20070131 portolio credit derivatives based on rating migrations

Electronic copy available at: http://ssrn.com/abstract=1348685Electronic copy available at: http://ssrn.com/abstract=1348685

PORTFOLIO CREDIT DERIVATIVES

BASED ON RATING MIGRATION

Nicolas Gisiger1

Thesis submitted to

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

and UNIVERSITY OF ZURICH

for the degree of

MASTER OF ADVANCED STUDIES IN FINANCE

January 2007

under the supervision of Prof. Dr. Paolo Vanini

Director of Financial Engineering at Zurich Cantonal Bank

[email protected]

Electronic copy available at: http://ssrn.com/abstract=1348685Electronic copy available at: http://ssrn.com/abstract=1348685

Abstract

This thesis discusses portfolio credit derivatives which offer rating migra-

tion protection on a portfolio of assets. Credit rating migrations are in-

teresting, both from an originator’s and an investor’s point of view. After

discussing the possibilities and problems of expressing a view on rating mi-

gration through default-sensitive instruments, we introduce a new portfolio

credit derivative. Applications and payoff examples are provided, as well as

a chapter on the modelling and pricing of the product.

2

Acknowledgement1

I especially thank Prof. Dr. Paolo Vanini for his motivating way of super-

vising this thesis and his important input.

I am very grateful for information, comments and fruitful discussions to:

Dr. Johannes Burgi (Walder Wyss & Partners), Oliver Gasser (UBS), An-

dreas Johansson (JP Morgan), Dr. Markus Kroll (Palomar Capital Advi-

sors), Stefan Kruchen (ZKB), Prof. Dr. Markus Leippold (Uni Zurich),

Stefan Lenz (ZKB), Alessandro Materni (Capital Efficiency Group), Bruno

Oberson (ZKB), King Yut Quan (BNP Paribas), Jurg Schnider (UBS), Prof.

Dr. Philipp Schonbucher (ETH Zurich), Jurg Syz (ZKB), Eric Wragge (JP

Morgan).

Finally, I want to thank Miret for her friendly help in setting up the thesis

in LATEX.

1Disclaimer: The views expressed in this thesis are the ones of the author, and are not

necessarily endorsed by any other individual or institution. The thesis has been written

as a research paper in the course of a master’s program. Under no circumstances does it

represent a recommendation to engage in any kind of financial transactions.

Contents

1 Introduction 6

2 Credit Risk Transfer 8

2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 European SME CDO Market . . . . . . . . . . . . . . . . . . 16

2.3 Swiss SME CDO: HAT II . . . . . . . . . . . . . . . . . . . . 18

3 Rating Migration 23

3.1 Rationale for taking a position on rating migration . . . . . . 23

3.2 Expressing a view on rating migration . . . . . . . . . . . . . 27

3.3 Issues related to default-based products . . . . . . . . . . . . 31

4 Rating Migration Derivative 38

4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Rating Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Problems confronted . . . . . . . . . . . . . . . . . . . . . . . 52

5 Modelling and Pricing 55

5.1 The basic setting of the model . . . . . . . . . . . . . . . . . 56

1

CONTENTS 2

5.2 Stochastic Model Time . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Conclusions 65

A Securitized asset classes 67

B Sample rating migration matrices 69

C Average annual rating migration probabilities 71

List of Figures

2.1 Credit Risk Transfer Instruments . . . . . . . . . . . . . . . . 9

2.2 Unfunded - Credit Default Swap (CDS) . . . . . . . . . . . . 11

2.3 Funded - Credit Linked Note (CLN) . . . . . . . . . . . . . . 11

2.4 True sale vs. synthetic transaction . . . . . . . . . . . . . . . 14

4.1 Rating migration matrix . . . . . . . . . . . . . . . . . . . . . 39

4.2 Asymmetric payoff function . . . . . . . . . . . . . . . . . . . 41

4.3 Payoff with asymmetric payoff function . . . . . . . . . . . . 42

4.4 Symmetric payoff function . . . . . . . . . . . . . . . . . . . . 43

4.5 Payoff with symmetric payoff function . . . . . . . . . . . . . 43

4.6 Payoff for re-hedging . . . . . . . . . . . . . . . . . . . . . . . 44

4.7 Payoff on rating drift . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 Payoff on rating activity . . . . . . . . . . . . . . . . . . . . . 47

5.1 Transition probabilities . . . . . . . . . . . . . . . . . . . . . 57

5.2 Generator: transition intensities . . . . . . . . . . . . . . . . . 59

B.1 Sample rating migrations from 1995 to the indicated year . . 70

3

LIST OF FIGURES 4

C.1 S&P average annual rating transitions, 1981 - 2005 (condi-

tional on no rating withdrawal) . . . . . . . . . . . . . . . . . 72

C.2 Moody’s average annual rating transitions, 1920 - 1996 (con-

ditional on no rating withdrawal) . . . . . . . . . . . . . . . . 72

List of Tables

2.1 Swiss structured credit deals . . . . . . . . . . . . . . . . . . . 19

2.2 HAT II structure . . . . . . . . . . . . . . . . . . . . . . . . . 20

5

Chapter 1

Introduction

The aim of this thesis is to discuss a new portfolio credit derivative where

the payoff to investors is based on the rating migration in a pool of assets

rather than on the defaults of these assets.1 The discussion is embedded

in the theme of transferring credit risk on a portfolio of small- to medium-

sized enterprise (SME ) loans. Such a product does not yet exist; in fact, no

portfolio credit derivative has based its payoff on rating migrations yet. It is

therefore interesting to see how one could be structured, how it compares to

a more traditional default-based product and whether it can be replicated

with traditional default-based products.

The motivation for such an instrument is largely, but not solely, origina-

tion driven. The thesis will discuss why a bank has an interest in hedging

its credit portfolio against rating migrations rather than plain defaults, and

under what circumstances it cannot use default-based products for that pur-

pose. The topic is presented from the perspective of transferring Swiss SME1Even though a default is a rating migration to the default-state, rating migration is

subsequently always used for rating migrations other than defaults

6

CHAPTER 1. INTRODUCTION 7

portfolio risk for two reasons. Firstly, the absence of a liquid market for

default-based products adds to the difficulty in using these for expressing a

view on rating migration. Secondly, several banks in Switzerland are cur-

rently looking at the issue of SME credit risk transfer.

The structure of the thesis is as follows. The sections of the second chap-

ter introduce the terminology and main instruments for credit risk transfer,

expose the underlying motivations for such transfers, as well as give a glance

into the European market of SME CDOs, including an illustration of a par-

ticular SME CDO: HAT II.2 The third chapter reviews the motivation and

possibilities of hedging rating migration, and the problems confronted when

using already existing products for this purpose. A product definition for a

migration-based product is presented in the fourth chapter, which includes

a discussion of possible applications with payoff examples and the required

rating mechanism, as well as the involved problems with such a new prod-

uct. The fifth chapter identifies a stochastic process driving the underlying

rating migration matrix of the rating migration derivative which is the ba-

sis for subsequent analysis and pricing of the product. Finally, chapter five

concludes.

2Helvetic Asset Trust II

Chapter 2

Credit Risk Transfer

2.1 Terminology

Over the course of the last decade, the growth in credit risk markets has

been tremendous. Figure 2.1 (Jobst, 2005) gives an overview of credit risk

transfer instruments.

The two major asset classes in capital market products for the transfer

of credit risk are credit derivatives and securitization products. The aim here

is to give a quick overview moving from credit derivatives to securitization

in order to finally highlight a combination of both (a hybrid product) which

is represented by a synthetic CDO. The rating migration based product will

also be such a hybrid product, but using a different credit derivative than

credit default swaps (which are the underlying asset of standard synthetic

CDOs).

Credit derivatives are financial instruments that are designed to transfer

the credit risk of an underlying asset or a portfolio of assets between two

parties. In the case of a credit default swap, it is designed to swap the default

8

CHAPTER 2. CREDIT RISK TRANSFER 9

Figure 2.1: Credit Risk Transfer Instruments

Source: Jobst (2005)


exposure towards a specific fixed income product between two parties. A

credit event at a reference entity leads to an obligation of the protection

seller to the protection buyer. In return, the protection buyer makes regular

payments to the protection seller (usually defined in terms of LIBOR plus

a spread). The contract specifies the fixed-income product relevant for the

contingent payment, such as a bond. Three types of credit events are mainly

used in standardized ISDA contracts,1 which can be used individually or in

combination as default triggers: bankruptcy, failure to pay (for instance for

90 days) and restructuring. Restructuring leads to most disputes on whether

the event has actually taken place, since it is more difficult to measure

objectively than the other two events. There are two main settlement types

when default occurs: physical delivery (reference security) or cash settlement

(reference security, valuation or fixed payout).

A credit derivative can be funded or unfunded. Figures 2.2 and 2.3

illustrate each an example for both possibilities. The protection buyer bears

the counterparty risk of the protection seller when the derivative is unfunded.

This implies that the protection buyer should consider the possibility of joint

default of the reference entity and the protection seller. The protection seller

bears the counterparty risk of the protection buyer when the derivative is

funded. Credit-linked Notes are also known as funded credit default swaps

because the entire notional is paid up-front by the protection seller. They

are so-called hybrid securities since they basically combine a credit derivative

with a vanilla bond (Choudhry, 2005).

