implied RISK NEUTRAL probability of default, redux

| 10/09/2008 |
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Update: we have newer posts on the topic, too, including Risk Neutral Valuation: There Are at Least Two Expected Values, that describes the difference between real and risk neutral distributions.  We also have: Price Implied Default Rates that provides an example more like a risky bond, and a multi-period example: Multi-period Bond Price Implied Default Rates and CDS.

The Wall Street Journal has an article about Iceland’s financial problems in today’s paper: Aftershocks Felt From Iceland.  It turns out that the country has more problems than being a small, cold island in the middle of the North Atlantic.

Any way, we’re not writing about its climate, especially since Western PA’s is probably worse and we have no beaches and few tall blonds.  No, we’re writing about the graph in the article and the blurb that states, “Trading in the credit default swap market puts the probability of a default by Iceland on its debt at a little over 50%.”

As presented, that statement is highly misleading and nonsense, and the purpose of this post is to explain why.

We’ve written about Implied RISK NEUTRAL probabilities of default a few times.  In the aptly titled Implied Risk Neutral Probabilities (of Default) we provided an example that illustrated the difference between the actual probability of default, which is never known in the real world, and the model-implied probability of default, which could be calculated from ANY model–regardless of its validity–that permits at least two outcomes, e.g., survival and failure of the entity.  Such a model may or may not assume risk neutrality, but risk neutrality makes the calculation simpler.

Regardless of whether Iceland goes bankrupt or not, we provide several examples that distinguish the risk-neutral, implied default rate from the true default rate.

In our earlier post, Implied Default Probabilities and Risk Neutral Models, we commented on a similar graph in another WSJ article from last June, and mentioned many of the factors that would be involved in such a calculation.  Unfortunately, we recently and accidentally deleted a very nice comment about that post, which expanded the analysis to include counterparty credit risk: the risk that the purchaser of a CDS contract would not get paid (the insurance proceeds) in case of bankruptcy because the insurer or CDS writer was also insolvent–kind of like AIG.

In this post, we’ll provide another numerical example with a different assumed, risk-averse, utility function for the insurance buyer.

We’ll again assume a single period, but we will not use the 50% probability of bankruptcy that we did in the earlier post; it would be too confusing. In fact, the 50% probability of default mentioned in the article is likely the cumulative probability of default over the five years.  It may or may not be based on equal marginal probabilities of default for each of the five years, regardless the annual marginal probability of default is not 10%; the 50% mentioned for five years was not found by multiplying five years times 10%.

Readers interested in an example of a discrete-time, multi-period survival problem that illustrates these issues should see Good Column, Bad Math. Readers interested in a calculation-intensive, similarly-structured, discrete-time problem, should see our research paper on moral hazard: Deadlines as Management Control Devices, which is based upon our dissertation.  In that paper, the game ends with success, rather than failure, but the outcome tree is very similar.

So, will provide a couple examples similar to our square root problem in August.

Case 1:  Assume that the person has natural logarithmic utility, which is strictly concave funtion and makes him risk-averse.  We’ll also assume that the person has an initial endowment of $75.858, which we choose for convenience as you’ll see below.  We’ll ignore time-value-money calculations and interest rates today; they’re inessential.

Assume that a firm will be worth $100 if it survives and $10 if if fails.  That makes the loss given default (LGD) $90, and the loss given default rate $90/$100 equal to 90%.  In the real world, we don’t know the loss given default until after a default occurs, the firm’s assets are liquidated, and the residual cash is paid to the debtholders.  LGD rate is always assumed in CDS and other similar calculations and, from our experience, seems to be considered much less than implied default rates.

Assume that the actual probability of default is 12%, i.e., the probability of getting $10 from the investment is 12%.  REMEMBER, two items that we never know in real life are the market participants preferences–expressed here as a ln(·) utility function–and the actual probability of default, 12%.  It is crucial never to forget this ignorance.

Also, we generally don’t know the person’s entire endowment, specifically his other wealth independent of the gamble.  In this first case, we cleverly chose the person’s endowment so that his other wealth, not tied up in this particular investment, is zero.  (You’ll that fact below.)

We’ll do what we need to do to calculate the risk-neutral probability of default and then later we’ll change a few assumptions to see how those changes affect the answer.

First, we’ll calculate the person’s expected utility with the investment.  Now, with logarithmic utility it is:

10% × ln($10) + 90% × $ln($100) = 4.375 utils.

Now, to get the same 4.375 utils of satisfaction from a certain gamble (involving no risk), the person should be willing to spend up to:

exp(4.375) = $75.858.

So, that $75.858 is his certainty equivalent, or the most he would pay for the uncertain investment.  (That’s why we cleverly set his initial wealth at the same $75.858, so there would be no money left-over after the investment.)  With the same hand-waving (about market interactions) that we performed in August, we’ll suppose that the $75.858 is also the price, i.e., competition among similarly-preferenced and endowed buyers drive the price to the break-even point; technically, it is an indifference point but only pedantics like ourselves care.

Now, a risk neutral person could–but need not–be modeled as caring only about expected cash flows on a dollar-for-dollar basis; so, for a risk-neutral person, we could set his utility equal to dollar values and expected dollar values.  In other words, he would value $10, $75.858, and $100 as 10 utils, 75.858 utils, and 100 utils, respectively.  (We wrote “but need not” above, because we could add a constant and multiply by a positive number without changing the essence of the analysis.)

