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Account for how the choice of benchmark shifts measured alpha and set out what makes a benchmark good for measuring it.

FRM® Part II · Risk Management and Investment Management · Alpha (and the Low-Risk Anomaly) · FRM2.IM.3.c

The idea

Alpha is always measured against something, so changing the benchmark changes the measured alpha even though the portfolio is untouched. Suppose a small-cap value manager is judged against a broad large-cap index, and small value stocks happen to outperform. The measured alpha then contains the reward for the size and value exposures, which a style-matched benchmark would strip out. A good benchmark is specified in advance, unambiguous and transparent. It is investable at low cost, measurable at the same frequency as the portfolio, and matched to the manager's style and risk, with a beta close to the portfolio's. It must also be free of ex post selection, because choosing, after the fact, whichever index flatters the manager biases alpha upward. Exam point: alpha is benchmark-relative, and a mismatched benchmark credits factor returns to skill.

Formulas

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

Adjusts the benchmark for the fund's beta, so a high-beta fund is not credited with return it earned simply by carrying more market risk.

A practice question

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

Which description fits a good benchmark for measuring alpha?

  1. It is the broadest available index, whatever the manager's style, because breadth makes it neutral.Why not: Breadth does not make it appropriate; a benchmark that ignores the manager's style misattributes factor returns to alpha.
  2. It is chosen after the period ends as the index that best explains the manager's returns.Why not: Ex post selection violates the requirement that the benchmark be specified in advance and biases alpha.
  3. It is specified in advance, transparent, investable and matched to the manager's investment style and risk. - correct
  4. It is the index that gives the manager the highest measured alpha, since this credits full skill.Why not: Selecting the benchmark by its alpha overstates skill; the benchmark must be set independently of results.

A good benchmark is unambiguous, specified in advance, investable, measurable and appropriate to the manager's style. Picking a benchmark after the fact, or to maximise alpha, invalidates the measure.

A harder one

A manager holds about 40 large-capitalisation US value stocks, rebalanced annually, and the client wants a benchmark for alpha. Which candidate is best?

  1. The broadest global equity index, because it is the most diversified and so the most neutral.Why not: Diversification does not give a matching style or risk; the index would credit the value tilt as alpha.
  2. A published, investable large-capitalisation value index fixed in advance. - correct
  3. The three-month Treasury bill return, because alpha is the return above the risk-free rate.Why not: Confuses alpha with excess return; a T-bill has none of the equity risk the manager takes.
  4. A custom blend chosen after reviewing the year's returns to fit what the manager actually delivered.Why not: Chosen ex post to fit the results, so it would bias alpha toward zero or whatever the fit dictates.

The benchmark should match the manager's style and risk, be investable and be set before the performance period. A style-matched large-cap value index meets all of these; the others fail on style, risk or timing.

The hardest tier

A fund with a beta of 1.3 returned 14% in a year when the risk-free rate was 3% and the benchmark index returned 10%. Adjusting the benchmark for the fund's beta, which is closest to the fund's alpha?

  1. 4.0%Why not: Ignores beta altogether and subtracts the unadjusted benchmark return (14% - 10%), which credits the fund's extra market risk as skill.
  2. 1.0%Why not: Scales the full benchmark return by beta without adding back the risk-free rate (14 - 1.3 x 10).
  3. 1.9% - correct
  4. -2.0%Why not: Scales the benchmark's total return by beta instead of its 7% excess return, then subtracts the risk-free rate.

Beta-adjusted benchmark = 3 + 1.3 x (10 - 3) = 12.1%. Alpha = 14.0 - 12.1 = 1.9%. Risk-adjusting the benchmark removes the return the fund earned simply by carrying more beta.

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