Instead of using credit derivatives, one can simply transfer the credit

risky asset itself (referred to as ”true-sale” transaction as opposed to ”syn-1International Swaps and Derivatives Association


Figure 2.2: Unfunded - Credit Default Swap (CDS)

Figure 2.3: Funded - Credit Linked Note (CLN)


thetic” transaction). This is achieved through securitization. Fabozzi (2004)

defines securitization as “a procedure by which financial assets such as loans,

consumer instalment contracts, leases, receivables, and other relatively illiq-

uid assets with common features that are held on the balance sheet of a

bank, financial institution, or other corporate entity are used as a collateral

backing for a package of securities that are issued to investors.” While the

idea is rather straightforward, a transaction can be very complex and involve

the services of many third parties.

The originator, the investor and the servicer are the three main parties

involved in a securitization transaction. They are related by means of a

special-purpose vehicle or SPV. In the course of its business activities, the

originator generates receivables from debtors. The ownership of a portfolio

of receivables is then transferred to the bankruptcy-remote SPV established

specifically for this purpose. The SPV structures the risks and payments

involved in the transaction, after which it passes them on to the investors

by issuing securities. The servicer is commissioned by the SPV to handle

the ongoing management of the assets.

In addition, a large number of other parties also have their share in a

securitization transaction. The arranger structures the securitization deal

and usually also evaluates the pool of receivables. During the structuring

process it may be decided that some of the risks will be transferred to credit

enhancers (i.e. providers of credit risk mitigation). The SPV is founded

by the sponsor, which may be the originator or the trustee. The trustee

monitors on behalf of the investors whether the execution of the transaction

is done properly as well as the business activities of the SPV and the servicer.

Securities are placed on the market by an underwriting syndicate, and the

credit quality of the securities is assessed by rating agencies in case they


are placed publicly. A liquidity facility provider is generally used in order

to streamline the cash flows on a transaction. Legal advisors are heavily

involved in the documentation of transactions. Usually, a ruling from the

tax authority is obtained for legal security concerning the tax treatment of

the deal.

A large variety of underlying assets has been used for securitizations.

Basically all receivables with reasonably predictable cash flows can be secu-

ritized.2

A special form of securitization products are Collateralized Debt Obli-

gations (CDOs), where the SPV issues debt and equity-like instruments

secured against a pool of assets. A seniority structure is introduced among

the issued classes of securities (the liability side of the SPV) by dividing

them into tranches. This leads to an internal credit enhancement for pri-

oritized tranches, since their credit quality improves the more subordinated

tranches there are. A tranche only starts bearing defaults in the underlying

portfolio once its subordinate tranches are completely wiped out, i.e. obli-

gations are assumed as soon as those to all prioritized tranches have been

fulfilled. Tranches are called (from lower to higher risk): 1) Super Senior

(AAA rated), 2) Senior (A to AAA rated), 3) Subordinated/Mezzanine (B

to BBB rated), and 4) Equity-Tranche (usually unrated; also called First

Loss, or Credit Enhancement). The pool of assets can be the same over the

life of the deal, or it can be managed by an asset manager. This is referred to

as a static, respectively revolving (or managed) structure. Forms of CDOs

that represent a combination of securitization techniques and credit deriva-

tives are synthetic CDOs. In a standard synthetic CDO, the issued notes2See appendix A for a non-exhaustive list of asset classes that have been securitized.


Figure 2.4: True sale vs. synthetic transaction

are backed by a pool of protection seller positions in credit default swaps.

Figure 2.4 illustrates a true-sale and a synthetic transaction. The true-sale

type is usually linked to funding needs, whereas the synthetic type is of-

ten used for pure risk management. The issued notes of synthetic deals are

backed by unfunded credit default swaps. The notional is therefore invested

in a high credit quality collateral during the life of the transaction.

Transaction structures vary significantly. The reason for that are on one

hand the different motivations behind the deals, and on the other hand the

different and evolving jurisdictions that are applicable to the deals. Since

large sums are usually at stake, considerable effort is put into optimizing a

structure for a specific purpose. This leads to increasing product diversity.

Increasingly, the CDO market moves from being originator driven to investor

driven. Single-tranche synthetic CDOs are pure investment products, where

the investor is allowed to define a reference portfolio for his protection seller


position, attachment and detachment points of his tranche, and the bank

then hedges its protection buyer position with credit default swaps (Fabozzi,

2004). Different motives for taking a position in a portfolio credit deriva-

tive from an originator’s as well as an investor’s perspective are given below.

A) Originator’s perspective:

1. Risk mitigation and diversification: By transferring credit risk in

a securitization transaction, a bank can restructure its credit portfo-

lio, thereby changing its risk/return profile. This might be useful, for

example, in cases where a bank’s credit portfolio accumulates consid-

erable concentration risks due to its regional sales strength. A bank

can also base its whole business model on securitization by concen-

trating on selling and cultivating client relationships, and leave the

subsequent credit risk management to the market.

2. Access to liquidity: Funded structures provide the securitizing bank

with additional funds. By isolating the pool of receivables and refi-

nancing it separately, the bank might also be able to obtain more

favourable terms than in the case of on-balance-sheet refinancing. This

is especially attractive to banks whose ratings would only allow less

favourable refinancing terms on the capital or interbank market.

3. Reduction of capital requirements: By transferring credit risks to

third parties, banks can reduce their regulatory and economic capital

requirements. Therefore, securitization allows the reduction of tied-up

capital, thus making it available for other business opportunities.

4. Product range enhancement: Banks frequently offer securitization


platforms to their corporate clients. This gives companies an alterna-

tive source of capital market financing in addition to conventional bank

loans.

B) Investor’s perspective:

1. Spread pick-up and leverage: Securities issued in securitization

transactions frequently offer attractive yields to investors. CDO notes

have historically given a pick-up in spreads as compared to similarly

rated debt instruments (Ahluwalia et al., 2006). Also, subordinated

tranches allow making investments on a leveraged basis.

2. Risk diversification: Whole new asset classes, such as bank loans,

can be made available through securitization which were not accessible

before to many investors. This can be interesting for diversification

purposes if the correlation of the new asset class’ credit risk and the

credit risk of already existing investment opportunities is low.

2.2 European SME CDO Market

As SME credit risk drives the underlying of the rating migration instrument

presented in this thesis, a quick overview of the European SME CDO market

is given here.

The European Commission defines a small- to medium-sized enterprise

(SME) as a firm with total staff of less than 250 individuals, an annual

turnover of less than €50 million (or a balance sheet size less than €43

million), and equity ownership by a company that is not an SME less than

25%. SMEs account for nearly 50% of the European Union’s GDP (EU

report, 2004).


The companies included in SME CDOs are, however, frequently larger

than the above stated definition indicates. A quantitative definition is very

useful for counting the entities. Yet when it comes to business, less clear-cut

criteria, such as sophistication, ownership structure, geographic reach, cor-

porate needs etc. can be more useful for customer segmentation. Obviously,

SME does then not directly refer to a size, but to other criteria frequently

shared by firms of small and medium size.

A currently important impact on the financing of SMEs comes from Basel

II, which may limit the banks’ willingness to lend to sub-investment grade

rated, thinly capitalized SMEs due to higher capital charges. This raises on

average, on one hand, the motivation of SMEs to tab directly the capital

market for their funding needs, and on the other hand, the banks’ tendency

to sell off their SME credit risks. Banks increasingly adopt risk-based pricing

of loans to SMEs.

The transfer of exposure to SME credit risk has recently been addressed

in Europe by the means of CDOs backed by SME loans. Transactions are

used for risk management purposes by originators, or for funding needs by

SMEs. According to Fitch, European SME transaction can be divided into

those with a promotional (often government-related) SME loan sponsor,

and those without (Fitch report, 2001). The steady increase of government

sponsored securitization schemes has led to an increasing issuance of SME

CDOs. The two main promotional sponsors have been Kreditanstalt fuer

Wiederaufbau (KfW) in Germany and Instituto de Credito Official in Spain.

Consequently, Germany and Spain are the two largest European markets

for SME CDOs. In 2005, the German SME CDO market was by far the

largest in Europe (57% outstanding notional), second was the Spanish mar-

ket (24%), followed by the UK market (8%) (Loizou, 2006). Most deals


so far featured synthetic exposure to SME risk. In Spain, true-sale trans-

actions are predominant, whereas the German market is largely synthetic.

There are examples of both, funded and unfunded deals, as well as static

and revolving structures.

The issued volume of European SME CDOs totalled €26 billion in

2005 which represented 38% of total European CDO issuance in that year

(Ahluwalia et al., 2006). The first transactions occurred in the end of the

90’s, so one can say that it is a fairly new market. European SME CDOs have

historically offered a spread pick-up compared to other European CDOs for

all debt tranches. However, this additional spread appears to have narrowed

over time (EU report, 2004).

2.3 Swiss SME CDO: HAT II

In order to highlight a single transaction which differentiates itself from the

SME rating migration product only by the use of a different credit derivative,

this section gives an overview of the HAT II CDO.3

In Switzerland there have been two synthetic securitizations of SME

credit risks until the end of 2006. Both were issued by UBS: HAT I in 2000

and HAT II in 2003. No true-sale SME deals have been conducted yet.

Table 2.1 shows the most prominent structured credit deals that have been

conducted in Switzerland between 1998 and 2006. The motivation behind

both HAT I and II was to reduce the credit risk exposure of UBS towards

the business cycle of the Swiss economy due to a high concentration of the

bank’s business in its original home market. The bank’s internal effect was3Information from meeting with Jurg Schnider and Oliver Gasser (UBS - Credit Port-

folio Management), and Offering Circular HAT II


Year Transaction Structure Volume Originator

1998 Tell RMBS DM 314 m SBV

2000 HAT I CDO CHF 350 m UBS

2001 Swissact RMBS Euro 355 m ZKB

2003 HAT II CDO Euro 2,500 m UBS

2003 Eiger CMBS Euro 699 m WTF

2003 Chalet I CDO CHF 3,026 m CS

2003 Chalet II CDO CHF 4,250 m CS

Table 2.1: Swiss structured credit deals

a reduction in the required allocation of economic capital to the manage-

ment of the credit portfolio. Exposures to large companies can be hedged

individually with the already existing CDS market and were therefore not

included in these transactions.

