# 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... Commented Apr 27, 2018 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. Commented Apr 28, 2018 at 11:54
• You don't know how I would calculate the VaR of this example do you? Commented Apr 28, 2018 at 12:39
• how did you get 0.0246?I believe it should be 3.92-2.5=1.42% Commented Jul 25, 2020 at 17:30
• 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. Commented Jul 26, 2020 at 20:04