Factor Models

MIT OCW 18.S096 + 15.450 (CC BY-NC-SA 4.0)

A factor model decomposes an asset's return into exposure to a few common risk factors plus idiosyncratic noise. If you can identify the factors, you can separate the risk you're paid for from the risk you can diversify away.

Fama-French three-factor model

CAPM says one factor (the market) explains all expected returns. Fama and French showed two more matter: SMB (small minus big, the size premium) and HML (high minus low book-to-market, the value premium). A stock's expected excess return is β₁MKT + β₂SMB + β₃HML.

Scheme

; Fama-French three-factor model
; R_i - Rf = alpha + beta1*MKT + beta2*SMB + beta3*HML + epsilon

(define (ff3-expected-return rf beta-mkt beta-smb beta-hml
                              mkt-premium smb-premium hml-premium)
  (+ rf
     (* beta-mkt mkt-premium)
     (* beta-smb smb-premium)
     (* beta-hml hml-premium)))

; Historical average premia (annualized, approximate)
(define mkt-prem 0.06)   ; 6% market risk premium
(define smb-prem 0.02)   ; 2% size premium
(define hml-prem 0.035)  ; 3.5% value premium
(define rf 0.04)          ; 4% risk-free rate

; Small value stock: high exposure to all three factors
(define small-value (ff3-expected-return rf 1.2 0.8 0.7
                                          mkt-prem smb-prem hml-prem))
(display "Small value expected return: ")
(display (exact->inexact small-value)) (newline)

; Large growth stock: market exposure, negative SMB and HML
(define large-growth (ff3-expected-return rf 1.0 -0.3 -0.4
                                           mkt-prem smb-prem hml-prem))
(display "Large growth expected return: ")
(display (exact->inexact large-growth)) (newline)

; The difference is the factor premium, not alpha
(display "Spread: ")
(display (exact->inexact (- small-value large-growth)))

; Fama-French three-factor model
; R_i - Rf = alpha + beta1*MKT + beta2*SMB + beta3*HML + epsilon

(define (ff3-expected-return rf beta-mkt beta-smb beta-hml
                              mkt-premium smb-premium hml-premium)
  (+ rf
     (* beta-mkt mkt-premium)
     (* beta-smb smb-premium)
     (* beta-hml hml-premium)))

; Historical average premia (annualized, approximate)
(define mkt-prem 0.06)   ; 6% market risk premium
(define smb-prem 0.02)   ; 2% size premium
(define hml-prem 0.035)  ; 3.5% value premium
(define rf 0.04)          ; 4% risk-free rate

; Small value stock: high exposure to all three factors
(define small-value (ff3-expected-return rf 1.2 0.8 0.7
                                          mkt-prem smb-prem hml-prem))
(display "Small value expected return: ")
(display (exact->inexact small-value)) (newline)

; Large growth stock: market exposure, negative SMB and HML
(define large-growth (ff3-expected-return rf 1.0 -0.3 -0.4
                                           mkt-prem smb-prem hml-prem))
(display "Large growth expected return: ")
(display (exact->inexact large-growth)) (newline)

; The difference is the factor premium, not alpha
(display "Spread: ")
(display (exact->inexact (- small-value large-growth)))

PCA on returns

Principal component analysis extracts factors from the data itself. Compute the covariance matrix of returns, then eigendecompose. The first eigenvector is usually the market; subsequent ones capture sector, size, and other patterns. PCA factors are statistical, not economic—they maximize variance explained, not interpretability.