The structure of HAT II can be seen in table 2.2. UBS kept the equity

tranche with a thickness of 6% of the reference portfolio on its own books,

since the spreads for buying protection for the first loss piece was judged

to be too expensive at the time of issuance, meaning that the deal would

have turned unprofitable in terms of opportunity cost of saved economic

capital minus protection payments. The super senior tranche (attachment

at 16%) was equally kept on UBS’ books. The bank assessed the default

risk of this tranche to be below 0.02%, which is lower than their estimated

default risk of the whole bank itself. Therefore, it seemed reasonable to bear

the risk instead. There was no intention to save regulatory capital through

the transaction. To achieve this purpose, there would have been additional

constraints on the structuring which would not allow optimizing the trans-

action for the purpose of economic capital relief. To UBS it seemed more


Notes Rating (Moody’s) Tranche Size 3m EUR

SUPER SENIOR 84.59%

Class A Aaa 5.17% +60bps

Class B Aa2 2.00% +110bps

Class C A1 1.06% +250bps

Class D Baa1 1.18% +500bps

LOSS THRESHOLD 6.00%

Table 2.2: HAT II structure

reasonable to put all the effort into achieving a structure which optimizes

the risk transfer, instead of taking constraints from external bodies into ac-

count. For example, the EBK would require UBS to issue also the super

senior tranche, implying that the bank buys credit protection on very low

risk assets.4

The transaction was therefore only partially funded. The notional amounts

of the issued securities were paid by investors up-front. Even though fund-

ing was no objective due to cheaply available funds for a bank with good

credit rating, zero funding would have considerably decreased the circle of

potential investors, since many of them require cash instruments. An im-

portant class of investors was Structured Investment Vehicles (SIVs) which

bought a significant share of the HAT II tranches and earned a margin by

refinancing themselves below the HAT II spreads. Such a structure relies on

the protection seller position being funded. Funding also transfers the credit

risk arising from the derivative transaction itself from UBS to the investor.

Most investors had a buy-and-hold strategy.4EBK (the Swiss Federal Banking Commission) is the supervisor of the Swiss banking

sector.


The default triggers used for HAT II are bankruptcy and failure to pay

for 90 days. For regulatory capital relief, the EBK would have required to

include the trigger restructuring as well, even though UBS felt adequately

protected with the first two triggers. In the credit default swap market,

additional protection on restructuring has been estimated to increase CDS

spreads on average by around 6-8% (Berndt et al., 2006). UBS assumed

that by including this trigger, the spreads on their deal would therefore

have been higher, also amplified by investor uncertainty on the issue of

objectively measuring restructuring events at SMEs.

The recovery rate in case of default was fixed at 45%. This was an

estimated average for UBS’ SME loan portfolio. Using a predetermined

figure avoids the time lag between the credit event and the determination

of the correct real recovery rate.

The issued HAT II tranches were rated by Moody’s and Fitch. To obtain

these ratings, UBS has conducted a mapping of its own SME ratings onto

the rating scales of the two rating agencies. For this purpose, the loss

history of SME loans at UBS was analyzed, and ratings were matched with

probabilities of default or expected losses, which are the link between the

rating systems. The whole rating package cost approximately CHF 500,000

up-front and CHF 30,000 annually per agency. UBS is allowed to re-calibrate

their internal rating system, but has to consult the rating agencies in case

they would like to change the rating method and process themselves.

A delicate point in both HAT deals is their revolving structure. A SME

loan at UBS has on average a maturity of between 12 and 24 months, whereas

the maturity of HAT I and II is five years. Maturing loans in the reference

portfolio are therefore substituted by UBS. In fact, UBS is allowed to substi-

tute loans in the reference portfolio independently of any event; the timing


is fully discretionary. Investors therefore might fear that the bank has an

interest to substitute loans with worse credit quality loans. In order to avoid

or limit moral hazard in this process, the rating agencies define criteria for

the substitution such as minimum debtor rating, industry distribution of the

reference portfolio, etc. The adherence to these guidelines is then monitored

by a portfolio auditor. Substitution guidelines are of outmost importance

for a managed portfolio credit derivative.

Chapter 3

Rating Migration

Before introducing the new product, this chapter discusses in the first section

why an originator or an investor might want to express a view on rating

migration rather than on defaults. The second section explains how existing

default-based products can be used for this purpose, before section three

exposes the problems involved.

3.1 Rationale for taking a position on rating mi-

gration

A) Protection buyer’s perspective:

• The manager of a credit portfolio worries about the margin between

the regular instalments paid by the debtor and the cost of funds as-

sociated with the loans. The cost of funds depends on the risk in-

volved in the loans. Since the loan conditions are locked-in for a pe-

riod of time, they might not stay risk adequate. To hedge the margin

against deteriorating credit quality, the manager would be interested

23

CHAPTER 3. RATING MIGRATION 24

in a product which covers exactly this risk. Some banks define inter-

nal prices in such a way that the responsibility for bearing this risk is

very clearly allocated. For example at UBS, the front business (origi-

nation) sells a loan, receives then regular instalments and pays a fixed

spread to the credit portfolio management during the whole life of the

loan. The fixed price is based solely on the initial credit quality of

the borrower. The credit portfolio management in return receives this

constant spread, but pays a variable spread to the corporate centre.

Here the price is based on the evolving actual credit quality of the

borrower. It is therefore obvious that the credit portfolio management

bears the credit risk and is interested in all rating migrations, and not

just defaults.

• An approaching default can be more easily dealt with in the course

of a loan’s life than rating migration. When a default becomes more

likely according to the bank’s monitoring, the exposure of the bank

towards the corresponding company can be reduced. Since a default

is usually the most remote credit deterioration, there is more room for

manoeuvre before the event than for other bad migrations. Efficient

monitoring might be cheaper than buying default protection. In this

case a down step in credit rating might not increase the cost of a

loan by increasing the probability of its default, but by increasing the

monitoring cost incurred by the bank.

• Hedging against rating migration only is a reasonable hedge for credit

risk managers during times when the credit quality of the portfolio is

very good and the worst event to be considered by the risk manager

are not defaults, but a deterioration of the portfolio’s credit quality


reflected in rating migrations.

• Instead of viewing a single loan as the asset, the whole customer re-

lationship can be perceived as the bank’s main asset. Even if the

maturity of the loan is short, the bank might not have an interest to

adapt loan conditions too aggressively with changing credit quality for

the purpose of cultivating a beneficial customer relationship.

• On the product side of the company, compensation for bad migration

gives the possibility to offer long-term financing to clients at fixed rates

without keeping potential migration risk and allowing the reduction

of financial covenants. Covenants give the bank the right to reduce its

exposure if the borrower does not fulfil some specific criteria. Since

covenants represent uncertainty for future funding, they are costly for

the borrower. Simpler loan conditions are perceived as more customer

friendly and transparent.1

• When the credit risk of a portfolio is hedged over a certain period,

the maturity of the hedging device does not necessarily match with

the risky assets. It is possible that the average credit quality of the

portfolio becomes worse, but no defaults are experienced. In this case,

the default-based product expires without any protection payments.

When the risk manager wants to hedge the credit risk again after ex-

piry, he might be faced with significantly higher protection spreads due

to the deteriorated credit quality or due to spread risk. However, in

this case the rating migration derivative is not used as an alternative

for default protection, because it must also include default as a credit1However, the fulfilment of covenants has disciplining effects and can be beneficial to

the credit quality of the borrower.


event. The payments due to defaults would then be the same as in

standard CDOs, and those due to other rating migrations should de-

pend on the market price of credit risk at expiry in order to cover the

risk of incurring higher cost when acquiring new default protection.

B) Protection seller’s perspective:

• An investor can have an opinion on the performance of an economy.

He then faces several possibilities to express his view by trading on

the financial markets. One possibility is to bet on the change in credit

quality in a portfolio of companies which are representative of the

economy of interest.2 A default is usually a low probability event (i.e. a

tail event) and can be very difficult to forecast. Even in a portfolio of a

thousand different entities, there might be only few defaults happening

within a few years. The number of defaults can therefore be seen as

quite a crude measure of a portfolio’s overall credit quality. In contrast,

rating changes of a company are much more probable. Many more

rating changes can be expected than defaults. An index based on all

rating changes in a portfolio can be seen as a more refined measure of

credit quality changes than the aggregate number of defaults, since it

incorporates changes on a larger credit spectrum.

• Defaults represent only downside risk. Rating migration risk is usually

upside as well as downside risk. Through rating migration, credit risk

behaves more like market risk, and views in both directions can be

expressed.2There might of course also be other reasons why an investor forms a direct or indirect

opinion on the development of a portfolio’s credit quality.


• Protection payments for defaults can be very high, up to 100% of the

protected notional. Moreover, defaults occur with a non-negligible

probability. This gives rise to a highly skewed loss distribution. With-

out protection payments occurring at defaults, but including more

moderate payments for less extreme credit quality deteriorations, the

loss distribution for the investor is much less skewed. Hence, effective

diversification in a pool of a few hundred loans becomes much easier

and the required credit risk premium is probably reduced. This issue

is explained more in depth in section three.

