Value at Risk

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

Value at Risk (VaR) answers one question: what is the worst loss you can expect over a given horizon at a given confidence level? Expected Shortfall (CVaR) asks the harder follow-up: given that you exceed VaR, how bad is it on average?

Historical simulation VaR

The simplest VaR method: sort historical returns, find the percentile. No distributional assumptions. The 95% VaR is the 5th percentile of the return distribution. Strengths: captures fat tails and skewness. Weakness: assumes the past is representative of the future.

Scheme

; Historical simulation VaR
; Sort returns, pick the alpha-percentile

(define returns '(-0.05 -0.03 -0.02 -0.01 -0.008
                  0.001 0.005 0.01 0.015 0.02
                  0.025 0.03 0.035 0.04 0.05
                  0.06 0.07 0.08 0.09 0.10))

(define (list-sort lst)
  (if (or (null? lst) (null? (cdr lst))) lst
      (let* ((pivot (car lst))
             (rest (cdr lst))
             (lo (filter (lambda (x) (< x pivot)) rest))
             (hi (filter (lambda (x) (>= x pivot)) rest)))
        (append (list-sort lo) (list pivot) (list-sort hi)))))

(define (nth lst n)
  (if (= n 0) (car lst) (nth (cdr lst) (- n 1))))

(define (historical-var returns alpha)
  (let* ((sorted (list-sort returns))
         (n (length sorted))
         (idx (- (ceiling (* alpha n)) 1)))
    (- (nth sorted idx))))

(define var-95 (historical-var returns 0.05))
(display "95% VaR: ") (display (* var-95 100)) (display "%") (newline)

(define var-99 (historical-var returns 0.01))
(display "99% VaR: ") (display (* var-99 100)) (display "%")

; Historical simulation VaR
; Sort returns, pick the alpha-percentile

(define returns '(-0.05 -0.03 -0.02 -0.01 -0.008
                  0.001 0.005 0.01 0.015 0.02
                  0.025 0.03 0.035 0.04 0.05
                  0.06 0.07 0.08 0.09 0.10))

(define (list-sort lst)
  (if (or (null? lst) (null? (cdr lst))) lst
      (let* ((pivot (car lst))
             (rest (cdr lst))
             (lo (filter (lambda (x) (< x pivot)) rest))
             (hi (filter (lambda (x) (>= x pivot)) rest)))
        (append (list-sort lo) (list pivot) (list-sort hi)))))

(define (nth lst n)
  (if (= n 0) (car lst) (nth (cdr lst) (- n 1))))

(define (historical-var returns alpha)
  (let* ((sorted (list-sort returns))
         (n (length sorted))
         (idx (- (ceiling (* alpha n)) 1)))
    (- (nth sorted idx))))

(define var-95 (historical-var returns 0.05))
(display "95% VaR: ") (display (* var-95 100)) (display "%") (newline)

(define var-99 (historical-var returns 0.01))
(display "99% VaR: ") (display (* var-99 100)) (display "%")

Parametric (variance-covariance) VaR

Assume returns are normally distributed. Then VaR = μ − z_α × σ. Fast to compute and extends naturally to portfolios via the covariance matrix. The catch: real returns have fat tails, so parametric VaR systematically underestimates extreme losses.

Scheme

; Parametric VaR under normality
; VaR = -(mu - z_alpha * sigma)

; z-scores for common confidence levels
(define z-95 1.645)
(define z-99 2.326)

(define (parametric-var mu sigma z)
  (- (* z sigma) mu))

; Portfolio: mean daily return 0.05%, volatility 1.5%
(define mu 0.0005)
(define sigma 0.015)

(define var95 (parametric-var mu sigma z-95))
(display "95% parametric VaR: ")
(display (* var95 100)) (display "%") (newline)

(define var99 (parametric-var mu sigma z-99))
(display "99% parametric VaR: ")
(display (* var99 100)) (display "%") (newline)

; Scale to 10-day horizon (sqrt-time rule)
(define var99-10d (* var99 (sqrt 10)))
(display "99% 10-day VaR: ")
(display (* var99-10d 100)) (display "%")

Expected shortfall (CVaR)

VaR tells you the threshold; expected shortfall tells you the average loss beyond that threshold. ES is always larger than VaR and is a coherent risk measure (it satisfies subadditivity, meaning diversification always reduces measured risk). Basel III requires banks to use ES rather than VaR.

