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Work out Jensen's alpha, M², the Treynor ratio and the Sharpe ratio and explain what each shows

CFA® Level I · Portfolio Construction · Portfolio Risk and Return: Part II · PM.8.a

The idea

Four measures adjust return for risk, and they differ in which risk they count. The Sharpe ratio divides excess return, Rp minus Rf, by total standard deviation, so it suits a portfolio that is the investor's whole holding. The Treynor ratio divides the same excess return by beta, counting only systematic risk, which is right when the portfolio is one part of a well-diversified whole. M² rescales the portfolio to the market's standard deviation, giving Rf plus the Sharpe ratio times the market standard deviation, so the result reads as a percentage return directly comparable with the market's. Jensen's alpha is the actual return minus the CAPM return, Rp minus [Rf plus beta times (Rm minus Rf)]. A positive alpha means the manager beat what the portfolio's beta alone would have earned. Sharpe and M² always rank portfolios identically, whereas Sharpe and Treynor can disagree when a portfolio carries a lot of diversifiable risk.

Formulas

Sharpe ratio

Sharpe = (Rp − rf) ÷ σp

Rp
the portfolio's return
rf
the risk-free rate
σp
the standard deviation of the portfolio's return

Return earned per unit of total risk, and the slope of the capital allocation line drawn through that portfolio. The steepest line is the best one available, which is what picking the tangency portfolio means.

M squared

M² = rf + Sharpe ratio × σm

rf
the risk-free rate
Sharpe ratio
the portfolio's excess return divided by its standard deviation
σm
the standard deviation of the market

The return the portfolio would have earned if it had the market's risk, mixed with cash or leveraged to get there. Compare it with the market's return: it ranks portfolios exactly as the Sharpe ratio does.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Rp
the portfolio's return
Rf
the risk-free rate
βp
the portfolio's beta

Excess return per unit of systematic risk. Dividing by σp instead, or forgetting to subtract Rf, is the common slip.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Rp
the portfolio's return
Rf
the risk-free rate
βp
the portfolio's beta
Rm
the market's return over the same period

Return above what the CAPM requires for the portfolio's beta. Positive means outperformance for the systematic risk taken.

A practice question

From the question bank, with the reason each wrong option is wrong.

A portfolio returned 11% with a standard deviation of 16% and a beta of 1.2. The risk-free rate was 3%. What is its Sharpe ratio?

  1. 0.50 - correct
  2. 0.69Why not: Divides the whole 11% return by standard deviation without first subtracting the 3% risk-free rate.
  3. 6.67Why not: Divides the 8% excess return by beta, which is the Treynor ratio rather than the Sharpe ratio.

Sharpe = (11% - 3%) / 16% = 8 / 16 = 0.50. Beta is not used, because the Sharpe ratio counts total risk.

A harder one

A portfolio returned 12% with a standard deviation of 20%. The market returned 9% with a standard deviation of 15%, and the risk-free rate was 2%. What is the portfolio's M² (risk-adjusted return)?

  1. 15.33%Why not: Scales the excess return by 20/15 instead of 15/20, levering the portfolio up rather than down to market risk.
  2. 7.50%Why not: Rescales the 10% excess return correctly but forgets to add the 2% risk-free rate back.
  3. 9.50% - correct

Sharpe = (12 - 2) / 20 = 0.5. M² = 2% + 0.5 x 15% = 9.50%, which exceeds the market's 9%, so the portfolio outperformed on a risk-adjusted basis by 0.50%.

The hardest tier

A portfolio returned 9% with a standard deviation of 12%. The market returned 10% with a standard deviation of 18%, and the risk-free rate was 3%. What is the portfolio's M² alpha relative to the market?

  1. -1.00%Why not: Compares raw returns, 9% against 10%, without adjusting the portfolio to the market's level of risk.
  2. -3.00%Why not: Scales the 6% excess return by 12/18 instead of 18/12, which shrinks rather than levers up the portfolio.
  3. +2.00% - correct

Sharpe = (9 - 3) / 12 = 0.5. M² = 3% + 0.5 x 18% = 12%. M² alpha = 12% - 10% = +2.00%: despite the lower raw return, the portfolio beat the market per unit of total risk.

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