Sept 2007, Q7

[LastMile]

Question:
Consider a corporate bond that will return £1 per bond to an investor at the end of a year provided the borrower does not default during the year. The constant annual probability of default is 4%.
Investor 1 holds one thousand such bonds that depend on the same borrower.
Investor 2 holds one thousand such bonds, each of which depends on a different borrower. Each borrower defaults (or not) independently of the other borrowers, but with the same probability of 4%.
All bonds were purchased at par. (i) For each investor calculate (using suitable approximations if necessary):
(b) 95% Value at risk
................

I am clear with solution for investor 1.
Can someone explain how VaR be calculated for investor 2.

Thanks!

 

[Mark Mitchell]

Based on the solution in the examiners report...

The amount lost by Investor 2 follows a binomial distribution : Bin(1,000, 0.04) (as each of the 1,000 bonds has a probability of default of 0.04). This can be approximated by a normal distribution with the same mean and variance. So the amount lost follows:

N(1,000*0.04, 1,000*0.04*0.96) = N(40, 38.4)

The 95% VaR is the upper 5% point of this loss distribution (the point that 5% of losses exceed):

40+1.6449*sqrt(38.4) = 50.19

where 1.6449 is the upper 5% point of N(0,1) (see pg 162 of the Tables).