Using historical losses to estimate the Expected Shortfall (ES) of a crypto portfolio comes down to one core action: sort the historical daily portfolio gains and losses from worst to best, take the worst small portion, and calculate their average. ES answers the question: "When losses exceed the VaR threshold, how bad is the average loss?" VaR only tells you that "there is a 95% confidence that the single-day loss will not exceed X," but it does not say how severe things get beyond X. ES fills that gap.
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Step-by-Step Historical Simulation for ES
Suppose you have a crypto portfolio that includes BTC, ETH, and several other coins, and you want to calculate the single-day ES at a 95% confidence level. The 95% confidence level means you are focusing on the worst 5% of trading days.
Step 1: Prepare the historical profit and loss series. Take the daily portfolio returns (or daily profit and loss amounts) over a past period. The window length depends on your data availability and your judgment about whether history is representative of the future. A common choice is 1 to 2 years, but the crypto market structure changes quickly, so a window that is too long may mix in market conditions that are no longer similar. Multiply daily returns by the current portfolio market value to get daily dollar profit and loss. Losses are recorded as negative values.
Step 2: Sort the profit and loss from worst to best. Suppose you have 500 trading days of data. Sort from the day with the largest loss to the day with the largest gain. The worst 5% corresponds to the first 25 days (500 × 5% = 25).
Step 3: Find the VaR threshold. The 95% VaR is the 25th loss value after sorting. This number tells you that over the past 500 days, 5% of trading days had losses exceeding this amount.
Step 4: Calculate the tail average. Take all the days whose losses exceed the VaR threshold and calculate their average loss. That is the 95% ES. In mathematical terms: if VaR is 8%, and the loss days exceeding 8% are 12%, 15%, and 20%, then ES is the average of those three numbers, about 15.67%.
This method does not rely on a normal distribution assumption, and it does not require fitting GARCH or extreme value theory models. It is simply "on the bad days that actually happened in history, how bad was the average outcome?" This is why historical simulation is the most directly usable approach for crypto portfolios: crypto asset return distributions are clearly fat-tailed, so using a normal distribution to calculate ES would systematically underestimate tail risk, while historical data at least preserves the extreme values that actually occurred.
A Reproducible Crypto Portfolio Example
Suppose a portfolio is currently worth 100,000 USD, and you use the daily return series from the past 250 trading days to calculate ES.
After sorting, the worst 5% corresponds to about 12.5 days. Under the common treatment in historical simulation, take the 13th worst day as the boundary for 95% VaR ([(1-0.95)×250]+1 = 13).
Suppose the daily return on the 13th worst day is -6.2%. Then the dollar loss corresponding to 95% VaR is 6,200 USD. This means that over the past 250 days, about 5% of trading days had losses exceeding 6,200 USD.
Next, take all the days whose daily returns were below -6.2% (that is, the 1st through 12th worst days). Suppose their daily returns are -18.5%, -14.2%, -12.8%, -11.0%, -10.3%, -9.1%, -8.4%, -7.9%, -7.5%, -7.1%, -6.8%, and -6.5%. The average return of these 12 days is about -10.0%.
Therefore, the dollar amount corresponding to 95% ES is about 10,000 USD. The meaning is: on those extreme days when losses exceeded VaR, the average single-day loss of this portfolio was about 10,000 USD, not 6,200 USD. ES is about 61% higher than VaR, and that gap is the part of tail risk that VaR hides.
The numbers in this example are hypothetical and are meant to show the calculation logic. When you use your own data, the sorting and averaging steps are the same, but the results will change depending on the portfolio composition and the historical window.
Four Key Limitations of Using Historical Simulation for Crypto Portfolio ES
The choice of historical window directly determines the size of ES. If your selected window happens to cover the LUNA collapse in May 2022 or the FTX event in November 2022, ES will be significantly higher. If the window only covers the relatively calm period of 2023 to 2024, ES will be lower. This is not a flaw in the method itself, but the nature of historical simulation: it assumes that "the magnitude and frequency of past extreme events are useful references for the future." Whether the frequency and magnitude of extreme events in the crypto market are stable is itself a judgment you need to make. A more honest approach is to calculate ES over multiple windows, such as 1 year and 2 years, and see how stable the ES estimate is.
