Price Implied Default Rates

Update: December 12, 2008.  While none of our analysis or calculations was incorrect, we did have a minor error in the penultimate paragraph. We should of said “first” not “last.” To make amends, here is a multi-period problem, Multi-period Bond Price Implied Default Rates and CDS, but it won’t make sense without reading this one first. We also added a few paragraphs below, which should help explain the multi-period case.

Further update: April 14, 2008. We also have a new, related post on default rates. It is Calculating Counterparty Credit Reserves from April 8, 2009. Much of that post involves default rates, too.

We see that we’re getting a number of hits from search engines for folks looking for information about price-implied default rates–possibly college students with homework assignments or people trying to understand the various types of default rates they may encounter in their jobs or readings.

We have a number of posts on risk-neutral default rates, including Implied Risk Neutral Probabilities (of Default) , implied RISK NEUTRAL probability of default, redux, Risk Neutral Valuation: There Are at Least Two Expected Values, but we doubt if those settings are the ones that all guests want to see, especially those looking for help on their homework.  (Of course, we think they are all worth reading.)  So, as a public service, we offer an example of a simple, one-period bond problem.  (It is single-period because it is gratis, after all.)

Suppose that a zero-coupon, risky bond with a face value of $1,000 matures in exactly one year.  (Yeah, we said it was simple.) We’ll ignore compounding issues and assume that the annual risk-free rate is 5%.  We’ll also assume that this risky bond’s yield-to-maturity is 8%.

Let’s calculate and discuss a few things before we provide additional assumptions.

We’ll calculate the bond’s price that corresponds to an 8% yield, and we’ll calculate the bond’s price if it were riskless; of course, by riskless we mean free of default risk or credit risk, only.  Our simple one-period model doesn’t really permit interest rate risk, which is a type of market risk.

The bond’s price with a 8% annual yield is: $1,000 ÷ (1 + 0.08) = $925.93.

Now, if the bond were risk-free, its price would be $1,000 ÷ (1 + 0.05) = $952.38,

which is $26.45 higher.  So, the price drops and the yield increases (over their risk-free equivalents) because the owner(s) of the bond is forced to bear some type of credit risk or probability of loss.

That $26.45 will appear again later, but at this point we can’t say much more than it is the difference in the prices of a one-period risk-free bond and our one-period risky bond.

The problem with simple calculations–whether in one or multiple periods–is that they ignore all of the factors that actually affect and determine prices.  In other words, we’ve completely ignored the market dynamics and factors that would cause the price to be $925.93.

The market-clearing price would depend upon supply and demand considerations.¹ Those considerations would depend upon the preferences, beliefs, and endowments of actual and potential sellers and buyers.  In our simple setting, the important preferences would be risk and time preferences, which could possibly be expressed as utility functions; beliefs would involve the probability of default as well as other probabilities associated with each agent’s wealth in other assets if they exist–i.e., their endowments.

So, we can think of the price of $925.93 as a “function” of preferences, U(·); beliefs, f(·); and endowments, w.²  Unfortunately, in real life, we don’t know those factors; so, we’ll never be able to solve the actual problem, but we can solve a substitute problem.

All we know is that the price is $925.93, and it can be expressed as a yield-to-maturity–or a yield curve for multi-period problems–of (our assumed) 8%.  So, the yield could be viewed as a function of the price if you want, but they’re really determined simultaneously.

As we’ve written many times before in related posts, because of several clever researchers in economics and finance, we can actually do more than just discuss the tautologies of price and yield.

In certain cases, we can assume that market participants are risk-neutral–that takes care of U(·) and makes the w irrelevant–and we can assume a particular form of a density or distribution function of outcomes, f(·).  Very importantly, with those assumptions, if we don’t know one of the parameters of f(·) we can solve for it if we know everything else.  That would be like solving for the misnamed implied vol or implied default rate, which is what we will do here.³

Here’s the key to all risk-neutral pricing: under certain assumptions, if agents are (assumed to be) risk-neutral, then we can treat prices as equal to the expected value of the asset’s cash flows according to an associated density function.  That’s the only time we can treat prices as expected cash flows, rather than expected utilities, but depending upon the level of the course, some profs are pretty bad at explaining that fact.⁴

So, there are three things to consider.  First, if agents are risk neutral, we can assume that they care only about expected values.

Second, if agents are risk neutral, then they won’t pay a premium for taking risk like risk-lovers would, nor will they need to be paid a premium for taking risk like risk-averse agents would need to be paid.

