Tail risk: reading VaR and CVaR

Tail risk asks roughly how much a bad day or month could cost, and how deep the worst days or months were on average. Value at Risk (VaR) marks a loss boundary at a chosen confidence level. Conditional Value at Risk (CVaR), also called Expected Shortfall, describes the average loss in that tail. Both measures here come from the same replay of today’s holdings. They describe a distribution of historical changes, rather than a forecast for the next day or month.

How to read it: at 95%, the example’s one-day VaR is 1.14%. On about 95 of every 100 days in the replay the portfolio lost less than 1.14%. On the other 5 days it lost more, and CVaR (1.66%) is their average loss.

Each bar counts days in a band of daily return. Losses are on the left and gains on the right; the bin width is 0.005. The example has 1,254 daily returns. The marked VaR and CVaR lines refer to the 95% one-day tail. The dotted line is a separate questionnaire reference, drawn from a hypothetical conservative answer. Bar heights describe frequency, while distance to the left describes depth of loss. A small bar far to the left can represent a rare but deep observation.

The table expresses losses as percentages, with example amounts on a hypothetical USD 100,000 portfolio. The SPY rows provide the same historical calculation for the comparison series, with percentages only. Each confidence level has its own reference values. The one-day and one-month columns come from different sets of returns, so a monthly number describes a complete trading-day window rather than a daily result repeated across a month.

The 95% one-day VaR is 1.14%, yet the worst observed day lost 3.88% on Apr 4, 2025. The worst day is already deeper than the VaR boundary inside this example’s own history. VaR is therefore not the largest possible loss. It separates most observations from a smaller tail, leaving the size of losses in that tail open. Events absent from the window cannot be represented by its historical observations.

CVaR averages the losses in the selected tail; at 95% for one day it is 1.66%. The historical calculation sorts returns, takes the tail count rounded upward, and uses the loss at the boundary for VaR and the average of that tail for CVaR. The two appear together because a boundary alone does not describe the depth of the observations beyond it. Expected Shortfall is another name for this tail average, not a prediction of the next loss.

At 95%, the daily tail has 63 observations. At 99%, it has just 13 out of 1,254. The narrower tail rests on fewer observations even though both levels use the same history. A single deep day can therefore have more influence on its average. The product displays 99% only when there are at least 500 daily returns. That sample gate describes data availability, without judging a confidence level or the resulting loss.

The one-month horizon uses actual overlapping returns across 21 trading days. Here there are 1,234 windows, but only about 59 non-overlapping months. Adjacent windows share most of their days, so they are not independent samples. The tool does not multiply the daily figure by the square root of 21. That shortcut assumes independent, identically distributed days; the historical monthly calculation instead retains the changes observed across each full window.

Historical simulation uses the observed returns directly. The normal model fits a bell-shaped distribution to their mean and sample standard deviation. At 95%, the one-day VaR figures are close: 1.14% historically and 1.18% under the normal model. At 99%, they differ: 2.01% and 1.69%. There were 9 days below the mean minus three standard deviations, against 1.7 expected under a normal distribution. The table also compares CVaR. The product presents the normal model only as a one-line comparison inside calculation details.

An exchange-traded fund (ETF) tracks an index and combines many assets. When it represents a single holding without usable history, its path is usually smoother than that holding’s own path. The substituted history can consequently understate tail losses. The card shows a notice whenever any proxied share is positive. The example’s own SPY and AGG history illustrates the method; it does not remove this limitation for other holdings.

The reference starts with the annualized-volatility threshold at which Overview Risk shows ‘above your stated range’: 20%, 30% or 40% for conservative, balanced or aggressive answers. A short horizon or capital-preservation objective shifts it down by 5 percentage points, floored at 20%. It converts that volatility to a daily and monthly loss under a zero-mean normal assumption. Historical VaR uses the actual distribution, so the bases differ. The line is a comparison only, with no ‘within’ or ‘beyond’ judgement; what is acceptable differs by person and over time. It appears as a table row and a dotted daily-histogram line, and is absent when the questionnaire is unanswered.

Historical VaR is exceeded on about five percent of the days used to estimate its 95% boundary, by construction. Counting those same observations again adds no independent information about performance on new data. An out-of-sample rolling test would estimate from one window and measure subsequent observations. That test is not offered at present, so this guide does not present an in-sample exceedance count as validation.

The results describe only the five-year replay window and today’s holdings. Events outside that window are absent, and today’s collection brings hindsight into earlier dates. The measures do not describe every possible market event or your actual trading history. A historical tail is a summary of the observations available to this calculation, with the disclosed proxy and currency assumptions, rather than a complete account of future losses.

J.P. Morgan/Reuters, RiskMetrics — Technical Document (1996). Artzner, Delbaen, Eber & Heath, Coherent Measures of Risk, Mathematical Finance 9(3), 1999. Rockafellar & Uryasev, Optimization of Conditional Value-at-Risk, Journal of Risk 2(3), 2000. Acerbi & Tasche, On the Coherence of Expected Shortfall, Journal of Banking & Finance 26(7), 2002.