ETF5952 · Quantitative Methods for Risk Analysis
Downside Risk Measures: Value-at-Risk and Expected Shortfall
This chapter of Monash University's ETF5952 Quantitative Methods for Risk Analysis quantifies downside risk with the two measures the unit builds the rest of the course on: Value-at-Risk (VaR) and Expected Shortfall (ES). Both start from a quantile of the return distribution — VaR reports that quantile as a positive loss, VaR(p) = −q_p, and ES averages the losses beyond it, so ES(p) ≥ VaR(p) always. You compute them three ways — parametrically (normal or Student-t), by historical simulation, and by bootstrap — skills examined by hand in the closed-book final and rehearsed in the R workshops.
What this chapter covers
- 01Quantiles of the return distribution: q_p = F⁻¹(p), with small p giving a negative (bad) return
- 02The VaR definition VaR(p) = −q_p, read as a positive loss magnitude
- 03Parametric VaR in location-scale form: −(μ + σ z_p), with z_0.05 = −1.645, z_0.01 = −2.326
- 04Student-t (fat-tailed) VaR: fatter tails give a larger VaR than the normal for the same μ, σ
- 05Historical-simulation VaR: the ⌈pT⌉-th worst sorted return, negated (round pT up)
- 06Bootstrap VaR: resample returns with replacement, take the p-quantile, negate
- 07Expected Shortfall ES(p) = −E[R | R ≤ q_p], the average loss in the worst-p tail
- 08The ordering ES(p) ≥ VaR(p), why ES is coherent, and the Basel move to 2.5% ES for capital
Historical-simulation VaR and Expected Shortfall at the 10% level
- +1Find the tail index: ⌈pT⌉ = ⌈0.10 × 20⌉ = ⌈2⌉ = 2. The empirical quantile is the 2nd smallest return.
- +1Read the empirical quantile: q̂_0.10 = r₍₂₎ = −0.044.
- +1Historical-simulation VaR = −q̂_0.10 = −(−0.044) = 0.044 = 4.4%. In dollars: 0.044 × $500,000 = $22,000.
- +1Average the worst ⌈pT⌉ = 2 returns (the tail): mean = (−0.062 − 0.044) / 2 = −0.106 / 2 = −0.053.
- +1Expected Shortfall = −(−0.053) = 0.053 = 5.3%. In dollars: 0.053 × $500,000 = $26,500.
- +1Check the ordering: ES (5.3%) > VaR (4.4%). ES averages the whole tail, which lies beyond the VaR threshold, so ES ≥ VaR holds.
Key terms
- Quantile (q_p)
- The return value the distribution falls below with probability p: F(q_p) = p, so q_p = F⁻¹(p). For downside risk p is small (0.01, 0.05), so q_p is a negative return — a bad outcome.
- Value-at-Risk (VaR)
- The loss threshold breached only with probability p over the horizon, reported as a positive magnitude: VaR(p) = −q_p. Equivalently, Pr(R ≤ −VaR(p)) = p.
- Expected Shortfall (ES)
- The average loss given the loss lands in the worst-p tail: ES(p) = −E[R | R ≤ q_p]. Also called conditional VaR; it always satisfies ES(p) ≥ VaR(p).
- Parametric VaR
- VaR from an assumed distribution in location-scale form R = μ + σZ, giving VaR(p) = −(μ + σ z_p). The normal uses z_0.05 = −1.645; a Student-t uses a more negative quantile, raising VaR.
- Historical-simulation VaR
- A non-parametric estimate that reads the quantile straight off the sorted observed returns: VaR_HS(p) = − r₍⌈pT⌉₎, where ⌈·⌉ rounds up. It assumes the past tail repeats and needs no distribution assumption.
- Bootstrap VaR
- A simulation estimate that resamples returns with replacement to build a large synthetic sample, then takes the p-quantile and negates it; repeating the resample also gives a sampling error for VaR.
- Coverage level (p)
- The tail probability defining the measure (e.g. 5% or 1%). A smaller p means a deeper tail, so both VaR and ES increase as p falls.
Downside Risk Measures: Value-at-Risk and Expected Shortfall FAQ
Why is Expected Shortfall always at least as large as Value-at-Risk?
Because ES averages returns that are all at or below the quantile q_p, the average sits at least as far into the tail as q_p itself. Negated, the tail mean is greater than or equal to the VaR threshold, so ES(p) ≥ VaR(p), with equality only in degenerate cases. That is also why the Basel framework now uses a 2.5% ES for bank capital — it captures the size of tail losses, not just how often they occur.
What is the difference between parametric, historical-simulation and bootstrap VaR?
All three estimate the same quantile q_p; they differ in where the distribution comes from. Parametric VaR assumes a family (normal or Student-t) and plugs into −(μ + σ z_p). Historical simulation reads the quantile off the actual sorted returns with no assumed shape. Bootstrap VaR resamples the observed returns with replacement, then takes the quantile — a simulation cousin that also gives a sampling error.
Can AI help me with Value-at-Risk and Expected Shortfall in ETF5952?
Yes — Sia can explain the method step by step: how to turn a quantile into VaR(p) = −q_p, when to use the normal versus Student-t parametric form, how to round ⌈pT⌉ up for a historical-simulation estimate, and why ES ≥ VaR. It works through practice problems with you and checks your sign and quantile, but it does not do graded assessments for you or promise a particular mark — always follow Monash's assessment and academic-integrity rules and confirm details on Moodle.
Exam move
Master the sign and the quantile first: VaR(p) = −q_p is a positive loss, and a lower p (a deeper tail) always gives a larger figure — most lost marks come from a dropped minus, a positive z_p, or a 1% VaR reported smaller than the 5% VaR. Drill the three routes on one dataset: parametric normal via −(μ + σ z_p), historical simulation via the ⌈pT⌉-th worst sorted return (round up), and the bootstrap. For ES, remember VaR uses one return while ES averages the whole tail, so ES ≥ VaR is a free sanity check. The final exam is worth 40%, is closed book and centrally scheduled in the Monash Semester-1 period (~June 2027 — confirm the exact date and length on Moodle); because the duration is not published in the unit materials, budget your time in proportion to the marks and bank the one-line definitional marks before grinding the arithmetic. Practise on Assignment 2, which is set in the style of the final.
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