Scheme

; PCA: eigendecomposition of covariance matrix
; 2x2 covariance matrix for two correlated assets

(define var-a 0.04)    ; variance of asset A (20% vol)
(define var-b 0.09)    ; variance of asset B (30% vol)
(define cov-ab 0.036)  ; covariance (correlation = 0.6)

; For a 2x2 symmetric matrix [[a,b],[b,c]]:
; eigenvalues = ((a+c) +/- sqrt((a-c)^2 + 4b^2)) / 2
(define trace (+ var-a var-b))
(define det (- (* var-a var-b) (* cov-ab cov-ab)))
(define discriminant (sqrt (- (* trace trace) (* 4 det))))

(define lambda1 (/ (+ trace discriminant) 2))
(define lambda2 (/ (- trace discriminant) 2))

(display "Eigenvalue 1 (PC1): ") (display (exact->inexact lambda1)) (newline)
(display "Eigenvalue 2 (PC2): ") (display (exact->inexact lambda2)) (newline)

; Variance explained by first component
(define var-explained (/ lambda1 (+ lambda1 lambda2)))
(display "Variance explained by PC1: ")
(display (exact->inexact var-explained)) (newline)

; Eigenvector for lambda1: [cov-ab, lambda1 - var-a]
(define v1-x cov-ab)
(define v1-y (- lambda1 var-a))
(define norm (sqrt (+ (* v1-x v1-x) (* v1-y v1-y))))
(display "PC1 direction: [")
(display (exact->inexact (/ v1-x norm)))
(display ", ")
(display (exact->inexact (/ v1-y norm)))
(display "]")

; PCA: eigendecomposition of covariance matrix
; 2x2 covariance matrix for two correlated assets

(define var-a 0.04)    ; variance of asset A (20% vol)
(define var-b 0.09)    ; variance of asset B (30% vol)
(define cov-ab 0.036)  ; covariance (correlation = 0.6)

; For a 2x2 symmetric matrix [[a,b],[b,c]]:
; eigenvalues = ((a+c) +/- sqrt((a-c)^2 + 4b^2)) / 2
(define trace (+ var-a var-b))
(define det (- (* var-a var-b) (* cov-ab cov-ab)))
(define discriminant (sqrt (- (* trace trace) (* 4 det))))

(define lambda1 (/ (+ trace discriminant) 2))
(define lambda2 (/ (- trace discriminant) 2))

(display "Eigenvalue 1 (PC1): ") (display (exact->inexact lambda1)) (newline)
(display "Eigenvalue 2 (PC2): ") (display (exact->inexact lambda2)) (newline)

; Variance explained by first component
(define var-explained (/ lambda1 (+ lambda1 lambda2)))
(display "Variance explained by PC1: ")
(display (exact->inexact var-explained)) (newline)

; Eigenvector for lambda1: [cov-ab, lambda1 - var-a]
(define v1-x cov-ab)
(define v1-y (- lambda1 var-a))
(define norm (sqrt (+ (* v1-x v1-x) (* v1-y v1-y))))
(display "PC1 direction: [")
(display (exact->inexact (/ v1-x norm)))
(display ", ")
(display (exact->inexact (/ v1-y norm)))
(display "]")

Systematic vs idiosyncratic risk

Total variance = systematic variance + idiosyncratic variance. Systematic risk comes from factor exposures and can't be diversified away. Idiosyncratic risk is stock-specific noise that vanishes in a large portfolio. R-squared from the factor regression tells you how much is systematic.

Scheme

; Risk decomposition: total = systematic + idiosyncratic
; For a single-factor model: Var(R) = beta^2 * Var(F) + Var(epsilon)

(define (risk-decomposition beta factor-vol idio-vol)
  (let* ((systematic-var (* beta beta factor-vol factor-vol))
         (idio-var (* idio-vol idio-vol))
         (total-var (+ systematic-var idio-var))
         (r-squared (/ systematic-var total-var)))
    (list total-var systematic-var idio-var r-squared)))

; Stock with beta=1.3, market vol=16%, idiosyncratic vol=25%
(define result (risk-decomposition 1.3 0.16 0.25))

(display "Total variance:       ") (display (exact->inexact (car result))) (newline)
(display "Systematic variance:  ") (display (exact->inexact (cadr result))) (newline)
(display "Idiosyncratic variance: ") (display (exact->inexact (caddr result))) (newline)
(display "R-squared:            ") (display (exact->inexact (cadddr result))) (newline)

