How CDS Par Spreads Determine Upfront Payments

# How to interpret the 'price' of a CDS?


# How to interpret the 'price' of a CDS?












I'm looking for an intuitive explanation of how to understand the 'price'/trade spread of a CDS.

Looking say at a current CDS on Santander, the index states that it is currently at 132. As I understand it, this is the trade spread.

I also understand that the 100bp coupon gets paid quarterly (i.e. 25k per quarter).

What is the actual price I would have to pay today to own this CDS? This is the bit I am not understanding. How does a value of 132 translate into a price? (I originally thought that 132bp * notional would be the price?)

## Answer by Dom (score 12)

https://quant.stackexchange.com/a/30263

A CDS is a contract with a protection leg that pays (100%-Recovery) immediately following a credit event if it happens before maturity, and a premium leg in which a coupon of 100 bps is paid until a credit event or maturity. Hence the value of $1 a short protection (receiving spread) contract is

V = 100 bps x PV of $1 per year paid to sooner of a credit event or maturity - Protection Leg PV

We call the PV of $1 per year paid to a credit event or maturity, whichever occurs sooner, the risky PV01 or RPV01 for short.

The CDS par spread is the spread that would make the value of the contract with the same maturity equal to zero right now. In your example this is 132 basis points. So we can write

0 = 132 bps x RPV01 - Protection Leg PV

The value of the Protection leg is therefore

Protection Leg PV = 132 bps x RPV01

It is the same protection leg as in V. Substituting this into V we have

V = (100 - 132 ) bps x RPV01 = - 32 bps x RPV01

This is negative because we are receiving 100bps to assume a risk for which we should be receiving 132bps. The 132bps is a measure of the credit risk of the CDS.

To compensate us for the fact that the contract has a negative value we must be paid an upfront amount equal to

U = 32 bps x RPV01

to get us to enter into it. We then put this cash into our cash account. The derivative has the opposite sign so its value offsets this cash amount and so we have not made anything on Day 1.

As for the RPV01, this is calculated using a model that extracts the probability of default from CDS spreads. But it is close to the PV of \$1 per year for the remaining life of the CDS and so for a 5 year CDS expect it to be 4-4.5.

A simple model for the value of a short protection CDS can be found if you write

V = (C-S) x RPV01

where

RPV01 = $\left (1-\exp\left(-gT\right)\right)/g$

and $C$ is the coupon, $S$ is the par CDS spread, $T$ is the remaining life in years and

$g=r+S/(1-R)$

where $r$ is the risk-free (Libor) rate and $R$ is the expected recovery rate, usually set to 40%.

If I set $r=0.02$ and $T=5$ for a notional of \$10m then I get $V$ equal to -\$144,317. So to enter into this contract I would receive an upfront payment of \$144,317.

## Answer by M. Jeunesse (score 6)

https://quant.stackexchange.com/a/30259

A spread of 132 means that buying the protection will cost you 132 bps per year up to the default or the maturity with no upfront.

Because of standardisation of the coupons, there is an upfront. So if the spread is 132 and if the coupon is 100bps, and if you buy protection you will pay something upfront because 132bps is what you should have paid per year to buy the protection with no upfront.