On Estimates of Long-Run Rates of Return: A Note

In deciding among these estimators for a particular application, gains in efficiency must be weighed against additional computational effort. On these grounds, there would seem to be good reason in most circumstances for preferring the wrap-around overlap estimator to the simple unbiased one. The gain in efficiency is achieved at a small cost in additional computation, and, in addition, no data need be discarded if the sample size (T) does not happen to be an integral multiple of N. In going beyond the wrap-around estimator to the combinatorial one, however, the decision is less clear. The gain in efficiency is relatively small and the number of return relatives to be computed may become quite large. For example, with T=30 and N=10, the last parameter set for which computations are made in Table 1, evaluation of the combinatorial estimator requires computation of (= approximately 3×108) ten-period return relatives. In many applications, this additional effort is probably not worthwhile. Significantly, the ordinary overlap estimator seems to fare worst, probably due to its asymmetric use of the data, and, therefore, should not be used.It also should be noted that when relatively large amounts of data are available, the simple unbiased estimator will in most cases perform quite adequately. A common application in financial research involves computing annual stock returns from monthly data. Representative relative efficiency computations using μ = 1.01, σ = .05, and N = 12 are given in panel B of Table 1 for T = 60, 120, and 360 months. The incremental improvement of the wrap-around estimator relative to the simple one is quite small. Figures for the combinatorial estimator are not presented because the number of permutations would be prohibitively large.