Solving Models with External Habit

Habit utility has been the focus of a large and growing body of literature in financial economics. This study investigates ways of accurately and efficiently solving the Campbell and Cochrane (1999) external habit model. Solutions for this model based on a grid of values for the state variable are shown to converge as the grid becomes increasingly fine. Convergence is substantially faster if the price-dividend ratio is computed as a series of ``zero-coupon equity'' claims rather than as the fixed-point of the Euler equation. Fitting the model to the term structure as well as to equity moments (as in Wachter (2005)) also results in faster convergence.

Why is Long-Horizon Equity Less Risky? A Duration-Based Explanation of the Value Premium

This paper proposes a dynamic risk-based model that captures the high expected returns on value stocks relative to growth stocks, and the failure of the capital asset pricing model to explain these expected returns. To model the difference between value and growth stocks, we introduce a cross-section of long-lived firms distinguished by the timing of their cash flows. Firms with cash flows weighted more to the future have high price ratios, while firms with cash flows weighted more to the present have low price ratios. We model how investors perceive the risks of these cash flows by specifying a stochastic discount factor for the economy. The stochastic discount factor implies that shocks to aggregate dividends are priced, but that shocks to the time-varying price of risk are not. As long-horizon equity, growth stocks covary more with this time-varying price of risk than value stocks, which covary more with shocks to cash flows. When the model is calibrated to explain aggregate stock market behavior, we find that it can also account for the observed value premium, the high Sharpe ratios on value stocks relative to growth stocks, and the outperformance of value (and underperformance of growth) relative to the CAPM.

Does the Failure of the Expectations Hypothesis Matter for Long-Term Investors

We consider the consumption and portfolio choice problem of a long-run investor when the term structure is affine and when the investor has access to nominal bonds and a stock portfolio. In the presence of unhedgeable inflation risk, there exist multiple pricing kernels that produce the same bond prices, but a unique pricing kernel equal to the marginal utility of the investor. We apply our method to a three-factor Gaussian model with a time-varying price of risk that captures the failure of the expectations hypothesis seen in the data. We extend this model to account for time-varying expected inflation, and estimate the model with both inflation and term structure data. The estimates imply that the bond portfolio for the long-run investor looks very different from the portfolio of a mean-variance optimizer. In particular, the desire to hedge changes in term premia generates large hedging demands for long-term bonds.

What is the Chance that the Equity Premium Varies over Time? Evidence from Regressions on the Dividend-Price Ratio

We examine the evidence on excess stock return predictability in a Bayesian setting in which the investor faces uncertainty about both the existence and strength of predictability. When we apply our methods to the dividend-price ratio, we find that even investors who are quite skeptical about the existence of predictability sharply modify their views in favor of predictability when confronted by the historical time series of returns and predictor variables. Correctly taking into account the stochastic properties of the regressor has a dramatic impact on inference, particularly over the 2000-2005 period.

Bayesian Performance Evaluation

This paper proposes a Bayesian method of performance evaluation for investment managers. We begin with a flexible set of prior beliefs that can be elicited without any reference to probability distributions or their parameters. We then combine these prior beliefs with a general multi-factor model and derive an analytical solution for the posterior expectation of alpha', the intercept term from the model. This solution can be computed using only a few extra steps beyond maximum likelihood estimation and does not require a comprehensive or bias-free database. We then apply our methodology to a sample of domestic diversified equity mutual funds and ask what prior beliefs would imply zero investment in active managers?' To justify such a zero-investment strategy, we find that a mean-variance investor would need to believe that less than 1 out of every 100,000 managers has an expected alpha greater than 25 basis points per month. Overall, our analysis suggests that even when the average manager is expected to underperform passive benchmarks, it requires very strong prior beliefs to imply zero investment in managers with the best past performance.

Optimal Consumption and Portfolio Allocation under Mean-Reverting Returns: An Exact Solution for Complete Markets

This paper solves, in closed form, the optimal portfolio choice problem for an investor with utility over consumption under mean-reverting re- turns. Previous solutions either require approximations, numerical methods, or the assumption that the investor does not consume over his life time. This paper breaks the impasse by assuming that markets are complete. The solution leads to a new understanding of hedging demand and of the behavior of the approximate log-linear solution. The portfolio allocation takes the form of a weighted average and is shown to be analogous to duration for coupon bonds. Through this analogy, the notion of invest- ment horizon is extended to that of an investor who consumes at multiple points in time

Asset Allocation

This review article describes recent literature on asset allocation, covering both static and dynamic models. The article focuses on the bond--stock decision and on the implications of return predictability. In the static setting, investors are assumed to be Bayesian, and the role of various prior beliefs and specifications of the likelihood are explored. In the dynamic setting, recursive utility is assumed, and attention is paid to obtaining analytical results when possible. Results under both full and limited-information assumptions are discussed.

Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?

Why is the equity premium so high, and why are stocks so volatile? Why are stock returns in excess of government bill rates predictable? This paper proposes an answer to these questions based on a time-varying probability of a consumption disaster. In the model, aggregate consumption follows a normal distribution with low volatility most of the time, but with some probability of a consumption realization far out in the left tail. The possibility of this poor outcome substantially increases the equity premium, while time-variation in the probability of this outcome drives high stock market volatility and excess return predictability.

Portfolio and Consumption Decisions Under Mean-Reverting Returns: An Exact Solution for Complete Markets

This paper solves, in closed form, the optimal portfolio choice problem for an investor with utility over consumption under mean-reverting returns. Previous solutions either require approximations, numerical methods, or the assumption that the investor does not consume over his lifetime. This paper breaks the impasse by assuming that markets are complete. The solution leads to a new understanding of hedging demand and of the behavior of the approximate log-linear solution. The portfolio allocation takes the form of a weighted average and is shown to be analogous to duration for coupon bonds. Through this analogy, the notion of investment horizon is extended to that of an investor who consumes at multiple points in time