Tail Risk Measurement Beyond VaR: Expected Shortfall and Stress Scenarios for Proxy Overlays
Abstract
The $1.70M proxy basket carries a one-day 95% VaR of approximately $56,000. On March 12, 2020, the equivalent basket lost $204,000 in a single session — 3.6× VaR. VaR answered the wrong question: what is the minimum loss on a bad day. Expected shortfall answers the right one: what is the average loss once that threshold is breached. This note covers why VaR fails on proxy baskets, how to compute conditional VaR, what three historical stress replays imply, and the gap scenario where the put spread stops containing losses.
1. The Gap: What VaR Said Versus What Arrived
Parametric VaR on the example basket, calibrated to 32% blended annual volatility:
- Daily volatility: ~2.02%.
- One-day 95% VaR: 1.645 × 2.02% × $1.70M ≈ $56,000 (3.3% of basket).
- Meaning: on the worst 5 days out of 100, expect to lose at least $56,000.
The same factor footprint printed these single sessions:
- October 15, 2008: −7.8% (~$133,000).
- March 12, 2020: −12.0% (~$204,000).
- June 13, 2022: −5.9% (~$100,000).
Each breached VaR by 2–4×. That is structural, not miscalibration. VaR marks the threshold and says nothing about the distribution past it. Small-cap proxy baskets have fat left tails: correlations spike, single-names gap, and the micro-cap prints its co-move late and all at once. The loss that matters lives beyond the line VaR draws.
2. Why VaR Fails on Proxy Baskets
Stable correlations assumed. VaR from trailing 252-day covariance treats the 1.35 market and 0.48 sector loadings as fixed. In stress, small-cap correlations compress toward 1.0 and the ETF-plus-two-names diversification evaporates in one session. Realized basket volatility in March 2020 ran 2.1× its trailing estimate.
Basis ignored. VaR on the basket measures the basket. The exposure is the $2M micro-cap. At R² of 0.78 the proxy tracks in normal conditions; in a liquidity vacuum the micro-cap gaps on no volume while the proxy prints continuously, or vice versa. A 5% basket loss can coincide with a 9% position loss. Basket VaR captures neither leg.
Kinked hedge linearized. The put spread — long 6% OTM, short 14% OTM, 60 DTE — pays nothing to −6%, dollar-for-dollar to −14%, then flatlines. VaR either excludes it or delta-adjusts it. Neither describes a −12% session.
Keep VaR as a daily position limit. It is not a tail risk measure.
3. Conditional VaR: Measuring the Tail Itself
Expected shortfall (conditional VaR) is the mean loss conditional on being in the worst 5% of outcomes. It captures the skew and correlation-spike behavior VaR discards.
Historical simulation over 10 years of factor-mapped basket returns:
- One-day 95% VaR (historical): $61,000 (3.6%).
- One-day 95% CVaR / expected shortfall: $89,000 (5.2%).
- One-day 99% CVaR: $141,000 (8.3%).
The ratio is the diagnostic. Under normality, CVaR 95% runs ~1.25× VaR. Here it is 1.46× — a fat left tail, driven by the single-name legs gapping where the ETF leg does not.
Sizing rule: strike the spread off CVaR, not VaR. The long 6% OTM strike sits just beyond one-day 95% CVaR (5.2%), so the spread engages only in genuine tail territory. The 8-point width (6% to 14%) spans roughly CVaR 95% to CVaR 99%. Recompute monthly; if CVaR 95% drifts above 6.5%, the long strike is too far out and must be re-struck closer.
4. Historical Stress Replay: 2008, 2020, 2022
Stress replay — the core historical stress test for a proxy overlay — forces the current basket through realized paths. Three episodes, $1.70M basket with the put spread in place:
| Stress episode | Window | Unhedged basket loss | Put spread payout | Net proxy loss | Micro-cap implied loss |
|---|---|---|---|---|---|
| 2008 GFC (Lehman) | Sep 8 – Oct 27, 2008 | −34% (~$578,000) | +$136,000 (max value, rolled once) | −26% (~$442,000) | −38% (~$760,000) |
| 2020 COVID crash | Feb 19 – Mar 23, 2020 | −28% (~$476,000) | +$136,000 (one spread cycle covers the fall) | −20% (~$340,000) | −31% (~$620,000) |
| 2022 rate bear market | Jan 3 – Oct 12, 2022 | −24% (~$408,000) | +$41,000 (three spreads bought, none reach full value) | −21.6% (~$367,000) | −27% (~$540,000) |
Three readings.
Fast crashes pay. In 2008 and 2020 the basket blows through both strikes in weeks and the spread pays its ~$136,000 maximum, cutting the basket loss by roughly a quarter.
Slow grinds bleed. 2022 never delivers a clean −14% window inside one 60-day cycle. Three spreads purchased at ~$11,500/month on the ETF leg expire near worthless against 3–5% leg-downs. Hedge cost over the episode exceeds $100,000 against $41,000 of payout. Spreads built from 60-DTE tenors protect against gaps, not nine months of multiple compression.
The position loss is always larger. Implied micro-cap losses use the 1.35 loading plus a 3–4 point illiquidity adder — the observed excess of sub-$150M names over factor prediction during forced selling. The hedge offsets $136,000 at best against $620,000–$760,000 of position drawdown in a fast crash. It covers the basket’s tail. The position’s tail is larger. Re-run this replay quarterly; if any net proxy loss exceeds 25% of basket, widen the spread or cut single-name weight.
5. The Gap Scenario: Where the Spread Stops Containing
The mechanical break: an overnight gap that clears both strikes, combined with proxy-to-position divergence.
A single-name leg gaps −22% on a restatement while the ETF leg opens −6%. The blended basket opens −12.4% and the spread opens near maximum value. That leg works. The micro-cap then prints three stale closes and gaps −25% on $90,000 of volume when a holder capitulates. The hedge has paid its capped $136,000 against a $500,000 position loss. No further response is available — the short 14% strike caps payout, and rolling post-gap means buying 70–90% implied volatility with skew fully priced.
The worse variant is a basket-level −18% gap landing beyond the short strike. Every point from 14% to 18% is naked on the basket, with micro-cap basis on top.
Mitigations are partial. Widening to a 6–18% spread roughly doubles amortized cost from $11,500 to ~$20,000/month, erasing a third of call premium income. Outright puts remove the cap at higher cost. Shifting weight to the ETF leg narrows the gap distribution but weakens the factor match. The operative control is a hard stop: if the basket gaps beyond the short strike, or micro-cap-to-proxy divergence exceeds 10 points in a stress window, cut the short-call layer, freeze rolling, and reassess the hedge ratio.
6. What to Monitor
Three numbers, checked monthly with the factor re-estimation. One-day 95% expected shortfall versus long-strike distance — re-strike closer if CVaR exceeds strike distance. Net loss under the 2020 replay versus 25% of basket — revise width or single-name weight on breach. Trailing 60-day micro-cap-to-proxy divergence — if annualized tracking error exceeds the model residual by 10 points, the hedge ratio is stale and every tail number above is overstated.