Crypto value at risk (VaR) is a loss quantile for a stated horizon and confidence level. Expected shortfall (ES) averages the worst tail probability mass at that level. VaR locates a cutoff; ES describes tail severity. Both depend on the loss definition and estimation method, and neither is a maximum-loss limit or a guarantee about future outcomes.
A report labeled “99% confidence” can look authoritative while deriving its tail estimate from only a few observations. The first useful questions are what counts as a loss, how the sample quantile was selected and exactly which probability mass entered the average.
Define the loss before the percentile
Use a positive number for a loss and a negative number for a gain. For a simple unlevered one-period return, this can be loss = −return. Keep gains in the distribution; flooring them at zero changes the quantity being estimated.
Record the horizon, valuation currency, sampling time, holdings and included costs. A one-day percentage loss and a one-hour dollar loss cannot be compared directly. Neither can historical strategy returns and historical market shocks applied to today's holdings: they describe different portfolio experiments.
The Basel Committee's market-risk terminology distinguishes VaR at a specified horizon and confidence level from average losses in the tail. That banking terminology supplies a definition, not an appropriate confidence level or capital allowance for a retail crypto account.
For confidence level c, the lower loss quantile is the smallest loss whose cumulative probability reaches c. A 95% VaR does not say losses cannot exceed it. ES asks about the average of the worst 5% probability mass, with boundary weighting when observations are discrete.
This is separate from maximum drawdown and recovery, which depend on the order of equity observations. Sorting one-period losses removes their sequence; it cannot reveal how long capital stayed underwater.
Reproduce a 20-observation estimate
Consider these invented one-day net loss percentages, sorted from smallest to largest. They are teaching values, not observed crypto returns or KLINEO results. Each observation receives equal probability, 1/20.
| Sorted rank | Loss percentages |
|---|---|
| 1–8 | −5, −4, −3, −2, −1, −0.5, 0, 0 |
| 9–16 | 0.5, 0.7, 1, 1.2, 1.5, 1.8, 2, 2.2 |
| 17 | 2.5 |
| 18 | 3 |
| 19 | 6 |
| 20 | 12 |
Choose 90% confidence solely to make the small example readable. Use the inverse empirical cumulative distribution, often called a type-1 quantile. The VaR rank is ceil(N × c): with N = 20 and c = 0.90, rank 18 gives VaR = 3%.
The worst 10% contains exactly two equally weighted observations. ES is therefore (6% + 12%) / 2 = 9%. For a hypothetical $10,000 exposure with losses scaling linearly to that fixed amount, these become $300 VaR and $900 ES. Neither is a dollar loss cap.
Now increase only the largest loss from 12% to 30%. The 90% VaR stays 3%, because rank 18 did not change. ES becomes (6% + 30%) / 2 = 18%. The cutoff can remain unchanged while the tail becomes much more severe.
A wider discussion of distribution shape belongs in return skewness and kurtosis. Those moments describe different features; they do not replace a specified VaR or ES calculation.
Keep the tail weight exact
For an equal-weight sample, let q = N × (1 − c) be the tail size measured in observation units. Starting with the largest loss, include whole observations and then any fraction of the next observation needed to reach q. Divide that weighted loss sum by q.
Using the original fixture:
| Confidence c | VaR rank and loss | Tail observation units q | Empirical ES |
|---|---|---|---|
| 90% | 18: 3% | 2 | 9% |
| 92.5% | 19: 6% | 1.5 | 10% |
| 99% | 20: 12% | 0.2 | 12% |
The middle row uses (12 + 0.5 × 6) / 1.5 = 10%. The last row uses only part of the largest observation's probability mass. Its 12% estimate does not mean twenty observations identified the future 99% tail well.
Require a nonempty sample and 0 < c < 1 for this worksheet. At c = 1 its tail denominator is zero; report that input as undefined here. Unequal scenario weights require cumulative probability weighting, not simply counting rows.
A particularly useful failure case is a tied cutoff. Replace ranks 17–19 with 3%, keeping the largest loss at 12%. At 90% confidence, VaR remains 3%, and the exact worst two observation units average to 7.5%. Averaging all losses strictly above VaR gives 12%; averaging all losses at or above it gives (3 + 3 + 3 + 12) / 4 = 5.25%. Neither preserves the requested 10% mass.
Acerbi and Tasche's original research on expected shortfall explains why discrete distributions require care with conditional-tail definitions. The practical lesson is to retain the tail probability, not choose a greater-than operator by habit.
Name the quantile algorithm
Two tools can receive identical data and return different VaR values without either misreading the file. R's sample-quantile documentation lists nine algorithms, including the inverse empirical distribution and interpolated alternatives.
For the original fixture, a type-7 90th percentile interpolates at position 1 + 19 × 0.90 = 18.1. That gives 3 + 0.1 × (6 − 3) = 3.3%, rather than the type-1 result of 3%.
Save the algorithm, software version and ES weighting rule together. Using an interpolated threshold and then averaging rows that cross it is not automatically the same empirical ES estimator used above. Resolve the convention before interpreting a discrepancy as a change in risk.
Separate risk confidence from estimation confidence
The 99% in “99% VaR” selects a distribution quantile. It is not a 99% confidence interval for the estimated VaR. With 250 equally weighted observations, the worst 1% has only 2.5 observation units. Increasing the confidence level concentrates the estimate on less tail evidence.
Dependencies further limit what those rows establish. The trade-count and independent-evidence guide explains why many related observations can add less information than their count suggests.
An original hypothetical validation check makes the distinction concrete. Suppose 100 independent forward observations have a constant unknown probability p of exceeding a prespecified loss threshold, and none does. The one-sided 95% exact binomial upper bound solves (1 − p)^100 = 0.05, giving p = 1 − 0.05^(1/100), approximately 2.95%.
Zero observed breaches therefore does not establish a breach probability below 1%. This bound concerns exception frequency, not the ES loss magnitude or uncertainty in the VaR estimate. It assumes independent trials with constant probability and a threshold fixed before these observations; choosing the threshold from the same data invalidates this interpretation. NIST's exact binomial confidence-limit guidance describes the underlying interval method.
For intervals on VaR or ES themselves, document a suitable estimation procedure and its assumptions. The bootstrap confidence-interval guide introduces resampling and dependence limits, but its mean-return example is not a validated tail-quantile interval recipe.
Attach a tail-risk review record
| Field | Evidence to retain |
|---|---|
| Loss contract | Horizon, currency, holdings, marks and cost treatment |
| Input sample | Dates, count, missing-data policy and immutable file hash |
| Estimator | Confidence level, quantile algorithm, scenario weights and ES boundary rule |
| Tail evidence | Tail observation units, largest losses and concentration by episode |
| Validation | Forward threshold protocol, breach count, severity and uncertainty assumptions |
| Stress limits | Unobserved shocks, liquidity assumptions and excluded liabilities |
A historical price-loss distribution can omit the cost of exiting size during stress. Review backtest volume participation when fills or exit capacity are part of the loss model. A precisely computed percentile cannot repair an implausible execution assumption.
Method and limits
This article is AI-assisted educational synthesis of the linked primary sources. The loss fixtures, tail-weight calculations and zero-breach illustration are original hypothetical examples with recomputed arithmetic. No market dataset, KLINEO VaR/ES feature or live risk-control behavior was tested. Historical estimates and statistical intervals remain conditional on their data and assumptions.