Implied Under the Assumption of Risk Neutrality
We have several posts related to the calculation of price-implied default rates under the assumption of risk neutrality and several posts related to simple CDS calculations.
Those posts have involved discrete, single-period problems, where there are only two dates of interest: today and a future date where an uncertain claim or cash flow will be realized, i.e., when bankruptcy would occur.
We’ve focused on binary models and will continue to do so here. In fact, to analyze a two-period problem, we’ll just build upon our latest post from December 2: Price Implied Default Rates.
We think that needless detail obfuscates the central points while providing no marginal explanatory power: either in a statistical or pedagogical sense. So, we like to keep things simple.
Note that we’re providing examples of simple, reduced-form models à la Jarrow and Turnbull (1995) or Hull and White (2000), not a structural Merton model like KMV. We’ll do that when we have the time.
In our December 2nd post, we considered a risky, one-year, zero-coupon bond. We assumed a face value of $1,000, a risk-free rate of 5%, and the risky bond’s yield to be 8%. We could have stated that last assumption as the bond has a price of $925.93.
From those assumptions, and the additional assumption that the owner of the bond would recover 60% of the face value, we calculated the risk-neutral-model-implied default rate of 6.94%.
Now the calculation of that default rate depends upon all of the assumptions, and obviously the answer will vary with changes in any of the assumed variables: the bond’s price or yield, the risk-free rate, and the loss given default rate.
Obviously, it also depends upon the applicability of risk-neutral valuation, which allows us to impose two very important considerations (versus reality). It allows us to (1) treat the bond’s price as the expected value of its cash flows, which is only valid if the creditor (in the model, not in real life) is risk-neutral, and (2) use the risk-free rate as the proper discount rate for a risk-neutral person. Those assumptions allow us to work with expected cash flows, rather than curvy preferences. We’ll focus on calculations in this post and not on applicability.
Finally, the answer also depends upon our choice of probability functions. Here, the only uncertainty involves full payment or not; so, that credit risk is easily modeled as a binary function, but it is important to note that risk-neutrality does not imply a particular probability function. Once the analyst has chosen from a family of distribution functions, the assumption of risk neutrality will determine (imply) particular parameter values, but that is all. For the more mathematically inclined, that is the change-of-measure that is referred to in the texts. (Probabilities are weights. Different parameter values within a distribution cause possible events to be weighed differently; ergo, the measure is changed.)
In this problem, we’ll keep the same assumptions as in our previous post for the first of our two periods. So, here is the setting: We have two zero-coupon, risky bonds issued by the same firm and each with a face value of $1,000: one matures in one-year and the other matures in two years. Imagine that there are two risk-free bonds, too.
The one-year risky bond is described as above; so, it will have a price of $925.93. If that bond were risk-free, it would have a price of $952.93. In a risk-neutral model, the difference in prices is the present value of the expected loss (of the risky bond, of course).
The risk-free rate in the second period is 7%. Note that there is no market risk—that is, no interest rate risk—so there is no evolution of interest rates or any type of rate process in our humble, little example. (We’re just making up numbers to illustrate a few basic ideas.)
The bond that matures in two years has a yield-to-maturity of 9.982%, which for all intents and purposes—and for everyone except the truly anal—is 10%.
As an aside, with our two sets of interest rates, we can calculate an overall yield-to-maturity from our term structure of forward, risk-free rates, and for risky rates, we can determine the structure of forward rates from our risky yield curve.
Risk-free yield-to-maturity: we don’t really need to calculate this, so you can skip it is you want, but if the risk-free bonds are priced to earn 5% in the first year, and a two-year bond is priced to earn 7% in the second year, then the geometric average return for the zero-coupon, risk-free bond better be close to the arithmetic mean of 6%. That yield-to-maturity is simply:
[(1 + r1)·(1 + r2)]1/2 - 1 = [1.05·1.07]1/2 - 1 = 5.995%
So, the yield on a two-year, zero-coupon, riskless bond is about 6%: just like we knew before we did the calculation.
Risky forward rate: now, given the risky yield-to-maturity is about 10% on the two-year, zero coupon, bond, and given a first-year risky rate of 8%, then the implied forward rate for the second period must be:
[(1 + 0.08)·(1 + r2)]1/2 - 1 ≈ 10% implies r2 = 1.12 /1.08 - 1 = 12%
So, if (and only if) the two-year, risky bond yields (about) 10%, then its price is:
$1,000 ÷ 1.12 = $826.45 ≈ $826.72.
By the way, we’re off by 26¢ by using the easy 10% instead of the more precise 9.982%, but the lesson is free; so, the reader really shouldn’t complain.
Notice that credit spread increased from 3% (8% - 5%) in the first year to 5% (12% - 7%) in the second. All things equal, we should expect that the risk-neutral, price-implied, default rate will increase, too. Let’s see if that happens.
Three Probabilities of Default (or default rates): when we move to a multi-period problem, we have to be careful to specify the default rate to which we’re referring. There are conditional, marginal, and cumulative probabilities of default, and that is true whether we’re discussing actual (but unknown) probabilities of default or risk-neutral-implied probabilities of default like we’re doing here.
