The exact distribution of the Hansen-Jagannathan bound

Under the assumption of multivariate normality of asset returns, this paper presents a geometrical interpretation and the finite-sample distributions of the sample Hansen-Jagannathan (1991) bounds on the variance of admissible stochastic discount factors, with and without the nonnegativity constraint on the stochastic discount factors. In addition, since the sample Hansen-Jagannathan bounds can be very volatile, we propose a simple method to construct confidence intervals for the population Hansen-Jagannathan bounds. Finally, we show that the analytical results in the paper are robust to departures from the normality assumption.

The Exact Distribution of the Hansen-Jagannathan Bound

Under the assumption of multivariate normality of asset returns, this paper presents a geometric interpretation and the finite-sample distributions of the sample Hansen–Jagannathan bounds on the variance of admissible stochastic discount factors, with and without the nonnegativity constraint on the stochastic discount factors. In addition, since the sample Hansen–Jagannathan bounds can be very volatile, we propose a simple method to construct confidence intervals for the population Hansen–Jagannathan bounds. Finally, we show that the analytical results in the paper are robust to departures from the normality assumption

Stochastic Discount Factor Bounds with Conditioning Information

Hansen and Jagannathan (HJ, 1991) describe restrictions on the volatility of stochastic discount factors (SDFs) that price a given set of asset returns. This paper compares the sampling properties of different versions of HJ bounds that use conditioning information in the form of a given set of lagged instruments. HJ describe one way to use conditioning information. Their approach is to multiply the original returns by the lagged variables, and much of the asset pricing literature to date has followed this ihmultiplicativel. approach. We also study two versions of optimized HJ bounds with conditioning information. One is from Gallant, Hansen and Tauchen (1990) and the second is based on the unconditionally-efficient portfolios derived in Ferson and Siegel (2000). We document finite-sample biases in the HJ bounds, where the biased bounds reject asset-pricing models too often. We provide useful correction factors for the bias. We also evaluate the asymptotic standard errors for the HJ bounds, from Hansen, Heaton and Luttmer (1995).

Evaluating Asset-Pricing Models Using The Hansen-Jagannathan Bound: A Monte Carlo Investigation

We conduct Monte Carlo experiments to examine whether the bound proposed by Hansen and Jagannathan (1991) is a useful device for evaluating asset pricing models. Specifically, we use recently developed statistical tests, which are based on a 'distance' between the model and the Hansen-Jagannathan bound, to compute the rejection rates of true models. We provide finite-sample critical values for asset pricing models with time separable preferences, and show how they depend upon nuisance parameters—risk aversion and the rate of time preference. Further, we show that the finite-sample distribution of the test statistic associated with the risk-neutral case is extreme, in the sense that critical values based on this distribution will deliver type I errors no larger than intended—regardless of risk aversion or the rate of time preference. Extending the analysis to accommodate other preferences, we show that in the state non-separable case, the small-sample distributions of the test statistics are influenced significantly by the degree of intertemporal substitution, but not by attitudes toward risk. For habit formation preferences, the small-sample distributions are strongly influenced by the habit parameter. However, the maximal-size critical values for time-separable preferences are appropriate for habit formation as well as state non-separable preferences. We conclude that with these critical values the HJ bound is indeed a useful evaluation device. We then use the critical values to evaluate three asset pricing models using U.S. data. We find evidence against the time-separable model and mixed evidence on the remaining two models.

On the Hansen-Jagannathan distance with a no-arbitrage constraint

We provide an in-depth analysis of the theoretical and statistical properties of the Hansen-Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. We show that for stochastic discount factors (SDF) that are spanned by the returns on the test assets, testing the equality of HJ distances with no-arbitrage constraints is the same as testing the equality of HJ distances without no-arbitrage constraints. A discrepancy can exist only when at least one SDF is a function of factors that are poorly mimicked by the returns on the test assets. Under a joint normality assumption on the SDF and the returns, we derive explicit solutions for the HJ distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the SDF parameters in the case of linear SDFs. This solution allows us to show that nontrivial differences between HJ distances with and without no-arbitrage constraints can arise only when the volatility of the unspanned component of an SDF is large and the Sharpe ratio of the tangency portfolio of the test assets is very high. Finally, we present the appropriate limiting theory for estimation, testing, and comparison of SDFs using the HJ distance with a no-arbitrage constraint.

Multifactor Models Do Not Explain Deviations from the CAPM

A number of studies have presented evidence rejecting the validity of the Capital Asset Pricing Model (CAPM). This evidence has spawned research into possible explanations. These explanations can be divided into two main categories - the risk based alternatives and the nonrisk based alternatives. The risk based category includes multifactor asset pricing models developed under the assumptions of investor rationality and perfect capital markets. The nonrisk based category includes biases introduced in the empirical methodology, the existence of market frictions, or explanations arising from the presence of irrational investors. The distinction between the two categories is important for asset pricing applications such as estimation of the cost of capital. This paper proposes to distinguish between the two categories using ex ante analysis. A framework is developed showing that ex ante one should expect that CAPM deviations due to missing risk factors will be very difficult to statistically detect. In contrast, deviations resulting from nonrisk based sources will be easy to detect. Examination of empirical results leads to the conclusion that the risk based alternatives is not the whole story for the CAPM deviations. The implication of this conclusion is that the adoption of empirically developed multifactor asset pricing models may be premature.

Reverse Engineering the Yield Curve

Prices of riskfree bonds in any arbitrage-free environment are governed by a pricing kernel: given a kernel, we can compute prices of bonds of any maturity we like. We use observed prices of multi-period bonds to estimate, in a log-linear theoretical setting, the pricing kernel that gave rise to them. The high-order dynamics of our estimated kernel help to explain why first-order, one-factor models of the term structure have had difficulty reconciling the shape of the yield curve with the persistence of the short rate. We use the estimated kernel to provide a new perspective on Hansen-Jagannathan bounds, the price of risk, and the pricing of bond options and futures.