Portfolio Migration

Based on the techniques presented in the last sections we can now tackle the problem of portfolio migration, i. e. we can assess the distribution ofn(t) credits over thedrating categories and its evolution over periodst∈ {1, . . . m}. Here, a stationary transition matrixPis assumed. The randomly changing number of credits in categoryj at timet is labeled by ˜nj(t) and allows to define non-

Fromj pˆ(1)_jd Stdd ˆ p∗_jd(1) pˆ(5)_jd Stdd ˆ p∗_jd(5) pˆ(10)_jd dStd ˆ p∗_jd(10) 1 0.00 0.000 0.004 0.003 0.037 0.015 2 0.00 0.000 0.011 0.007 0.057 0.022 3 0.00 0.000 0.012 0.005 0.070 0.025 4 0.00 0.000 0.038 0.015 0.122 0.041 5 0.00 0.000 0.079 0.031 0.181 0.061 6 0.12 0.042 0.354 0.106 0.465 0.123 Table 4.2. Estimatedm-period default probabilities and the bootstrap estimator of their standard deviations form= 1,5,10 periods

negativeportfolio weights

˜

wj(t) def

= n˜j(t)

n(t), j= 1, . . . , d,

which are also random variables. They can be related to migration counts ˜ cjk(t) of periodt by ˜ wk(t+ 1) = 1 n(t) d X j=1 ˜ cjk(t) (4.22)

counting all migrations going from any category to the rating categoryk. Given the weights ˜wj(t) = wj(t) at t, the migration counts ˜cjk(t) are binomially

distributed

˜

cjk(t)|w˜j(t) =wj(t)∼B (n(t)wj(t), pjk). (4.23)

The non-negative weights are aggregated in a row vector ˜

w(t) = ( ˜w1(t), . . . ,w˜d(t))

and sum up to one

d

X

j=1

wj(t) = 1.

In the case ofindependent rating migrations, the expected portfolio weights at

t+ 1 given the weights attresult from (4.22) and (4.23) as E[ ˜w(t+ 1)|w˜(t) =w(t)] =w(t)P

and the conditional covariance matrixV[ ˜w(t+ 1)|w˜(t) =w(t)] has elements vkl def =      1 n(t) Pd j=1wj(t)pjk(1−pjk) k=l for − 1 n(t) Pd j=1wj(t)pjkpjl k6=l. (4.24)

Form periods the multi-period transition matrix P(m)_=Pm _{has to be used,}

see Section 4.3.1. Hence, (4.22) and (4.23) are modified to

˜ wk(t+m) = 1 n(t) d X j=1 ˜ c(_jkm)(t) and ˜ c(_jkm)(t)|w˜j(t) =wj(t)∼B n(t)wj(t), p (m) jk .

Here, c(_jkm)(t) denotes the number of credits migrating from j to k over m

periods starting in t. The conditional mean of the portfolio weights is now given by

E[ ˜w(t+m)|w˜(t) =w(t)] =w(t)P(m)

and the elements of the conditional covariance matrixV[ ˜w(t+m)|w˜(t) =w(t)] result by replacingpjk andpjl in (4.24) byp

(m) jk andp

(m) jl .

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portfolio models

R¨udiger Kiesel andTorsten Kleinow

To assess the riskiness of credit-risky portfolios is one of the most challenging tasks in contemporary finance. The decision by the Basel Committee for Bank- ing Supervision to allow sophisticated banks to use their own internal credit portfolio risk models has further highlighted the importance of a critical eval- uation of such models. A crucial input for a model of credit-risky portfolios is the dependence structure of the underlying obligors. We study two widely used approaches, namely a factor structure and the direct specification of a copula, within the framework of a default-based credit risk model. Using the powerful simulation tools ofXploRewe generate portfolio default distributions and study the sensitivity of commonly used risk measures with respect to the approach in modelling the dependence structure of the portfolio.

5.1

Introduction

Understanding the principal components of portfolio credit risk and their in- teraction is of considerable importance. Investment banks use risk-adjusted capital ratios such as risk-adjusted return on capital (RAROC) to allocate eco- nomic capital and measure performance of business units and trading desks. The current attempt by the Basel Committee for Banking Supervision in its Basel II proposals to develop an appropriate framework for a global financial regulation system emphasizes the need for an accurate understanding of credit risk; see BIS (2001). Thus bankers, regulators and academics have put con- siderable effort into attempts to study and model the contribution of various ingredients of credit risk to overall credit portfolio risk. A key development has been the introduction of credit portfolio models to obtain portfolio loss distributions either analytically or by simulation. These models can roughly

be classified as based on credit rating systems, on Merton’s contingent claim approach or on actuarial techniques; see Crouhy, Galai and Mark (2001) for exact description and discussion of the various models.

However, each model contains parameters that effect the risk measures pro- duced, but which, because of a lack of suitable data, must be set on a judge- mental basis. There are several empirical studies investigating these effects: Gordy (2000) and Koyluoglu and Hickmann (1998) show that parametrisation of various models can be harmonized, but use only default-driven versions (a related study with more emphasis on the mathematical side of the models is Frey and McNeil (2001)). Crouhy, Galai and Mark (2000) compare models on benchmark portfolio and find that the highest VaR estimate is 50 per cent larger than the lowest. Finally, Nickell, Perraudin and Varotto (1998) find that models yield too many exceptions by analyzing VaRs for portfolios over rolling twelve-month periods.

Despite these shortcomings credit risk portfolio models are regarded as valu- able tools to measure the relative riskiness of credit risky portfolios – not least since measures such as e.g. the spread over default-free interest rate or default probabilities calculated from long runs of historical data suffer from other in- trinsic drawbacks – and are established as benchmark tools in measuring credit risk.

The calculation of risk capital based on the internal rating approach, currently favored by the Basel Supervisors Committee, can be subsumed within the class of ratings-based models. To implement such an approach an accurate under- standing of various relevant portfolio characteristics within such a model is required and, in particular, the sensitivity of the risk measures to changes in input parameters needs to be evaluated. However, few studies have attempted to investigate aspects of portfolio risk based on rating-based credit risk models thoroughly. In Carey (1998) the default experience and loss distribution for privately placed US bonds is discussed. VaRs for portfolios of public bonds, using a bootstrap-like approach, are calculated in Carey (2000). While these two papers utilize a ”default-mode” (abstracting from changes in portfolio value due to changes in credit standing), Kiesel, Perraudin and Taylor (1999) employ a ”mark-to-market” model and stress the importance of stochastic changes in credit spreads associated with market values – an aspect also highlighted in Hirtle, Levonian, Saidenberg, Walter and Wright (2001).

The aim of this chapter is to contribute to the understanding of the performance of rating-based credit portfolio models. Our emphasis is on comparing the effect of the different approaches to modelling the dependence structure of

the individual obligors within a credit-risky portfolio. We use a default-mode model (which can easily be extended) to investigate the effect of changing dependence structure within the portfolio. We start in Section 5.2 by reviewing the construction of a rating-based credit portfolio risk model. In Section 5.3 we discuss approaches to modelling dependence within the portfolio. In Section 5.4 we comment on the implementation inXploReand present results from our simulations.