Question

Consider a portfolio of three bonds A, B and C. The credit exposure (CE) and default probabilities (p) are given in the following table: Issue CE

p A 30 0.12

B 20 0.05

C 50 0.1

Assume that the defaults are independent across the three bonds. Calculate the mean and variance of the expected losses. Comment on the results.

Answer 1

Given Information:

Issue	Credit Exposure (CE)	Default Probabilities (p)
A	30	0.12
B	20	0.05
C	50	0.1

Notes:

1. Credit Exposure is a measurement of the maximum potential loss to a lender if the borrower defaults on payment. e.g. If a bank has issued a total loan of $ 10 million, its credit exposure is $ 10 million.

2. Default probability is the probability of a borrower defaulting on payments. The probability is applied to the credit exposure to arrive at the expected loss i.e. Expected Loss = Credit Exposure * Default probability

Hence, expected loss for the 3 issues are:

A = 30 * 0.12 = 3.6

B = 20 * 0.05 = 1

C = 50 * 0.1 = 5

a. Mean of expected Loss (EL):

Mean of expected loss is the average of expected losses calculate above.

Mean (μ) =     

Mean (μ) = (3.6 + 1 + 5) / 3

Mean (μ) = 3.2

b. Variance of expected loss (Var):

Variance is the measure of variability of returns. It denotes the spread of returns i.e. how far is each measure from its mean. A large variance denotes greater risk, as in a bad economy, the expected loss can surpass the risk appetite of the lender.

Variance (Var) =

Expected Loss (EL)	EL - μ	(EL - μ)2
3.6	0.4 (3.6 - 3.2)	0.16
1	-2.2 (1 - 3.2)	4.84
5	1.8 (5 - 3.2)	3.24
		8.24

Variance = 8.24 / 3

Variance = 2.747