# If a CDS is x%, why does this mean $(1 - x)\%$ probability of default?

At 11:52, Tim Bennett says:

Well let's have a look at 11:52 some prices. Currently the cost of 11:57 insuring Greek debt is in excess of 12:02 $$\color{green}{2,000}$$ basis points. Another way of 12:07 looking at that is that you are paying as a 12:09 premium annually $$1/5$$th of the value 12:12 of the debt in question. So another way 12:16 of looking at that is: the market is 12:19 effectively giving the chance of Greeks 12:20 going bust within five years over $$\color{red}{80\%}$$ probability.

Doesn't the cost a Credit Default Swap (CDS) for Greek debt is $$\color{green}{20\% (= 2000}$$ bp), entail that the chance of default by Greece is also 20%? Why $$\color{red}{80\%}$$?

• Your "$1-x$" interpretation may not be what this person had in mind.
– user18
Apr 10 '20 at 3:53

He might have thought that $$1 - Prob(\text{no default for five years running}) = 1 - (1 - 0.2)^5 > 0.8$$ but this is not true as $$1 - (1 - 0.2)^5 \approx 0.67$$
If x is a Bernoulli trial with probability $$p$$ then we can use the binomial distribution to calculate, as @Giskard does, the probability that something doesn't happen in n trials (here five) as $$1-(1-p)^5$$. And, as he or she points out, this implies a probability of about 67% if p=.2.
$$E[L] = p_{default} \cdot E[Loss|default]$$
You can see that if the $$E[Loss|default] = LGD = 1$$, then the price of the CDS is the expected loss and also the probability of default. If so, then the remainder of the argument is as @Giskard makes it. However, if the $$LGD<1$$ is lower then this implies the the probability of default is higher at a given expected loss. For example, at a 55% recovery rate (at a standard assumed LGD for sovereigns of 45%) and expected loss of 20% per year, the probability of default per year is 44.4%. An annual probability of default of 44.4% implies a 5 year default rate of 94.7%.