unable to exactly duplicate the calculations that yielded the tables here but should get close enough to determine whether they are doing them correctly. 8. Restricting investors to using stocks and non-inflation- indexed bonds, Siegel derived a mean–variance-efficient stock allocation of 115 percent for an investor with a 30-year horizon and what he described as “moderate risk tolerance” (p. 38). This high percentage occurred because his 200-year series of inflation-adjusted stock returns has some of the properties typical of series generated from a mean-reverting process rather than an IID process. As a result, the 30-year cumulative real returns presented in his series have lower volatility than would occur if returns were IID and were thus more favorably appraised by his mean–variance criterion than IID returns would be. When the possibility of allocating assets to TIPS (earning an assumed real rate of 3.5 percent a year) was added, the mean–variance-efficient “tangency” portfolio (of only the stocks and non-inflation-indexed bonds) for an investor with a horizon of 10 years still allo-

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• Winter '10
• Acharya,Kandarp

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