# Conditional Value At Risk - CVaR

## What is 'Conditional Value At Risk - CVaR'

Conditional value at risk (CVaR) is a risk assessment technique often used to reduce the probability that a portfolio will incur large losses. This is performed by assessing the likelihood (at a specific confidence level) that a specific loss will exceed the value at risk. Mathematically speaking, CVaR is derived by taking a weighted average between the value at risk and losses exceeding the value at risk.

## BREAKING DOWN 'Conditional Value At Risk - CVaR'

CVaR is also known as mean excess loss, mean shortfall, tail Var, average value at risk or expected shortfall. CVaR was created to serve as an extension of value at risk (VaR). The VaR model allows managers to limit the likelihood of incurring losses caused by certain types of risk, but not all risks. The problem with relying solely on the VaR model is that the scope of risk assessed is limited, since the tail end of the distribution of loss is not typically assessed. Therefore, if losses are incurred, the amount of the losses will be substantial in value.

CVaR was created to calculate the average of the losses that occur beyond the VaR cutoff point in the distribution. The smaller the value of the CVaR, the better.

## Conditional Value at Risk Calculation and Example

Though the formula for CVaR uses calculus, it is still straightforward. The CVaR is calculated as:

CVaR = (1 / (1 - c)) x the integral of xp(x)dx from -1 to VaR

Where

p(x)dx = is the probability density of getting a return x

c = the cut-off point on the distribution where the analyst sets the VaR breakpoint

VaR = the agreed-upon VaR level

As a simplified example, assume a \$500,000 portfolio with the possible gains and losses (along with the probability of them happening) below:

10% of the time, a loss of \$500,000

30% of the time, a loss of \$100,000

40% of the time, a gain of \$0

20% of the time, a gain of \$250,000

Given a probability of occurrence, q, the expected shortfall for this portfolio is:

5% = \$500,000

10% = \$500,000

20% = \$300,000

30% = \$233,300

40% = \$200,000

50% = \$160,000

60% = \$133,300

70% = \$114,300

80% = \$100,000

90% = \$61,100

100% = \$30,000

This is calculated by probability-weighting the loss for given chance of the loss occurring. For example, knowing that 10% of the time, the portfolio will lose all of its value, the expected shortfall for q=5% and q=10% are both \$500,000. For higher values of q, an analyst would continue down the expected outcomes and weight them according. For example, consider q=40%. An analyst would use the following formula:

Expected shortfall (40%) = ((10% x -\$500,000) + (30% x -\$100,000)) / 40% = \$200,000

Similarly, using interpolation, for q=90%, the expected shortfall is:

Expected shortfall(90%) = (((10% x -\$500,000) + (30% x -\$100,000) + (40% x \$0) + (10% x \$250,000)) / 90% = \$61,100