3.2 Expressing a view on rating migration

This section reviews how already existing default-based products can be

used to express a view on credit rating migration. The question basically

boils down to how a migration product can be composed of default-based

products. After discussing the basic arguments and applications, a general

strategy for hedging rating migrations is given.

The standard product in the credit market is the credit default swap.

Protection payments are made once the reference entity fulfils the definition

of having defaulted on its bonds (or other specified fixed-income product).

However, a CDS can also be used to hedge against unfavourable rating

migrations. A credit rating expresses an expected probability of default (or

an expected loss, if the expected recovery rate is assumed to be uncertain as

well). If the CDS spread is fair, then the present value of the fee leg equals

the value of the contingent leg. According to a widespread practitioner’s

method, the spread therefore directly implies an expected probability of

default.


PVfee leg = N × (1− P(d))×DF × Scds

PVcontingent leg = N × (1−R)× P(d)×DF

PVfee leg = PVcontingent leg ⇒ P(d) =Scds

Scds + (1−R)

where N is the protected notional, P(d) the default probability, DF the

discount factor, Scds the CDS spread and R the recovery rate.3

The expected recovery amount also directly impacts the equilibrium CDS

spread. If the recovery amount is fixed however, as in the HAT II CDO for

instance, no uncertainty remains in this parameter.

If the CDS spread moves, it implies therefore a revaluation of the market

participants’ expected probability of default. Logically, the CDS spread for

a company is negatively related to its credit rating. Hull et al. (2004) find

that rating announcements by Moody’s are anticipated by CDS spreads, as

the spreads are useful in estimating the probability of a rating event.

From the viewpoint of an investor, who does not want to hedge any-

thing, but simply take a position based on his opinion on where ratings will

migrate, the strategy is simply to buy default protection for betting on wors-

ening ratings, and to sell protection for betting on improving ratings. If the

expected credit risk goes up, the market spread widens, causing the deriva-

tive with a lower locked-in spread to gain market value which compensates

for the higher risk; the reverse applies for improving credit quality.

A credit risk manager is more interested in the downside for possible

rating migrations. In order to hedge its loans, the bank buys default protec-

tion for the required maturity and uses changes in the value of the derivative3The above formulas make several simplifying assumptions, such as a flat credit term

structure, no interest rate risk, etc.


due to differences between the market spread and the locked-in spread as

compensation for changing credit risk.

Just like a bond, the value of a CDS experiences a pull-to-par effect

towards expiry.4 The closer the contract is to expiry, the more moderate

is the effect of changing credit quality on the value of the contract, since a

difference between the locked-in spread and actual market spread matters

less the closer the CDS is to its maturity. Since the same effect happens to

a loan, the position in the CDS is a static hedge for rating migration and

does not need to be adjusted over time.

The same arguments apply for a portfolio credit derivative. However,

the parameter which comes into play additionally with portfolios is default-

correlation. Li (2000) defines default correlation as linear survival time

correlation.5 TA is the time length that firm A is surviving. The survival

time correlation between firm A and B is then given by:

ρAB =cov(TA, TB)√

var(TA) var(TA)

The market spreads of the portfolio derivative tranches also depend on

expected default correlation. Higher default correlation is good for equity

tranche holders, since it increases the probability that no default will occur.

The opposite is true for super-senior tranches where the probability of de-

fault becomes more likely with higher default correlation. The sensitivity

of both equity and super senior tranches with respect to default correla-

tion is monotonic; for tranches in between, the relation is not that clear-cut

(Elizalde, 2005). In general, if the same risk-averse entity bought all tranches4Pull-to-zero effect for an unfunded credit default swap.5It is important to note that linear correlation only fully captures the dependance

between two random variables when they are elliptically distributed. See McNeil et al.

(2005).


of a deal, higher correlation is punitive for the protection seller, because it

diminishes the diversification gained by pooling the assets. Hence, if default

correlation increases, the sum of tranche-notional-weighted market spreads

for the deal rises as well. Since lower spreads are locked-in, the deal gains

value for the protection buyer. It is obvious that hedging through marking-

to-market is affected by default correlation risk. Since the same effect occurs

with a portfolio of loans, it is not unwished.

Default protection can theoretically be acquired for any maturity. In

practice, there is much more liquidity for some standard maturities. Hence,

it is possible to decide to split a hedge into several time steps. At each

roll-over, the market spreads for new protection can change, either due to

a different credit quality of the underlying, or changing market spreads for

the same level of risk. We term this risk of incurring higher market spreads

at roll-over re-hedge risk. To hedge this risk, the hedging device needs to

take into account the market spread differential between the initial spread

of the initial credit risk level and the spread at expiry for the credit risk level

at expiry to compensate for higher hedging cost. If the credit risk or the

market price of credit risk jumps a level higher just before expiry, a jump in

the payoff should be caused by that in order to compensate for higher re-

hedge cost. To use a CDS for hedging the re-hedge risk, its maturity must be

longer than the first hedging period in order to have a value which is sensitive

to the market spread at the end of the first hedging period. The closer a

CDS moves towards expiry, the lower its sensitivity to credit spread changes.

The required number of credit defaults swaps towards expiry would rise to

infinity. However, hedging the re-hedge risk only becomes an issue precisely

when there is either no default protection available for the maturity of the

risky asset, or the risk manager does not want to use the same maturity. A


new product is therefore needed for this application.

We now expose a general strategy for hedging rating migration according

to Schonbucher (2003). To hedge a credit-sensitive instrument V against

rating migration, we need as many hedge instruments Fk as there are rating

classes (K). The hedge weights αk have to satisfy

V(r,R1)−V(r,R0) =K∑

k=1

αk(Fk(r,R)− Fk(r,R0)), ∀1 ≤ R ≤ K

where V is supposed to migrate from rating class R0 to rating class R1.

Since there are K − 1 possibilities for migrating to another rating class,

the K hedging instruments must satisfy K − 1 equations. To obtain a

unique solution for the hedge weights, a Kth equation comes from hedging

the continuous interest rate risk, as follows:

∂

∂rV (r,R0) =

K∑

k=1

αk∂

∂rFk(r,R0)

There are several assumptions involved in this strategy. It is assumed

that the impact of a rating change on the value of V and all hedging in-

struments is known in advance. Furthermore, this impact must be differ-

ent for all hedging instruments, so that all rating change possibilities can

be hedged simultaneously. In addition to the difficulty of finding enough

hedging instruments, there is still the possibility that credit spreads change

independently of credit risk, which makes an effective implementation of this

hedging strategy very unlikely.

3.3 Issues related to default-based products

This section explains the basis risk involved in using default-based instru-

ments for hedging rating migration and the non-suitability of these instru-


ments for expressing a view on rating migration in general.

1) Other risk factors driving the credit spreads

Credit ratings change discretely, whereas credit spreads can change almost

continuously. This would suggest that rating changes represent more nu-

anced expressions of credit risk. There are several reasons why this argument

is misleading.

Most importantly, CDS spreads are not only a reflection of the market’s

expectations of a probability of default. For companies with a given credit

rating, quite a variation in the CDS spread can be observed (Hull et al.,

2004). One could argue that a credit rating is a discrete estimation of

risk and that there are still risk differences within a rating class. However,

there seem to be also other drivers of credit spreads. A CDS spread is a

market price: it is established by supply and demand which can take into

consideration other aspects than default probability.

For the bond market, Delianedis et al. (2001) conclude that expected

default risk only explains 5% (22%) of credit spreads for AAA (BBB) -rated

firms.6 The wide gap between corporate debt spreads and default proba-

bilities is known as the credit spread puzzle. They estimate that recovery

risk can also not explain this puzzle, and attribute a major role to liquidity

risk by stating that increased trading volume in a firm’s bond significantly

reduces the residual credit spread. Taxes are another relevant driving factor,

since government bonds often enjoy special tax treatments.

Elton et al. (2001) claim that probabilities of default cannot entirely

explain the credit spreads of corporate bonds. They explain this difference6By defining the credit spread as the difference between the yield on corporate debt

and government bonds.


with the fact that bondholders bear systemic, non-diversifiable credit risk

and ask comprehensibly for a credit risk premium.

Also D’Amato et al. (2003) argue that the answer to the credit spread

puzzle lies in the difficulty of diversifying default risk. The loss distribution

for defaultable-bonds is heavily positive skewed, meaning that it exhibits a

fat right tail. The probability for huge losses is high enough that it becomes

very difficult to diversify the unexpected default risk away in a portfolio that

is not extraordinarily large. Defaults are not counterbalanced by an equally

small chance to make similarly large gains. They claim that in practice

portfolios cannot be large enough to avoid a heavily skewed loss distribution

even on a portfolio level. Such undiversified risk is consequently priced into

credit spreads. For example, between 1997 and 2003, BBB-rated corporate

bonds offered on average a yield of 170 basis points annually with three to

five years to maturity. The average yearly loss during this period amounted,

however, to only 20 basis points annually.

It is, however, not suitable to directly compare corporate bond spreads to

CDS spreads. Longstaff et al. (2005) computed bond-implied CDS spreads

and concluded that CDS spreads were consistently lower than bond-implied

spreads due to tax and liquidity effects.

Using credit default swap data, Berndt et al. (2004) estimate actual

and risk neutral default probabilities. On average, the risk neutral default

probabilities are approximately twice the actual default probabilities, which

is a way of recognising the risk premia in the market additional to the time

value of money. Additionally, they conclude that for a given probability of

default, there is substantial variation in credit spreads over time.

Since the loss distribution of a CDS is as skewed as the one of a bond,

it can be expected that part of the risk premia is a credit risk premium for


non-diversifiable unexpected default risk.

