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

CFA® Level I · Portfolio Management · 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. M-squared 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 M-squared always rank portfolios identically, whereas Sharpe and Treynor can disagree when a portfolio carries a lot of diversifiable risk.

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 M-squared (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. M-squared = 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

The risk-free rate is 3%. Portfolio A returned 12% with a standard deviation of 24% and a beta of 0.8; Portfolio B returned 14% with a standard deviation of 20% and a beta of 1.2. An investor will hold only one of them as her entire portfolio. Which should she prefer?

  1. A, with a Treynor ratio of 11.25 against 9.17Why not: Treynor ignores the unsystematic risk that a sole holding leaves undiversified, so it is the wrong measure here.
  2. B, with a Sharpe ratio of 0.55 against 0.38 — correct
  3. A, as its lower beta makes it the safer holdingWhy not: Beta misses A's large diversifiable risk; as a sole holding A is the riskier, with a 24% standard deviation against 20%.

As a stand-alone holding, total risk matters, so use Sharpe: A = 9/24 = 0.375, B = 11/20 = 0.55. Treynor would favour A (9/0.8 = 11.25 vs 11/1.2 = 9.17) because A carries heavy diversifiable risk that beta does not see.

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