mathematics

Hansen–Jagannathan bound

Hansen–Jagannathan bound is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hansen–Jagannathan bound rather than just read about it. In short: 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.

Key takeaways

Hansen–Jagannathan bound belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hansen–Jagannathan bound to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hansen–Jagannathan bound from memory before moving on to harder problems.

Reference excerpt

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 plots as 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. S2CID 155085294. 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. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)CS1 maint: multiple names: authors list (link)

External links Hansen and Jagannathan bounds

Worked examples

Example 1 — a first encounter with Hansen–Jagannathan bound

Start with the simplest possible case. Write down what Hansen–Jagannathan bound claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hansen–Jagannathan bound before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hansen–Jagannathan bound ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hansen–Jagannathan bound

In research: Hansen–Jagannathan bound appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hansen–Jagannathan bound in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hansen–Jagannathan bound is common in secondary-school and first-year university syllabi. It links to neighbouring topics Finance stubs, Financial economics, Financial ratios, so understanding it makes those chapters shorter.
In everyday life: Look for Hansen–Jagannathan bound outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hansen–Jagannathan bound in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hansen–Jagannathan bound means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hansen–Jagannathan bound out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hansen–Jagannathan bound in simple terms?

Why does Hansen–Jagannathan bound matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hansen–Jagannathan bound?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hansen–Jagannathan bound.