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Using Fama-French as the worked example, account for the role that time-varying risk factors play inside a multifactor return model.

FRM® Part II · Risk Management and Investment Management · Factors · FRM2.IM.2.d

The idea

The Fama-French model explains a portfolio's excess return with three factor returns. MKT is the market return minus the risk-free rate, SMB is the small-cap minus large-cap return, and HML is the high book-to-market minus low book-to-market return. Written out for one portfolio, the model says r = α + βmkt × MKT + βSMB × SMB + βHML × HML + ε. Expected excess return is the sum of each beta times that factor's premium, and alpha is whatever is left. The factors are time-varying: their premia fluctuate and can be negative for years, so an average estimated from a short sample is noisy. Let σ be a factor's annual volatility and n the number of years of data. Then the standard error of the average premium is SE = σ ÷ √n. Betas can drift too, and a constant-beta regression misstates alpha when exposure moves with factor returns. Judge alpha only after controlling for factors, and expect long dry spells even for sound factors.

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

Factor premia vary over time, so a premium averaged over a short sample is noisy. Keep the sign of a negative loading.

Standard error of an average factor premium

SE = σ ÷ √n

σ
the population standard deviation
n
the sample size

Assumes independent annual returns. Premia are small relative to their volatility, so the error shrinks only slowly with the sample.

Years of data needed to establish a factor premium

n = (t × σ ÷ μ)²

t
the t-statistic wanted, usually 2
σ
the factor's annual volatility
μ
the factor's true average annual premium

From t = μ × √n ÷ σ. Remember to square: the answer is usually decades.

A practice question

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

In the Fama-French three-factor model, SMB and HML are defined respectively as the return on:

  1. small-capitalization stocks minus large-capitalization stocks, and high book-to-market stocks minus low book-to-market stocks. - correct
  2. large-capitalization stocks minus small-capitalization stocks, and low book-to-market stocks minus high book-to-market stocks.Why not: Reverses both spreads, so signs of the factor premia and betas would flip.
  3. small-capitalization stocks minus large-capitalization stocks, and past winners minus past losers.Why not: Confuses HML with the momentum factor, which sorts on past returns.
  4. high book-to-market stocks minus low book-to-market stocks, and small-capitalization stocks minus large-capitalization stocks.Why not: Swaps the two definitions: HML is the value spread and SMB is the size spread.

SMB = small minus big (size); HML = high minus low book-to-market (value). Together with the market excess return they form the three factors.

A harder one

In a Fama-French three-factor model the risk-free rate is 2.0%, the market premium 5.0%, the SMB premium 2.0% and the HML premium 3.0%. A fund has betas of 1.1 on the market, 0.4 on SMB and -0.3 on HML. Its expected return is closest to:

  1. 9.2%Why not: Flips the sign of the negative HML contribution (adds 0.9 instead of subtracting it).
  2. 5.4%Why not: Omits the risk-free rate, reporting the expected excess return only.
  3. 8.3%Why not: Leaves out the HML term entirely, ignoring the negative value exposure.
  4. 7.4% - correct

Expected return = 2.0 + 1.1(5.0) + 0.4(2.0) + (-0.3)(3.0) = 2.0 + 5.5 + 0.8 - 0.9 = 7.4%.

The hardest tier

The value factor has a true average premium of 4% a year with annual volatility of 12%, and returns are independent across years. Roughly how many years of data are needed before the sample mean is two standard errors above zero (t-statistic of 2)?

  1. 6 yearsWhy not: Computes t x sigma / mu = 6 but forgets that the sample size is the square of that ratio.
  2. 9 yearsWhy not: Squares sigma/mu but uses a t-statistic of 1 instead of 2.
  3. 18 yearsWhy not: Multiplies t by (sigma/mu)² instead of squaring t x sigma/mu; the factor 2 should be squared to 4.
  4. 36 years - correct

Standard error = sigma/sqrt(n); t = mu x sqrt(n)/sigma = 2, so n = (2 x 12/4)² = 36 years. Because factor premia are small relative to their volatility, establishing them takes decades.

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