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. Under the law of one price, Hansen and Jagannathan (1991) derive a lower volatility bound (unconstrained HJ-bound hereafter) that every valid stochastic discount factor (SDF) must satisfy. In addition, Hansen and Jagannathan (1991) propose a tighter volatility bound (constrained HJ-bound hereafter) that is applicable to nonnegative SDFs. The unconstrained HJ-bound has received wide attention in the literature. Examples include Snow (1991), Bekaert and Hodrick (1992), Ferson and Harvey (1992), Backus, Gregory, and Telmer (1993), Cecchetti, Lam, and Mark (1994), Burnside (1994), Heaton (1995), and Epstein and Zin (2001), among many others. In addition, Ferson and Siegel (2003) and Bekaert and Liu (2004) show how conditioning information can be used to optimally tighten the unconstrained HJ-bound; while Kan and Zhou (2006) tighten the unconstrained HJ-bound by making the SDF explicitly a function of a set of state variables. Although the constrained HJ-bound is sharper than the unconstrained HJ-bound and is theoretically appealing, the constrained HJ-bound has not received nearly as much attention as the unconstrained HJ-bound in empirical work. The few empirical papers that use the constrained bound besides Hansen and Jagannathan (1991) are Cecchetti, Lam, and Mark (1994), Burnside (1994), He and Modest (1995), Balduzzi and Kallal (1997), and Hagiwara and Herce (1997). In this paper, we provide a geometrical interpretation and the finite-sample distributions of both the unconstrained and constrained HJ-bounds.1 While there is a well-known mapping between the unconstrained HJ-bound and the mean-variance frontier of portfolio returns, the mapping between the constrained HJ-bound and the mean-variance frontier that we provide in this paper is new. We show that the linkage between the unconstrained HJ-bound and the mean-variance frontier also exists for the case of the constrained HJ-bound, except that we need to replace the mean and variance of the portfolio returns by the truncated mean and truncated variance of portfolio returns. As we mentioned above, the constrained HJ-bound has not been very popular in the literature. We suspect that the lack of popularity of the constrained HJ-bound is due to its computational difficulty. This is because when there are N assets, one has to solve N nonlinear equations in order to obtain the constrained HJ-bound. In this paper, we show that under the assumption that returns are multivariate normally distributed, the constrained HJ-bound has a very simple analytical expression. This analytical expression allows us to obtain a maximum likelihood estimator of the constrained HJ-bound which is simpler and more precise than the traditional nonparametric Hansen, Heaton, and Luttmer (1995) provide the asymptotic distributions of the unconstrained and constrained sample HJ-bounds.