A new approach is proposed to imposing economic constraints on time-series forecasts of the equity premium. Empirically, economic constraints systematically reduce the risk of selecting a poor forecasting model and improve measures of out-of-sample forecast performance.
The authors develop a new methodology for imposing constraints that rules out negative equity premiums and places a bound on the conditional Sharpe ratio (SR) from above and below. Imposing economic constraints on the equity premium forecast improves the accuracy of the prediction models. When used to select portfolio weights, the constrained forecasts are found to yield higher certainty equivalent returns than the unconstrained counterparts.
How Is This Research Useful to Practitioners?
Equity premium forecasts play a central role in areas as diverse as asset pricing, portfolio allocation, and performance evaluation of investment managers. The authors propose a new approach for incorporating economic information via inequality constraints on moments of the predictive distribution of the equity premium. They consider two types of economic constraints: an equity premium (EP) constraint and an SR constraint.
Under broad conditions, the conditional equity risk premium can be expected to be positive. By imposing the nonnegative EP constraint, the parameters in the prediction model should be estimated subject to the constraint at all times. The conditional SR has to lie between zero and one. The zero lower bound is identical to the EP constraint, and the upper bound rules out the price of risk becoming too high.
Three key variables are used to predict the equity premium: (1) ratio of log dividend to price, (2) T-bill rate, and (3) default yield spread. After imposing economic constraints on the estimation models, the parameter uncertainty is reduced. For example, for the ratio of log dividend to price, which is always negative, the EP constraint rules out large positive values of the beta parameter, which could induce a negative equity premium. In short, the economic constraints tighten the predictive ability of the equity premium. Comparing the out-of-sample predictive performance, the R2 of the EP- and SR-constrained forecasts generally performs much better than the unconstrained approach at the monthly, quarterly, and annual horizons. The constraints also reduce the risk of selecting a bad forecast model.
The authors also investigate the optimal asset allocations of a representative investor with power utility. The EP- and SR-constrained models deliver higher certainty equivalent return (CER) values than their unconstrained counterparts. The benefits appear to be present at monthly, quarterly, and annual horizons.
How Did the Authors Conduct This Research?
The authors adopt the empirical Bayesian approach to compute the predictive density of the equity premium subject to economic constraints.
For their empirical analysis, they use data on stock returns along with a set of 17 predictor variables. The sample is from January 1927 to December 2010. Stock returns are computed from the S&P 500 Index and include dividends. A short T-bill rate is subtracted from stock returns to capture excess returns.
The authors use the first 20 years of data as a training sample. For example, for the monthly data, they initially estimate the regression models over the period from January 1927 to December 1946. They use the estimated coefficients to forecast excess returns for January 1947 and use the corresponding estimates to predict excess returns for February 1947. They proceed in this recursive fashion until the last observation in the sample.
The predictor variables fall into three broad categories: (1) valuation ratios, (2) measures of bond yields, and (3) estimates of equity risk. The authors focus the analysis on three predictors: the ratio of log dividend to price, the T-bill rate, and the default yield spread.
In Bayesian statistical inference, a prior probability distribution has to be chosen before evidence is taken into consideration. The authors start with the case in which no constraints are imposed. Next, they impose the economic constraints on the model parameters to modify the priors. The predictive moments of the return distribution get updated as new data arrive, and so the inequality constraints give rise to dynamic learning effects.
The authors test the model empirically by comparing the out-of-sample predictive performance and CER with the unconstrained models.
The noise generated by large variations of predictor variables can be reduced by imposing the economic constraints. The approach can help investment managers improve their accuracy in predicting the equity premium.