What is the difference between VaR and CVaR?

VaR is a threshold: at 95%, the loss exceeded on only 5% of days. CVaR is the average loss on those 5% of days, so at the same level it is never smaller than VaR. VaR says nothing about how large losses become once past the threshold; CVaR is built from exactly those losses. Two portfolios can share a VaR and still differ widely in CVaR.

Why is VaR not subadditive?

Because a quantile can ignore a risk that sits just beyond it. Take two independent loans of $100, each with a 4% chance of total loss. Alone, each has a 95% VaR of zero, because a loss happens less than 5% of the time. Together, the chance that at least one defaults is 7.84%, so the pair's 95% VaR is $100, more than the sum of the parts. Their 95% CVaR is $80 each alone and $103.20 together, below the $160 sum.

How do you calculate each position's contribution to CVaR?

With the Euler allocation. Because CVaR grows in proportion when every position is scaled up, it equals the sum over positions of weight times marginal CVaR. In a set of historical or simulated scenarios, each position's contribution is its average loss in the scenarios where the portfolio is in its tail. The contributions add up exactly to the total, which is what lets a CVaR budget be charged position by position, and a hedge's contribution is negative.

How does Opulence Alpha check its VaR and CVaR estimates?

By backtesting them against realised returns. VaR breaches are tested for the right frequency (Kupiec) and for clustering (Christoffersen) and placed in the Basel traffic-light zones. CVaR is tested with the Acerbi–Székely Z2 statistic, reported as an effect size rather than a p-value. A backtest can only tighten: when the VaR backtest shows losses being under-predicted, a multiplier of at least one scales up both VaR and CVaR, so limits bind sooner, never later.

FAQ · Conditional value at risk (CVaR)

What is conditional value at risk (CVaR)?

Reviewed against the platform's code on Sep 26, 2026

Conditional value at risk (CVaR), also called expected shortfall, is the average loss in the worst outcomes beyond a chosen confidence level. Where 95% value at risk (VaR) is the loss exceeded on only 5% of days, 95% CVaR is the average loss on those days: it measures how deep the tail goes, not only where it begins.

Why it matters

VaR says where the bad days begin, not how bad they get. Two portfolios with the same 95% VaR can lose very different amounts once past it, and a strategy that sells far out-of-the-money options can take on a large tail loss while barely moving its VaR. VaR can also penalise diversification. It is not subadditive: two positions together can show a higher VaR than the sum of their separate VaRs, so a budget written in VaR can be used up by a trade that diversifies. CVaR avoids both problems, one reason the Basel Committee's revised market-risk rules (FRTB) replaced 99% VaR with 97.5% expected shortfall in banks' internal models.

How it works

At a confidence level such as 95%, VaR is that quantile of the loss distribution and CVaR is the average of the losses beyond it. Artzner, Delbaen, Eber and Heath (1999) set out the properties a coherent risk measure should have; VaR fails subadditivity, and CVaR satisfies them all. Rockafellar and Uryasev (2000) showed that CVaR can be minimised by linear programming over a set of scenarios. Because CVaR is positively homogeneous, doubling every position doubles it, and Euler's theorem splits it exactly into position contributions. Each contribution is the position's weight times its marginal CVaR; in a set of scenarios it equals that position's average loss in the scenarios where the portfolio is in its tail (Tasche, 1999).

How Opulence Alpha applies it

Opulence Alpha's risk layer sets its tail budget in CVaR, not VaR. Every proposed allocation is measured before any instruction exists: one-day VaR and CVaR at 95% and 99%, as a fraction of portfolio value. Volatility comes from a factor model of style and GICS sector factors estimated across the whole universe. The tail's shape comes from filtered historical simulation: up to two years of the universe's equal-weighted daily returns, standardised by their own volatility and rescaled to the portfolio's current volatility. CVaR is then a multiple of volatility, so each position's Euler share of it is its share of variance, and the shares sum exactly to one. A proposal that would breach the budget is cut back to fit; a portfolio already over its tail budget before trading may only reduce positions.

Why the risk budget is CVaR, not VaR →

Related concepts

Kelly criterionThe Kelly criterion sizes a position to maximise long-run capital growth. How the formula works, why practitioners bet a fraction of it, and how we test it.Regime-aware portfolio constructionRegime-aware portfolio construction lets the market regime shape how a portfolio is built and how much risk it takes — tightening in stress, never loosening automatically.Probabilistic forecastingProbabilistic forecasting publishes a distribution of possible returns, with calibrated intervals, instead of one expected number — so a position can be sized from its uncertainty.

All concepts →Sector behaviour in every market regime →

Questions

What is the difference between VaR and CVaR?: VaR is a threshold: at 95%, the loss exceeded on only 5% of days. CVaR is the average loss on those 5% of days, so at the same level it is never smaller than VaR. VaR says nothing about how large losses become once past the threshold; CVaR is built from exactly those losses. Two portfolios can share a VaR and still differ widely in CVaR.
Why is VaR not subadditive?: Because a quantile can ignore a risk that sits just beyond it. Take two independent loans of $100, each with a 4% chance of total loss. Alone, each has a 95% VaR of zero, because a loss happens less than 5% of the time. Together, the chance that at least one defaults is 7.84%, so the pair's 95% VaR is $100, more than the sum of the parts. Their 95% CVaR is $80 each alone and $103.20 together, below the $160 sum.
How do you calculate each position's contribution to CVaR?: With the Euler allocation. Because CVaR grows in proportion when every position is scaled up, it equals the sum over positions of weight times marginal CVaR. In a set of historical or simulated scenarios, each position's contribution is its average loss in the scenarios where the portfolio is in its tail. The contributions add up exactly to the total, which is what lets a CVaR budget be charged position by position, and a hedge's contribution is negative.
How does Opulence Alpha check its VaR and CVaR estimates?: By backtesting them against realised returns. VaR breaches are tested for the right frequency (Kupiec) and for clustering (Christoffersen) and placed in the Basel traffic-light zones. CVaR is tested with the Acerbi–Székely Z2 statistic, reported as an effect size rather than a p-value. A backtest can only tighten: when the VaR backtest shows losses being under-predicted, a multiplier of at least one scales up both VaR and CVaR, so limits bind sooner, never later.

References

Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance 9(3), 203–228.
Rockafellar, R. T. & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk 2(3), 21–41.
Tasche, D. (1999). Risk Contributions and Performance Measurement. Working paper, Technische Universität München.
Barone-Adesi, G., Giannopoulos, K. & Vosper, L. (1999). VaR without Correlations for Portfolios of Derivative Securities. Journal of Futures Markets 19(5), 583–602.

Educational content about research methods. Not investment advice.