The Exact Distribution of the Hansen-Jagannathan Bound

Raymond Kan and Cesare Robotti
Working Paper 2008-9
February 2008

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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.

JEL classification: G12

Key words: Hansen-Jagannathan bound, exact distribution, no-arbitrage

Raymond Kan gratefully acknowledges financial support from the National Bank Financial of Canada. The authors also thank Wayne Ferson, Sergei Sarkissian, Tim Simin, Chu Zhang, Guofu Zhou, and seminar participants at the Federal Reserve Bank of Atlanta for helpful comments. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility.

Please address questions regarding content to Raymond Kan, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6, 416-978-4291,, or Cesare Robotti (contact author), Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470, 404-498-8543,

For further information, contact the Public Affairs Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, Georgia 30309-4470, 404-498-8020.