Third, that means we can assume that risk neutral agents are satisfied earning the risk-free rate. ⁵ So, given all of our words above, that means that risk neutral agents would value assets at the discounted value of the expected cash flows–discounted at the risk-free rate.

So, as we showed above, if the bond were actually risk-free, then price would have been $952.38, but the price is $925.93.  That means that market participants must expect to receive less than the face value of $1,000 at least some percentage of the time, and that percentage is the probability of default.

Let’s see exactly how much less than $1,000, but first note that we could write the price of a risk-free bond in a slightly expanded way.  Risk-free means 100% chance of getting $1,000; so,

Equation A:

$952.38 = 100% × ($1,000 ÷ (1 + 0.05)) + 0% × (value given default ÷ (1 + 0.05))

We did nothing but add zero to our previous calculation of a risk-free bond.

Let’s make it risky.  Let p represent the probability of default, then for a risk-neutral person, we could write that same line as:

price = (1 – p) × ($1,000 ÷ (1 + 0.05)) + p × (value given default ÷ (1 + 0.05))

Thus, with a price of $925.93, we could write:

$925.93 = (1 – p) × ($1,000 ÷ (1 + 0.05)) + p × (value given default ÷ (1 + 0.05))

There are two unknowns: the probability of default, p, and the value of the bond given default, which has to be less than $1,000.  In fact, we could put a determine a upper bound that is less than $1,000 if we wanted to do so.  (How?)

Now, look at the last equation.  Once we know or assume the value given default, we could find the probability of default, p, or vice versa.

Usually, one assumes the value given default and solves for p.  There’s not really a good reason for doing it other than that’s what just about everyone does.  (Don’t let anyone attempt to fool you with some lame justification. It’s tradition, custom, convention.  Regardless of the word, it is arbitrary.)

So, let’s make-up–er, we mean assume–a value given default.  This is often given in terms of a loss given default, a loss given default rate, or a recovery rate, but they’re all equivalent as one can see in the following relationships.

value given default = $1,000 – loss given default

value given default = $1,000 – loss given default rate × $1,000 = $1,000 × (1 – loss given default rate)

value given default = $1,000 × (1 – loss given default rate) = $1,000 × recovery rate

The loss given default is often abbreviated LGD.  Unfortunately, the loss given default rate is sometimes abbreviated as LGD.  Don’t let the bad notation fool you.  Now, where were we?

That’s right.  Let’s suppose that the loss given default rate is 40%.  That means the recovery rate is 60%, which is its complement.  Regardless, of how that assumption is stated, that means that the value given default is $600.  So, now we have another number to put into our equation:

$925.93 = (1 – p) × ($1,000 ÷ (1 + 0.05)) + p × (600 ÷ (1 + 0.05))

or,

Equation B:

$925.93 = (1 – p) × $952.38 + p × 571.43.

If we did the arithmetic correctly, then solving for p gives a probability of default of almost 7%: 6.94%.  Clearly, all things equal, which means holding everything else constant, as the loss given default increases, the probability of default decreases.  One can make a graph of that relationship as we did in Implied Default Probabilities and Risk Neutral Models in June, 2008.

Now, under the assumption of risk-neutral agents, the difference between the two bond prices of $26.45 can be express as the difference in the present value of their expected cash flows.  The difference in the present values of the expected cash flows in Equations A and B is the present value of the expected loss.  The loss given default is $400.  The undiscounted expected loss is: 0.0694 × $400 = $27.76.  The present value of the expected loss is–not surprisingly–$27.76 ÷ 1.05 = $26.45.

That’s not the most someone would spend for insurance.  That insurance premium depends upon the person’s risk-aversion.

Multi-period problems aren’t that much different, but they require bonds of multiple maturities if one is attempting to derive a credit curve, and one works for from the last first period forward solving maturity-by-maturity.  Otherwise, one can find an “average” annual marginal probability of default.  (We talk about a similar issue in Good Column, Bad Math.)  So, in our multi-period example, we’ll explain the price of a two-year bond as the difference in present values between a risky and risk-free two-year bond.  Then we’ll say much much of that can be attributed to the first period and then the second period.

Note: WE”VE SAID ABSOLUTELY NOTHING ABOUT THE REAL PROBABILITY OF DEFAULT!  If all of the agents are risk-averse, then the unknown real probability of default will be less than the risk-neutral rate, but that’s not too helpful, is it?  Some of our older posts do illustrate this idea.

Good luck with the assignment.

Copyright ©2008 Spero Consulting.

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