Portfolio Theory

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

Diversification is the only free lunch in finance. By combining assets that don't move in lockstep, you can reduce portfolio risk below the weighted average of individual risks.

Two-asset portfolio return and risk

A portfolio's expected return is the weighted average of its components. But portfolio variance depends on the covariance between assets. With weight w in asset A and (1-w) in asset B: σ²_p = w²σ²_A + (1-w)²σ²_B + 2w(1-w)σ_Aσ_Bρ.

Scheme

; Two-asset portfolio: return and risk
; E[R_p] = w*E[R_A] + (1-w)*E[R_B]
; Var_p = w^2*var_A + (1-w)^2*var_B + 2*w*(1-w)*sigma_A*sigma_B*rho

(define (port-return w er-a er-b)
  (+ (* w er-a) (* (- 1 w) er-b)))

(define (port-risk w sig-a sig-b rho)
  (sqrt (+ (* w w sig-a sig-a)
           (* (- 1 w) (- 1 w) sig-b sig-b)
           (* 2 w (- 1 w) sig-a sig-b rho))))

(define er-a 0.10) (define sig-a 0.15)
(define er-b 0.16) (define sig-b 0.25)
(define rho 0.3)

; 60% A, 40% B
(define w 0.6)
(display "E[R_p] = ") (display (port-return w er-a er-b)) (newline)
(display "sigma_p = ") (display (port-risk w sig-a sig-b rho)) (newline)

; Notice: sigma_p < weighted average of sigmas
(define weighted-avg (+ (* w sig-a) (* (- 1 w) sig-b)))
(display "Weighted avg sigma = ") (display weighted-avg) (newline)
(display "Diversification benefit!")

Correlation and diversification

When correlation ρ = +1, diversification does nothing. When ρ = -1, a zero-variance portfolio is possible at the right weights. Real-world correlations fall between, so diversification reduces but doesn't eliminate risk. The lower the correlation, the bigger the benefit.

Scheme

; Effect of correlation on portfolio risk
; Same assets, vary rho from -1 to +1

(define (port-risk w sig-a sig-b rho)
  (sqrt (+ (* w w sig-a sig-a)
           (* (- 1 w) (- 1 w) sig-b sig-b)
           (* 2 w (- 1 w) sig-a sig-b rho))))

(define sig-a 0.15)
(define sig-b 0.25)
(define w 0.5)

(define (show-rho rho)
  (display "rho=") (display rho)
  (display "  sigma_p=")
  (display (port-risk w sig-a sig-b rho))
  (newline))

(show-rho  1.0)   ; no diversification benefit
(show-rho  0.5)
(show-rho  0.0)
(show-rho -0.5)
(show-rho -1.0)   ; zero-variance portfolio possible

Efficient frontier

The efficient frontier is the set of portfolios offering the highest return for each level of risk. Any portfolio below the frontier is dominated: you can get more return for the same risk, or less risk for the same return.

Scheme

; Trace the efficient frontier by varying weights
(define (port-return w er-a er-b)
  (+ (* w er-a) (* (- 1 w) er-b)))

(define (port-risk w sig-a sig-b rho)
  (sqrt (+ (* w w sig-a sig-a)
           (* (- 1 w) (- 1 w) sig-b sig-b)
           (* 2 w (- 1 w) sig-a sig-b rho))))

(define er-a 0.10) (define sig-a 0.15)
(define er-b 0.16) (define sig-b 0.25)
(define rho 0.3)

(define (show-port w)
  (let ((ret (port-return w er-a er-b))
        (risk (port-risk w sig-a sig-b rho)))
    (display "w=") (display w)
    (display "  E[R]=") (display ret)
    (display "  sigma=") (display risk)
    (newline)))

(display "--- Frontier (w = weight in A) ---") (newline)
(show-port 1.0)
(show-port 0.8)
(show-port 0.6)
(show-port 0.4)
(show-port 0.2)
(show-port 0.0)

Minimum variance portfolio

The minimum variance portfolio (MVP) has the lowest possible risk. For two assets, the optimal weight in A is: w* = (σ²_B - σ_Aσ_Bρ) / (σ²_A + σ²_B - 2σ_Aσ_Bρ). This is a closed-form solution from setting the derivative of portfolio variance to zero.

Scheme

; Minimum variance portfolio: closed-form weight
; w* = (sig_B^2 - sig_A*sig_B*rho) / (sig_A^2 + sig_B^2 - 2*sig_A*sig_B*rho)

(define (mvp-weight sig-a sig-b rho)
  (/ (- (* sig-b sig-b) (* sig-a sig-b rho))
     (+ (* sig-a sig-a) (* sig-b sig-b) (* -2 sig-a sig-b rho))))

(define (port-risk w sig-a sig-b rho)
  (sqrt (+ (* w w sig-a sig-a)
           (* (- 1 w) (- 1 w) sig-b sig-b)
           (* 2 w (- 1 w) sig-a sig-b rho))))

(define (port-return w er-a er-b)
  (+ (* w er-a) (* (- 1 w) er-b)))

(define sig-a 0.15) (define sig-b 0.25) (define rho 0.3)
(define er-a 0.10) (define er-b 0.16)

(define w* (mvp-weight sig-a sig-b rho))
(display "MVP weight in A: ") (display w*) (newline)
(display "MVP weight in B: ") (display (- 1 w*)) (newline)
(display "MVP return: ") (display (port-return w* er-a er-b)) (newline)
(display "MVP risk:   ") (display (port-risk w* sig-a sig-b rho))

Neighbors

📈 Risk and Return — individual asset risk concepts that portfolio theory builds on
📈 CAPM and APT — equilibrium pricing models that follow from portfolio theory
🎰 Probability — covariance and correlation are the mathematical engine here
📊 Statistics — estimating covariance matrices from historical data

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