## Solution to Question 1 from UoL exam 2016, Zone A (FN3142)

Frankly, this is a crazy exercise. I deserve full 100 marks for this solution but I wouldn't be able to solve this during an exam. In the problem statement, I took the liberty to change some terminology and notation. In particular, I use the definitions of the Value at Risk and expected shortfall that give negative values. You can go ahead and redo everything with definitions that give positive values.

**Problem statement**

Assume daily returns that are normally distributed with constant mean (equal to zero) and variance, i.e., they are given by

where

where the time increment is 1-day.

(a) [25 marks] Derive the following formula for the Value-at-Risk at the % critical level and 1-day horizon

(1)

where is the standard normal cumulative density function.

(b) [25 marks] The expected shortfall at the critical level % and 1-day horizon can be defined as

.

Using the VaR formula from part (a) derive the following formula for the 1-day expected shortfall

(2)

where is the standard normal probability density function.

(c) [50 marks] Prove that the relative difference between the 1-day expected shortfall and 1-day Value-at-Risk, as a proportion of the 1-day Value-at-Risk, converges to zero when goes to zero, i.e., show that

(3)

## Solution

(a) The answer is contained in this post.

(b) This part has been solved here.

(c) Plug (1) and (2) in (3):

(crossing out and multiplying everything by )

At this point it helps to replace and note that is equivalent to . Then we get

This is indeterminacy of type . In such cases people use the L'Hôpital's Rule. The above expression has the same limit as

(we know that )

This is again indeterminacy of type and by the L'Hôpital's Rule this expression has the same limit as

(replacing the derivative)

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