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 cutoff point on the distribution where the analyst sets the VaR breakpoint
VaR = the agreedupon 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 probabilityweighting 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

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