As well as governments, companies also need to borrow money and one way of achieving this is to issue bonds. Like government bonds, these pay a fixed rate of interest each year. Of course, this rate of interest will typically be quite a bit higher than government bonds in order to justify the increased credit risk associated with them. The difference between the two rates of interest is called the credit spread.

As we did before, today I’m just going to look at pricing a ZCB, but coupon-bearing bonds can be constructed as a product of these so it’s a relatively straight-forward extension.

Consider a ZCB issued by a risky company. It’s going to pay £1 at expiry, and nothing in-between. However, if the company defaults in the mean-time we get nothing. In fact, we’ll extend this a bit – instead of getting nothing we say that we will receive a fraction R of £1, since the company’s assets will be run down and some of their liabilities will get paid. This also allows us to distinguish between different classes of debt based on their seniority (we’re getting dangerously close to corporate finance here so let’s steer back to the maths).

In order to protect against the risk of loss, we might enter a Credit Default Swap (CDS) with a third party. In exchange for us making regular payments of £K/year to them (here we’re assuming that the payment accrues continuously for mathematical ease), they will pay us if and when the counterparty defaults. There are a few varieties of CDS – they might take the defaulted bond from us and reimburse what we originally paid for it, or they might just pay out an amount equal to our loss. We’ll assume the second variety here, so in the event of default they pay us £(1-R), since this is what we lose (up to some discount factors, but we’re keeping it simple here!). Also, if default occurs we stop making the regular payments, since we no longer need protection, so this product is essentially an insurance policy.

We value this product in the normal way. The value of the fixed leg is just the discounted cash flow of each payment multiplied by the probability that the counterparty hasn’t defaulted (because we stop paying in that case) – and remember that we said the payments are accruing continuously, so we use an integral instead of a sum:

and the value of the insurance leg is the value of the payment times the chance that it defaults in any given interval

where in keeping with our notation from before, is the probability of default at a given time and is the chance that the counterparty hasn’t defaulted by a given time.

The contract has a fair value if these two legs are of equal value, which happens when

At this point we refer to our exponential default model from Part I of this post.

In the exponential default model described in the previous post, we postulated a probability of default which led to Survival Function . Substituting these in to the equation above we have

which is only zero when the integrand is also zero, so

This is called the credit triangle, and it tells us the price of insuring a risky bond against default if we have it’s hazard rate. If the expected lifetime of the firm increases (ie. decreases) then the spreads will fall, as insurance is less likely to be required. If the expected recovery after default increases ( increases) then spreads also fall, as the payout will be smaller if it is required.

Although a little care is needed when moving from ZCBs to coupon-bearing bonds, it can be seen that the payments (normally called the spreads) paid to insure the bond against default should be essentially the difference in interest payments between government (‘risk-free’) bonds and the risky bonds we are considering.

The default probabilities we have used here can be calibrated from market observed values for spreads K between government and corporate bonds. This resembles the process used to calculate volatility – the observable quantity on the market is actually the price, not the volatility/hazard rate, and we back this out from market prices using simple products and then use these parameters to price more exotic products to give consistent prices. This is the market’s view of the default probabilities in the risk-neutral measure, which means that they may not actually bear any resemblance to real world default probabilities (eg. historical default probabilities) and in fact are usually larger and often several times larger. See for example this paper for more on the subject.

Of course, throughout this analysis we have ignored the risk of the insurer themselves defaulting – indeed, credit derivatives rapidly become a nightmare when considered in general, and the Lehman Brothers default was largely down to some fairly heroic approximations that were made in pricing credit products that ultimately didn’t hold up in extreme environments. I’ll explore some of this behaviour soon in a post on correlated defaults and the gaussian copula model.