How to Combine Long and Short Return Histories Efficiently

Abstract

A common challenge in portfolio risk analysis is that certain assets have shorter return histories than others. Unfortunately, many standard portfolio risk analysis techniques—including historical tail risk measurement, regime-dependent risk analysis, and bootstrapping simulations—require full return histories for all assets or risk factors. The author presents easy instructions on how to efficiently combine data for investments whose histories differ in length and offers a new model to better account for non-normal distributions.

A common challenge in portfolio risk analysis is that certain assets have shorter return histories than others. Unfortunately, many standard portfolio risk analysis techniques—including historical tail risk measurement, regime-dependent risk analysis, and bootstrapping simulations—require full return histories for all assets or risk factors.

To address this problem, analysts often discard potentially valuable data on assets with long histories. For example, the history of returns for bonds and equities in the United States is a rich one, going back to the 1920s. This data sample includes the Great Depression, wars, several business cycles, periods of hyperinflation and deflation, and various financial crises. Analysts often exclude most of this rich data sample because returns for the other assets in the portfolio are not available as far back in history. Consequently, their models may not capture important risk dynamics for both assets with long return histories and assets with short return histories.

In this article, the author presents easy instructions on how to efficiently combine data for investments whose histories differ in length and suggests a new model to better account for non-normal distributions. The new model backfills the missing data for assets with short return histories. (Of course, the author does not claim to offer a model that magically transforms missing data into additional information. The model merely makes it easier to incorporate information from the long sample and to analyze more regimes than those included in the short sample.) Importantly, unlike conventional approaches that rely on the normal distribution, the model samples empirical residuals from the short sample to model uncertainty around the backfilled returns. Essentially, the model represents a hybrid between maximum likelihood estimation and bootstrapping. It provides an easy, simple, relatively assumption-free approach to account for fat tails—and other features of the distribution—in the return-backfilling process beyond means and covariances.