Hi
@RushilChulani I think I do understand why that's not obviously intuitive. After all, it's different than how the contingent payoff is treated which may be more intuitive in contrast. After all, the contingent payoff probabilities are unconditional (aka, joint) default probabilities which means they are mutually exclusive and naturally additive; for example, the year 5 payoff pd is only 1.84% because it is a joint probability given by 92.2% cumulative probability thru the end of year 4 (the prior year) * 2.0% conditional probability. On this payoff side, perhaps it is more intuitive because the 2.0% is a conditional PD such that we don't perceive the double-counting.
But the difference on the payment side is that the
protection buyer pays the spread every year as long as the reference survives (as opposed to the protection seller who only pays once, if at all); i.e., if the reference never defaults, there will be five (not one!) spread payments. If there were only one spread payment at the end of five years, we would be double-counting. But in this case of a "stream of spreads" we are merely weighting the stream of payments by their probability of occurring. Let A(t) be the annuity factor; i.e., the sum of discount factors, which is 4.317. Imagine that survival is guaranteed, in which case five (5) spreads are paid with a present value = 4.317*S. Implicitly that would be to assign 100% to each of the five cumulative probabilities because each spread is guaranteed: S*100%*0.951 + S*100%*0.905 ... S*100%*0.729. Notice how that isn't double-counting? So these probabilities are just weighting each year's spread payment. I hope that's helpful!
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