# What is the connection between default probabilities calculated using the credit rating and the price of a CDS?

I'm working on a tool to price Credit Default Swaps. I've already done the standard pricing tools. I'm working on a pricing tool which uses the credit rating for the default probabilities used in the pricing of CDS. What is the relation between these probabilities and the price?

• Hi David, welcome to quant.SE and thank you for posting your question. Please consider registering so that the site can track your contributions regardless of where you log in from. Also, can you please elaborate on what kind of relationship between probability and price you are looking for? If you have a set of default probabilities as a function of time, then finding the price is simply a matter of setting the NPVs of premium payments equal to default payments. – Tal Fishman Sep 13 '11 at 14:42
• Hi Tal. Thank you for your answer. I've registered now, sorry. I've a set of default probabilities per rating as a function of time. But can you give more details to find the price using these probabilities please? Many thanks. – David Sep 13 '11 at 14:51
• Because I'd like to use also the cds spread. Because if I don't, the price of the CDS will be the same for corporates with the same rating... – David Sep 13 '11 at 14:53
• I do not understand what is your distinction between the CDS price and the CDS spread. CDS are typically quoted in terms of either spread of points upfront + 500 bp running. The value of CDS, like any other swap, is zero at inception. Are you looking to find the current value of an existing CDS contract should it be unwound? – Tal Fishman Sep 13 '11 at 15:09
• Exactly, that's what I want to do. I've used the paper of Hull & White for the standard pricing using the cds spread curve to have my price. Now I've calculated my default probabilities using the credit rating and I want to price existing CDS but I don't know the relation between the price and these probabilities – David Sep 13 '11 at 15:13

Your transition matrix $M$ has a time horizon associated with it, typically one year but sometimes 3 months or 5 years. Assume for convenience the horizon is 3 months. If it is not, you may wish to take a matrix square root to turn it into a 3 month matrix.

Now the 6 month transition probabilities are formed by multiplying the matrix with itself, $M \cdot M$ and the process can be repeated. So $N$ quarters into the future, the appropriate matrix is $M^N$. Let us take the convention that the first row $\{d_{1,j}\}_{j=1}^R$ of $M$ represents default. Let's say the current rating corresponds to row $I$.

A CDS has 2 kinds of cashflows, coupon payments $c_n$ and a default payment $D_n$ (normally $c_n$ and $D_n$ are constant). The coupon payment $n$ quarters from now has probability $p_n$ of occurring that you can read as the non-default entries for the initial rating's row, $\sum_{j=2}^R (M^N)_{I,j}$, or more easily as $p_n=1-(M^N)_{I,1}$ since the probabilities must sum to 1.

The coupon leg $L_C = \sum_{n=1}^N PV_n p_n c_n$ of the CDS has value corresponding to the present value of these cashflows times their probabilities of occurring.

The default leg is priced similarly. A default payment $D_n$ occurs only on the occurrence of a fresh default at iteration $n$. The probability of this occurring is the sum of probabilities of achieving various ratings in the iteration just before default, times their respective probabilities of freshly defaulting in one more iteration. That is to say, the total probability is is $q_n=\sum_{j=2}^R (M^{n-1})_{I,j} M_{j,1}$. This payment may occur at any iteration $n$ occurring before default, so you have to total up the default value contributions for all $n$ prior to the contract expiration.

The default leg $L_D = \sum_{n=1}^N PV_n q_n D_n$ has value equal to the sum of these payments times their probabilities of occurring.

The overall contract value can now be written as $L_D-L_C$.

Technically, this is all known as a ratings migration model, and is used a lot for risk control. The ratings paths form something called a Markov chain.

• Hello Brian. Firstable, thanks a lot for your answer. The point is that since 2009, the coupon payments are fixed (for example 100bps) and don't take into account the value of the 5y spread CDS curve. So the problem is that for a same rating, the price will be the same and that's my problem and my question. Is there a way to add the CDS 5Y level into my calcul (for example to price a 5Y cds) please?? Many thanks – David Sep 19 '11 at 6:26
• You will have to alter your transition matrix to achieve that. For example, you could take probability off columns 2...K and put it onto column 1 (the default column) -- or vice versa as necessary -- until the transition matrix price agrees with the 5Y CDS rate. – Brian B Sep 19 '11 at 18:07

One could say that a CDS price is determined by the physical default probability and the risk premium.

The physical PD (PPD) is the actual probability of company defaulting within the given period of time. It is purely a theoretical concept as no one really knows what this probability is. We could estimate it using some models or credit ratings, but those are just guesses.

In other words, if you'd known a PPD precisely you'd be able to calculate a break-even CDS price. If you write a lot of CDSs at break-even prices (on different underlyings), some will be triggered, some won't - but on average you will not make or lose money.

Of course, there is no point in doing this. So you would actually add a margin to each break-even price, so that you will make money on average (again assuming that PPDs are perfectly known to you). This margin is exactly the risk premium.

In reality, you are financially constrained. If all underlyings default you'll have to default yourself. If you are responsible you'll make your best to ensure this risk is tiny.

In the real world there's peer pressure. If you are a CEO of a company A and you see the company B is getting tons of money by writing CDSs you can start thinking along the lines of "Why are we not doing it already?". Best case you'll make money and become a great CEO, worst case you'll get your golden parachute...

Getting back to the original question. If you use credit ratings to calculate physical PDs you will find that a lot of variation in CDS prices is due to changes in risk premiums. Is it actually the case or is it that the credit ratings-based PDs are inaccurate and do not reflect all the information available up to date?

Risk premium filtered in such a way are not easily explained by other macroeconomic or financial indicators. For instance, it is highly correlated with VIX, but still is quite different.

This is a valid area in academic research in Finance. You could check "Measuring Default Risk Premia from Default Swap Rates and EDFs" by Berndt, Douglas et. al. There are slides and the paper itself online (look for the latest versions).

You should use the ratings-based default probabilities to derive the "fair" spreads on a set of hypothetical new contracts and compare this result to the market spreads. Each could then be used independently to also derive the price for an existing CDS. There is no set way to combine the two prices, as these are two completely different and independent approaches to solving the same problem.

Well I m affraid that there is a little bit of confusion here. Ratings are ... Ratings usually when used by notation agencies they imply a definite fixed once for all default probability (or transition matrix to some other rating) and then issuers are classified among those ratings usually by using some historical data. When using CDS spread then you get market implied default probability for some period and you bootstrap it from CDS quotes by using standard procedure, you can have a look at Brigo and Mercurio's book for the details. Hope this helps Regards

• Thanks for your answer. The point is that I've already made a pricer using the market implied default probabilities. Now I'm working on a different version of the pricing using a transition matrix (from AAA to default) to generate default probabilities. And what I want to do now is a way to use these probabilities to have a price for a CDS. I know that it's not the way the CDS are priced on the market, but I'd like to try this method. Perhaps you can help me. I'd like to use the default probabilities as a function of time and also the CDS curve to price the CDS please. Many thanks – David Sep 14 '11 at 14:47