Scheme

; Expected Shortfall from historical data
; Average of all losses beyond VaR

(define returns '(-0.05 -0.03 -0.02 -0.01 -0.008
                  0.001 0.005 0.01 0.015 0.02
                  0.025 0.03 0.035 0.04 0.05
                  0.06 0.07 0.08 0.09 0.10))

(define (list-sort lst)
  (if (or (null? lst) (null? (cdr lst))) lst
      (let* ((pivot (car lst))
             (rest (cdr lst))
             (lo (filter (lambda (x) (< x pivot)) rest))
             (hi (filter (lambda (x) (>= x pivot)) rest)))
        (append (list-sort lo) (list pivot) (list-sort hi)))))

(define (expected-shortfall returns alpha)
  (let* ((sorted (list-sort returns))
         (n (length sorted))
         (cutoff (ceiling (* alpha n)))
         (tail (let loop ((lst sorted) (i 0) (acc '()))
                 (if (= i cutoff) acc
                     (loop (cdr lst) (+ i 1) (cons (car lst) acc))))))
    (- (/ (apply + tail) (length tail)))))

(define es-95 (expected-shortfall returns 0.05))
(display "95% Expected Shortfall: ")
(display (* es-95 100)) (display "%") (newline)

; Compare: VaR < ES always
(display "95% VaR: 5.0%") (newline)
(display "ES captures how bad the tail really is")

; Expected Shortfall from historical data
; Average of all losses beyond VaR

(define returns '(-0.05 -0.03 -0.02 -0.01 -0.008
                  0.001 0.005 0.01 0.015 0.02
                  0.025 0.03 0.035 0.04 0.05
                  0.06 0.07 0.08 0.09 0.10))

(define (expected-shortfall returns alpha)
  (let* ((sorted (list-sort returns))
         (n (length sorted))
         (cutoff (ceiling (* alpha n)))
         (tail (let loop ((lst sorted) (i 0) (acc '()))
                 (if (= i cutoff) acc
                     (loop (cdr lst) (+ i 1) (cons (car lst) acc))))))
    (- (/ (apply + tail) (length tail)))))

(define es-95 (expected-shortfall returns 0.05))
(display "95% Expected Shortfall: ")
(display (* es-95 100)) (display "%") (newline)

; Compare: VaR < ES always
(display "95% VaR: 5.0%") (newline)
(display "ES captures how bad the tail really is")

Backtesting and model validation

A 95% VaR model should be violated about 5% of the time. Too many violations means the model underestimates risk. Too few means capital is wasted. The Kupiec test checks whether the observed violation rate is statistically consistent with the model's confidence level. Regulators use a traffic-light system: green (0–4 violations in 250 days), yellow (5–9), red (10+).

Scheme

; Kupiec backtest: likelihood ratio test for VaR violations

(define (kupiec-lr n violations alpha)
  ; Expected violation rate = alpha
  ; Observed violation rate = violations / n
  (let* ((p-hat (/ violations n))
         (p0 alpha)
         ; LR = -2 * ln[(p0^v * (1-p0)^(n-v)) / (p-hat^v * (1-p-hat)^(n-v))]
         (log-lr (* -2
                    (- (+ (* violations (log p0))
                          (* (- n violations) (log (- 1 p0))))
                       (+ (* violations (log p-hat))
                          (* (- n violations) (log (- 1 p-hat))))))))
    log-lr))

; 250 trading days, 95% VaR
(define n 250)
(define alpha 0.05)

; Expected: ~12.5 violations
(display "Expected violations: ") (display (* n alpha)) (newline)

; Test with 8 violations (acceptable)
(define lr-8 (kupiec-lr n 8 alpha))
(display "LR stat (8 violations): ") (display lr-8) (newline)

; Test with 20 violations (model failure)
(define lr-20 (kupiec-lr n 20 alpha))
(display "LR stat (20 violations): ") (display lr-20) (newline)
(display "Critical value (chi2, 1df, 5%): 3.84")

; Kupiec backtest: likelihood ratio test for VaR violations

(define (kupiec-lr n violations alpha)
  ; Expected violation rate = alpha
  ; Observed violation rate = violations / n
  (let* ((p-hat (/ violations n))
         (p0 alpha)
         ; LR = -2 * ln[(p0^v * (1-p0)^(n-v)) / (p-hat^v * (1-p-hat)^(n-v))]
         (log-lr (* -2
                    (- (+ (* violations (log p0))
                          (* (- n violations) (log (- 1 p0))))
                       (+ (* violations (log p-hat))
                          (* (- n violations) (log (- 1 p-hat))))))))
    log-lr))

; 250 trading days, 95% VaR
(define n 250)
(define alpha 0.05)

; Expected: ~12.5 violations
(display "Expected violations: ") (display (* n alpha)) (newline)

; Test with 8 violations (acceptable)
(define lr-8 (kupiec-lr n 8 alpha))
(display "LR stat (8 violations): ") (display lr-8) (newline)

; Test with 20 violations (model failure)
(define lr-20 (kupiec-lr n 20 alpha))
(display "LR stat (20 violations): ") (display lr-20) (newline)
(display "Critical value (chi2, 1df, 5%): 3.84")

Neighbors

📉 Finance II Ch.10 — credit derivatives create the tail risk VaR tries to measure
📉 Finance II Ch.12 — dynamic portfolio choice uses risk measures to set constraints
🎲 Probability Ch.5 — normal distribution and tail probabilities underpin parametric VaR
📊 Statistics Ch.7 — hypothesis testing drives the Kupiec backtest

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