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Factor loadings of a fund from 60 months of returns: one-factor line and three-factor regression (illustrative data)

Tool	Factor Exposure
Source	Illustrative data: 60 synthetic monthly returns from a seeded random-number generator with known loadings (not market data). Method after Fama, E. F., and French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, Vol. 33, No. 1, pp. 3–56.
Level	Undergraduate

Problem

A fund's monthly returns are given beside three factor returns (Mkt-RF, SMB, HML) and the risk-free return RF for 60 months. The data were generated with alpha 0.10 % per month and loadings 1.05, 0.30 and −0.20. Find the single-factor beta and alpha, then the three-factor loadings with their standard errors, the annualised alpha, R² and the share of variance of each factor.

Inputs to Enter

The tool's default table: 60 rows of Period, Fund, Mkt-RF, SMB, HML, RF in percent per month (synthetic, seed 322)
Monthly periods (12 a year)

Open the Factor Exposure tool (loaded with this example)

Answers

Excess return y = Fund − RF: mean 1.5415; Mkt-RF: mean 1.2788, variance 17.2619; covariance 18.0289; variance of y 21.0234
One factor: beta = 18.0289 / 17.2619 = 1.0444, alpha = 1.5415 − 1.0444 · 1.2788 = 0.2058 % per month, R² = 0.8957
Three factors: alpha 0.2207 (standard error 0.1634), Mkt-RF 1.0288 (0.0375), SMB 0.2896 (0.0494), HML −0.1052 (0.0621)
95 % intervals: Mkt-RF 0.9537 to 1.1039, SMB 0.1907 to 0.3885, HML −0.2295 to 0.0192; each contains the true loading
R² = 0.9358, adjusted 0.9323; annualised alpha 2.65 % per year (interval −1.28 to 6.58); residual volatility 1.193 · √12 = 4.13 % per year
Share of variance: Mkt-RF 88.22 %, SMB 5.06 %, HML 0.29 %, unexplained 6.42 %
Newey-West standard errors at lag 3 (Advanced): 0.1671, 0.0412, 0.0358, 0.0488

Notes

The tool's default inputs are this table; press Submit. The numbers are the tool's own calculation, checked against an independent numpy solution and, for the least-squares routine, against the NIST StRD Longley certified values (test_factor_exposure.php). The data are made up; the example is not a published one.