So I am reading lecture notes here:


The example is this:

We have two independent portfolios of bonds. They both have a probability of 0.02 of a loss of £10 million and a probability of 0.98 of a loss of £1 million over a 1-year time window.

To calculate the individual ES of 97.5% I know that it would be


that makes sense. However it is the figure of £11.4 for the combined portfolio. I don't understand how they came to that figure at all? I realise that the example is meant to be lower than the combined individual (8.2+8.2=16.4) but if you could explain the formula that was used for the 11.4 that would be great.


  • $\begingroup$ I took the liberty of correcting a typographical error in your calculation... $\endgroup$
    – Alex C
    Apr 27, 2018 at 21:05

1 Answer 1


Since the bonds are independent we have one of three things that can happen

(1) With probability 0.98*0.98 both bonds lose 1 Million, the total loss is 2 Million

(2) With probability 2*0.98*0.02 one bond loses 1 million and the other 10, for a combined loss of 11 million

(3) With probability 0.02*0.02 both bonds lose 10, overall loss 20

Now we need to find the bad outcomes that account for 2.5% of probability: this consists of 0.0004 probability of 20 loss and 0.0246 probability of 11 loss.

So we have ES = (0.0004*20+0.0246*11)/0.025 = 11.144

(=>It would appear that there is an error in the 11.4 figure in your document, note that 2 lines below it is given as 11.14, which is correct).

  • $\begingroup$ Thank you so much, can't believe I didn't see that. $\endgroup$ Apr 28, 2018 at 11:54
  • $\begingroup$ You don't know how I would calculate the VaR of this example do you? $\endgroup$ Apr 28, 2018 at 12:39
  • $\begingroup$ how did you get 0.0246?I believe it should be 3.92-2.5=1.42% $\endgroup$ Jul 25, 2020 at 17:30
  • $\begingroup$ We need two numbers that add up to 0.025 (the desired threshold for ES). Since the first number is 0.0004, it is clear what the second number has to be. $\endgroup$
    – nbbo2
    Jul 26, 2020 at 20:04

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