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A portfolio consists of one (long) 100 million asset and a default protection contract on this asset. The probability of default over the next year is 10% for the asset, 20% for the counterparty that wrote the default protection. The joint probability of default for the asset and the contract counterparty is 3%. Estimate the expected loss on this portfolio due to credit defaults over the next year assuming 40% recovery rate on the asset and 0% recovery rate for the contract counterparty.

If expected loss of a portfolio is:

Default Probability x Loss Given Default x Exposure at Default

How can I use this formula to solve this problem? Or can this equation be used if we are not given an exposure amount for the contract?

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    $\begingroup$ bad homework question. "100 million asset" what kind of asset? Commodity? Equity? FX? The context suggests it's a credit-risky bond. "default protection contract on this asset" do you mean standard credit default swap, or some variant like zero-recovery, or some other hedge like letter of credit or put option? Is its notional the same as the asset? $\endgroup$
    – Dimitri Vulis
    Dec 14, 2020 at 15:44
  • $\begingroup$ @DimitriVulis Thats all the information I was given. I assumed just for the question it means $100 million worth asset or an asset of 100 million dollar for example. It does say a 0% recovery rate for the contract counterparty. $\endgroup$
    – May
    Dec 14, 2020 at 15:51
  • $\begingroup$ If this the homework you got in some n-week quantitative finance program - your teacher is being sloppy. 0% is if the protection seller defaults, rendering the protection worthless. But if the asset (credit-risky bond) defaults, can you assume that you can put the bond to the protection seller and get the notional back, as in a standard credit default swap, or is this some other kind of "protection"? I will answer with these assumptions. $\endgroup$
    – Dimitri Vulis
    Dec 14, 2020 at 15:57

2 Answers 2

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Assuming "asset" means a credit-risky bond, and "protection" is a standard credit default swap on the same notional.

Ignoring the coupons and interest payments, there are 4 scenarios:

Probablity 10%-3% = 7% : the bond has a credit event. The credit protection seller has not defaulted. You put the defaulted bond to the credit protection seller and receive the notional. Your P&L is the notional (the face value of the bond) that you receive, minus the mark to market of the credit protection.

Probability 3%: both the bond and the credit protection seller have credit events simultaneously. Your defaulted bond is now worth 40% (recovery) * 100 mil (notional). Your protection is worthless, so you lose the mark to market of the protection with 0% recovery. We don't know how much that was worth. (In reality, they are unlikely to defalt simultaneously. What matters is which one of them defaults first.)

Probablity 20%-3% = 17% : only the credit protection seller has a credit event. You lose the mark to market of the protection. We don't know how much that was worth. You still have a performing credit-risky bond. We don't know what it may be trading at. You may want to buy replacement credit protection from someone else.

Probability 100%-10%-17%=73% neither the bond not the protection seller have a credit event, so no credit losses. We don't know what the asset may be trading at, nor the mark to market of the credit protection.

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  • $\begingroup$ If it helps the answer they have given as the correct one is 1.8 million as the estimated expected loss, I just don't know how to arrive at this answer from the information given. $\endgroup$
    – May
    Dec 14, 2020 at 19:01
  • $\begingroup$ I'm curious, where are you getting these? $\endgroup$
    – Dimitri Vulis
    Dec 14, 2020 at 19:06
  • $\begingroup$ This question was provided as part of my practice questions when revising for 'Professional Risk Management' PRM exam. $\endgroup$
    – May
    Dec 14, 2020 at 19:31
  • $\begingroup$ I see, thank you. It sounds like you should take your practice questions with a grain of salt. You may also find this career advice interesting. $\endgroup$
    – Dimitri Vulis
    Dec 14, 2020 at 19:53
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Some irrelevant and unclear info in the Q.

If asset defaults and counterparty does not, we are covered for the full value => no loss

If counterparty defaults and asset does not, we lose nothing bar the cover (as asset is fine) => no loss

We're only caught in the case that both default (3% chance).

Even in this case we would still recover 40% of the value of the asset (40,000,000).

Effectively, 3% of the time we lose $60,000,000

Expected loss = (100,000,000-40,000,000) * 0.03 = $1,800,000

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