The conditional probability of default for a period, t, is the easiest notion to understand: given that the firm has survived until the beginning of that period, it is the probability that the firm can’t pay its bills during the next interval of time; here, we’re using one year as the time interval. We’ll denote conditional probabilities as pt for every period t.
The marginal probability of default is the probability that the firm will default in period t. Now, the firm only has the opportunity to default in period t, if it hasn’t already defaulted; so, the marginal probability considers the probability of surviving until that point and the conditional probability of default. If p1 is the (marginal) probability of default in the first period, the (1 - p1), then the marginal probability of default is:
(1 - p1)·p2,
For our little problem, we won’t introduce any special notation for the marginal probabilities of default.
Finally, the cumulative probability of default is the sum of all the marginals: p1 + (1 - p1)·p2 in a two-period problem. We wrote about longer term cumulative probabilities of events in this post, Good Column, Bad Math, where we talk about 100-year floods.
So, let’s find the conditional probability of default in the second period. Given that there was no default at the end of the first period, what is the probability of default in the second period implied by the bond’s price?
Well, with one period remaining, the price of the only remaining bond is:
$1,000 ÷ 1.12 = $892.86.
So, we can find the conditional probability of default in the second-period, p2, the same way that we found the probability in our one-period problem.
price = $892.86= (1 - p2) × ($1,000 ÷ (1 + 0.07)) + p2 × (600 ÷ (1 + 0.07))
$892.86= (1 - p2) × $934.58 + p2 × 560.75.
So, if the firm survives the first period, there is an 11.16% conditional probability of default in the second period. That means that the marginal probability of default for the second period is the probability that the firm survives the first period multiplied by the conditional probability of default in the second:
(1 - p1) ·p2 = (1 - 0.0694) · 0.1116 = 10.385%
The cumulative probability of default is the sum of the two marginals: 6.94% + 10.39 = 17.33%.
Note that at the end of the first period the difference between the risk-free bond’s price of $934.58 and the risky bond’s price of $892.86 is $41.72. The $41.72 represents the risk-neutral, “present value” at the start of the second period of the conditional expected loss in the second period of the two-period bond. So, the $41.72 is related to the conditional probability of loss and the potential loss of $400:
($400 × 11.16%) ÷ 1.07.
But the second period will be experienced only if there was no default in the first period! So, in a risk-neutral world, a creditor will only experience the opportunity to lose (a discounted average) of $41.72 if there is no default in the first period: with probability (1 - 6.944%).
And the value of that today—at the start of it all—must be discounted by the first period’s risk-free rate of 5%. So, the present value of that expected loss that
$41.72 × (1 - 0.06944) ÷ 1.05 = $36.97.
Is our analysis correct? Let’s see. A two-year, risk-free, zero-coupon bond would have a price of $890.08. Our risky bond has a price of $826.45. That means that in a risk-neutral world—given all of our assumptions—the present value of the sum of the expected losses is the difference: $890.08 - $826.45 = $63.63.
In the first year, the present value of the expected loss on debt with a face value of $1,000 is $26.67. That means that the present value of the expected loss in the second period must be: $63.63 - $26.67 ≈ $36.97. Hey, where did we see that number before? That’s right—-a few inches above where we discounted the expected present value of the second-period loss.
What about CDS?
To protect against loss, the CDS should provide $400 in case of default at the end of each period.
If the CDS policy were sold period-by-period, i.e., one-year terms, the first year’s premium would have to be at least $26.67 and the second year’s if sold today would cost at least $36.97. The actual cost, like everything else in the real world, would depend upon how badly creditors want to protect against loss, but those values are actuarially fair in a risk-neutral setting.
Also note that if the CDS policy were sold at the start of the second period, the premium would have be to at least $41.72 to be actuarially fair in a risk-neutral world. So, if purchased consecutively, the insurance premiums would need to $26.67 today and $41.72 next year in our risk-neutral world.
What if the insurance were purchased for two periods? What would the constant premium be? In that case, there is a chance that one or both premiums will be received (or paid). If there is no bankruptcy in the first period, then the premium will be paid twice; so, we need:
premium + (1 - 0.06944) premium ÷ 1.05 = $63.63
premium (1.0 +0.93056 ÷ 1.05) = $63.63
premium = $33.74
We assumed that the premium was paid at the beginning of each period; so, it is like an “annuity due” and actually is like a random, annuity due. It’s random because it is a constant stream of cash flows, but the ending date is unknown. In this simple two-period example, the ”stream” could be one or two payments.
Also remember that risk-averse creditors should be willing to pay more than that, i.e., a risk premium, too.
And remember, we’ve said absolutely nothing about probabilities in the real world that our example represents. Risk neutral probabilities and default rates are derived from a set of assumptions that permits (relatively) easy calculation, but those probabilities and rates only work in our model, and they do not represent real frequencies. For more on that, please see our other posts on the topic.
As we hope that you can see, CDS is identical to term life insurance—except millions and millions of similar firms don’t die each year; so, there is little empirical evidence of various factors, including loss given default rates.
By the way, we’ve ignored counter-party risk and a host of other complicating assumptions.
As with many of our longer posts, we’ll likely edit this one in the near future.
Copyright © 2008 Spero Consulting.
Footnotes: