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Apply the three-factor model of Fama and French to estimate asset returns, and describe how it is set up.

FRM® Part I · Foundations of Risk Management · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return · FRM1.FR.6.e

The idea

The Fama-French three-factor model explains excess returns with three factors. Excess return = a + bm × (Rm − Rf) + bs × SMB + bh × HML + e. SMB (small minus big) is the return on small-cap stocks minus large-cap stocks. HML (high minus low) is the return on high book-to-market (value) stocks minus low book-to-market (growth) stocks. Both are zero-cost long-short portfolios, built from a two-by-three sort on size and book-to-market. SMB averages the three small portfolios minus the three big ones. HML averages the two high portfolios minus the two low ones. A positive HML loading signals value behaviour, and a negative one growth. Expected return is Rf plus each loading times its premium. A fund's alpha is its average return less that figure. Remember to add back the risk-free rate.

Formulas

Fama-French three-factor expected return

E(R) = Rf + bm × MRP + bs × SMB + bh × HML

Rf
the risk-free rate
bm
the loading on the market factor
MRP
the market risk premium, E(Rm) − Rf
bs
the loading on the size factor
SMB
the small-minus-big premium
bh
the loading on the value factor
HML
the high-minus-low book-to-market premium

Add back Rf, since the regression explains excess returns. A fund's alpha is its average return minus this figure.

A practice question

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

In the Fama-French three-factor model, the SMB factor is the return on:

  1. a portfolio of large-capitalisation stocks minus a portfolio of small-capitalisation stocks.Why not: Reverses the order; SMB is small minus big.
  2. a portfolio of small-capitalisation stocks minus a portfolio of large-capitalisation stocks. - correct
  3. a portfolio of small-capitalisation stocks minus the risk-free rate.Why not: SMB is a long-short portfolio of stocks; it is not an excess return over the risk-free rate.
  4. a portfolio of high book-to-market stocks minus a portfolio of low book-to-market stocks.Why not: That is HML, the value factor, not the size factor.

SMB (small minus big) is the return on small-cap stocks minus large-cap stocks. HML (high minus low) is the return on high book-to-market (value) minus low book-to-market (growth) stocks. Both are zero-investment long-short portfolios.

A harder one

Over a sample period a fund earned an average return of 13.0% while the risk-free rate was 2.0%. Its Fama-French loadings are 1.2 on the market, 0.5 on SMB and 0.3 on HML. The average factor premiums were 6.0% for the market, 1.5% for SMB and 3.0% for HML. The fund's alpha relative to the three-factor model is closest to:

  1. 4.15%Why not: Subtracts the model's excess return from the fund's total return; either both must be excess returns or Rf must be added to the model.
  2. -2.15%Why not: Reverses the sign: alpha is realised return minus model-predicted return.
  3. 3.80%Why not: Uses a one-factor (market-only) benchmark and attributes the SMB and HML premiums to alpha.
  4. 2.15% - correct

Model-expected return = 2.0% + 1.2 x 6.0% + 0.5 x 1.5% + 0.3 x 3.0% = 10.85%. Alpha = 13.0% - 10.85% = 2.15%, the part of return not explained by market, size and value exposure.

The hardest tier

Using a two-by-three sort on size (small, big) and book-to-market (low, medium, high), a researcher records the monthly returns of the six portfolios: small-low 6%, small-medium 8%, small-high 11%, big-low 5%, big-medium 7%, big-high 9%. Following the Fama-French construction, SMB and HML are closest to:

  1. SMB -1.33%, HML 4.50%Why not: Computes SMB as big minus small; the factor is small minus big.
  2. SMB 1.33%, HML 4.50% - correct
  3. SMB 1.33%, HML 5.00%Why not: Calculates HML from the small-cap row only (high minus low), rather than averaging across both size groups.
  4. SMB 2.00%, HML 4.50%Why not: Builds SMB from the high book-to-market column only; SMB should average over all three book-to-market groups to remove the value effect.

SMB = average of the three small portfolios minus average of the three big: (6+8+11)/3 - (5+7+9)/3 = 8.33 - 7.00 = 1.33%. HML = average of the two high book-to-market portfolios minus the two low: (11+9)/2 - (6+5)/2 = 10.0 - 5.5 = 4.50%.

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