October 2011 Q8

Hi Jasmin - have you by any chance been on the ActEd tutorial for SP9? This question is in the Day 1 handout and there is a detailed solution. Let me know if not and I will post something more detailed here for you.
Anna

 

HI Darshan

With VaR questions, I find it helpful to start by writing down the definition from Sweeting:

VaR(97.5%) = inf {l: P(L>l) ≤ 0.025}

(Note 'inf' means we are looking for the smallest loss that satisfies the criteria in the curly brackets.)

Then consider the distribution for the losses and work from there.

Portfolio A

The distribution of losses, L, is:

• £2m prob 0.02 (if the bond defaults, we lose 20 * £100K = £2m)
• –£100K prob 0.98 (if the bond doesn't default, we gain 20 * £5K = £100K, or a loss of –£100K)

Then go through the values of the losses in turn to see which meets the criteria in the definition:

• if l = £2m, then P(L>l) = 0 ≤ 0.025 and the criteria is met
• if l = –£100K, then P(L>l) = 0.02 ≤ 0.025 and the criteria is met

(Note that the inequality is strictly greater than in the probability.)

Of these two bullets, the smallest loss that satisfies the criteria is l = –£100K.

ie VaR(97.5%) = –£100K.

Since VaR is expressed as a 'loss', this result is equivalent to a gain of £100K.
We can interpret the result by saying there is a 97.5% chance that profits will be at least as great as £100K.

Portfolio B

The distribution of losses, L, is:

• –£100K prob 0.6676 (if no bonds default, we gain 20 * £5K = £100K, or a loss of –£100K)
• £5K prob 0.2725 (if one bond defaults, we gain 19 * £5K – 1* £100K = –£5K, or a loss of £5K)
• £110K prob 0.0528 (if two bonds default, we gain 18 * £5K – 2* £100K = –£110K, or a loss of £110K)
• £215K prob 0.0065 ...
• £320K prob 0.0006 ...
• £X prob 0

Then go through the values of the losses in turn to see which meets the criteria in the definition:

• if l = –£100K, then P(L>l) = 0.3324 > 0.025 and the criteria is NOT met
• if l = £5K, then P(L>l) = 0.0599 > 0.025 and the criteria is NOT met
• if l = £110K, then P(L>l) = 0.0071 ≤ 0.025 and the criteria is met
• if l = £215K, then P(L>l) = 0.0006 ≤ 0.025 and the criteria is met
• if l = £320K, then P(L>l) = 0 ≤ 0.025 and the criteria is met

Of these bullets, the smallest loss that satisfies the criteria is l = £110K.

ie VaR(97.5%) = £110K.

We can interpret the result by saying there is a 97.5% chance that losses will be no greater than £110K.

Re Tail Value at Risk, we can use a similar approach, ie start with the definition from Sweeting:

TVaR(97.5%) = E[L | L > VaR(97.5%)]

Portfolio A

VaR(97.5%) = –£100K

Losses, L, are:

• £2m prob 0.02
• –£100K prob 0.98

We want E[L | L > –£100K]

Lots of ways of doing this, I've done a sort of weighted average below:

E[L | L > –£100K] = (£2m * 0.02 + –£100K * 0.005) / 0.025 = £1.58m

Portfolio B

VaR(97.5%) = £110K.

Losses, L, are:

• –£100K prob 0.6676
• £5K prob 0.2725
• £110K prob 0.0528
• £215K prob 0.0065
• £320K prob 0.0006
• £X prob 0

We want E[L | L > £110K]

Lots of ways of doing this, I've done a sort of weighted average below:

E[L | L > –£100K] = (£320K * 0.0006 + £215K * 0.0065 + £110K prob 0.0179) / 0.025 = £142.34K

Hope this helps.
Anna