Calibration of PD Term Structures: To Be Markov Or Not To Be

1 Calibration of PD Term Structures: To Be Markov Or Not To BeChristian Bluhm (Credit Suisse) and Ludger Overbeck (Universityof Giessen)November 19, structures of default probabilities are omnipresent in credit risk modeling: time-dynamic creditportfolio models, default times, and multi-year pricing models, they all need the time evolution of default proba-bilities as a basic model input. Although people tend to believe that from an economic point of view the Markovproperty as underlying model assumption is kind of questionable it seems to be common market practice to modelPD term structures via Markov chain techniques.

2 In this paper we illustrate that the Markov assumption carries usquite far if we allow fornonhomogeneoustime behaviour of the Markov chain generating the PD term a proof of concept we calibrate anonhomogeneoustime-continuous Markov chain to observed one-year ratingmigrationsandmulti-year default frequencies, hereby achieving convincing approximation Markov Chains in Credit Risk ModelingThe probability of default (PD) for a client is a fundamental risk parameter in credit risk man-agement. It is common practice to assign to every rating grade in a bank s masterscale a one-yearPD in line with regulatory requirements; see [1].

3 Table 1 shows an example for default frequenciesassigned to rating grades from Standard and Poor s (S&P). 1: One-year default frequencies (D) assigned to S&P ratings; see [17], Table , credit risk modeling concepts like dependent default times, multi-year credit pricing,and multi-horizon economic capital require more than just one-year PDs. For multi-year creditrisk modeling, banks need a wholeterm structure (p(t)R)t 0of (cumulative) PDs for every ratinggradeR; see, , [2] for an introduction to PD term structures and [3] for their application tostructured credit bank has its own (proprietary) way to calibrate PD term structures1to bank-internal andexternal data.

4 A look into the literature reveals that for the generation of PD term structuresvarious Markov chain approaches, often based on time-homogeneous chains, dominate currentmarket practice. A landmarking paper in this direction is the work byJarrow, Lando,andTurnbull[7]. Further research has been done by various authors, see, ,Kadam[8],Lando[10],Sarfaraz et al.[12],SchuermannandJafry[14, 15],TrueckandOezturkmen[18],just to mention a few examples. A new approach via Markov mixtures has been presented recentlybyFrydmanandSchuermann[5].

5 In Markov chain theory (see [11]) one distinguishes between time-discrete and time-continuouschains. For instance, a time-discrete chain can be specified by a one-year migration or transition1In the literature, PD term structures are sometimes calledcredit multi-year transitions via powers (Mk)k 1ofM. The corresponding (yearly)time-discrete PD term structures are given byp(k)R= (Mk)row(R),8(k= 1,2,3, ..)whererow(R) denotes the row in the migration matrixMcorresponding to ratingR. Time-continuous chains are specified by a Q-matrix2 Qsuch that exp(tQ) defines the migration matrixfor the time interval [0, t], where exp( ) denotes the matrix exponential.

6 Time-continuous PD termstructures corresponding to a generatorQare given byp(t)R= (exp(tQ))row(R),8(t 0).(1)Time-continuous Markov chains are superior to time-discrete Markov chains because they allowfor a consistent way to measure migrations and PDs for time horizons betweenyearly time gridpoints. If for a discrete chain defined by a one-year migration matrixMwe find a generatorQsuch thatM= exp(Q),(2)one says that thetime-discrete chain can be embedded into a continuous-time chain. In general,we can only expect to find approximative embeddings; seeIsrael, Rosenthal, andWei[6],Jarrow, Lando, andTurnbull[7],KreininandSidelnikova[9], and [2], Chapter 6.

7 In [3],Section , we discuss an example of a generatorQalmost perfectly fitted to a given one-yearmigration matrix from S&P; see Appendix problem is that we find thata well-fitted generator nevertheless can generate model-implied PDterm structures significantly deviating from observed multi-year default frequencies. In this paper,we address this problem,not by rejecting the Markov assumption but by dropping the homogeneity3assumption time. Our results in Figure 2 show that in the context of PD term structurecalibration the Markov assumption indeed is not as wrong as people sometimes claim.

8 In fact,dropping the time-homogeneity assumption provides sufficient flexibility to calibrate a Markovprocess to empirical migration and default frequencies with convincing quality. Therefore, weclaim that the anser to the question raised in the title of this paper is to be Markov , but nottime-homogeneous .2 Calibration of a NHCTMC for PD term structuresIn the sequel, we construct a NHCTMC, which we use for the generation of PD term Appendix I we provide some comments on the stochastic rationale of the point for our construction is the generatorQ= (qij)1 i,j 8from Table 4.

9 But now wedo no longer assume that the transition ratesqijare constant over time, leading to a , we replace the time-homogeneous generatorQleading to migration matrices exp(tQ) forthe time interval [0, t] by the time-dependent generatorQt= (t) Q(3)where denotes matrix multiplication and (t) = ( ij(t))1 i,j 8is the diagonal matrix inR8 8with ij(t) ={0ifi6=j i, i(t) ifi=j(4)Because (t) is a diagonal matrix,Qtis a Q-matrix (scaling rows of aQ-matrix gives aQ-matrix).The functions , parameters and are defined as follows.}

10 Set , : [0, ) [0, ), t7 , (t) = (1 e t)t 12A square matrixQis a Q-matrix/generator if Nj=1qij= 0 i, 0 qii< i, andqij 0 i6= Markov chain istime-homogeneousif transition probabilities (the generator) do not depend on nonnegative constants and . We want to normalize the functions in a way such that at timet= 1 the functions take on the value 1. Therefore, we define , as , : [0, ) [0, ), t7 , (t) = , (t) , (1).Figure 1 illustrates the functionst7 t , (t). They have the following properties:1.]]]]