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September 2007 Q7

K

kika258

Member
Can someone please explain in greater detail the answer to this question?

Investor 1

Why is the variance 1000^2 x 0.04 x0.96, isn't the variance for a binomial, npq? so why 1000^2 and not 1000?

Investor 2

how is the probability of shortfall equal to c.100%?
 
Yes, the variance of a binomial is n*p*q. For Investor 1 , n =1 as there is only one "trial" or "event" as all the bonds either default or they don't. So, we can in effect treat the 1,000 bonds a s a single bond with a payoff of 1,000.

So, the variance if there was a single bond worth £1 would be: n*p*q = 1*0.94*0.04 = 0.0384

However, here the payoff if they don't default is £1,000 and so we need to times by 1,000^2 to get the overall variance, which therefore equals:

1,000^2*0.0384 = £38,400

For Investor 2, the probability that each individual bond does not default is 0.96.

So, the probability that none of the 1,000 independent bonds doesn't default is:

0.96^1,000 = 0 (approximately).

So, the probability that at least one defaults (and so Investor 2 gets less than £1,000 back) is approximately 1.
 
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