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Use a factor regression to build a multifactor benchmark, gauge how sensitive a portfolio is to those factors, and measure alpha relative to the benchmark.

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

The idea

A factor regression regresses the portfolio's excess return on the excess returns of a set of factors. Writing r for the portfolio's excess return and F for a factor's excess return, the model is r = α + b1 × F1 + b2 × F2 + … + ε. The slope coefficients, or factor loadings, measure the portfolio's sensitivity to each factor, and the intercept is alpha, the average return left unexplained by those exposures. The multifactor benchmark is the factor portfolio weighted by the estimated loadings, so it reproduces the portfolio's systematic risk. Alpha against that benchmark is the portfolio's excess return minus the sum of each loading times its factor return. Its reliability is judged by the t-statistic, which is alpha divided by its standard error, with about 2 as the usual threshold. Adding factors can shrink alpha, because apparent skill turns out to be factor exposure. Common errors are forgetting to subtract the risk-free rate and ignoring the sign of a negative loading.

Formulas

Alpha against a multifactor benchmark

α = (Rp − Rf) − (b1 × F1 + b2 × F2 + … + bk × Fk)

Rp
the portfolio's return
Rf
the risk-free rate
b1, b2, bk
the portfolio's estimated loading on each factor
F1, F2, Fk
each factor's return over the period

Subtract the risk-free rate from the portfolio's return first, and keep the sign of any negative loading or factor return.

t-statistic of a regression alpha

t = α ÷ SE

α
the estimated alpha
SE
its standard error

About 2 is the usual threshold; below it, zero alpha cannot be rejected however positive the estimate.

A practice question

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

In a multifactor regression of portfolio excess returns, what does the coefficient on a particular factor measure?

  1. The portfolio's average return that is not explained by any factor.Why not: This is the intercept, alpha, not a factor coefficient.
  2. The sensitivity of the portfolio's return to that factor. - correct
  3. The share of the portfolio's return variance explained by the whole regression.Why not: This is R², not a coefficient on one factor.
  4. The average annual return earned by the factor itself.Why not: This is the factor risk premium; the coefficient measures the portfolio's exposure to it.

The coefficient (loading) says how many units the portfolio's return moves per unit move in the factor. Multiplied by the factor return it gives that factor's contribution to the portfolio.

A harder one

A fund returned 11% in a year when the risk-free rate was 2%. Its estimated loadings are 1.1 on the market, 0.4 on SMB and -0.2 on HML. The factor premiums that year were 6% (market), 2% (SMB) and 3% (HML). What is the fund's alpha against its multifactor benchmark?

  1. 2.4%Why not: Uses only the market factor (CAPM) and ignores the SMB and HML contributions.
  2. 4.2%Why not: Does not subtract the risk-free rate, comparing the total return with the factor-based excess return.
  3. 2.2% - correct
  4. 1.0%Why not: Ignores the negative sign of the HML loading and adds 0.6% instead of subtracting it.

Benchmark excess return = 1.1 x 6 + 0.4 x 2 + (-0.2) x 3 = 6.6 + 0.8 - 0.6 = 6.8%. Fund excess return = 11 - 2 = 9%. Alpha = 9 - 6.8 = 2.2%.

The hardest tier

A fund returned 12% in a year when the risk-free rate was 1%. Its regression loadings are 1.2 on the market and 0.5 on SMB. That year the market excess return was 10% and the SMB return was -4%. What is the fund's alpha against the multifactor benchmark?

  1. 1.0% - correct
  2. 2.0%Why not: Compares the raw 12% return with the benchmark's excess return, forgetting to subtract the risk-free rate.
  3. -3.0%Why not: Adds the SMB contribution (+2.0%) the wrong way round: the negative factor return times a positive loading reduces the benchmark return.
  4. -1.0%Why not: Uses only the market factor and leaves out the SMB contribution entirely.

Benchmark excess return = 1.2 x 10 + 0.5 x (-4) = 12 - 2 = 10%. Fund excess return = 12 - 1 = 11%. Alpha = 11 - 10 = 1.0%. The SMB factor lowered the benchmark's return, so ignoring it or flipping its sign gives the wrong alpha.

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