CAPM and APT

OpenStax Principles of Finance (CC BY 4.0) · MIT OCW 15.401 (CC BY-NC-SA 4.0)

Only systematic risk is rewarded. The CAPM prices assets by their beta to the market. APT generalizes this to multiple risk factors. Both say: diversifiable risk earns no premium.

Beta calculation

Beta measures how much an asset moves with the market. β = Cov(R_i, R_m) / Var(R_m). A beta of 1.5 means the asset amplifies market moves by 50%. A beta below 1 dampens them. Beta captures systematic risk only.

Scheme

; Beta = Cov(R_i, R_m) / Var(R_m)
; Using sample data

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

(define (covariance xs ys)
  (let ((mx (mean xs)) (my (mean ys)) (n (length xs)))
    (/ (apply + (map (lambda (x y) (* (- x mx) (- y my))) xs ys))
       (- n 1))))

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

(define (beta asset-returns market-returns)
  (/ (covariance asset-returns market-returns)
     (variance market-returns)))

; Monthly returns: asset vs market
(define r-asset  '(0.05 -0.02 0.08 -0.04 0.06 0.03))
(define r-market '(0.03 -0.01 0.05 -0.03 0.04 0.02))

(display "Cov(R_i, R_m) = ") (display (covariance r-asset r-market)) (newline)
(display "Var(R_m)       = ") (display (variance r-market)) (newline)
(display "Beta           = ") (display (beta r-asset r-market))

Security market line

The security market line (SML) is the CAPM equation graphed: E[R_i] = r_f + β_i(E[R_m] - r_f). Under CAPM assumptions, expected returns should lie on this line. Assets above it are underpriced (positive alpha), assets below are overpriced (negative alpha).

Scheme

; Security Market Line: E[R_i] = r_f + beta * (E[R_m] - r_f)
(define (capm rf beta er-m)
  (+ rf (* beta (- er-m rf))))

(define rf 0.03)     ; risk-free rate 3%
(define er-m 0.10)   ; market expected return 10%

(display "--- CAPM Expected Returns ---") (newline)
(display "beta=0.0: ") (display (capm rf 0.0 er-m)) (newline)  ; r_f
(display "beta=0.5: ") (display (capm rf 0.5 er-m)) (newline)
(display "beta=1.0: ") (display (capm rf 1.0 er-m)) (newline)  ; market
(display "beta=1.5: ") (display (capm rf 1.5 er-m)) (newline)
(display "beta=2.0: ") (display (capm rf 2.0 er-m)) (newline)

; Alpha: actual return minus CAPM prediction
(define actual 0.12)
(define beta-i 1.2)
(define predicted (capm rf beta-i er-m))
(display "Predicted: ") (display predicted) (newline)
(display "Actual:    ") (display actual) (newline)
(display "Alpha:     ") (display (- actual predicted))

Capital market line

The capital market line (CML) applies only to efficient portfolios (combinations of the risk-free asset and the market portfolio). E[R_p] = r_f + (E[R_m] - r_f) / σ_m * σ_p. The slope is the market's Sharpe ratio. The SML uses beta; the CML uses total risk.

Scheme

; Capital Market Line: E[R_p] = r_f + ((E[R_m] - r_f) / sigma_m) * sigma_p
; Only applies to efficient portfolios

(define (cml rf er-m sig-m sig-p)
  (+ rf (* (/ (- er-m rf) sig-m) sig-p)))

(define rf 0.03)
(define er-m 0.10)
(define sig-m 0.18)   ; market std dev 18%

(display "Market Sharpe ratio = ")
(display (/ (- er-m rf) sig-m)) (newline)

(display "--- CML: expected return by portfolio risk ---") (newline)
(define (show-cml sig-p)
  (display "sigma_p=") (display sig-p)
  (display "  E[R_p]=") (display (cml rf er-m sig-m sig-p))
  (newline))

(show-cml 0.00)   ; 100% risk-free
(show-cml 0.09)   ; 50/50 mix
(show-cml 0.18)   ; 100% market
(show-cml 0.27)   ; levered 150%
(show-cml 0.36)   ; levered 200%

Arbitrage pricing theory

The APT generalizes CAPM from one factor (market) to multiple factors. E[R_i] = r_f + β_1λ_1 + β_2λ_2 + ... Each β measures exposure to a factor; each λ is that factor's risk premium. No arbitrage means prices adjust until this holds.

Scheme

; APT: E[R_i] = r_f + sum(beta_k * lambda_k)
; Multi-factor model

(define (apt rf betas lambdas)
  (+ rf (apply + (map * betas lambdas))))

(define rf 0.03)

; Three factors: market, size, value
(define factor-names '("Market" "Size" "Value"))
(define lambdas '(0.06 0.03 0.04))  ; factor risk premiums

; Stock A: high market beta, small, value
(define betas-a '(1.2 0.8 0.5))
(display "Stock A E[R] = ")
(display (apt rf betas-a lambdas)) (newline)

; Stock B: low market beta, large, growth
(define betas-b '(0.7 -0.2 -0.3))
(display "Stock B E[R] = ")
(display (apt rf betas-b lambdas)) (newline)

; Decompose Stock A's risk premium
(display "--- Factor contributions (Stock A) ---") (newline)
(for-each (lambda (name b l)
            (display name) (display ": ")
            (display (* b l)) (newline))
          factor-names betas-a lambdas)

Neighbors

📈 Portfolio Theory — CAPM is the equilibrium result of Markowitz optimization
📈 Risk and Return — beta decomposes total risk into systematic and idiosyncratic
📊 Statistics — regression estimates beta from return data
💰 Economics — equilibrium pricing and no-arbitrage are economic principles

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