; In a portfolio of N independent stocks, idio risk shrinks by 1/N
(display "--- Diversification ---") (newline)
(define n-stocks 50)
(define port-idio-var (/ (caddr result) n-stocks))
(display "Portfolio idio var (50 stocks): ")
(display (exact->inexact port-idio-var)) (newline)
(display "Portfolio idio vol: ")
(display (exact->inexact (sqrt port-idio-var)))

; Risk decomposition: total = systematic + idiosyncratic
; For a single-factor model: Var(R) = beta^2 * Var(F) + Var(epsilon)

(define (risk-decomposition beta factor-vol idio-vol)
  (let* ((systematic-var (* beta beta factor-vol factor-vol))
         (idio-var (* idio-vol idio-vol))
         (total-var (+ systematic-var idio-var))
         (r-squared (/ systematic-var total-var)))
    (list total-var systematic-var idio-var r-squared)))

; Stock with beta=1.3, market vol=16%, idiosyncratic vol=25%
(define result (risk-decomposition 1.3 0.16 0.25))

(display "Total variance:       ") (display (exact->inexact (car result))) (newline)
(display "Systematic variance:  ") (display (exact->inexact (cadr result))) (newline)
(display "Idiosyncratic variance: ") (display (exact->inexact (caddr result))) (newline)
(display "R-squared:            ") (display (exact->inexact (cadddr result))) (newline)

; In a portfolio of N independent stocks, idio risk shrinks by 1/N
(display "--- Diversification ---") (newline)
(define n-stocks 50)
(define port-idio-var (/ (caddr result) n-stocks))
(display "Portfolio idio var (50 stocks): ")
(display (exact->inexact port-idio-var)) (newline)
(display "Portfolio idio vol: ")
(display (exact->inexact (sqrt port-idio-var)))

Factor mimicking portfolios

A factor mimicking portfolio is a long-short portfolio whose return tracks a given factor. For SMB: go long small caps, short large caps in equal dollar amounts. The portfolio has unit exposure to the size factor and zero net market exposure. This lets you trade abstract risk factors as concrete portfolios.

Scheme

; Factor mimicking portfolio construction
; SMB: long small-cap basket, short large-cap basket

(define (portfolio-return weights returns)
  (if (null? weights) 0
      (+ (* (car weights) (car returns))
         (portfolio-return (cdr weights) (cdr returns)))))

; 3 small stocks, 3 large stocks
; Weights: +1/3 each small, -1/3 each large (dollar neutral)
(define smb-weights (list 1/3 1/3 1/3 -1/3 -1/3 -1/3))

; Monthly returns: small stocks outperformed
(define month-returns (list 0.05 0.03 0.04   ; small caps
                            0.01 0.02 0.015)) ; large caps

(define smb-return (portfolio-return smb-weights month-returns))
(display "SMB factor return: ")
(display (exact->inexact smb-return)) (newline)

; Net investment is zero (long-short)
(define net-investment (portfolio-return
                         (list 1 1 1 1 1 1)
                         smb-weights))
(display "Net dollar exposure: ")
(display (exact->inexact net-investment)) (newline)

; HML: long high B/M, short low B/M
(define hml-weights (list 1/3 1/3 1/3 -1/3 -1/3 -1/3))
(define hml-returns (list 0.04 0.035 0.03    ; value stocks
                          0.01 0.015 0.02))  ; growth stocks
(define hml-return (portfolio-return hml-weights hml-returns))
(display "HML factor return: ")
(display (exact->inexact hml-return))

; Factor mimicking portfolio construction
; SMB: long small-cap basket, short large-cap basket

(define (portfolio-return weights returns)
  (if (null? weights) 0
      (+ (* (car weights) (car returns))
         (portfolio-return (cdr weights) (cdr returns)))))

; 3 small stocks, 3 large stocks
; Weights: +1/3 each small, -1/3 each large (dollar neutral)
(define smb-weights (list 1/3 1/3 1/3 -1/3 -1/3 -1/3))

; Monthly returns: small stocks outperformed
(define month-returns (list 0.05 0.03 0.04   ; small caps
                            0.01 0.02 0.015)) ; large caps

(define smb-return (portfolio-return smb-weights month-returns))
(display "SMB factor return: ")
(display (exact->inexact smb-return)) (newline)

