# Calculating Expected Shortfall of combined portfolios

So I am reading lecture notes here:

https://courses.edx.org/c4x/DelftX/TW3421x/asset/Week3_var_3_slides.pdf

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

((0.02*10)+(0.005*1))/(0.025)=8.2

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.

thanks

• I took the liberty of correcting a typographical error in your calculation... – Alex C Apr 27 '18 at 21:05

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).

• Thank you so much, can't believe I didn't see that. – chocolatekeyboard Apr 28 '18 at 11:54
• You don't know how I would calculate the VaR of this example do you? – chocolatekeyboard Apr 28 '18 at 12:39