mathematics

Kurtosis risk

Kurtosis risk 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 Kurtosis risk rather than just read about it. In short: In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations which have a tendency to occasionally be much further (in terms of number of standard deviations) from the average than is expected for a normal distribution. Overview Kurtosis risk applies to any kurtosis-related quantitative model that assumes the normal…

Key takeaways

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

Reference excerpt

In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations which have a tendency to occasionally be much further (in terms of number of standard deviations) from the average than is expected for a normal distribution.

Overview Kurtosis risk applies to any kurtosis-related quantitative model that assumes the normal distribution for certain of its independent variables when the latter may in fact have kurtosis much greater than does the normal distribution. Kurtosis risk is commonly referred to as "fat tail" risk. The "fat tail" metaphor explicitly describes the situation of having more observations at either extreme than the tails of the normal distribution would suggest; therefore, the tails are "fatter". Ignoring kurtosis risk will cause any model to understate the risk of variables with high kurtosis. For instance, Long-Term Capital Management, a hedge fund cofounded by Myron Scholes, ignored kurtosis risk to its detriment. After four successful years, this hedge fund had to be bailed out by major investment banks in the late 1990s because it understated the kurtosis of many financial securities underlying the fund's own trading positions.

Research by Mandelbrot Benoit Mandelbrot, a French mathematician, extensively researched this issue. He felt that the extensive reliance on the normal distribution for much of the body of modern finance and investment theory is a serious flaw of any related models including the Black–Scholes option model developed by Myron Scholes and Fischer Black, and the capital asset pricing model developed by William F. Sharpe. Mandelbrot explained his views and alternative finance theory in his book: The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward published on August 3, 2004.

See also Kurtosis Skewness risk Stochastic volatility Holy grail distribution Taleb distribution The Black Swan: The Impact of the Highly Improbable by Nassim Nicholas Taleb

Notes

References Mandelbrot, Benoit; Hudson, Richard L. (2004). The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books. ISBN 0-465-04355-0. Premaratne, G., Bera, A. K. (2000). Modeling Asymmetry and Excess Kurtosis in Stock Return Data. Office of Research Working Paper Number 00-0123, University of Illinois

Worked examples

Example 1 — a first encounter with Kurtosis risk

Start with the simplest possible case. Write down what Kurtosis risk 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 Kurtosis risk 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 Kurtosis risk 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 Kurtosis risk

In research: Kurtosis risk 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 Kurtosis risk 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: Kurtosis risk is common in secondary-school and first-year university syllabi. It links to neighbouring topics Investment, Mathematical finance, Normal distribution, so understanding it makes those chapters shorter.
In everyday life: Look for Kurtosis risk 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 Kurtosis risk in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kurtosis risk 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 Kurtosis risk out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kurtosis risk in simple terms?

Why does Kurtosis risk 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 Kurtosis risk?

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 Kurtosis risk.