Survival probabilities starting from CDS spreads

Question

How is that possible to get survival probabilities starting from CDS spread? Could you please provide me with a demonstration?

What is more, is that true that CDS Zero type is necessary so as to get survival probabilities? If so, why?

Answer 1

OK, here is a simplified demonstration:

Before we consider swaps, let us consider very simple bonds. Suppose that you have a choice of two zero-coupon bonds. A riskless one costs 95 and is certain to pay 100 in 1 year. A risky one costs 90, is expected to also pay 100 in 1 year, but with some probability $p$ will default and only pay some $R<100$ on the dollar. Making some assumption about what $R$ is, you can solve for $p$, which depends on $R$. ($p$ is the risk-neutral probablity, thanks @Kermittfrog ).

Now let us consider swaps. Suppose you can buy 1 year of CDS protection by paying $U$ upfront for each dollar of notional. Suppose it's 30 basis points, i.e. 0.003 of the notional. This is the entire fixed leg (no running spread for now).

The CDS contract says if a credit event happens, then you receive \$1 notional notional and deliver $R$, the recovery value of the post-default debt (floating leg contingent on a credit event). But if a credit event does not happen, then you get nothing.

Let $p$ denote the probability of the credit event.

You don't know $R$ until after the credit event. Let us assume that it'll be 40%, i.e. the defaulted bond will be worth 40 cents on a dollar. (These are some commonly used $R$ assumptions, like 40% for corporates, 25% for emerging markets, etc.)

So you pay 0.003 now and with probability $p$ you receive 1 and pay 0.40 later, i.e. you receive net 0.60. You don't know when "later" is, so let's assume you might get it in 1 year, at your contract's maturity. With probability $1-p$, you get nothing on this leg.

Suppose that as a credit risk free alternative, you can invest money for 1 year and earn $r=1\%$ interest.

So, putting this all together, you invest $U$ and the expected value of what you receive is $p * (1-R)/(1+r)$. This should earn you the same interest $r$ on your invetment $U$ as a risk-free investment. Now you can solve for (riks-neutral) $p$ using closed form.

As you see, this $p$ depends on your assumption about $R$. If in addition to the kind of CDS where the protection buyer receives notional minus recovery, you could also observe quotes for "zero-recovery" CDS variant, where the protection buyer receives the entire notional, and doesn't need to deliver the value of the defaulted bond, then you could solve for both risk-neutral $p$ and risk-neutral $R$. Such CDS used to trade, but pretty much stopped after 2008, so their quotes are impossible to observe.

In reality, CDS contracts are a little more complicated. As part of the fixed leg, in addition to the upfront, the protection buyer pays running spread (most commonly 1% of the notional a year, paid quarterly), which accrues until the day of the credit event (and so, depends on $p$ a little). Also if the credit event happens, the buyer gets paid soon after that, not at maturity. Risk-free interest rates have term structure, athough in reality they have very little effect on any of these credit numbers. A few people even have term structure for $R$, although most don't. All these bells and whistles don't actually make the math much more complicated. Instead of closed form, you'll probably use some numeric iterative solver to get $p$'s. But this is the general idea - solve for $p$ that gives the two legs of the swap the same value.

Furthermore, only a few CDS (usually very distressed names) are quoted as upfronts (usually corresponding to 5% annual running spread). Most CDS are quoted as a running spread that you'd pay annually if the upfront were 0. But when you execute the trade, it actually uses the upfront and the standard running spread.

Obviously, $p$ is not a single number, but has term structure - the probability of default within 2 years is likely to be more than the probability of default within 1 year. To interpolate bbetween the dates where risk-free $p$ can be derived from observed CDS quotes, most people assume that the hazard rate (the thing that causes the default probability to go up) is constant. This is similar to assuming that for interest rate curves, forward rate are constant between observable quotes and leads to similar problems, but fewer people use some kind of smoothing for survival curves.

My advice to you is to look at the ISDA CDS Standard Model, which implements all these bells and whistles. It's adequately documented and open-source. Walk through the code and understand the calculations.