CVaR Calculator

A Conditional Value at Risk (CVaR) calculator measures the expected loss in the worst-case scenarios beyond your Value at Risk threshold. Also known as Expected Shortfall, it is used by portfolio managers, risk analysts, and sophisticated investors who need a more complete picture of tail risk than VaR alone provides.

How to Use the CVaR Calculator

1. Enter your portfolio value in pounds or your chosen currency.
2. Set your confidence level, typically 95% or 99%.
3. Input your portfolio's expected return and standard deviation (annualised).
4. Choose your time horizon, such as one day or one month.
5. Read the CVaR output, which shows the average loss you can expect in the worst-case tail scenarios.

The Formula

CVaR is calculated as the expected value of losses that exceed the VaR threshold.

For a normal distribution: CVaR = Mean - (Standard Deviation multiplied by the probability density function of the z-score, divided by one minus the confidence level).

Where:

• Mean is the expected portfolio return over the period
• Standard Deviation is the volatility of returns
• The z-score corresponds to your chosen confidence level (1.645 for 95%, 2.326 for 99%)
• The probability density function (PDF) is evaluated at that z-score

CVaR always produces a larger loss estimate than VaR at the same confidence level, because it averages all losses in the tail rather than just identifying the threshold.

Real-World Example

A portfolio is worth £500,000. The daily expected return is 0.04% and daily standard deviation is 1.2%. At a 95% confidence level:

First, calculate VaR: z-score for 95% is 1.645. Daily VaR = 0.0004 minus (0.012 multiplied by 1.645) = 0.0004 minus 0.01974 = negative 1.934%, meaning a loss of £9,670.

Then calculate CVaR: PDF at z = 1.645 is approximately 0.1031. CVaR = 0.0004 minus (0.012 multiplied by 0.1031 divided by 0.05) = 0.0004 minus 0.02474 = negative 2.434%, meaning an expected loss of £12,170.

So while VaR says losses will not exceed £9,670 on 95% of days, CVaR tells you that on the 5% of worst days, the average loss will be £12,170.

Why CVaR Is Preferred Over VaR

CVaR satisfies the mathematical property of sub-additivity, meaning the combined CVaR of two portfolios is always less than or equal to the sum of their individual CVaRs. This makes it a coherent risk measure, unlike VaR, which can violate this property. Regulators under Basel III and many institutional risk frameworks now require CVaR reporting alongside VaR. CVaR is also more sensitive to extreme events and fat tails in return distributions, making it more realistic for assets like options or emerging market equities where large losses cluster in the tail.

Frequently Asked Questions

What is the difference between VaR and CVaR? VaR identifies the loss threshold that will not be exceeded with a given probability. CVaR goes further and calculates the average loss when that threshold is breached. CVaR is always equal to or worse than VaR at the same confidence level.

What confidence level should I use? 95% is standard for internal risk management and daily reporting. 99% is common for regulatory reporting and stress testing. Higher confidence levels produce larger CVaR estimates and are more conservative.

Can CVaR be applied to individual securities? Yes. CVaR can be calculated for any asset with a return history or a modelled return distribution. It is particularly useful for options, bonds with credit risk, and illiquid assets where tail losses can be severe.

Does CVaR assume a normal distribution? The formula above assumes normally distributed returns, which underestimates tail risk for many assets. More advanced approaches use historical simulation or Monte Carlo methods to capture fat tails and skewness in real return distributions.

The arithmetic behind the worked example

The example on this page uses a 500,000 pound portfolio, a daily expected return of 0.04 percent and a daily standard deviation of 1.2 percent, tested at 95 percent confidence. Working both measures through in full shows where each number comes from.

The 95 percent z-score is 1.645. Value at Risk subtracts the standard deviation multiplied by that z-score from the mean:

CVaR adds one more term. The standard normal probability density at z equals 1.645 is 0.103111, and dividing that by one minus the confidence level scales it into the tail:

The page above reports the CVaR loss as 12,170 pounds, which comes from rounding the return to minus 2.434 percent before multiplying: 0.02434 multiplied by 500,000 gives 12,170.00. Carrying the unrounded return gives 12,173.30, so the rounding step accounts for 3.30 pounds. Carry the unrounded value through the multiplication and round once at the end when the figure feeds a limit or a report.

