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**Hansen–Jagannathan bound** is a theorem in financial economics that says that the ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe ratio attained by any portfolio. This result applies, among others, the Cauchy–Schwarz inequality. The Hansen-Jagannathan (H-J) bound is a type of mean-variance frontier. The main contribution is that it allows us to say something about moments of the stochastic discount factor, which is unobservable, in terms of moments of returns, which can be (in principle) observed. Specifically, given the observed Sharpe ratio (say, around 0.4), the bound tells us that the SDF must be at least just as volatile.

## References

- Hansen, Lars Peter; Jagannathan, Ravi (1991). "Implications of Security Market Data for Models of Dynamic Economies" (PDF).
*Journal of Political Economy*.**99**(2): 225–262. doi:10.1086/261749. - Otrok, C., Ravikumar, B., Whiteman C.H. (2002). "Evaluating Asset-Pricing Models Using The Hansen-Jagannathan Bound: A Monte Carlo Investigation".
*Journal of Applied Econometrics*.**17**(2): 149–174. CiteSeerX 10.1.1.15.6332. doi:10.1002/jae.640.CS1 maint: multiple names: authors list (link)

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