Advanced Financial Management · Portfolio Management

Portfolio Performance Evaluation: Sharpe, Treynor, Jensen and Fama (CA Final AFM)

Updated 5 October 2026 · Fact-checked

Portfolio performance evaluation checks whether a portfolio's return was good for the risk taken. Sharpe = (Rp − Rf) ÷ σp, Treynor = (Rp − Rf) ÷ βp, Jensen alpha = Rp − [Rf + βp(Rm − Rf)]. Compute each, then rank portfolios: higher is better.

Understand Portfolio Performance Evaluation

A higher return does not mean a better portfolio. A manager may simply have taken more risk. Performance evaluation adjusts return for risk, so you can compare portfolios fairly.

All the measures start with the risk-free return (Rf). The portfolio's excess return is Rp − Rf. This is the reward earned for taking risk. The measures differ in how they define the risk.

Sharpe ratio divides excess return by total risk (standard deviation, σp). Use it when the portfolio is your whole investment, or when comparing portfolios that are not well diversified. Treynor ratio divides excess return by systematic risk (beta, βp). It suits a well-diversified portfolio, where unsystematic risk has been diversified away.

Jensen's alpha works differently. It compares the actual return with the return CAPM says the portfolio should earn for its beta. A positive alpha means the manager beat the benchmark for that risk. A negative alpha means the manager underperformed. Sharpe and Treynor are ratios and are used to rank. Alpha is an absolute return difference.

Fama's decomposition splits the excess return into parts, to show where the manager added value. The total excess return (Rp − Rf) is broken into the return for the risk the investor chose to accept, (σp ÷ σm)(Rm − Rf), and the manager's selectivity. Selectivity is the extra return over what the portfolio's own total risk justifies.

Diversification is the extra return required for incomplete diversification, (σp ÷ σm − βp)(Rm − Rf). Keep these two relations separate:

Jensen's alpha = Rp − Rf − βp(Rm − Rf) = Selectivity + Diversification. Alpha uses beta only, so it includes the diversification term.
Net selectivity = Selectivity − Diversification. It is the part of the manager's selection skill left after covering the extra return that incomplete diversification demanded.

A positive net selectivity means the manager's selection more than paid for the lack of diversification. A negative net selectivity means the selection return was less than the extra return required for the incomplete diversification. The parts are built on the capital market line and the security market line.

Key rules to remember

Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Uses total risk. Higher is better. Related to the Capital Market Line.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Uses systematic risk. Higher is better. Related to the Security Market Line.
Jensen's alpha: α = Rp − [Rf + βp(Rm − Rf)]; Positive means outperformance. Zero means in line with CAPM. Negative means underperformance.
CAPM expected return: Expected Rp = Rf + βp(Rm − Rf); This is the benchmark return used inside Jensen's alpha.
Excess return: Excess return = Rp − Rf; The numerator of Sharpe and Treynor.
Fama: selectivity: Selectivity = (Rp − Rf) − (σp ÷ σm)(Rm − Rf); Return for the investor's risk = (σp ÷ σm)(Rm − Rf). Selectivity is the excess return left after deducting it.
Fama: selectivity split: Diversification = (σp ÷ σm − βp) × (Rm − Rf); Net selectivity = Selectivity − Diversification; Two separate relations. Jensen's alpha = Selectivity + Diversification (in the worked example, 1.4 = 0.5 + 0.9). Net selectivity = Selectivity − Diversification (0.5 − 0.9 = −0.4), so it is not equal to Jensen's alpha. A negative net selectivity means the selection return was less than the extra return required for incomplete diversification.

How to solve Portfolio Performance Evaluation questions

Use this method for any question that asks you to evaluate or rank portfolios.

1List the data for each portfolio: return, standard deviation, beta. Note Rf and the market return (Rm) and σm if given.
2Check whether returns are given for the same period (annual, or the same time unit). Convert if they are not.
3Compute the excess return Rp − Rf for every portfolio.
4Compute the measure the question asks for: Sharpe, Treynor, Jensen alpha, or all three. Keep the same decimal places for all portfolios.
5If asked for Fama's decomposition, compute the return for the investor's risk, then selectivity, then the diversification part, then net selectivity.
6Rank the portfolios on each measure. Higher is better.
7State the conclusion in a line. Mention if rankings differ and say why: Sharpe uses total risk and Treynor and Jensen use beta.
8Include the market in the comparison if the question gives market data. The market's Treynor ratio is Rm − Rf (beta is 1).

Quickest way: Compute excess return once, then divide

When to use it: Use this when the question gives several portfolios and asks for the best one under time pressure.

Write one column of Rp − Rf for all portfolios.
Divide that column by σ for Sharpe and by β for Treynor.
For Jensen, compute βp(Rm − Rf) next to each portfolio, add Rf, and subtract from Rp.
Rank, then write the conclusion. Do not redo the working in words.