The same portfolio at 99 percent confidence

Changing the confidence level changes both measures, and it changes them by different amounts. On the same 500,000 pound portfolio:

The VaR term is the standard deviation multiplied by the z-score, which is 0.019740 at 95 percent and 0.027912 at 99 percent. The CVaR term adds the density divided by the tail size, which is 0.0247466 at 95 percent and 0.0320085 at 99 percent. The gap between the two measures therefore moves from 2,503.30 pounds at 95 percent to 2,048.23 pounds at 99 percent, and the ratio of CVaR to VaR falls from 1.259 to 1.149.

That direction surprises people who expect the tail to get more expensive all the way up the confidence scale. The reason is that the density at the z-score falls faster than the tail divisor shrinks. Between the two levels above, the density drops by a factor of 3.9 while the tail divisor grows by a factor of 2. The CVaR term grows as a result, but more slowly than the VaR term, so the two measures move closer together in proportion even as both get larger.

Scaling the inputs to a longer horizon

The worked example uses daily inputs. Setting the tool to a one month horizon means scaling both the mean and the volatility, and the two scale differently: the mean grows in proportion to the number of days, while the volatility grows with the square root of the number of days.

At a daily standard deviation of 1.2 percent, a 10 day horizon gives 1.2 multiplied by the square root of 10, which is 3.7947 percent. A 21 day horizon gives 1.2 multiplied by the square root of 21, which is 5.4991 percent. With the mean also scaled, the results on the same portfolio are:

Note the shape of that growth. Moving from 1 day to 21 days multiplies the horizon by 21 and the volatility by about 4.58, which is the square root of 21. Risk does not scale linearly with time, and a model that multiplies a one day figure by 21 overstates the 21 day loss by a wide margin.

Why regulators replaced VaR with expected shortfall

The Basel framework's Fundamental Review of the Trading Book drops the 10 day 99 percent VaR in favour of expected shortfall at a 97.5 percent confidence level. The two are not the same measure, and the choice of 97.5 percent rather than 99 percent is deliberate.

For a normal distribution, the sigma multiple on 99 percent VaR is 2.326, and the sigma multiple on 97.5 percent expected shortfall is 2.3376, which is the density at a z-score of 1.96 divided by 0.025. The two land within half a percent of each other. A bank moving from one measure to the other therefore keeps a comparable capital number while gaining a measure that satisfies sub-additivity, the property VaR can violate when two books are combined.

The gap between the measures matters most where returns are not normal. Under fat tails, VaR at 99 percent quotes a threshold that the worst days sail past, while expected shortfall keeps averaging the losses beyond it. That is the same reason CVaR on this page reads higher than VaR on the same inputs.

Assumptions and what the model leaves out

Three assumptions are built into the closed-form calculation, and each one has a stated direction of error.

1. Returns are normally distributed. Real return series have fatter tails and skew, so the formula understates tail risk for assets such as options, credit and emerging market equities. Historical simulation or a Monte Carlo approach handles those cases without the normal assumption.
2. Volatility is constant over the horizon. The square root of time scaling above depends on that, and volatility clusters in practice. A period that starts calm and turns violent will breach a limit set on the calm reading.
3. The loss distribution is continuous and the portfolio is static. The formula says nothing about positions that change during the horizon, and nothing about gaps in prices. A position that cannot be closed during a fast market produces a realised loss beyond the modelled tail.

The formula is also sensitive to the input that is hardest to estimate. Doubling the volatility from 1.2 percent to 2.4 percent doubles both VaR and CVaR, so the accuracy of the standard deviation estimate sets the accuracy of the whole answer. Compute that input from a stated sample period, state the period alongside the result, and keep the sample consistent between measures you intend to compare.

One further caution on the tool's input labels. The steps above describe the expected return and standard deviation as annualised while the worked example uses daily figures. Check which the field expects before you enter a number, and convert deliberately: a daily volatility of 1.2 percent is roughly 19.0 percent annualised at 252 trading days, which is the square root of 252 multiplied by 1.2. Feeding an annualised figure into a field that expects a daily one overstates the result by a factor of about 16.

Also try these free tools:

• Value at Risk Calculator
• Portfolio Volatility Calculator
• Sharpe Ratio Calculator