Hello, does any one have breakdown of the calculation for VaR and TVaR in October 2011 exam's Q8 that they can share please? Examiner's report is not so detailed.
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
It does help a lot! Thanks Anna, happy to know that there are many ways that the examiner would have accepted but your method seems okayish for me.