mathematics

Treynor ratio

Treynor ratio 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 Treynor ratio rather than just read about it. In short: In finance, the Treynor reward-to-volatility model (sometimes called the reward-to-volatility ratio or Treynor measure), named after American economist Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has no risk that can be diversified (e.g., Treasury bills or a completely diversified portfolio), per unit of market risk assumed.

Key takeaways

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

Reference excerpt

In finance, the Treynor reward-to-volatility model (sometimes called the reward-to-volatility ratio or Treynor measure), named after American economist Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has no risk that can be diversified (e.g., Treasury bills or a completely diversified portfolio), per unit of market risk assumed. The Treynor ratio relates excess return over the risk-free rate to the additional risk taken; however, systematic risk is used instead of total risk. The higher the Treynor ratio, the better the performance of the portfolio under analysis.

Formula

T = r i − r f β i , {\displaystyle T={\frac {r_{i}-r_{f}}{\beta _{i}}},}

where T {\textstyle T} is the Treynor ratio, r i {\textstyle r_{i}} is the return of portfolio i {\textstyle i} , r f {\textstyle r_{f}} is the risk free rate, and β i {\textstyle \beta _{i}} is the beta of portfolio i {\textstyle i} .

Example Taking the equation detailed above, let us assume that the expected portfolio return is 20%, the risk free rate is 5%, and the beta of the portfolio is 1.5. Substituting these values, we get the following:

T = 0.2 − 0.05 1.5 = 0.1. {\displaystyle T={\frac {0.2-0.05}{1.5}}=0.1.}

Limitations Like the Sharpe ratio, the Treynor ratio (T) does not quantify the value added, if any, of active portfolio management. It is a ranking criterion only. A ranking of portfolios based on the Treynor Ratio is only useful if the portfolios under consideration are sub-portfolios of a broader, fully diversified portfolio. If this is not the case, portfolios with identical systematic risk, but different total risk, will be rated the same. But the portfolio with a higher total risk is less diversified and therefore has a higher unsystematic risk which is not priced in the market. An alternative method of ranking portfolio management is Jensen's alpha, which quantifies the added return as the excess return above the security market line in the capital asset pricing model. As these two methods both determine rankings based on systematic risk alone, they will rank portfolios identically.

See also Bias ratio (finance) Hansen-Jagannathan bound Jensen's alpha Modern portfolio theory Modigliani risk-adjusted performance Omega ratio Sharpe ratio Sortino ratio Upside potential ratio V2 ratio

References

Worked examples

Example 1 — a first encounter with Treynor ratio

Start with the simplest possible case. Write down what Treynor ratio 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 Treynor ratio 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 Treynor ratio 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 Treynor ratio

In research: Treynor ratio 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 Treynor ratio 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: Treynor ratio is common in secondary-school and first-year university syllabi. It links to neighbouring topics Finance stubs, Financial ratios, Investment indicators, so understanding it makes those chapters shorter.
In everyday life: Look for Treynor ratio 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 Treynor ratio in 20 minutes

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

Frequently asked questions

What is Treynor ratio in simple terms?

Why does Treynor ratio 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 Treynor ratio?

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 Treynor ratio.