Question

The probability that the loss from a portfolio will be greater than $10 million in one month is estimated to be 1%.

a) What is the loss that has a 0.1% chance of being exceeded, assuming the change in value of the portfolio is normally distributed with zero mean?

b) What is the loss that has a 0.1% chance of being exceeded, assuming the power law applies with α = 3?

Answer 1

For this question, we need to use the critical value of z to calculate the loss values.

The 1% tail of the distribution starts 2.33 standard deviations from the mean. The standard deviation is therefore $10/2.33= $4.292 million.

a) The one-month loss (i.e. value at risk) with 0.1% chance of exceeding will use the z-value for 0.1%, which is 3.29.

Therefore, loss value = $4.292 × 3.29 = $14.120 million

b) We know that Probability of loss exceeding X = K * ?^ (−?), where K is a constant

In this case, equation becomes: 0.01 =K×10^-3.

So, K = 10.

To find the 99.9% VaR, we use the above formula: 10*?^(−3) =0.001

=> x^ (-3) = 0.001/10 = 0.0001

=> x^3 = 1/0.0001 = 10000

x= (10000)^(1/3) = $21.54million