Now let's consider how we might price a credit default swap by imagining we are faced with providing protection to somebody for a one year period in the deal.

You are required to pay out a million dollars if the reference entity defaults or zero if they don't.

The question is how do you decide how much you need to receive upfront? To answer this question, we need to understand the probability of default.

That is how likely is it that we are asked to pay out the million dollars? Let's say we assess this probability to be 5%.

That means that on average we would pay out $50,000.

This average payout is calculated by multiplying the probability of default by the amount paid in the event of default.

If you imagine running this trade 100 times in 95 instances, you pay out nothing but five times.

You pay out a million.

So over 100 trials, you would pay out 5 million giving an average of 50,000 per trial.

In finance, we call this average outcome the expected loss, and the $1 million.

Here is our loss given default.

So the expected loss is the default probability times the loss given default note.

In practice, we would need to discount our future loss back to today, but we are ignoring that here for simplicity.

So pricing credit risk begins with calculating your expected loss in an effort not to overcomplicate matters.

We will work on the assumption that you are happy to receive an amount which exactly covers your expected loss, and you require no extra risk premium in order to enter into this trade.

This is what is known in finance as risk neutral pricing.

In practice, this is not how people behave, but for now, we will park the implications of this and proceed mathematically on the risk neutral basis.

Now, let's extend our example into a second year.

The payout in the event of default remains the same at $1 million, but could now happen in year one or in year two, or not at all.

The amount we receive is now X at the start of year one plus another X at the start of year two.

As long as the reference entity does not default in year one, we keep the branching probability of defaulting or not in any year at 5%.

But note that in the second year, this Is now a conditional probability.

That is it tells us the likelihood of a default in year two, given the reference entity survives year one.

From this information, we can work out the probabilities of the three possible outcomes.

Firstly, the probability of surviving both years is 95% squared, which is 90.25%.

The probability of defaulting in year one we already know is 5%.

Therefore, the remaining probability of defaulting in year two must be the number which makes the sum equal to 100%, which is 4.75%.

Note, we could also have calculated this by multiplying 95% by 5%.

To illustrate why these calculations work, consider 10,000 companies all start year one.

At the end of year one, nine and half thousand will still be alive, and 500 that is 5% of 10,000 will have defaulted.

Now, as we go through year two, 95% of the nine and half thousand companies that survived year one will also survive year two, giving an amount equal to 95% of nine and 5,000, which is 9,025 companies, or 90.25% of those which started of the nine and 5,000 companies that survived year one, 5% will default in year two, which as an amount is given by 5% times nine and 5,000, which is 475 or 4.75% of the number of companies we started with.

As you can see, as long as we know the branching probability in each year, we can calculate the probability of survival for the whole maturity and the probability of defaulting in each year.

Note that the 4.75% probability of defaulting in year two is an unconditional probability.

It describes the probability that this company defaults in year two only.

Now that we have calculated our probabilities, we can work out our expected loss, which is 5% times 1 million, plus 4.75% times 1 million, which equals $97,500.

We can add these amounts together because the outcomes are independent.

They can't both happen, and we pay out 1 million if we get a default in year one or in year two.

This then has to be balanced against the amount we need to receive to compensate us for this expected loss.

We will receive X at the start guaranteed, and then we have a 95% chance of receiving X again at the start of year two.

So in total, we received 1.95 x.

If we solve for X, we get $50,000, the same answer to our one year example.

This gives rise to an interesting result.

If the branching probability stays constant in this case at 5%, then the amount we need per period remains the same regardless of how long the deal lasts.

A five year version of this trade would give rise to an annual payment of 50,000 per year to offset the expected loss.

This branching probability, the probability of defaulting in year end, given that the reference entity survives to the start of year end is called either the default intensity or the hazard rate, and is often represented by the Greek letter Lambda.

We will use the name default intensity going forward.

It has hopefully become obvious that what we are pricing in the preceding simple examples was a credit default swap, albeit a simplified version.

In practice, we need to match the exact details to those of CDS, which trade in the market.

For example, they tend to have quarterly payments rather than annual, and the premium leg will accrue up to the point of default rather than discreetly each year.

The equation on screen shows the formula for pricing the CDS, even though it may look complicated, all it is saying is that at inception, the present value of the premium leg must equal the present value of the default leg.

The expected payment on each leg is weighted by the probability of it happening.

Survival probabilities for the premium leg and default probabilities for the default leg.

Then all projected payments are discounted back to today to compute a present value.