A liquidity risk comes into play. Tang et al. (2006) estimate an implied

illiquidity premium in CDS spreads between five and eleven basis points.

Spreads also adjust for changing expectations on the availability of deliver-

able obligations in case of default (Stuttard, 2006).

These drivers of credit spreads additional to credit risk are neither con-

stant, nor are the associated risk premia constant. For a risk manager, these

factors represent basis risk, and for an investor interested in rating migra-

tion, they are unwished noise. In the case of Swiss SME credit risk, one must

acknowledge that there is basically no secondary market and consequently

no secondary market credit spreads.

2) Loss of sensitivity

For a risk manager interested in hedging the margin of the bank’s loan

business, the sensitivity loss over time with respect to rating migration is

welcome as it coincides with the same effect of the value of loans with limited

maturity. For an investor interested in expressing his view on migration, this

effect is however not really useful. Quite the contrary, his instrument loses

sensitivity towards the factor of interest.

3) Cost and default risk

The party betting on credit deterioration is short the default risk and pays

regularly a premium for a protection that is not required. It is therefore a

very costly strategy.

On the other hand, gaining with improving credit quality necessitates

being long the credit event risk, which implies bearing the risk of losing large

amounts when a default happens. This is of course a very risky business.


The risk involved in rating migration is more moderate than in defaults,

meaning that the loss distribution is more centred.

4) Liquidity risk

Another important requirement is to have a liquid market for the CDS, so

that the position can be closed out when the rating migration has happened.

Liquidity assures that the investor can close-out at a market price which is

built efficiently. Quoted CDS spreads are usually quoted for a standardized

trading volume of $10 million and are not guaranteed for any further depth

(Delianedis et al., 2001). If the default-based instrument cannot be closed-

out efficiently at discretionary timing, this gives rise to a liquidity risk, which

is different from the risk of a changing illiquidity premium incorporated in

the CDS spread.

For CDOs, liquidity might not be given to obtain fair spreads very

shortly. Especially for equity tranches, which are the most sensitive to

credit quality changes in the underlying reference portfolio, potential in-

vestors spend some time analyzing the deal before any investment is made.

According to CDO traders, the secondary market of SME CDO tranches is

relatively illiquid, especially in the lower rated tranches.

5) Correlation risk

Portfolio instruments are influenced by the default correlation of the assets

in the reference portfolio. It is important to note that default correlation

does not capture the joint behaviour of rating migrations, and is therefore

not an adequate concept for rating migration correlation in a portfolio of

assets. Denoting Yi,t as the rating class of asset i at time t, Gagliardini et al.

(2004) define the joint probability of two assets to migrate within a system


of discrete rating classes as

Pkk∗,ll∗ = P[Yi,t+1 = k∗, Yj,t+1 = l∗|Yi,t = k, Yj,t = l]

They further define rating migration correlation as

ρkk∗,ll∗ =Pkk∗,ll∗ − αkk∗ αll∗√

αkk∗(1− αkk∗)√

αll∗(1− αll∗),

where αkk∗ is the expected transition probability for each asset to migrate

from rating class k to rating class k∗.7

A CDO is sensitive to default correlation. Hence, rating migration corre-

lation might not be adequately captured by existing default-based portfolio

credit derivatives.

By using the whole dataset on annual rating migrations between 1920

and 1996, a study conducted by Moody’s rejects the hypothesis of inde-

pendent rating transitions for Baa-rated firms at the 99% confidence level.

They conclude in particular that credit quality correlation seems to be very

much associated with the firms’ industry and geographic domain (Moody’s

report, 1997).

6) Accounting problem for protection buyer

The financial accounting for credit derivatives has changed with the adop-

tion of IFRS by many European companies. A mark-to-market valuation

(replacement value) is used and gains and losses in the market value have

to be accounted for in the profit and loss statement (trading income).

The basis for the appropriate accounting of credit derivatives is:

• IFRS (International Financial Reporting Standards) via IAS 39 (recog-

nition and measurement of financial instruments)7It should be noted that this is again a linear correlation.


• US GAAP via FAS 133 (Accounting for derivative instruments and

hedging activities)

Loans are usually booked at cost, whereas credit derivatives are recog-

nized at fair value. This leads to a timing mismatch which causes unwanted

volatility in the profit and loss statement and the balance sheet.

In the absence of defaults, a default-based instrument can only be val-

ued by using a fair market spread. The basis risk discussed in this section

can heavily influence the P&L volatility of the bank. A migration product

which gets triggered immediately when rating migrations occur, and caus-

ing further obligations with more rating migrations until the expiry of the

instrument, offers another base for marking the instrument’s value to mar-

ket. Instead of using market spreads and comparing them to the locked-in

spreads, it is possible to simply take the promised payoff up to date into

account, and base a valuation on this figure. Thereby, one could also avoid

letting the pull-to-par effect influence the profit and loss statement. Such a

migration-based instrument is presented in the next chapter.

If the intention is to use the derivative for hedging the re-hedge risk, the

problem of a timing mismatch between the derivative and the risk cannot

be avoided since it is not allowed to build reserves during the course of the

derivative for the purpose of hedging this risk, and the promised payoffs of

the derivative must be accounted for as they occur.

Chapter 4

Rating Migration Derivative

All the problems associated with default-linked products would be even en-

hanced when using them for Swiss SME loans. Hence, this chapter intro-

duces a rating migration-linked portfolio credit derivative; after an initial

definition, several applications are discussed. Moreover, payoff examples are

given on the basis of historical S&P rating transitions within a portfolio of

fictitious loans. The last section discusses the major problems confronted

when implementing such an instrument.

4.1 Definition

By assuming that all loans in a defined portfolio are assigned to one of eight

rating classes (AAA, AA, A, BBB, BB, B, CCC, D), we can define a 8x8

rating migration matrix M(t) (see figure 4.1), where each element mi,j is the

sum over all loan notional values with an initial rating i and a rating j at

time t.

with mi,j = 0, at t = 0, ∀i, j, i 6= j and mK,j = 0, ∀j

38

CHAPTER 4. RATING MIGRATION DERIVATIVE 39

M(t) =

m1,1 m1,2 . . . . . . . . . m1,K

m2,1 m2,2 . . . . . . . . . m2,K

......

. . ....

...... mi,j

...

mK−1,1 mK−1,2 . . . . . .. . . mK−1,K

0 0 . . . . . . . . . 0

Figure 4.1: Rating migration matrix

This matrix is the underlying of our derivative.1 It is now possible to assign

different (possibly time-dependant) payoff functions to this matrix depend-

ing on the aim of entering into the transaction. For this purpose we define a

8x8 payoff matrix A(t) which allocates a payoff to the structure of the rating

migration matrix at time t. Each column of A corresponds to a different

starting rating, and the different elements of a column specify the payoff

linked to a specific final rating. To obtain the payoff x at time t ≤ T , we

multiply the rating migration matrix M(t) with A(t) and take the sum over

all diagonal elements of the resulting matrix.

x(t) = trace[M(t)A(t)]

The obligation to deliver the payoff can also be tranched in case it seems

favourable to introduce a priority structure among protection sellers. This

would be the case if the required aggregate protection spreads are lower

when different priority classes are introduced than when there is only equal

treatment in terms of risk and return.1We keep the Kth row in the matrix even though it does not contain any information,

because it facilitates manipulations of the matrix in the modelling chapter.


4.2 Applications

1) Hedging the credit margin

To hedge the downside of possible rating migrations, each negative transition

is weighted by the credit spread difference between the initial rating class

and the rating class at time t. An example for A(t) is shown in figure 4.2.

The values are expressed in terms of basis points of the notional values

and are for illustrative purposes only. In practice, they should correspond

to the differences between risk-adequate costs of funds within the bank.

Often, internal credit spreads are fixed over some time interval and get

adjusted only every couple of years. This fact is reflected in the determined

payoff weights for particular migrations over the life of the derivative. It is

important to note that the spread differentials are expressed on the basis

of one year. The payoff matrix is thus applied in annual time steps to the

rating migration matrix. This makes sense because SMEs usually hand-in

their financial reports once per year, and migrations will happen mostly at

this time of the year.

The first column specifies the annual payoffs in basis points of the sum of

loan notional values which were rated AAA at initiation. Each row further

down corresponds to a downgrading in the rating. The last row corresponds

to the default state. We selected no payoff for defaults, which means that

there is no default protection. It might make sense to define a relatively low

payoff for defaults as well, so that CCC-assets can be included right from

the beginning.

To illustrate an example of the payoffs, we randomly created a portfolio

of 1,000 names rated by S&P, with the only condition that each name is

continuously assigned a rating during the life of the transaction. The deal


A(t) =

0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0

30 20 0 0 0 0 0 0

70 60 40 0 0 0 0 0

150 140 120 80 0 0 0 0

500 490 470 430 350 0 0 0

1000 990 970 930 850 500 0 0

0 0 0 0 0 0 0 0

Figure 4.2: Asymmetric payoff function

shall be active from the beginning of 1995 to the beginning of 2005. 90%

of the companies in the portfolio turned out to be located in the US, the

rest are from Canada, Western Europe, Japan, Australia and New Zealand.

During ten years, rating migrations are recorded once per year. The notional

value of each loan is CHF 1,000,000. Accordingly, the total notional value

of the reference portfolio is CHF 1,000,000,000.

It is important to note that this illustration might not show the same

patterns if it was applied to Swiss SME ratings.2 SMEs can be expected to

show a less stable credit performance than larger, publicly rated companies,

as the latter are more diversified internally. Moreover, the leading rating

agencies do not adjust their ratings very aggressively, so that the probability

of default of a rating class can temporarily change through a business cycle.

