Strategic Financial Management · Portfolio Performance Evaluation and Portfolio Revision

Sharpe and Treynor Ratio: Formulas and Numericals

Updated 11 October 2026 · Fact-checked

The Sharpe ratio is (portfolio return − risk-free return) ÷ standard deviation of the portfolio. The Treynor ratio is (portfolio return − risk-free return) ÷ beta. Both measure excess return per unit of risk. Calculate each for every portfolio, then rank them: the higher the ratio, the better the performance.

Understand Sharpe and Treynor Performance Measures

A portfolio with a high return is not necessarily well managed. It may simply have taken more risk. Risk-adjusted measures fix this by asking how much extra return you earned for each unit of risk taken.

The starting point is the excess return: portfolio return minus the risk-free return. The risk-free return is what you would have earned without taking any risk, for example the return on treasury bills. Only the part above it is a reward for taking risk.

The Sharpe ratio (reward-to-variability) divides excess return by the standard deviation, which is total risk. It includes both systematic and unsystematic risk. Use it when the portfolio is your whole investment, or when you compare portfolios that are not fully diversified.

The Treynor ratio (reward-to-volatility) divides excess return by beta, which is systematic risk only. It assumes unsystematic risk has been diversified away. Use it for a well-diversified portfolio, or for one of several portfolios held by an investor.

Because the two use different risk measures, they can rank the same portfolios differently. A portfolio with high unsystematic risk looks worse on Sharpe than on Treynor. Neither ratio is meaningful alone. It is useful only for comparing portfolios, or comparing a portfolio with the market.

Key rules to remember

Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Rp = portfolio return, Rf = risk-free return, σp = standard deviation of portfolio returns. Uses total risk.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; βp = portfolio beta. Uses systematic risk only. Meaningful for ranking when betas are positive.
Market benchmark Sharpe: Sharpe (market) = (Rm − Rf) ÷ σm; Compare the portfolio's Sharpe with this. Higher means it beat the market on a risk-adjusted basis.
Market benchmark Treynor: Treynor (market) = (Rm − Rf) ÷ 1 = Rm − Rf; The market beta is 1, so the market's Treynor ratio equals the market risk premium.
Decision rule: Higher ratio = better risk-adjusted performance; Rank portfolios in descending order of the ratio.

How to solve Sharpe and Treynor Performance Measures questions

Use this method for any question that asks you to compute, compare or rank portfolios using Sharpe or Treynor.

1Write down the risk-free rate Rf and the market return Rm if given. Convert all percentages to the same basis, for example annual.
2For each portfolio, note the return, standard deviation and beta. Check which risk measure the question gives or asks for.
3Compute excess return for each portfolio: Rp − Rf.
4Divide by standard deviation for Sharpe, and by beta for Treynor. Keep at least 3 or 4 decimals so ranks do not tie by rounding.
5Compute the same ratio for the market if a benchmark comparison is needed.
6Rank portfolios from highest to lowest ratio. Show the ranking in a small table of your own in the answer.
7Write a conclusion: name the best portfolio and say whether it beat the market. If Sharpe and Treynor ranks differ, explain it by unsystematic risk.

Quickest way: Excess return first, then one division per portfolio

When to use it: Use this in the MCQ section or when a question lists three or more portfolios and you have little time.

Subtract Rf from every return in one pass and write the excess returns in a column.
Divide each by the given risk figure (σ for Sharpe, β for Treynor).
Compare using rough decimals. Check the two nearest values more precisely only if they are close.
Pick the highest. For an MCQ on difference, remember: Sharpe uses total risk, Treynor uses systematic risk.

Common mistakes in Sharpe and Treynor Performance Measures

Dividing the portfolio return by risk without subtracting the risk-free rate.
Students rush and treat the ratio as return per unit of risk.
Fix: Always write Rp − Rf as the first step, before any division.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio.
The two formulas look alike and the names are easy to mix up.
Fix: Remember: Sharpe starts with S, for standard deviation. Treynor uses beta.
Ranking by the highest return instead of the highest ratio.
Return is the most visible number in the table.
Fix: Rank only on the computed ratio. A high-return portfolio can rank last.
Mixing units, such as a 12% return with a risk-free rate written as 0.06.
Data in the question is given in different forms.
Fix: Convert everything to percentages or everything to decimals before calculating.
Not explaining why Sharpe and Treynor ranks differ.
Students stop after the numbers.
Fix: Add one line: the difference comes from unsystematic risk, which Sharpe includes and Treynor ignores.

