Expected Shortfall for a Portfolio of Defaultable Bonds

Summary

The document sets up a two-bond credit portfolio and asks how to calculate its Expected Shortfall across confidence levels. Each bond has a stated market price, default probability, recovery amount, and independent default event. Those assumptions define a small discrete set of portfolio outcomes: neither bond defaults, one defaults, or both default. Expected Shortfall can be found by ordering the resulting portfolio losses and averaging the worst tail, with care over probability mass at the quantile boundary.

The text supplies the inputs for such a calculation but does not provide a solution, specify a precise Expected Shortfall convention, or state a holding-period loss definition beyond the one-year horizon. In a discrete distribution, alternative conventions for fractional tail mass can affect the result, so the chosen definition matters. The setup is a compact illustration of credit risk aggregation and tail-risk measurement, rather than empirical evidence about actual bonds or default dependence.

Key ideas

• Independent defaults create a discrete distribution of portfolio losses.
• Expected Shortfall is based on the average loss in the worst tail of that distribution.
• A discrete loss distribution requires care when the confidence threshold cuts through an outcome's probability mass.
• The example assumes fixed recovery and independence, which limits its realism for credit portfolios.

Tags

# Methods for calculating Expected shortfall


# Methods for calculating Expected shortfall












Let B1, B2 be two defaultable zero-coupon bonds maturing in 1 year, each with a face value of $100. Assume:

- each bond is priced at 90 dollars

- each bond has a 4% probability to default within 1 year

- the events of default are independent

- recovery on default is 30% of face value

Let $\Pi$ denote a portfolio consisting of a long position of 100 dollars face value in each bond, i.e. $\Pi$ has a value of 2 × 90 = 180.

I am asked to calculate the Expected Shorfall for the portfolio $\Pi$ as a function of $\alpha\in(0,0.5)$. How can it be done?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.