Therefore, more volatility can be expected when the used rating classes

correspond to constant probabilities of default.

The rating migration matrices for each year are shown in appendix B.2SME ratings were not available for this study.


Figure 4.3: Payoff with asymmetric payoff function

It can easily be seen that the dispersion of the rating increases over time

in most initial rating classes. Surprisingly, there is no migration happening

in the lowest non-default rating class CCC after the first year. Our data

from other years suggests that this is coincidence. Figure 4.3 illustrates

the development of the payoff over time, when we apply payoff function

A(t). The annual payoff is increasing each year, reflecting the increasing

probability of downgradings from the initial rating classes over time.

Next we define a symmetric payoff function B(t), which compensates

downgradings with upgradings. This turns the protection seller for down-

gradings into the protection buyer of upgradings, and vice versa. The credit

risk managing bank forgoes the benefit of upgradings, but it will thereby

receive its downgrading protection cheaper.

Figure 4.5 shows that resulting payoffs for each year. Until the year

2000, credit migrations in each direction balance each other more or less.

After that year, protection payments for credit quality deterioration start


B(t) =

0 −10 −30 −70 −150 −500 −1000 0

10 0 −20 −60 −140 −490 −990 0

30 20 0 −40 −120 −470 −970 0

70 60 40 0 −80 −430 −930 0

150 140 120 80 0 −350 −850 0

500 490 470 430 350 0 −500 0

1000 990 970 930 850 500 0 0

0 0 0 0 0 0 0 0

Figure 4.4: Symmetric payoff function

Figure 4.5: Payoff with symmetric payoff function


C(T ) =

c1,1 c1,2 . . . . . . c1,K−1 0

c2,1 c2,2 . . . . . . c2,K−1 0...

.... . .

......

... ci,j...

cK−1,1 cK−1,2 . . . . . . cK−1,K−1 0

0 0 . . . . . . 0 0

Figure 4.6: Payoff for re-hedging

dominating rapidly. It appears to us that this effect can be attributed to the

economic downturn in the US and Western Europe, causing systematically

more downgradings than up-gradings. However, we also believe that a gen-

eral downward drift in credit ratings lies in the nature of business, since a

default is an absorbing state and can be reached by each firm with a positive

probability. It is therefore interesting to note that the economic expansion

of the late 90s in the Western world did not seem to result in an important

overweight of credit quality improvements.

2) Re-hedge risk

Re-hedge risk is defined in section 3.2. To hedge this risk, the payoff must

occur at maturity only, and depend on credit spread differentials between

the market spreads at initiation and spreads at expiry. Formally, we de-

fine the payoff structure C(T) (see figure 4.6), where ci,j is the difference

between the market credit spread between rating class i at time 0 and the

market credit spread of rating class j at time T.

Critically, one can note that portfolio default protection supposedly costs

less than the notional-weighted spread average for protecting its elements.


This is an effect of diversification. With the suggested payoff structure in

figure 4.6, the protection seller pays therefore more than is required for

hedging the portfolio. In the absence of a liquid market for exactly this

portfolio, one cannot resort to other observable market credit spreads.

Since the payoff relies on the observability of market credit spreads for

all rating classes K, this instrument is certainly not applicable to SME credit

risks.

3) Index products

International investors might be quite hesitant when it comes to investing

in a very local market, such as a Swiss Canton for instance. A bank which

has a regionally narrow credit risk exposure could ally with other regional

banks. Each participating bank then defines a portfolio of loans that it

wants to protect against rating migrations. All the loans are communicated

to a calculation agent, who determines the credit events and allocates re-

sulting protection payments to the banks. Each bank gets protection on its

loans. Thereby, the product fulfils its objective for risk management, and

additionally it becomes more interesting for an investor whose knowledge or

investment-target is on a more macro-level than the exposure of the local

bank.

When the united reference portfolio becomes geographically big enough,

it can again be divided into segments such as different industries or other

criteria. Moreover, if the sustainable volume of new loan issuance of the al-

lied banks is high enough, a new reference portfolio can be launched reliably

in equal time steps. Such resulting rating migration matrices could then be

used, not only for risk management, but as an underlying for any imaginable

payoff structure. The banks are then no longer necessarily taking a position


in further transactions, but simply provide the calculation basis for the pay-

off structures agreed between the long and short parties. For example the

iTraxx Europe is an index provided on the basis of the 125 most actively

traded credit default swaps and is re-launched every six months. This index

is also provided in a tranched structure, allowing investors to apply different

leverage to their exposure.

The demand and supply of protection would either be matched directly

by some auction mechanism, or they pass through market makers, as it is

the case for iTraxx Europe derivatives. The existence of a pool of market

makers which can guarantee a tight bid-ask spread would be essential for

investors to trade confidently in the market.

An advantage of using a portfolio index is that it provides basically

immediate ramp-up, whereas other portfolio credit derivatives often require

several months to build up their exposure. Different payoff structures can be

specifically invented to correspond to the needs of different investor groups

such as asset managers, hedge funds, risk managers and correlation traders.

We present here two payoff possibilities which can be used to bet on

rating drift and on rating activity. Moody’s defines rating drift as the num-

ber of upgradings weighted by the number of jumped rating classes minus

downgradings weighted by the number of jumped rating classes, divided by

the number of unchanged ratings (Moody’s report, 1997). Rating activity

is defined similarly, except that down- and upgradings are added together.

Figure 4.7 shows the payoff for annual rating drift in the fictitious portfolio.

Figure 4.8 is based on annual rating activity. The payoffs are expressed in

basis points.


Figure 4.7: Payoff on rating drift

Figure 4.8: Payoff on rating activity


4.3 Rating Mechanism

This section discusses the requirements on the ratings used in the underlying

of the derivative. In the end, we make a suggestion for a possible implemen-

tation.

A) Requirements

Each firm included in the underlying of the derivative must be assigned a

credit rating. SMEs are usually not publicly rated. For our purpose, we

need point-in-time instead of through-the-cycle ratings. This means that

a rating class expresses an unconditional probability of default and, hence,

gets adjusted with changes of the latter. Conditioning on the business cycle

would largely remove the desired effect of rating migrations due to economic

up- and downturns, and introduce more stability.

There are a couple of other requirements. Ratings must be meaningful,

transparent, consistent and cheap. A rating has meaning when it exhibits

predictive power concerning the probability of default. This is achieved by

using a suitable credit risk model. Transparency is especially important for

the migration derivative because protection sellers and buyers have to un-

derstand the rating function in order to analyse the dynamics of migrations.

Hence, the rating function has to be public and completely hard-wired, so

that ambiguities in rating allocations can be excluded. A rating mechanism

is consistent when it is applied to firms, markets and conditions that are of

the same nature as those which were used for establishing and calibrating

the credit risk model. To satisfy this requirement, one probably has to draw

upon recent cross-sectional data for fitting the model. Last but not least,

the rating allocation must be a cost-efficient process. The outstanding loan


notionals to SMEs are on average much lower than those of public compa-

nies. Ideally, the protection buyer combines the rating procedure with its

already existing process used for credit approval and monitoring. Basel II

requests a 12 months point-in-time probability of default anyway (EU re-

port, 2004). Cost efficiency is an important requirement for such an exotic

derivative to be viable.

Modern credit risk models can be broadly divided into structural models

and reduced-form models (Altman et al., 1998). In a structural model a

default occurs as the value of assets falls below a certain threshold. The

inspiration of all structural models is Merton’s model which is based on

observable equity prices. It perceives the equity holder of a firm as a holder

of a call option on the firm’s assets, where the strike price is in relation with

the firm’s liabilities.3 From the equity value and its volatility, it is possible

to derive the level and volatility of the firm’s asset value. This leads to a

distance-to-default measure, which is simply the difference of the asset value

and the default trigger. This difference is then mapped to a probability of

default either by assuming a particular probability distribution, or by using

historical data. Such an approach is, however, not useful for rating SMEs

as they rarely place equity publicly. Thus, there is no secondary market

delivering observable equity prices.

Reduced-form models leave the process of default unspecified. Instead,

they assign directly intensities to the default possibility. Defaults occur ran-

domly in this approach. Intensity-based models decompose observable credit

spreads into an expected loss (incorporating probability of default and loss

given default). As SMEs usually do not issue any bonds, there is again no3Due to different maturities of a firm’s debt and usual recovery rates below 100%, the

strike price is not simply equal to the total level of debt. Adjustments must be made.


secondary market revealing the market’s expectation of a firm’s default risk.

B) Implementation suggestion

Without going into much detail, we suggest a simple credit scoring method.

Such a model is not necessarily based on economic theory, but simply identi-

fies factors which showed statistically an explanatory power in differentiating

defaulting firms from non-defaulting firms (Allen et al., 2003).

Thus, the first step consists in finding appropriate risk factors. The

model should be parsimonious; overfitted models often show bad out-of-

sample predicting power. Studies by Pinches et al. (1973) and Libby (1975)

indicate that the optimal number of factors used for prediction is probably

between five to ten. From Moody’s RiskCalc methodology for rating private

firms, we list the following risk factors with corresponding measures:

1. Profitability measures the distance from incurring losses (measure:

net income or EBITDA).

2. Leverage reduces the potential for absorbing negative shocks (mea-

sure: liability / asset).

3. Debt coverage: a higher ratio is associated with a lower probability

of default (measure: cash flow / interest charges).

4. Growth has a non-monotonic relationship with probability of default.

Both low and high growth is associated with increased risk (measure:

sales growth).