When the portfolio composition changes, historical returns cannot be applied directly. If your current portfolio is 60% BTC, 30% ETH, and 10% others, but during the past 250 days you once held a large position in a small coin that has since been sold off, the ES calculated directly from the historical portfolio returns reflects your past holdings, not your current holdings. The correct approach is to multiply the current weights by the historical daily returns of each asset and reconstruct a virtual historical profit and loss series for "if I had always held the current portfolio." This reconstruction assumes that the historical correlations among assets are consistent with the future, but crypto asset correlations tend to spike sharply during extreme market conditions, which can make ES too low during calm periods.
ES is an estimate, not a promise. The fact that the average loss of the worst 5% of trading days in history was 10,000 USD does not mean the next extreme day will produce a single-day loss of 10,000 USD. It could be smaller, or it could be much larger. Historical simulation can only tell you "what happened within the sample," not "how bad the future can get." New extreme events in the crypto market, such as new protocol failures, new regulatory shocks, or new patterns of liquidity drying up, may fall outside the range of the historical sample.
When the tail sample is too small, the average is unstable. If you only have 100 days of data, the 95% ES is based on the average of only the worst 5 days. An average of 5 numbers can easily be pulled off course by one day that was especially bad or especially mild. The shorter the data window, the larger the estimation error of ES. The crypto market trades 24/7, so in theory you could use hourly data to increase the sample size, but the correlation of hourly tail events is different from that of daily tail events, so mixing them requires additional judgment.
Difference from Parametric ES: When Not to Use Historical Simulation
The parametric approach, such as assuming a normal distribution or a t-distribution, calculates ES using formulas. The advantage is that you only need to estimate the mean and volatility to get a number. The disadvantage is that the distribution assumption may be wrong. The 99% ES calculated under a normal distribution tends to be too low in the crypto market because the frequency of extreme losses in crypto assets is far higher than what a normal distribution would predict. A t-distribution can adjust the degrees of freedom to fit fat tails, but the degrees of freedom parameter itself is also estimated and is unreliable when the sample is limited.
The advantage of historical simulation is that it does not assume a distribution shape and directly uses real data. The cost is that it depends entirely on whether the period of history you selected is "representative." For crypto portfolios, a reasonable compromise is to use historical simulation as the primary reference for ES, while also calculating a value from a parametric method, such as a t-distribution, for comparison. If the two values differ greatly, it means that your historical window contains extreme events that the parametric method cannot capture, or that the parametric distribution assumption is not suitable. That gap itself is information and does not need to be forcibly eliminated.
A Practical Verification Checklist
After calculating ES, ask yourself three questions: Does the window cover at least one major tail event in the crypto market? If not, your ES most likely underestimates the true tail risk. Are the current portfolio weights consistent with the weights during the historical window? If not, you are calculating the ES of "the past portfolio," not "the current portfolio." Is the ratio of ES to VaR reasonable? For crypto portfolios, the ES/VaR ratio is usually between 1.5 and 3. If the result is close to 1, check whether the tail sample is too small or whether there is an error in the sorting and averaging steps. If the ratio is unusually large, such as above 5, check whether one day of extreme loss dominates the entire tail average. In that case, ES is highly sensitive to a single sample.
The purpose of ES is not to give a precise "maximum loss," but to give you a historically grounded sense of magnitude for "how bad the average extreme day can be" when building a portfolio and setting risk budgets. Its value lies in laying out the truncated tail behind VaR for you to see, rather than promising that the future will not be worse.
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References
- Bangko Sentral ng Pilipinas·Confidence Interval for Expected Shortfall Using Bootstrap Methods，2026-10-30
- Bangko Sentral ng Pilipinas·Expected Shortfall Estimators（Historical Simulation Section），2026-10-30
- LMU Munich·VaR and ES: Historical Simulation and Parametric Approaches，2026-10-30
- Wiley·Forecasting Regime‐Dependent Tail Risk in Digital Asset Portfolios With a Mixture of Hidden Markov Factor Analyzers，2026-10-30
