Long Term Equity Returns

2004 Canadian Institute of Actuaries Stochastic Modelling Symposium.

Abstract The volatility of long-term equity returns is an important but often overlooked aspect of actuarial modeling. The volatility of compound returns has a significant impact on a number of important areas such as product pricing, reserving, economic capital and asset allocation. Our aim is to define a reasonable range for compound equity volatility estimates. Because the volatility of compound returns cannot be directly observed from history, we use a number of techniques to help develop this range.

First, we review an approximation formula in which the volatility of compound returns decreases with the square root of the time horizon. We then analyze historical data, but since the volatility of long-term returns cannot be directly calculated, we must rely on imperfect estimates. Next, we fit a number of statistical models to various historical periods to infer the volatility of long-term returns. We review some statistical tests that can help us evaluate the fit of the models and use the results to develop a reasonable range for long-term equity volatility. We find that good short-term models are not necessarily good long-term models; models that incorporate fat-tails and mean reversion provide superior fits to long-term data.

Finally, we explore areas for future research.