# Calculating expected shortfall

I'm trying to calculate the expected shortfall for the below scenario. I don't understand why the 1.04% probability of 0 bonds defaulting is used as a weight when calculating ES, since the binomial probability was 96.04%.

Problem

﻿Assume a two-bond portfolio where the probability of bond default is 2% for each and independent (i.i.d). The face value of each bond is $100 and recovery is zero. What is the 95% expected shortfall? Solution Given 1. Calculate the binomial probabilities of both bonds defaulting, only one defaulting and 0 bonds defaulting which are: P (0 defaults) = 96.04% P (1 defaults) = 3.92% P (2 defaults) = 0.04% 2. The 5% tail therefore contains the following: 0.04% probability of 2 defaults 3.92% probability of 1 default 1.04% probability of 0 defaults The 95% ES is given by: ((0.04% * 2) + (3.92% * 1) + (1.04% * 0))/5% = 0.8 * 100 = 80.00 The average loss in the 5% tail (of the binomial) is 0.8 defaults or \$80.

• To compute the 5% tail, you string events together (from worst outcome to best outcome) until they add up to 5% probability, so 3.92+0.04+1.04=5.00. We only take a small part (namely 1.04) of the no default case because that is all we need to get to our desired 5%. – noob2 Nov 9 '15 at 14:13
• How would I construct the probability weights if my binomial probabilities were each > 5%? – AfterWorkGuinness Nov 9 '15 at 16:06
• Then you take your 5% from the worst case scenario, I believe. – noob2 Nov 9 '15 at 16:27
• @noob2, I don't follow sorry. Assuming I have a similar 2 bond i.i.d. portfolio, but with a PD of 65%. My binomial probabilities for 0,1 and 2 defaults would 12.25%, 45.50% 42.25% respectively. Given each of these is well over 5%, how would I calculate a 95% ES as a probability weighted average? – AfterWorkGuinness Nov 9 '15 at 19:23
• In English, the 5% worst outcomes all involve double default so the loss is 200. Using math we have (5%*2)/5% multiplied by 100 is 200. In fact all K% ES for K from 100 to 57.75 are equal to 200 in this case. – noob2 Nov 9 '15 at 19:43