I examine two questions about the long-run performance of an equity portfolio. What is the relation between short-term measures of portfolio risk and reward and the portfolio’s long-run performance? Does this relationship depend on the degree of diversification?

To address these questions, I examine portfolio performance for a hypothetical investor over a 30-year accumulation period. By “short-term measure,” I mean a statistic, such as standard deviation of total portfolio return, that is determined by monthly returns and modeled with an asset pricing model, or the corresponding estimate of that statistic based on historical returns. I evaluate the relation between short-term measures of risk and the downside risk of ending real wealth. I also look at the relation between these measures of risk and level of ending real wealth. I analyze these relations under different levels of diversification (determined by number of stocks in the portfolio), type of portfolio weighting, method for making contributions, level of trading costs, and form of the asset pricing model. I also carry out the analysis for the Sharpe ratio calculated from the short-term measures of return and risk.

A general principle in finance is that long-run performance should be consistent with short-term measures of risk and reward. We expect that higher short-term risk should correspond to greater long- run risk. Moreover, we expect that higher short-term risk should be compensated by greater long-run ending real wealth. An exception is that, in an efficient market, the short-term diversifiable risk should not be compensated, because the investor easily can diversify it away.

I demonstrate that basic short-term measures often are not good predictors of the long-run performance of a portfolio of stocks. Short-term total portfolio risk and short-term diversifiable risk often are poor predictors of downside risk measured as the 10-percentile of ending real wealth. In addition, short-term total portfolio risk, short-term diversifiable risk, and the Sharpe ratio often are poor predictors of the median level of ending real wealth.

The exceptions are portfolios that are not well-diversified. For these portfolios, I observe a positive relation between short-term portfolio risk and long-run downside risk (as expected). Under some circumstances when portfolios are not well-diversified, the median levels of ending real wealth are higher when short-term total portfolio risk is greater (as expected) as well as when the Sharpe ratio is higher (since non-optimal portfolios are more likely to perform poorly). While this behavior is consistent with our expectations, it holds strongly only for portfolios of 100 stocks or fewer and, in some scenarios, only for 20 stocks or fewer. Moreover, the positive association between the short-term measures and long-run risk and reward declines substantially as portfolios become more diversified and, for all practical purposes, usually does not exist when portfolios reach the level of 500 stocks.

At first glance, it is puzzling that the expected positive relations between short-term measures of risk and reward and the long-run risk and reward hold well for the least diversified portfolios but less so as portfolios become more diversified and vanish for portfolios of large numbers of stocks. I propose that this behavior can be explained by the cross-sectional distributions of the short-term measures, conditional on the number of stocks in the portfolio. The cross-sectional distributions of total portfolio risk, diversifiable portfolio risk, and the Sharpe ratio are relatively wide when a portfolio has only a small number of stocks and shrink significantly as number of stocks in the portfolio increases. Portfolios with large numbers of stocks have about the same total portfolio risk, about the same diversifiable risk, and about the same Sharpe ratio. As a result, the ability of these short-term measures to discriminate between portfolios falls substantially as number of stocks in the portfolio increases.

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