Abstract
Unconditional default probabilities are the coin of the realm for credit and quantitative analysts. These assessments of relative creditworthiness associated with individual obligors are of enormous value. This information, however, does not come for free. There are, in fact, two broad approaches used in the determination of default probabilities: estimation and calibration. This chapter examines both. Estimation exploits credit-counterparty transition and default history by employing statistical techniques to approximate one’s desired values. The general framework is presented along with, quite importantly, two approaches used to assess the uncertainty in one’s estimates. Rarity of default and relatively modest historical data make interval estimation an essential practice; its efficacy is examined in a simulation study. Calibration, conversely, examines market instruments—such as bond obligations or credit-default swaps—and seeks to extract implied default probabilities from their observed prices. Using credit-default swaps, the theory and practice of this default-probability calibration method are carefully investigated. Although tempting, it is a mistake to treat these two approaches as equivalent. Reconciling these alternative default probability values involves grappling with risk preferences. The pitfalls and limits of this thorny task are the final topic of this chapter.
A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness.
(Alfred Korzybski)
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Notes
- 1.
- 2.
The reader familiar with fixed-income markets will, of course, see the strong similarities with the modified duration measure.
- 3.
Different agencies use slightly different classifications, but there is generally a fairly straightforward one-to-one mapping between these groupings.
- 4.
Appendix C also summarizes a number of useful technical aspects related to the Markov chain.
- 5.
- 6.
It is possible to force the columns of P to sum to unity, but this changes the interpretation of the matrix. Although either choice is correct, it is important to select a perspective and stay with it.
- 7.
A few important technical features of Markov chains, and elaboration of some of these ideas, are summarized in Appendix C.
- 8.
Facetiously, one might refer to this as the Hotel California state: “you can check-out any time you like, but you can never leave.”
- 9.
The multivariate analogue of the exponential function, the matrix exponential, is defined as,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} e^{At} = \sum_{k=0}^{\infty} \frac{(At)^k}{k!}. \vspace{-2pt}\end{array} \end{aligned} $$(9.35)See Golub and Loan (2012, Chapter 9) for a discussion of various algorithms for its computation.
- 10.
Since credit ratings are often associated with not just a current value, but also a positive or negative outlook, the effective number of credit states can become quite a bit more sizable than one might expect.
- 11.
Some datasets extend back as far as the 1930s. While the extra observations are certainly welcome, they do create other issues. In particular, it creates worries about the applicability of observations from very different economic settings to the current environment.
- 12.
We will relax this assumption in Chap. 10.
- 13.
There are faster and more complex ways to simulate a Markov chain—see, for example, Fishman (1983)—but this simple approach fully meets our needs.
- 14.
Starting all of the obligors in credit state A will reduce the observations in states B and C, which might increase the noise of their transition-probability estimates.
- 15.
Unlike some real-life datasets, we assume that each credit obligor is rated in every period.
- 16.
A lack of observations can, in principle, occur for any credit state. In our setting, this is highly improbable, but a more general implementation would take this possibility into consideration.
- 17.
The interested reader is referred to Savage (1976) for an overview of Fisher’s voluminous contributions to statistics.
- 18.
Typically, one fixes the other parameters at their maximum-likelihood value.
- 19.
Construction of Fig. 9.8 requires a bit of caution. One cannot simply fix the other parameters at their maximum-likelihood values, because of the constraint that all elements in a given row of \(\hat {P}\) must sum to unity. As we vary the value of a given parameter of interest, the remaining weight is thus allocated equally to each of the other fixed parameters. There may exist a more clever solution, but this potentially imperfect approach meets our purposes.
- 20.
For a rigorous treatment of these topics, please see Held and Bové (2014).
- 21.
For finance professionals, this is not to be confused with the recursive technique used for extracting zero-coupon rates from bond prices. As we’ll see in this discussion, it refers to an entirely different procedure.
- 22.
It appears to be common practice in the bootstrapping literature to refer to the bootstrap data, distribution or estimator, X, as X ∗. We will not deviate from this tradition.
- 23.
Each Bernoulli trial is simulated using \(\hat {p}\) and the uniform-random-variate trick summarized—albeit in the multinomial setting—in Fig. 9.4.
- 24.
\(\bar {p}^*\) and SE(p ∗) are computed precisely as one would expect. Given M simulations, they are defined as,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \bar{p}^* = \sum_{m=1}^M \frac{p^{*}(m)}{M}, \end{array} \end{aligned} $$(9.76)and
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \text{SE}(p^{*}) = \sqrt{\sum_{m=1}^M \frac{\left(p^{*}(m)-\bar{p}^*\right)^2}{M-1}}, \end{array} \end{aligned} $$(9.77)where p ∗(m) denotes the maximum-likelihood estimate from the mth simulated dataset in our bootstrap computation.
- 25.
This is, of course, conceptually quite odd. If we really knew the true parameters, we would not require any estimate of their uncertainty. At the same time, we can view it as another practical verification of the bootstrap approach.
- 26.
The state-variable correlation parameter, ρ, is set to 0.3 for both models, while the degrees-of-freedom parameter in the t-distributed implementation is 9.
- 27.
We have, in the interests of full disclosure, pinched some convenient notation from Brigo and Mercurio (2001).
- 28.
That is, to be explicit
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \delta_i \equiv \delta(t,T_i), \end{array} \end{aligned} $$(9.83)for i = 1, …, β.
- 29.
- 30.
Of course, if c i is constant, then independence follows trivially.
- 31.
This quantity is often referred to as the day-count convention. While the specific choice is certainly important in practice, it is neither complex nor stochastic. As such, we do not allocate it any particular attention.
- 32.
Often a fixed spread, s, is added to reflect the creditworthiness of the individual counterparty. We will set s = 0 without any loss of generality.
- 33.
- 34.
Duffie and Singleton (2003) do not make this assumption and develop an entire framework incorporating uncertainty with regard to both quantities. While both elegant and useful, we do not require this level of sophistication for our application.
- 35.
- 36.
- 37.
See Andersen et al. (1993) for a deep dive into the underlying theory of Poisson and Cox processes.
- 38.
Duffie and Singleton (2003) use stochastic models—such as the Ornstein-Uhlenbeck and CIR processes—to describe the infinitesimal dynamics of λ(t), which leads to a sophisticated theory combining interest-rate and default elements into a single framework.
- 39.
- 40.
This can be set to unity without any loss of generality.
- 41.
- 42.
The swap rates would be used to form the projection curve, whereas government securities (or overnight-indexed swaps) would be employed for the risk-free discount curve.
- 43.
This sub-routine, as the reader will recall, is essentially a wrapper around MINPACK’s hybrd algorithm.
- 44.
There are a number of complex details, but roughly speaking, it is useful to think of the credit-default-swap spread as being loosely equivalent to the difference between the obligor’s bond yield and the comparative risk-free rate in the economy. At par, we would expect agreement between credit-default-swaps spreads and coupons.
- 45.
One can use this trick, with R = 0.7 in Table 9.6, to verify its reasonableness.
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Bolder, D.J. (2018). Default Probabilities. In: Credit-Risk Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-94688-7_9
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