PRM 3 – Question 141 vs 143 – Calculate expected loss on 2 bond portfo

[Anonymous]

Hi,
Can you please explain the difference between these 2 questions?
In 141, it says to ignore the joint probability and just says to multiply the expected loss x each loan and sum it up.
In 143, it says to add up 3 different scenarios, i.e. 1) bond A defaults, while B survives, 2) vice versa, and 3) both default and add up those. Why is the math different in these 2 examples?

Thanks

Question 143 : There are two bonds in a portfolio, each with a market value of $50m. The probability of default of the two bonds over a one year horizon are 0.03 and 0.08 respectively. If the default correlation is zero, what is the one year expected loss on this portfolio?
(a) $1.38m
(b) $5.5m
(c) $5.26m
(d) $11m

Your Answer is Correct

The correct answer is choice ‘b’

The probabilities of default of the two bonds are independent (as indicated by a zero default correlation). The various possible states of the portfolio are as follows:

First bond defaults, and the second does not: Probability * Loss = 0.03*0.92 * $50m = $1.38m
Second bond defaults, and the first does not: Probability * Loss = 0.97*0.08 * $50m = $3.88m
Both bonds default: Probability * Loss = 0.03*0.08 * $100m = $0.24m

Thus total expected loss on this portfolio = $5.5m. Since recovery rates are not provided, those should be assumed to be zero.

Question 141 : A portfolio has two loans, A and B, each worth $1m. The probability of default of loan A is 10% and that of loan B is 15%. The probability of both loans defaulting together is 1%. Calculate the expected loss on the portfolio.
(a) $240,000
(b) $500,000
(c) $1,000,000
(d) $250,000

Your Answer is Incorrect

The correct answer is choice ‘d’

The easiest way to answer this question is to ignore the joint probability of default as that is irrelevant to expected losses. The joint probability of default impacts the volatility of the losses, but not the expected amount. One way to think about it is to think of asset portfolios, where diversification reduces risk (ie standard deviation) but the expected returns are nothing but the average of the expected returns in the portfolio. Just as the expected returns of the portfolio are not affected by the volatility or correlations (these affect standard deviation), in the same way the joint probability of default does not affect the expected losses. Therefore the expected losses for this portfolio are simply $1m x 10% + $1m x 15% = $250,000.

This can also be seen from the lens of a joint probability distribution as follows:

307.29.e

There are four possibilities for this portfolio:
– Only loan A defaults: loss of $1m: 9% probability
– Only loan B defaults: loss of $1m: 14% probability
– Both loan A and B default: loss of $2m: 1% probability
– Neither A nor B default: loss of $0m: 76% probability

Therefore the expected losses on the portfolio are ($1m x 9%) + ($1m x 14%) + ($2m x 1%) + ($0m x 76%) = $250,000.

[Jeffrey Chudy]

PRM 3 – Question 141 vs 143 – Calculate expected loss on 2 bond portfo

[Anonymous]

Whichever way you do it, you get the same result (and the q.141 solution shows it done both ways to verify this point). You can either assess the 3 scenarios, or just ignore the joint probability and sum the expected loss on each loan.
Let’s say that we have two loans: Loan1 worth $A and Loan2 worth $B. Let the (marginal) probability of Loan1 defaulting be P1 and that of Loan2 be P2. Let’s say the joint probability of both defaulting is P12.
This means that the Probability of Loan1 defaulting but Loan2 not defaulting is P1-P12 (for example,look at the 9% in q.141, which is 10%-1%). And the probability of Loan2 defaulting but Loan1 not defaulting is P2-P12 (for example,look at the 14% in q.141 which is 15%-1%). So the 3 scenarios and expected losses are:
Scenario1 – Both default: Expected Loss = P12*(A+ = P12*A+P12*B
Scenario2 – Only Loan1 defaults: Expected Loss = (P1-P12)*A = P1*A – P12*A
Scenario3 – Only Loan2 defaults: Expected Loss = (P2-P12)*B = P2*B – P12*B
When you add these three totals the terms involving P12 (the joint probability of default)cancel out, leaving a total expected loss of P1*A +P2*B. And this is the Sum of the Expected Loss on Loan1 plus the Expected Loss on Loan2.

Jim

[Anonymous]

Actually the math is the same in both the questions. The second question you have cited has explained both methods, but the first one has only one (and unfortunately only the more complex method).

You can use the easier way to solve this as well: default correlation does not affect expected losses, but their volatility. You can calculate the expected losses of the two bonds and add them up, ie, $50m*0.03 + $50m *0.08 = $5.5m

I have expanded the explanation for the first one to include the above. Thank you for pointing this out.

[Author]

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