calculating expected loss on a 2 bond portfolio

[Jeffrey Chudy]

calculating expected loss on a 2 bond portfolio

[Anonymous]

Can you please explain what the difference is between these two and why the calculation is not the same?

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) $500,000

(b) $250,000

(c) $1,000,000

(d) $240,000
The correct answer is choice ‘b’
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.

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) $5.26m

(b) $5.5m

(c) $1.38m

(d) $11m
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.

[Anonymous]

With these questions of default(D)/no-default(ND) I like to make out a simple table as follows:
A
D ND
B D 0.01 0.14 0.15
ND 0.09 0.76 0.85
0.10 0.90 1.00
The marginal probabilities are given in the question as 0.10 and 0.15. You are also given the value of 0.01 for both defaulting together. Once you have these numbers and the overall total, which must be 1.00, you can work out the rest. The 0.85 and 0.90 come from subtraction of the 0.15 and 0.10 from 1.00. And then the other cells with 0.14, 0.09 and 0.76 follow immediately.
So the expected loss is:
Probability both fail x 2,000,000 = 0.01 x 2,000,000 = 20,000, plus
Prob A defaults and B doesn’t x 1,000,000 = 0.09 x 1,000,000 = 90,000, plus
Prob B defaults and A doesn’t x 1,000,000 = 0.14 x 1,000,000 = 140,000
= $250,000

This method will work for the other question below – you just have to recognise the fact that zero correlation implies independence which implies that the probability of both defaulting together is 0.03×0.08 = 0.024. In the question above you were given this as part of the question. In this question they are NOT independent, i.e., 0.10 x 0.15 = 0.015 which is not equal to 1% as given in the question.

[Anonymous]

Sorry, but the formatting of the table seems to have been lost once I submitted it. So it’s a bit difficult to read. The table is essentially a 2 x 2 grid,with row and column totals making it 3 x 3.
The first row is 0.01, 0.14, 0.15, and so on for the other two rows.
The columns show probabilities for A defaulting and not defaulting, and the rows are for B.

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