# FIN40420 Fixed Income Securities-conditional default probabilities

1. A firm exhibits a constant hazard rate and has an expected recovery of 55% on its senior debt. The Libor curve is flat at 1% p.a., compounded annually. You calculate the appropriate five-year CDS spread as 11% p.a. However, you note that it is market practice to quote CDS to very highrisk entities on a ‘points upfront’ basis:

a. Calculate the ‘points upfront’ which must be paid in addition to 500bp p.a. there after for five years.

Points Upfront: For highly distressed credits, trading on spread can become problematic because the P&L distribution of buying or selling protection becomes much wider than the P&L distribution of buying or selling the underlying reference entity (even though the expected values are the same). In order to keep the distributions more in line, when spreads get very wide, the market convention is for the protection seller to require a 500bps premium plus an additional upfront payment from the protection buyer.

i. Plot the P&L distribution for the seller in both scenarios. Comment on the investment bank’s thesis.

2. A portfolio comprises five-year loans of €10m each to ten firms and the firms exhibit 20% asset correlation with one another. The conditional default probabilities, which have been extracted from the firms’ bond prices, are 2% p.a. out to five years for all the firms. The expected loss given default is 45% for each loan and the Libor curve is flat at 2% p.a., compounded annually:

a. Adopting a Monte Carlo approach with 1,000 simulations, calculate the cost of buying protection on the following tranches assuming LGD is fixed:

i. 0-3%
ii. 3-6%
iii. 6-12%
iv. 12-100%

b. If LGD is beta-distributed with a mean of 45% and a standard deviation of 25%, recalculate the cost of buying protection on the same tranches.

c. Plot the cost of the tranche protection for asset correlation in the range 0%-40% and for conditional default probabilities of 0.1%, 0.5%, 1%, 3%, 4% and 5% p.a.
i. When LGD is fixed, and
ii. When LGD is beta-distributed as described above