Hansen–Jagannathan bound is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

financial economics Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade".William F. Sharpe"Financial Economics", in Its co ...

that says that the ratio of the

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

of a

stochastic discount factor The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow \tilde_i by the stochastic factor \tilde ...

to its mean exceeds the

Sharpe ratio In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for ...

attained by any

portfolio Portfolio may refer to: Objects * Portfolio (briefcase), a type of briefcase Collections * Portfolio (finance), a collection of assets held by an institution or a private individual * Artist's portfolio, a sample of an artist's work or a ...

. This result applies, among others, the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

. 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.

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External links

Hansen and Jagannathan bounds
Financial economics Financial ratios Statistical ratios {{finance-stub