# Fixed coupon for CDS index

Here is the fixed coupon for CDS index in John Hull's book Options, Futures and Other Derivatives 9th page 580.

Actually, I don't much understand the goal of this part, here is some of my understanding, I am not sure whether it is right.

Assume there is only one company in the index, the buyer of CDS will pay the coupon $c$ to seller every quarter until the default occurs.

Seller will pay the the protection $(1-R)B$ when the default occurs. As the definition of spread, seller equivalently pays the spread $s$ to buyer every quarter until the default occurs.

$D$ is the present value of paying $1$ every quarter until the default occurs. So the present value of this contract of notional principle $1$ is $$D\times (s-c)$$

which is the amount buyer should pay for the seller at beginning.

Are the above statements right? But why do we need the coupon $c?$ We can set $c=0$ to simply the process i.e there is only one cash flow from the seller during the contract.

## 1 Answer

Coupons are there to reduce the counterparty risk between the seller and the buyer. If you didn't have that 500 bp coupon on a high yield bond the protection buyer would have to make a big payment upfront then wave it goodbye when the protection seller defaults. It also helps standardizing contracts (same quarterly payments whatever the spread you entered the contract at).

• thanks, is my understanding of the value of CDS index right? – A.Oreo Aug 30 '17 at 10:40
• No, the seller doesn't pay the spread to the buyer. The spread is just used to compute the initial cash outflow (and subsequently the mark-to-market). If the spread is 150 bps and the coupon is 100 bps, the initial price of the CDS is 50 bps * duration. – Lliane Aug 30 '17 at 14:03
• As mentionned in the book the regular quarterly payments are independent of the spread. – Lliane Aug 30 '17 at 14:10
• yeah,, I mean the seller only pays the protection when the default occurs, and the present value of this equals to pay the spread every quarter. – A.Oreo Aug 30 '17 at 14:20
• thank you for your help, I am recently reading credit products in John Hull's, maybe there are a lot of questions. – A.Oreo Aug 30 '17 at 17:22