5. Liquidity is not necessarily a sign of a firm’s health, but can absorb

a negative shock (measure: current ratio or quick ratio).


6. Activity is a sign of a firm’s fundamental health (measure: inventory

/ cost of goods sold).

7. Size is related to the business diversification within a company and

affects consequently the volatility of the firm’s value (measure: sales

or total assets).

Moody’s finds that most of these factors are not in a linear relationship

with default risk. Especially growth variables are not even monotonic. In

order not to lose the information in the non-linearity, we suggest to apply

a non-parametric method at the univariate level, linking each measure to

default probability individually.4 For this purpose we fit a transformation

to all possible values of a measure by local averaging between points of the

empirical dataset. From a vector y of input measures, we obtain then a

vector of transformations T(y).

The default probabilities indicated by each measure are then jointly

mapped to a single default probability. The outcome for each firm is bi-

nary: default or no default. As the predicted default probability has to lie

on the unit interval, we cannot use a linear OLS regression. Instead we

suggest estimating a probit model as follows:

P[d|T (x)] =∫ β

′T (x)

−∞

1√2π

exp−z2

2

dz

= Φ(β′T (x))

where β is the vector of weights allocated to the default probabilities pre-

dicted by the different measures. If the empirical data allows it, then the

weights should be industry-specific.4Such a method is applied within Moody’s RiskCalc.


According to Allen et al. (2003), credit scoring is quite inaccurate for

informationally opaque small businesses. However, for our purpose, calibra-

tion is more important than the power of the credit scoring method. Since

there can be a portfolio diversification effect if the inaccuracy of the model

is non-systematic. If the errors caused by weak prediction accuracy bal-

ance out on the portfolio level, then the level of the protection payments is

approximately right.

The most serious shortcomings of credit scoring models arise from data

limitations. This and other potential problems of the rating migration

derivative are addressed in the next section.

4.4 Problems confronted

• Financial reporting: SMEs which do not raise any funds publicly

usually do not do not offer extensive disclosure of their financial sit-

uation. Professional investors, however, require in general access to

corporate data in order to analyze the investment opportunity. Since

there is no contractual relationship between the SME and a possi-

ble investor, the only way for the investor to access data is through

the originating bank. The bank itself is tied to the banking secrecy

and cannot issue client-specific information. The only way to over-

come this hurdle is by giving the investor information on an aggregate

level about the whole portfolio. The services of a trustee would be

required to verify the content of the information. This adds to the

cost of the deal. Annual reporting periods are a very low informa-

tion frequency for investors. Moreover, talking to practitioners reveals

that Swiss SME financial statements are particulary inadequate for


efficient credit scoring. The degree of standardized reporting is on

a lower level than in countries such as Germany, Belgium or France.

The consequence is that more time is needed to prepare the state-

ments for scoring. Weak prediction power of the credit scoring model

is, however, not necessarily a problem if the portfolio is large enough.

Furthermore, as the rating function must be fixed over the life of the

derivative, there is a risk that accounting standards change over time

and that the derivative therefore bears some accounting risk.

• Substitution: SME loans have on average a maturity of around two

years. Hence, there would be on average only one or two reporting

periods during which a loan is in the reference portfolio. Often, how-

ever, loans are renewed and the new loan’s performance is still con-

tingent on the performance of the same debtor. For the derivative,

it is absolutely crucial to define substitution guidelines for maturing

or pre-paying loans. As opposed to CDOs, minimal credit quality

is not necessarily a good criteria for minimizing expected protection

payments in a rating migration derivative. A solution would be to

substitute loans always with loans of the same rating class. Depend-

ing on the application of the derivative, the new loan should enter on

the diagonal of the rating migration matrix and start off in a neutral

position.

• Basis risk: By respecting substitution constraints, the profile of the

reference portfolio might drift away from the average profile of the

bank’s loan portfolio. Moreover, if payoffs are defined in terms of loan

notional instead of number of firms, there is a risk that loan notionals

do not coincide after substitutions.


• Moral hazard: The bank has an incentive to include the loans in

the reference portfolio which promise the highest expected payoff. As

opposed to default-protection instruments, the bank does not lose its

incentive to monitor the debtor, however, since it still bears the default

risk. Therefore, moral hazard is probably a more important problem

when an instrument includes default protection. The rating process

and the monitoring of the respect of substitution guidlines has to be

outsourced to an independant calculation agent, or closely monitored

by external parties.

Chapter 5

Modelling and Pricing

“I am extremely sceptical about our current ability to capture socio-economic

randomness with models - but such information is vital in and by itself and

can be used to get out of trouble.”

- Nassim Nicholas Taleb

To analyse and subsequently price a portfolio credit derivative based on

rating migration, one needs to employ a model which describes the behaviour

of the derivative’s underlying. The main problem confronted in modelling

SME rating migrations is data availability. A consistent set of data is not

widespread in SME loan banks. Historical rating time series needed for

calibrating the model are affected by changes in the rating system and the

loan portfolio structure. One should not use published data from the leading

rating agencies if the underlying portfolio is not of the same nature as the

one to be modelled.

Wolf (2001) suggests that the performance of SME loan portfolios de-

pends strongly on the state of national economies. He claims to obtain very

good fits from regressions on an aggregate level. McNeil et al. (2005) con-

55

CHAPTER 5. MODELLING AND PRICING 56

firm that default rates are higher in recessions and lower during periods of

economic expansion.

In this chapter we suggest a model which originally stems from the work

of Jarrow et al. (1997) who develop a pricing method which is fully based

on rating migration. Ratings are assumed to follow a time-homogeneous

finite state space Markov chain, which is specified by empirical unconditional

transition probabilities of credit ratings.

5.1 The basic setting of the model

Credit quality is subdivided into K-1 classes of ascending default risk and

the default state which is represented by class K. Instead of modelling single-

name rating migrations and recombining all of them later with a correla-

tion structure, we directly model the whole rating migration matrix used as

the underlying of our migration derivative as an adapted stochastic process

M(ω, t) on a finite state space S, where each state is represented by a KxK

matrix with the elements mi,j representing the sum of loan notionals with

an initial rating in class i and a current rating in class j.1

The rating matrix is a function M : Ω × [0, T ] → S. The underlying

uncertainty is represented by a filtered probability space (Ω,P,Ft), where

P is the actual probability measure. The dependence of the process M(t)

on ω ∈ Ω is from now on suppressed. We further define a KxK transition

probability matrix P for the period [t,t+1] under the actual probability

measure P (see figure 5.1). The element pi,j is the probability for a rated

firm to migrate from rating class i at any time t to class j at time t+1. These

probabilities satisfy three conditions:1See section 4.1


P(t, t + 1) =

p1,1 p1,2 . . . . . . . . . p1,K

p2,1 p2,2 . . . . . . . . . p2,K

......

. . ....

...... pi,j

...

pK−1,1 pK−1,2 . . . . . .. . . pK−1,K

0 0 . . . . . . . . . 1

Figure 5.1: Transition probabilities

(i) 0 ≤ qi,j ≤ 1, ∀i, j

(ii)∑K

j=1 pi,j , ∀i

(iii) pi,K ≤ 1, ∀i 6= K

The shortest time interval for which a transition probability matrix is es-

timated is typically one year. By using empirical transition probabilities,

we are employing a backward looking model. If there are good reasons to

believe that the future looks substantially different, other estimations might

deliver better results.

Over zero time, no migration can happen. Therefore P(t,t) = I, where I

is the identity matrix. Time-homogeneity implies that the migration prob-

abilities do not change over time and are constant over the entire time

horizon.

P(t, T ) = P(T − t), ∀t ≤ T

This property does usually not hold, for example for the one-year rating

transition matrices reported by the leading rating agencies. We will, how-

ever, introduce time-inhomogeneity into our process by distinguishing model

time and real time as described in the next section.


The Markov property states that the probabilities of different future

credit ratings only depend on the current rating of the firm. The history

of how the firm has migrated into the current rating class, or how long it

has been assigned the current rating, does not matter. Empirical studies

show that this is certainly a simplifying assumption. Nickel et al. (2000),

Christensen et al. (2002) and Duffie et al. (2003) showed the presence

of non-Markovian behaviour such as rating drifts and time variation due

in particular to the business cycle. For this reason, we will condition the

migration process on the business cycle.

The Fundamental Theorem of Asset Pricing suggests that under the ab-

sence of arbitrage opportunities, there exists a physical-probability-equivalent

martingale measure on transition probabilities. So it is possible to obtain

a transition risk-neutral-probability matrix. The use of risk neutral proba-

bilities would allow avoiding the estimation of risk premia associated with

the uncertainties of different outcomes by implying no-arbitrage risk premia

in the probability weights. These probabilities would be extracted mean-

ingfully from the liquid market prices of instruments which bear the same

risk factors. However, since no payoffs of other instruments closely reflect

the risks inherent in Swiss SME credit rating migrations, an arbitrage-free

market would still allow a range of different measures, because the risk neu-

tral measure is non-unique when the market setting is incomplete. Since the

no-arbitrage assumption gives therefore not much pricing guidance under

such conditions, we will stick to the physical (or actual) measure on transi-

tion probabilities, and rather try to estimate a risk premium linked to the

randomness in our model.

Due to the time-homogeneity assumption, all one-period migration ma-

trices are identical, i.e. P(tn, tn+1) = P. In order to obtain the migration


Λ =

λ1,1 λ1,2 . . . . . . . . . λ1,K

λ2,1 λ2,2 . . . . . . . . . λ2,K

......

. . ....

...... λi,j

...

