What Will the Likely Range of My Wealth Be?

Abstract

The median is often a better measure than the mean in evaluating a portfolio’s long-term value. The standard plug-in estimate of the median, however, is too optimistic. It has a substantial upward bias that can easily exceed a factor of 2. This article provides an unbiased forecast of the median of a portfolio’s long-term value. It also provides an unbiased forecast of an arbitrary percentile of a portfolio’s long-term value distribution, which enables the construction of the likely range of a portfolio’s long-term value for any given confidence level. The article offers an unbiased forecast of the probability of a portfolio’s long-term value falling within a given interval. The article’s unbiased estimators give a more accurate assessment of a portfolio’s long-term value than do traditional estimators and are useful for long-term planning and investment.

Forecasting long-term portfolio value is of great interest to investors and fund managers. Earlier researchers showed that in estimating the expected terminal value of a portfolio, both geometric mean returns and arithmetic mean returns are substantially biased, and they offered an approach to correct the bias. Others pointed out that the median terminal wealth is often more useful than the expected terminal wealth in evaluating a portfolio’s long-term value. The usual plug-in median forecast, however, is overoptimistic. It has a substantial upward bias that can easily exceed a factor of 2.

In this article, we provide an unbiased forecast of the median of a portfolio’s long-term value. We also provide an unbiased forecast of an arbitrary percentile of a portfolio’s long-term value distribution, which enables the construction of the likely range of a portfolio’s long-term value for any given confidence level. We offer an unbiased forecast of the probability of a portfolio’s long-term value falling within a given interval. Our unbiased estimators give a more accurate assessment of a portfolio’s long-term value than do traditional estimators.

Using the U.S. equity market and seven international equity markets as the investment asset, we show that even with 456 months of available data (January 1970–December 2007), the usual estimates of the medians, ranges, and probabilities are substantially biased. In contrast, the unbiased forecasts provide more-accurate assessments of the terminal wealth and are much more useful for long-term planning and investment.