Valuation of Credit Default Swaps; in plain English
This article is primarily sourced from Modelling the Price of a Credit Default Swap by Agnieszka Zalewska.
In very simple terms, CDS is an agreement to insure the cash you will get in the future. Similar to a common health insurance, where you pay a premium to insurance company to cover your expenses when you have an accident, you have to make a payment to the financial institution to insure those cash receipts in case that borrower defaults. And when you had an accident, that means when your bond defaults, you can knock your insurers door, and receive the payment.
So let’s say you invest into a bond, and this bond will make semi-annual payments.Then, you go to an insurer and make an agreement to insure it; which is called Credit Default Swap, consequently you start paying a premium for each installments, at the same time receive the coupon payments from the borrower. At any time let’s say your borrower is not able to make a payment, the insurer will step in and make the remaining payments. Thus your cash flow has two legs, one coming from your borrower, two from your insurer when default occurs but second leg will start after the default event is triggered. So the value of your CDS is the difference between those two legs. If the borrower has strong credit quality, then the probability is low that it will default and insurer will step in and make those payments, also borrower will have enough assets that you can liquidate and recover your money, so CDS premium will be minimal. But if the borrower has low credit quality, rated below Baa1, and has junk assets, probability that it will default is high and the money you may recover is nearly zero (extreme example). So the value of CDS will skyrocket.
Now you got the basics, let’s move to Mme. Zalewska’s paper
The price of the credit default swap is described by the differences between the cash flow in both legs (the legs mentioned above). Assume that the par value is normalized to unity and in the credit event the buyer receives the par value minus recovery rate R.
T = tn: The life time of the credit default swap
Q(t): The probability of having no default
K: The fixed annual insurance payment to compensate for the risk of default
R: The recovery rate reducing the payment in the case of credit event
P(t; T): The discount function for a zero bond at time t maturing at T
C(s; t): The accrual function denoting the fraction of year between date s and t
Vfixed: The present value of the fixed leg
Vdefault: The present value of the default leg
VCDS: The present value of the credit default swap
Don’t get overwhelmed by the alphabet soup. Just replace the terms with the words and it will be simple like
Value of fixed is the total of all Cash Payments multiplied by the probability of payment, discounted.
Value of default is a cumulative function that any time the borrower will default times amount you can recover; discounted.
Thus the value of CDS is the difference between the Value of Default and Value of Fixed payments. Voila
the paper can be found there: