The Term Structure of the Risk–Return Trade-Off

Abstract

Expected excess returns on bonds and stocks, real interest rates, and risk shift over time in predictable ways. Furthermore, these shifts tend to persist for long periods. Changes in investment opportunities can alter the risk–return trade-off of bonds, stocks, and cash across investment horizons, thus creating a “term structure” of the risk–return trade-off. This term structure can be extracted from a parsimonious model of return dynamics, as is illustrated with data from the U.S. stock and bond markets.

Recent research in empirical finance has documented that expected excess returns on bonds and stocks, real interest rates, and risk shift over time in predictable ways. Furthermore, these shifts tend to persist over long periods of time. One important implication of time variation in expected returns is that investors, particularly aggressive investors, may want to engage in market-timing (or tactical asset allocation), based on the predictions of their return forecasting model, in order to maximize short-term return. There is considerable uncertainty, however, about the degree of asset return predictability, which makes it hard to identify the optimal market-timing strategy. A second, less obvious implication of asset return predictability is that risk—defined as the conditional variances and covariances per period of asset returns—may be significantly different for different investment horizons, thus creating a “term structure of the risk–return trade-off.” This article characterizes this trade-off and explores its implications for the asset allocation decisions of long-horizon investors.

We present an empirical model that captures the complex dynamics of expected returns and risk but is simple to apply. Specifically, we model interest rates and returns as a vector autoregressive (VAR) model. We show how to extract the term structure of risk using this parsimonious model of return dynamics, and we illustrate our approach with the use of quarterly data from the U.S. stock, T-bond, and T-bill markets for the period since World War II. In our empirical application, we use variables that have been identified as return predictors by past empirical research, such as the short-term interest rate, the dividend–price ratio, and the yield spread between long-term and short-term bonds. These variables enable us to capture horizon effects on stock market risk, inflation risk, and real interest rate risk.

Among our findings are the following: Mean reversion in stock returns decreases the volatility per period of real stock returns at long horizons, whereas reinvestment risk increases the volatility per period of real T-bill returns. Inflation risk increases the volatility per period of the real return on long-term nominal bonds held to maturity. Stocks and bonds exhibit relatively low positive correlation at both ends of the term structure of risk, but they are highly positively correlated at intermediate investment horizons. Inflation is negatively correlated with the real returns on bonds and stocks at short horizons but positively correlated at long horizons.

These patterns have important implications for the efficient mean–variance frontiers that investors face at different horizons and suggest that asset allocation recommendations based on short-term risk and return may not be adequate for long-horizon investors. For example, the composition of the global minimum variance (GMV) portfolio changes dramatically for different horizons. We calculated the GMV portfolio when predictor variables are at their unconditional means—that is, when market conditions are average—and found that at short horizons, the GMV portfolio consists almost exclusively of T-bills but at long horizons, reinvestment risk makes T-bills risky. Thus, long-term investors can achieve lower risk with a portfolio that consists predominantly of long-term bonds and stocks.

We also found that the composition of the tangency portfolio of bonds and stocks (calculated under the counterfactual assumption that a riskless long-term asset exists with a return equal to the average T-bill return) becomes increasingly biased toward stocks as the horizon increases. The reason is the increasing positive correlation between stocks and bonds at intermediate investment horizons and the decrease of the volatility per period of stock returns at long horizons.

To concentrate on horizon effects, we bypass several other considerations that may be important in practice—for example, changes in volatility through time—and, ignoring the possibility that investors care about other properties of the return distribution, we consider only the first two moments of returns. In addition, our results depend on the particular model of asset returns that we estimated. We treated the parameters of our VAR(1) model as known, whereas these parameters are highly uncertain, and investors should take this uncertainty into account in their portfolio decisions. Fortunately, our main conclusions hold up well when the model is estimated over subsamples or is extended to allow higher-order lags.

The technical details of this study are provided in “Long-Horizon Mean–Variance Analysis: User Guide,” which is available in the supplemental material.

We are grateful for suggestions and helpful comments from the seminar participants at the 2004 Three-Way Symposium of Inquire Europe, Inquire UK, and the Q-Group. Kristin Longhine and Li Wang provided helpful research assistance at different stages of this project. This article is based on work supported by the National Science Foundation under Grant No. 0214061 to John Campbell. Luis Viceira acknowledges the financial support of the Division of Research of the Harvard Business School.