Risk and Return

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

Investments that offer higher expected returns usually expose investors to more risk. Variance measures dispersion around the mean, and the Sharpe ratio measures excess return over the risk-free rate per unit of volatility.

Expected return

The expected return is the probability-weighted average of all possible outcomes. If an asset has three scenarios with known probabilities and returns, multiply each return by its probability and sum.

Scheme

; Expected return: E[R] = sum of p_i * r_i
; Three scenarios: boom (30%), normal (50%), bust (20%)

(define (expected-return probs returns)
  (apply + (map * probs returns)))

(define probs '(0.30 0.50 0.20))
(define returns '(0.25 0.10 -0.15))

(define er (expected-return probs returns))
(display "E[R] = ") (display er) (newline)
; 0.30*0.25 + 0.50*0.10 + 0.20*(-0.15)
; = 0.075 + 0.05 - 0.03 = 0.095

(display "Expected return: ")
(display (* er 100)) (display "%")

Variance and standard deviation

Variance is the expected squared deviation from the mean. Standard deviation is its square root, in the same units as returns. A higher standard deviation means wider dispersion of outcomes.

Scheme

; Variance: Var[R] = sum of p_i * (r_i - E[R])^2
; Standard deviation: sigma = sqrt(Var[R])

(define (variance probs returns er)
  (apply + (map (lambda (p r) (* p (* (- r er) (- r er))))
               probs returns)))

(define probs '(0.30 0.50 0.20))
(define returns '(0.25 0.10 -0.15))
(define er 0.095)

(define var-r (variance probs returns er))
(define sigma (sqrt var-r))

(display "Variance = ") (display var-r) (newline)
(display "Std dev  = ") (display sigma) (newline)
(display "Sigma    = ") (display (* sigma 100)) (display "%")

Sharpe ratio

The Sharpe ratio measures excess return per unit of total risk: (E[R] - r_f) / σ. A higher Sharpe means better risk-adjusted performance. The risk-free rate r_f is what you earn with zero risk (e.g., Treasury bills).

Scheme

; Sharpe ratio = (E[R] - r_f) / sigma
; r_f = risk-free rate

(define (sharpe-ratio er rf sigma)
  (/ (- er rf) sigma))

(define er 0.095)      ; expected return 9.5%
(define rf 0.03)       ; risk-free rate 3%
(define sigma 0.1375)  ; std dev 13.75%

(define sr (sharpe-ratio er rf sigma))
(display "Sharpe ratio = ") (display sr) (newline)

; Compare two assets
(display "Asset A: ") (display (sharpe-ratio 0.12 0.03 0.20)) (newline)
(display "Asset B: ") (display (sharpe-ratio 0.08 0.03 0.10)) (newline)
(display "B is better risk-adjusted despite lower return")

Historical vs expected returns

Historical returns use past data with equal weights (sample mean). Expected returns use forward-looking scenario probabilities. Historical data informs expectations but doesn't determine them. The sample standard deviation divides by (n-1), not n.

Scheme

; Historical (sample) statistics from observed returns
(define annual-returns '(0.12 -0.05 0.18 0.08 -0.02))
(define n (length annual-returns))

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

(define mu (mean annual-returns))
(display "Sample mean = ") (display mu) (newline)

; Sample variance (divide by n-1)
(define (sample-var xs mu)
  (/ (apply + (map (lambda (x) (* (- x mu) (- x mu))) xs))
     (- (length xs) 1)))

(define sv (sample-var annual-returns mu))
(display "Sample variance = ") (display sv) (newline)
(display "Sample std dev  = ") (display (sqrt sv)) (newline)

; Geometric mean (actual compound growth)
(define (geom-mean xs)
  (- (expt (apply * (map (lambda (r) (+ 1 r)) xs))
           (/ 1 (length xs)))
     1))

(display "Geometric mean  = ") (display (geom-mean annual-returns))

Risk-return tradeoff

Assets with higher expected returns tend to have higher volatility. This is the fundamental tradeoff: you cannot earn equity-like returns with bond-like risk. The question is always whether the extra return compensates for the extra risk.

Scheme

; Compare asset classes by risk and return
(define (describe name er sigma rf)
  (display name) (display ": ")
  (display "E[R]=") (display (* er 100)) (display "%, ")
  (display "sigma=") (display (* sigma 100)) (display "%, ")
  (display "Sharpe=")
  (display (/ (- er rf) sigma))
  (newline))

(define rf 0.03)

(describe "T-Bills " 0.035 0.03  rf)
(describe "Bonds   " 0.06  0.08  rf)
(describe "Lg Stocks" 0.10  0.20  rf)
(describe "Sm Stocks" 0.13  0.30  rf)

(newline)
(display "Higher return always comes with higher sigma.")
(newline)
(display "Sharpe ratio reveals which tradeoff is best.")

Neighbors

🎰 Probability — expected value and variance are probability concepts applied to prices
📊 Statistics — sample vs population statistics, hypothesis testing on returns
📈 Portfolio Theory — risk and return extend to portfolios of assets

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