Remember, in the real world, we don’t know the 12% or the actual market participant’s preferences, which we assumed to be logarithmic here, or his starting wealth, BUT if we assumed that he was risk neutral in our dollar-for-dollar way, then we solve for the corresponding probability of default, i.e., find p such that:

p × 10 + (1 – p) × 100 = 75.858.

Rearranging and solving for p, we get the risk neutral-implied probability of default, p, equals about 26.83% (versus the real probability of default of 12%, which, again, we never know in real life).

So, the WSJ writer or editor is calling that 26.83% the probability of default, when it is, in fact, the implied probability of default assuming that market participants were risk-neutral.  (Here, our “model” is so simple as to be innocuous, but in more robust settings–with more details–that’s not the case.)

That risk-neutrality, which provides linearity of preferences, is what allows the analyst to view the price and set it equal to the expected value of the cash flows in the possible outcomes, e.g., survive or fail, for a possible probability, p.  In real life, analysts would use different distributions to calculate an implied probability of default based upon their specific model in much the same way that they would calculate a model-implied volatility when using Black-Scholes or a variant.  (Provide market variables or guesses about those variables, provide a model, and solve for the last remaining unknown.  Notice that there are quite a lot of assumptions in such a process.)

(By the way, for those with a little knowledge of stochastic processes, setting the price equal to the expected value (under risk neutral valuation) is why the phrase Martingale Method is used.  That’s what a Martingale is: a process where the value today is equal to the expected value in the future, and it doesn’t really change if we add interest rates and discounting.)

Now please note, unlike in real-life, in this example, we know that the true probability of default is 12%.  To an outside observer, without our information to construct the calculations, there is no clear relationship between the 12% and the 26.83%.  In other words, knowing only the 26.83% says nothing about the true probability of default, and that is the error that the journalist makes in today’s article.

Because the 50% for Iceland is such a large number, the graph and the blurb seem almost designed to insight hysteria; however, actual–albeit unknown rate–could be substantially lower.

Case 2: let’s keep everything the same, but make the person “more” risk-averse.  In microeconomics, that has a particular, technical meaning having to do with the concavity (the curvedness) of the utility function, but here we’ll avoid the issue by reusing the natural logarthmic function recursively, i.e., our utility function is now ln(ln(·)).

In such a problem, the additional concavity reduces the certainty equivalent of the gamble, and possibly the price.  We’ll wave our hands again as a way to stay on course, and assume that the price falls to the new certainty equivalent.  To make it work, without trying to hard, we’ll arbitrary assume that as soon as the person purchases the firm, his preferences, via utility function, (and risk aversion) changes to the double-log thing, ie.,

12% × ln(ln($10)) + 88% × ln(ln($100)) = 1.444 utils.

For the changed person to get the same 1.444 utils of satisfaction for sure, he’d be willing to sell it for:

exp(1.444) = $69.243.

(As his risk aversion increases, the value of a gambles decreases.) Now, in the real world, a decrease in a potential seller’s reservation price doesn’t necessarily change the market price, but we’ll assume that it does.  So, immediately, the price is $69.243.  We can now find the revised risk-neutral probabilities:

p × 10 + (1 – p) × 100 = 69.243.

Solving for p yields a new, risk-neutral, implied probability of default of 34.175%.  So, a change in risk preferences will change the implied probability of default.  You may call it the market implied probability of default, but it is really the implied probability of default using the market price and assuming that buyers are risk-neutral, but that gets kind of long.  The real probability of default is still 12%.

Case 3: Now, let’s go back to our first case, where we used the natural log, ln, only once, not twice.  Let’s assume that right after the purchase, the new owner discovers that the loss given default is really $99 dollars, not the $90 that (it was assumed that) the market knows.

In that case, the new expected utility is 0 + .88 × ln(100) or 4.145 utils.  Taking the inverse gives exp(4.053) = 57.544.

Now, IF everyone knows that the two states are {$1, $100}, then the risk-neutral probability of default satisfies:

p × 1 + (1 – p) × 100 = 57.544,

and equals 42.9%.  Remember the actual probability of default is still 12%, but the low outcome is particularly low for a log utility function.  So, the implied, risk-neutral probability of default is more than 3.5 times the true probability of default.

Case 4: let’s take Case 3, and assume that the buyer knows that the loss given default has increased from $90 to $99, but a trader or analyst at another firm has not observed that change but has observed the new price of $57.544.  In that case, the analyst very likely keep the same LGD assumption and solve for a new implied probability of default of (using the erroneous, but assumed $10, rather than the correct $1:

p × 10 + (1 – p) × 100 = 57.544.

In that case, solving for p gives an model-implied, under the assumption of risk-neutrality probability of default of 47.2%.  Of course, once again, the real probability of default is 12%.

The difference between the 47.2% and 42.9% implied default rates is solely attributed to the (incorrect) assumption about the loss given default.  In our experience, the LGD is the least-challenged, least-investigated assumption used to price CDS and related products.  In real-life, it is extremely common to maintain that assumption in the face of falling prices.

We’ll probably refine this post in the coming days, but our four simple cases should be sufficient to cast deep suspicion on Iceland’s reported probability of default, when it is really a model-implied, default rate under the assumption of risk neutrality.  Remember in all of our cases, the real probability of default is 12%.  The models used to calculate that rates involve more variables and more calculations, but apply no more knowledge than do our simple examples here.

If you have any questions or comments, please write.

Copyright © 2008 Spero Consulting.

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