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How to Price Risky Bonds

In previous articles, we have laid out the basic properties of debt valuation in a setting where interest rates are constant (or at most, non-randomly time dependent) and the issuing party is deemed to be of high credit worthiness. A high credit rating is often a good indication that the risk to investors of the issuing party defaulting on the repayment of the principal is small. In this article we will consider the converse: a significant risk of default and how this impacts the pricing of bonds.

As a first step we can ask ourselves how the price of a risky bond (e.g. a corporate, emerging market or junk bond) should compare with a risk free bond (e.g.Treasury bills, large corporates, UK Gilts). In any rational market investors will demand a higher reward for taking on the risk of default, i.e. a higher yield. This signals that the price of risky bonds should be lower than the risk free counterparts. In general, the lower the issuing party's credit rating the lower the bond price.

In an earlier article we derived a derivative pricing scheme for counterparty credit risk using the counterparty share price as an indicator of default. Although the same methodology can be applied here, this is not usually adopted due to its weakness - The alternative approach that is more frequently used by practitioners is the 'intensity approach'. This follows from the Poisson process as a model for the occurrence of default; the first jump observed by the process signals the default event. Since the Poisson process is a random process default occurs instantaneously without warning. Sometimes this is not a reasonable model to use but at others it exactly mimics what is observed in the market (e.g. Enron, strong share price, sudden default).

Recapping default free bond pricing, we know that the price of a bond is the discounted future cash flows (coupons and notional). For a zero coupon bond (which only pays off a principal at redemption date T and no coupons) this becomes:

In all that follows we will use to denote the time of default. This is a random variable but it is not a process. Using the Poisson process, the time to default will be modeled as the waiting time until the first jump of a Poisson process; it is hence exponentially distributed with some parameter (see earlier article). In finance, the parameter is usually termed the 'intensity' of default.The bond pricing equation above is strictly for cases where default is not significant. Once this is not longer the case, the return of principal itself is no longer certain. In fact, we can talk probabilistically of there being a chance that default does not occur whereby the entire principal is returned. We also include the chance that default does occur whereby the bond does not pay out the principal, but the recovery rate (put in links). For simplicity we assume the recovery rate to be zero. Therefore in a loose sense, the price of the bond can be considered as the expected future cash flow:

Hence, once we have we can evaluate the price of the bond can be evaluated a number of ways. We can look at historical default rates or infer it from some economic study. But this suffers from the fact that it is investor dependent (the value will as different investors have different perceptions towards the credit worthiness of a company). Furthermore, it can not be a constant since this produces a very unfeasible structure for the probability of default. Recall that the pdf of default as a function of maturity is calculated as - this is just the pdf of the exponential distribution. This is exponentially decaying behavior implies that the highest risk of default is in the short term.

In fact, markets do not share this view in general. In general the view is this: In the short term (few months) a currently healthy company has little chance of default (this is true for all corporates of high rating).
In the long term (few years), the chance of default is also low; the intuition here being that if a company were to default it would have done so already. The highest risk of default is in the medium term.

This humped shaped default profile is clearly not reflected by the exponentially decaying structure derived thus far. Does this make the Poisson model redundant? Well, not really because we can do something to change it so that it produces this favorable structure. And we can use for this purpose.We will now demonstrate how we can use the technique 'bootstrapping' to deduce'market implied intensity parameters' which result in a humped shaped probability profile for default. In trying to determine (from now called because we accept that it can never be constant) we will use the following example.

The prices of risky bonds are those from another credit risky company for similar rating/size/sector to the issuing company. If the issuing party already has bonds outstanding then they can also be used.

The price of a bond now becomes

For the shortest maturity bonds for instance:

Here B1_free is the price of the risk free (Treasury) bond. Since we have no information about what the market thinks about default within year 1 we can only make the simplest of assumptions that is constant over this period:

We can now write the price of the second risky bond as

Making the assumption that is constant over the second time period we arrive at:

And so on... This is bootstrapping.

Strictly speaking, this calculated here is the intensity for the risky bonds that are already in circulation in the market. We are making the explicit assumption that this same profile holds for the company issuing the new bonds which we are seeking to price. Therefore, the closer the 2 companies resemble each other the better this assumption is likely to be (same size, sector, profile, etc.). Using the example in the table above we have bootstrapped out the following profile for the default probability as a function of time and it gives us a more favorable shape.

Note that the humped shape need not always be the case. It all depends on the values of the bonds in the market and thus on investor sentiment. Taking this approach we have completely stripped out any individual attitude towards default risk and considered only the general market opinion. There is nothing to stop the market from holding the general view that short term bears the highest risk of default, in which case the rate of default will resemble the first graph above.

The advanced reader will notice that this is in fact exactly what happens in the valuation of bonds under stochastic interest rates, where the 'market price of risk'  is used to adjust the model to the market's attitude towards interest rate risk. Here, we have implicitly incorporated a 'market price of default' risk. The analogy with interest rates is an important one and extends from the fact that default, like interest rate, is not a tangible asset that can be traded. You cannot short interest rate, in the same way that you cannot short default and therefore default hedging, like interest rate hedging is a tricky business.

Written by Samy Mohammed

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