Common mistakes in Portfolio Performance Evaluation

Using Rp instead of Rp − Rf in the numerator
Students rush and treat the ratio as return per unit of risk.
Fix: Always write the excess return as a separate first step before dividing.
Dividing by beta in Sharpe or by standard deviation in Treynor
The two formulas look alike and the names get confused.
Fix: Remember: Sharpe uses Standard deviation (S for S). Treynor uses beta, the systematic risk.
Treating Jensen's alpha as a ratio and ranking on the wrong basis
Students compare it with Sharpe and Treynor values directly.
Fix: Alpha is a percentage return difference. Rank by higher alpha, but do not compare it numerically with the ratios.
Mixing percentages and decimals
Returns given as 12% are used as 12 in one place and 0.12 in another.
Fix: Use one form throughout. Percentage form is fine, as long as the ratios are compared consistently.
Giving no conclusion after the numbers
Students think the calculation is the answer.
Fix: Write which portfolio is best on each measure and the reason for any difference in ranking.
Using the wrong market data in Fama's decomposition
Students use βp where σp ÷ σm is needed, or the reverse.
Fix: The risk the investor accepts uses σp ÷ σm. The diversification term uses (σp ÷ σm − βp).

Worked examples

Example 1

The risk-free rate is 6% and the market return is 13%. Portfolio A earned 15% with standard deviation 10% and beta 1.2. Portfolio B earned 14% with standard deviation 7% and beta 0.9. Evaluate both portfolios using Sharpe, Treynor and Jensen's alpha.

Show the solution

Excess return of A = 15 − 6 = 9%. Excess return of B = 14 − 6 = 8%.
Sharpe A = 9 ÷ 10 = 0.90. Sharpe B = 8 ÷ 7 = 1.1429.
Treynor A = 9 ÷ 1.2 = 7.50. Treynor B = 8 ÷ 0.9 = 8.89.
CAPM return for A = 6 + 1.2 × (13 − 6) = 6 + 8.4 = 14.4%. Alpha A = 15 − 14.4 = 0.6%.
CAPM return for B = 6 + 0.9 × 7 = 6 + 6.3 = 12.3%. Alpha B = 14 − 12.3 = 1.7%.
B ranks higher on all three measures.

Answer: Sharpe: A 0.90, B 1.14. Treynor: A 7.50, B 8.89. Jensen's alpha: A 0.6%, B 1.7%. Portfolio B performed better on every measure, so B is preferred.

Example 2

Rf is 5%, market return is 11% and market standard deviation is 12%. A fund returned 13% with standard deviation 15% and beta 1.1. Using Fama's decomposition, find the return for the investor's risk, the selectivity, the diversification component and the net selectivity.

Show the solution

Total excess return = Rp − Rf = 13 − 5 = 8%.
Market premium = 11 − 5 = 6%.
Return for investor's risk = (σp ÷ σm) × (Rm − Rf) = (15 ÷ 12) × 6 = 1.25 × 6 = 7.5%.
Selectivity = 8 − 7.5 = 0.5%.
Diversification = (σp ÷ σm − βp) × (Rm − Rf) = (1.25 − 1.1) × 6 = 0.15 × 6 = 0.9%.
Net selectivity = Selectivity − Diversification = 0.5 − 0.9 = −0.4%.
Jensen's alpha = Rp − [Rf + βp(Rm − Rf)] = 13 − (5 + 1.1 × 6) = 13 − 11.6 = 1.4%. Check: Selectivity + Diversification = 0.5 + 0.9 = 1.4%.
Interpretation: net selectivity of −0.4% means the manager's stock selection (0.5%) was less than the extra return required for the incomplete diversification (0.9%).

Answer: Return for the investor's risk is 7.5%. Selectivity is 0.5%. Diversification is 0.9%. Net selectivity is −0.4%, so the selection skill did not compensate for the incomplete diversification. Jensen's alpha is 1.4% (0.5% + 0.9%).

Exam tips

Write the formula first. Marks are given for the formula, working and conclusion.
Check whether the question asks for a ranking. If so, say which portfolio is best on each measure.
If the market data is given, include the market in the comparison. Its Treynor ratio is Rm − Rf, and its Sharpe is (Rm − Rf) ÷ σm.
Explain a difference in ranking in one line. A portfolio with high unsystematic risk can rank well on Treynor and poorly on Sharpe.
Keep all numbers to two decimal places and state units.

Practice questions from Portfolio Management

Portfolio Performance Evaluation in other exams

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

Portfolio Performance Evaluation: frequently asked questions

What is the difference between Sharpe ratio and Treynor ratio?

Both divide excess return by a risk measure. Sharpe uses total risk (standard deviation). Treynor uses systematic risk (beta). Use Sharpe for undiversified portfolios and Treynor for well-diversified ones.

How do I calculate Jensen's alpha?

First find the CAPM return: Rf + β × (Rm − Rf). Then subtract it from the actual portfolio return. A positive result means the portfolio beat what its beta justified.

Can Sharpe and Treynor rank portfolios differently?

Yes. If a portfolio holds a lot of unsystematic risk, its standard deviation is high but its beta may be low. It can then look good on Treynor and weak on Sharpe.

What does a negative Jensen's alpha mean?

The portfolio earned less than CAPM says it should for its level of systematic risk. The manager underperformed on a risk-adjusted basis.