Handbook example 24.4

[jcjc0602]

200m portfolio, one-year probability of default is 4%, recovery is 60%, defaults are uncorrelated over years. What is the cumulative expected credit loss on the portfolio?

I've seen two versions of solution:
1. year 1: 200*4%*(1-60%)=3.2
year 2: (200-3.2)*4%*(1-60%)=3.15
total is 3.2+3.15=6.35
2. cumulative probability of default=1-(1-4%)*(1-4%)=7.84%, then times 200 and 40%, I got 6.27

I don't know which one is correct. Anybody can help me ? THANKS!!

 

[aadityafrm]

I think the second method is accurate as it uses the cumulative default probability for two years. I visualize it as follows -

Expected value of cash flows in two years = X0*p+X1*p(1-p), where X0, X1 are cash flows in year one and two and p is the default probability. X0=Face value of portfolio*(1-recovery rate). now assuming a zero coupon payments, for both the years then X0=X1= 200*(1-0.0.6).
then expected value = 200*(1-0.6)*0.04+200*(1-0.6)*0.04*(1-0.04) = 6.27

If there are coupon payments then X0= 200*(1-0.6) and X1=(200+C)*d*(1-0.6) where d is discount factor. Hope this helps.

 

[sleepybird]

I think the first method is correct and more intuitive (but I could be wrong), i.e., what could go wrong if you do it year by year?
The difference between the 2 methods seems to come from the usage of recovery rate (or LGD).
The first method explicitly uses the recovery rate twice: in year 1 and then in year 2.
The second method seems to ignore the recovery rate in first year and only considers it in year 2.
I think the formula: cumulative PD =1- (1-PD1)(1-PD2|no default in Yr 1) only works when assuming LGD=100%.
Thanks.