Cds Valuation
For any CDS, if we know five of the six variables which are maturity, discount rate, spread, recovery rate, default risk, value, we can calculate the unknown.
In this question, to calculate the 3-year CDS we bought one year ago, we still have one unknown which is the default rate, so here I will use bootstrap to calculate the marginal default first.
I’ll use the 0.5 year maturity CDS to illustrate how we use bootstrap to get the marginal default risk.
First of all, we’ll calculate future cash flow. As a protection buyer, I should pay spread (which is half of the annual spread) when t = 0.5 if default events doesn’t happen. And if default happens at the middle of half year, I may get paid for (1- recovery rate) * notional amount but at the same time, I should still pay the accrual interests (which is 1/4 of the annual spread).
Secondly, we need to calculate the probability of defaults occurring at different time. Assume marginal semi-annual default rate is d, in this half year CDS, the default rate should be d, and the survival rate should be (1-d).
Then we can get the nominator which is the weighted average of cash flow for protection buyer, the formula is listed below:
At t = 0.25: accrued interest: (2.25%/4) * d
At t= 0.5: cash outflow: (2.25%/2) * (1-d)
cash inflow: (1- 75%) * d
Then we calculate the discount factor using semi-annual compounding to get the present value of each cash flow. Since the value of CDS is 0 at the time of issuing, we can use this equation to calculate the only unknown default rate.
After knowing the first period marginal default rate, we can use this and other variables given in the questions to get the second period marginal default