Factor Exposure: Method

Factor Exposure: Method

What it does

A factor is a return series that many investments share, such as the return of the stock market above the risk-free rate, or the return of small companies minus large ones. The page measures how strongly the returns of one fund or portfolio have moved with each factor (its loadings, also called exposures or betas), how much average return is left over once those movements are accounted for (its alpha), and how much of the ups and downs the factors explain (R²). The tool is a regression: it describes a sample of past returns.

The method

Symbols: n is the number of periods and k the number of factors; y_t is the return of the fund in period t, minus the risk-free return when a column RF is pasted (the excess return); F_tj is the return of factor j; all in percent per period. Averages are written m, and s_uv = Σ (u_t − m_u)(v_t − m_v) / (n − 1) is the covariance of two series (the variance when u = v).

1. One factor. With x the first factor column, the least-squares line y = α + β x has

β = s_xy / s_xx,   α = m_y − β m_x,   R² = s_xy² / (s_xx s_yy)

When x is the market's excess return this is the regression of the capital asset pricing model, and α is often called Jensen's alpha.

2. Several factors. The model is y_t = a + b₁ F_t1 + … + b_k F_tk + e_t. Ordinary least squares chooses a and b to minimise the sum of squared errors SSE = Σ e_t². In matrix form, with X the n × (k + 1) table whose first column is 1, the coefficients solve X′X b = X′y. Then

s_e² = SSE / (n − k − 1),   standard error of b_j = s_e √[(X′X)⁻¹]_jj,   t_j = b_j / standard error

The p-value is the two-sided tail probability of t_j under Student's t distribution with n − k − 1 degrees of freedom, and the 95 % interval is b_j ± t_0.975 standard errors.

3. Fit. R² = 1 − SSE / SST with SST = Σ (y_t − m_y)²; adjusted R² = 1 − (1 − R²)(n − 1)/(n − k − 1). Annualised alpha = a × p and annualised residual volatility = s_e √p, with p the number of periods in a year.

4. Share of variance. Because the errors are uncorrelated with every factor in the sample, s_yy = Σ_j b_j c_j + SSE/(n − 1), where c_j is the covariance of factor j with y. The share attributed to factor j is b_j c_j / s_yy; the shares add up to R².

5. Newey-West standard errors (Advanced). The coefficients are unchanged; their covariance matrix is (X′X)⁻¹ S (X′X)⁻¹ with S = Σ_t e_t² x_tx_t′ + Σ_l=1…L (1 − l/(L + 1)) Σ_t e_t e_t−l (x_tx_t−l′ + x_t−lx_t′), x_t being row t of X. The lag L is the whole number below 4 (n/100)^2/9 unless entered. No small-sample factor is applied, and the p-values use the same t distribution.

6. Charts. Growth of 1 is the running product of (1 + y_t/100), and the same for the fitted return a + Σ b_j F_tj. Rolling loadings repeat step 2 on every window of 36 consecutive periods (drawn from 60 periods).

What it assumes

The loadings and the alpha are constant over the sample. The relation is linear. For the classical standard errors, the errors have one variance and are unrelated across periods; the Newey-West errors relax both. Annualising multiplies by the number of periods and ignores compounding. The fund's and the factors' returns cover the same periods and are in the same units (percent).

When it misleads

A loading is a description of the past. It says how the returns moved together in the sample, not how they will move, and a fund that changes what it holds changes its loadings. The rolling chart shows how much they moved within the sample.

Alpha is measured with a wide margin. Monthly returns are noisy, so the 95 % interval of an annualised alpha from five years of data is typically several points wide. An alpha inside that margin cannot be told from zero, and one outside it may reflect a factor that is missing from the table rather than skill.

The answer depends on the factors chosen. Adding or removing a factor changes the other loadings when the factors are correlated, and the variance shares are an accounting convention that can then be negative. A high R² means the factors track the returns closely; it says nothing about whether the returns were high or low.

Smoothed or stale prices understate loadings. Holdings that are valued infrequently show returns that lag the market, which lowers the measured loading and raises the measured alpha.

Worked example

The default table: 60 made-up monthly returns of "Fund" generated with known loadings (alpha 0.10 % per month, Mkt-RF 1.05, SMB 0.30, HML −0.20, random error of 1.2 %). It is illustrative, not market data.

One factor: m_y = 1.5415, m_x = 1.2788, s_xx = 17.2619, s_yy = 21.0234, s_xy = 18.0289. β = 18.0289 / 17.2619 = 1.0444; α = 1.5415 − 1.0444 × 1.2788 = 0.2058 % per month; R² = 18.0289² / (17.2619 × 21.0234) = 0.8957.

Three factors: a = 0.2207 (standard error 0.1634), Mkt-RF 1.0288 (0.0375), SMB 0.2896 (0.0494), HML −0.1052 (0.0621); R² = 0.9358, adjusted 1 − (1 − 0.9358) × 59/56 = 0.9323. The 95 % intervals (± 2.003 standard errors) contain the true values 0.10, 1.05, 0.30 and −0.20; the HML interval, −0.2295 to 0.0192, also contains zero, so 60 months are not enough to detect a loading of that size.

Annualised alpha = 0.2207 × 12 = 2.65 % per year (interval −1.28 to 6.58); residual volatility = 1.193 × √12 = 4.13 % per year. Share of variance of Mkt-RF = 1.0288 × 18.0289 / 21.0234 = 88.2 %, SMB 5.1 %, HML 0.3 %, unexplained 6.4 %.

Real factor returns are published free of charge in the Kenneth R. French Data Library (Tuck School of Business, Dartmouth College) in percent per period. Paste them beside the returns of the fund, one row per period, with a header row such as: Date, Fund, Mkt-RF, SMB, HML, RF.

A calculator for education and analysis, not investment advice.