Question

In a previous example, the probability density function of Triangle Asset Management's daily profits was described by the following function:

For -10 ≤ x ≤ 0: p = (10/1) + (100/1)x
For 0 < x ≤ 10: p = (10/1) - (100/1)x

In that example, Triangle's one-day 95% VaR was calculated as a loss of 6.84. For the same confidence level and time horizon, what is the expected shortfall (ES)?

ES = (1/p) * ∫ from -L to VaR x * p dx

Please calculate the expected shortfall using the given VaR value and the probability density function. If you have any further questions or need clarification, please let me know.

Answer 1

To calculate the expected shortfall (ES) using the given VaR value and the probability density function, you first need to calculate the probability (p) for the VaR value. Then, you can use the formula ES = (1/p) * ∫ from -L to VaR x * p dx to calculate the expected shortfall.

Step-by-step explanation:

To calculate the expected shortfall (ES) using the given VaR value and the probability density function, we first need to calculate the probability (p) for the VaR value. The probability density function is given by:

For -10 ≤ x ≤ 0: p = (10/1) + (100/1)x

For 0 < x ≤ 10: p = (10/1) - (100/1)x

Given that the one-day 95% VaR was calculated as a loss of 6.84, we substitute this value into the equation to find the probability:

For -10 ≤ x ≤ 0: p = (10/1) + (100/1)(-6.84) = 0.316

For 0 < x ≤ 10: p = (10/1) - (100/1)(-6.84) = 0.684

Next, we calculate the expected shortfall using the formula:

ES = (1/p) * ∫ from -L to VaR x * p dx

For -10 ≤ x ≤ -6.84: ES = (1/0.316) * ∫ from -10 to -6.84 x * (10/1) + (100/1)x dx

For -6.84 < x ≤ 0: ES = (1/0.684) * ∫ from -6.84 to 0 x * (10/1) - (100/1)x dx

Solving these integrals will give us the expected shortfall for the given VaR value and probability density function.