Expected Shortfall for Two Independent Bond Portfolios
Summary
The note works through a 97.5% expected shortfall calculation for two independent bond portfolios. Each portfolio has two possible loss amounts, and independence lets the combined outcomes be enumerated: both portfolios incur the smaller loss, one incurs each loss, or both incur the larger loss. The probability of each combined outcome follows from multiplying the individual probabilities.
To obtain expected shortfall, the example averages losses across the worst 2.5% of outcomes, including only the needed fraction of probability mass at the cutoff. The calculation gives a combined-portfolio value of 11.144 in the stated monetary units and points out that a separate 11.4 figure in the referenced material appears inconsistent with its own calculation. This is a discrete illustration under the specified loss distribution and independence assumption; it does not establish how diversification behaves under dependence or more general loss distributions.
Key ideas
- Independence allows the joint loss scenarios to be formed by multiplying portfolio outcome probabilities.
- The combined portfolio has three possible total losses in this example.
- Expected shortfall averages losses in the worst tail, using only the probability mass needed to reach the confidence level.
- The worked combined result is 11.144, while the separate 11.4 figure is identified as an apparent error.
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# Calculating Expected Shortfall of combined portfolios # 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 ## Answer by Alex C (score 3, accepted) https://quant.stackexchange.com/a/39476 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).
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