Worked examples

Example 1

The risk-free return is 6%. Three portfolios have the following data. Portfolio A: return 14%, standard deviation 12%, beta 1.1. Portfolio B: return 16%, standard deviation 20%, beta 1.2. Portfolio C: return 12%, standard deviation 8%, beta 0.6. Compute Sharpe and Treynor ratios and rank the portfolios on each.

Show the solution

Excess returns: A = 14 − 6 = 8%. B = 16 − 6 = 10%. C = 12 − 6 = 6%.
Sharpe ratios: A = 8 ÷ 12 = 0.667. B = 10 ÷ 20 = 0.500. C = 6 ÷ 8 = 0.750.
Treynor ratios: A = 8 ÷ 1.1 = 7.27. B = 10 ÷ 1.2 = 8.33. C = 6 ÷ 0.6 = 10.00.
Sharpe ranking: C (0.750), A (0.667), B (0.500).
Treynor ranking: C (10.00), B (8.33), A (7.27).
Ranks of A and B differ. B has a high standard deviation relative to its beta, so it carries a lot of unsystematic risk. Sharpe penalises this, Treynor does not.

Answer: Portfolio C is best on both measures. Sharpe: C 0.750 > A 0.667 > B 0.500. Treynor: C 10.00 > B 8.33 > A 7.27. A and B swap places because B has more unsystematic risk.

Example 2

The market return is 13%, its standard deviation is 15% and the risk-free return is 7%. Fund X, an equity fund, earned 15% with a standard deviation of 14% and a beta of 0.9. Did Fund X beat the market on a Sharpe and a Treynor basis?

Show the solution

Market Sharpe = (13 − 7) ÷ 15 = 6 ÷ 15 = 0.400.
Market Treynor = (13 − 7) ÷ 1 = 6.00.
Fund X excess return = 15 − 7 = 8%.
Fund X Sharpe = 8 ÷ 14 = 0.571.
Fund X Treynor = 8 ÷ 0.9 = 8.89.
Compare: 0.571 > 0.400 and 8.89 > 6.00.

Answer: Fund X beat the market on both measures. Sharpe 0.571 against 0.400, and Treynor 8.89 against 6.00. It earned more excess return per unit of both total and systematic risk.

Exam tips

Check what the question gives. If only standard deviation is given, the answer is Sharpe. If only beta is given, it is Treynor.
Show the excess return column explicitly. Marks are given for method even if the final division has a slip.
Always end with a ranking and a one-line recommendation. Numerical questions in SFM expect a decision.
In MCQs on the difference, the key points are total versus systematic risk, and diversified versus undiversified portfolios.
Keep three decimals for Sharpe and two for Treynor, and check ties carefully before ranking.

Practice questions from Portfolio Performance Evaluation and Portfolio Revision

Sharpe and Treynor Performance Measures in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Sharpe and Treynor Performance Measures: frequently asked questions

What is the difference between the Sharpe ratio and the Treynor ratio?

Both give excess return per unit of risk. Sharpe divides by standard deviation, which is total risk. Treynor divides by beta, which is only systematic risk. Treynor suits well-diversified portfolios.

Can the Sharpe and Treynor ratios rank portfolios differently?

Yes. If a portfolio has a lot of unsystematic risk, its standard deviation is high compared with its beta. It then ranks lower on Sharpe than on Treynor.

Which ratio should I use if the question gives both beta and standard deviation?

Compute the one the question asks for. If it does not say, compute both and comment on any difference in ranking. This earns full marks in most cases.

What does a higher Sharpe or Treynor ratio mean?

It means the portfolio earned more excess return for each unit of risk taken. Among portfolios, the highest ratio is the best performer on that measure.