; Net investment is zero (long-short)
(define net-investment (portfolio-return
                         (list 1 1 1 1 1 1)
                         smb-weights))
(display "Net dollar exposure: ")
(display (exact->inexact net-investment)) (newline)

; HML: long high B/M, short low B/M
(define hml-weights (list 1/3 1/3 1/3 -1/3 -1/3 -1/3))
(define hml-returns (list 0.04 0.035 0.03    ; value stocks
                          0.01 0.015 0.02))  ; growth stocks
(define hml-return (portfolio-return hml-weights hml-returns))
(display "HML factor return: ")
(display (exact->inexact hml-return))

Running the regression

To estimate factor loadings, regress excess returns on the factor returns. The betas measure sensitivity; alpha measures unexplained return. A positive, statistically significant alpha means the asset outperforms its factor-predicted return—genuine skill or a missing factor.

Scheme

; Estimate alpha and beta from return data
; Simple OLS for single factor: beta = Cov(R,F)/Var(F), alpha = mean(R) - beta*mean(F)

(define (mean lst)
  (/ (apply + lst) (length lst)))

(define (covariance xs ys)
  (let ((mx (mean xs)) (my (mean ys)))
    (/ (apply + (map (lambda (x y) (* (- x mx) (- y my))) xs ys))
       (length xs))))

(define (variance xs)
  (covariance xs xs))

; 12 months of excess returns and market factor
(define stock-excess (list 0.03 -0.01 0.05 0.02 -0.03 0.04
                           0.01 0.06 -0.02 0.03 0.00 0.04))
(define mkt-factor   (list 0.02 -0.01 0.04 0.01 -0.02 0.03
                           0.01 0.04 -0.01 0.02 0.00 0.03))

(define beta (/ (covariance stock-excess mkt-factor)
                (variance mkt-factor)))
(define alpha (- (mean stock-excess) (* beta (mean mkt-factor))))

(display "Beta:  ") (display (exact->inexact beta)) (newline)
(display "Alpha: ") (display (exact->inexact alpha)) (newline)

; Residual variance
(define residuals (map (lambda (r f) (- r alpha (* beta f)))
                       stock-excess mkt-factor))
(define idio-var (variance residuals))
(display "Idiosyncratic vol: ")
(display (exact->inexact (sqrt idio-var)))

; Estimate alpha and beta from return data
; Simple OLS for single factor: beta = Cov(R,F)/Var(F), alpha = mean(R) - beta*mean(F)

(define (mean lst)
  (/ (apply + lst) (length lst)))

(define (covariance xs ys)
  (let ((mx (mean xs)) (my (mean ys)))
    (/ (apply + (map (lambda (x y) (* (- x mx) (- y my))) xs ys))
       (length xs))))

(define (variance xs)
  (covariance xs xs))

; 12 months of excess returns and market factor
(define stock-excess (list 0.03 -0.01 0.05 0.02 -0.03 0.04
                           0.01 0.06 -0.02 0.03 0.00 0.04))
(define mkt-factor   (list 0.02 -0.01 0.04 0.01 -0.02 0.03
                           0.01 0.04 -0.01 0.02 0.00 0.03))

(define beta (/ (covariance stock-excess mkt-factor)
                (variance mkt-factor)))
(define alpha (- (mean stock-excess) (* beta (mean mkt-factor))))

(display "Beta:  ") (display (exact->inexact beta)) (newline)
(display "Alpha: ") (display (exact->inexact alpha)) (newline)

; Residual variance
(define residuals (map (lambda (r f) (- r alpha (* beta f)))
                       stock-excess mkt-factor))
(define idio-var (variance residuals))
(display "Idiosyncratic vol: ")
(display (exact->inexact (sqrt idio-var)))

Neighbors

📐 Linear Algebra Ch.1 — eigenvectors underpin PCA and factor extraction
🎲 Probability Ch.1 — variance, covariance, and regression are the statistical toolkit
📉 Finance II Ch.1 — stochastic processes model the returns that factors decompose

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