λK−1,1 λK−1,2 . . . . . .. . . λK−1,K

0 0 . . . . . . . . . 0

Figure 5.2: Generator: transition intensities

probabilities for several years, we can simply take P to the power of the

number of years, i.e. P(tn, tm) = Pm−n, ∀n ≤ m.

However, we have to be able to price the derivative for any point in time

within a given year. Therefore, one has to move from a discrete to a contin-

uous time-homogeneous Markov chain. We rely on the results of Jarrow et

al. (1997), Lando (1998) and Israel et al. (2001) for deriving an infinitesimal

generator matrix Λ (see figure 5.2) with the following properties:

(i) expΛ = P

(ii) λij ≥ 0, ∀i, j, i 6= j

(iii) λii = −∑Kj=1, j 6=i λij , for i = 1, 2, . . . ,K

Finding such a generator is known as the embedding problem for continuous-

time Markov chains. By assuming that a generator matrix exists and that

there is never more than one rating transition per year for each firm, Jarrow

et al. (1997) approximate Λ by the formula:

λi,i = log pi,i and λi,j =pi,j log(pi,i)

pi,i − 1, ∀i 6= j


To obtain matrices for any time t ≥ 0, we set P(t) = exptΛ, which is

defined as:

exptΛ =∞∑

u=0

(tΛ)u

u!= I + tΛ +

(tΛ)2

2!+

(tΛ)3

3!+ . . .

Israel et al. (2001) calculate the first sixteen terms of the series to get

intensities with an accuracy of 10−8.

There are some problems associated with empirically estimated tran-

sition probabilities. Firstly, the frequency of low-probability events is of-

ten estimated with a very low number of observations, and therefore not

very significant. There might even be zero events of a particular migration,

which, however, does not mean that the probability for such an event is zero.

Secondly, historical rating migration frequencies are often not monotonous,

meaning that a more extreme migration happened more often than a less

extreme migration (Schonbucher, 2003). The remedy to such problems is to

find an approximate generator matrix, rather than the true generator.

Israel et al. (2001) identify the conditions under which a true generator

matrix does or does not exist and present several methods for estimating

Λ. They also show how to obtain an approximate generator when a true

generator does not exist, so that expΛ is approximately equal to P. Lando

et al. (2002) propose a maximum likelihood estimator of the generator

matrix.

The fact that we model the migration matrix directly as a stochastic

process means that we do not attempt to model pair-wise credit quality

correlations. While those might be needed for large international companies

that overlap in their markets and activities, credit ratings of SMEs are highly

correlated due to their dependance on the domestic economy.

In order to account for time-varying transition probabilities, we intro-


duce randomness by incorporating the future state of the local economy

in the determination of the rating migration process. This is achieved by

distinguishing model time and real time.

5.2 Stochastic Model Time

Empirical rating transition probabilities usually imply a downward drift.

This means that for any given rating class, the cumulative probability of

being downgraded is higher than being upgraded. This property is true, for

instance, for the empirical average annual transition probabilities for S&P

ratings from 1981 to 2005, or for Moody’s ratings from 1920 to 1996.2 This

means that the longer the migration intensity matrix is applied, the more

downgradings can be expected.

We distinguish between real time t and model time τ , and set P(t) =

expτ(γt)Λ, where τ is a function of an adapted stochastic process γt. If

the transition generator matrix is applied more intensely over real time, the

expected downward drift is stronger.

We propose to let model time depend on the state of the economy, in

order to induce an increased downward drift in the rating migrations during

recessions, and a weaker downward drift during expansion periods. Model

time density is equal to real time density when the expansion of the economy

at time t equals the long term growth. Given that model time is not equal

to real time, the rating migration matrix is no longer a time-homogeneous,

but a real-time-inhomogeneous stochastic process.

We prefer this approach because it avoids estimating the whole correla-

tion structure among single names, which would be difficult due to the lack2See appendix C.


of data in many SME loan banks. Instead of estimating joint correlation

figures for all pairs of loans, we are left with the estimation of a stochastic

process driving the model time. Firstly, this is easier because fewer parame-

ters come into play, and secondly, we think that the model is able to capture

the most relevant effect of SME rating migrations for a portfolio derivative,

which is the accumulation of downgradings during recessionary periods.

Assuming that we obtain an estimation of the generator matrix and that

it stays constant over time, empirical one year rating migration probability

matrices each imply a data point, which links model time τ to real time

t. For each empirical transition probability matrix, we choose the τ which

minimizes the distance between the matrix P(t) and expτΛ.3

The objective then is to fit either directly a stochastic process to this

series of implied model times, or to define a functional relationship between

model time and other processes which are Ft-measurable. Together, these

parameters span a model time surface. Since we want to condition the

migration process on the business cycle, we choose the latter approach.

Business cycles are, as their name indicates, mean-reverting. Therefore,

we define an Ornstein-Uhlenbeck process for the expansion η of the economy

at time t:

dηt = −θ(ηt − η)dt + σdWt

ηt = ηoe−θt + η(1− e−θt) +∫ t

0σeθ(s−t)dWs

where θ imposes the speed of mean-reversion, η indicates the average long

term growth of the economy, and σ scales the random shocks of the Brownian

Motion. The parameters θ and σ are fitted to the expected process for3See Jafry et al. (2003) for different metrics which can be used for measuring the

distance between credit migration matrices.


the time to maturity of the derivative instrument. Historical time series

might deliver good estimations. To obtain the model time intensity from

the economic expansion at time t, we set:

γt = expα(ηt − η), with α ≥ 0

α measures the impact of the business cycle on the model time density. It

can be calibrated by using the data set of implied γt from the empirical

rating migration probabilities, and the fitted business cycle process ηt. The

calibration can occur via OLS, for example. It might be necessary to choose

a different functional form for the equation above; this can be judged once

the implied series of τ is obtained. Qualitatively, this formula fulfils the

requirement of decreased model time density during bullish periods and in-

creased model time density during bearish times. Quantitatively, the values

adopted by ηt are always non-negative, between 0 and 1 for higher than av-

erage expansion and above 1 for below average expansion. During average

expansion, ηt is equal to 1, which means according to the formula below,

that model time density is equal to real time density.

τ(γt) =∫ t

0γzdz

5.3 Pricing

In chapter four we defined the payoff x(t), where M(t) is the rating migration

matrix, and A(t) the payoff structure.

x(t) = trace[M(t)A(t)]

To price the derivative, we first conduct a Monte Carlo simulation to esti-

mate the payoff, and secondly we try to estimate a risk premium associated


with the uncertainty in this payoff. The Monte Carlo simulation gives an

estimation of the expected payoff vector over the life of the derivative. The

whole randomness for the simulation comes from the Brownian motion driv-

ing the business cycle process. A particular view on the future state of the

economy can be integrated by modifying the probabilities assigned to dif-

ferent outcomes of the business cycle process.

EP[x(t)] = EP[trace[M(t)A(t)]]

= EP[trace[M(0)expτ(γt)ΛA(t)]]

The second element needed for pricing the derivative is to discount the

expected payoff with the time value of money and a risk premium associated

with the uncertainty structure of the payoff vector. An easy solution is to

estimate the β of the payoff in the CAPM framework using historical data.

The discount rate is then equal to:

rmig = rf + β(rm − rf )

rf is the risk-free rate, rm the expected return of the market portfolio, and

β =cov(rmig, rm)

var(rm)

There are several theoretical problematic assumptions underlying the CAPM

framework.4

Since the rating migration derivative is almost designed for bearing the

risk of business cycles and because of its portfolio structure (≥ 1000 names),

we do not believe that there remains an important part of heavy-tailed

idiosyncratic risk in the instrument. For practical purposes, and also due to

the lack of precision in valuation without waterproof replication, the CAPM

approach appears to be very useful and adequate enough.4See Sharpe (1964)

Chapter 6

Conclusions

With the further development of the markets for structured credit products,

we expect a development of credit derivatives which are triggered by rating

events. Especially since banks are forced by regulatory bodies to use risk-

adequate pricing, rating events have their impact on banks’ P&L, and what

gets measured gets recognized.

The process of developing migration derivatives will probably be pushed

by originator’s in the beginning, and once there is a well-functioning market,

become more investor driven. The use of a bank’s loan portfolio as an un-

derlying for purposes detached from risk management would be an example.

Default-sensitive instruments are sometimes thought to capture migration

risks in a way that these are hedgeable. We see that this is usually not the

case, since it is hard to single out the effect of rating migration alone in

market spreads. Important basis risk is involved.

The most important obstacles in introducing portfolio credit derivatives

on rating migration is probably the establishment of investor confidence in

the rating process.

65

CHAPTER 6. CONCLUSIONS 66

* * * * * * * * * * * *

Appendix A

Securitized asset classes

List of asset classes that have been securitized (non-exhaustive).

• Commercial mortgages

• Residential mortgages

• Credit card receivables

• Automobile loans

• Trade receivables

• Commercial real estate

• Corporate bank loans

• Pub lease receivables

• Bonds

• Consumer loans

• Tax receivables

67

APPENDIX A. SECURITIZED ASSET CLASSES 68

• Electricity receivables

• Student loans

• Telephone receivables

• Healthcare receivables

• Aircraft lease

• Future export receivables

• Airline ticket receivables

• Equipment lease receivables

• Lottery receipts

• Public sector housing receipts

Appendix B

Sample rating migration

matrices

The following matrices are rating migration matrices from a sample of 1,000

firms rated by S&P.

69

APPENDIX B. SAMPLE RATING MIGRATION MATRICES 70

Figure B.1: Sample rating migrations from 1995 to the indicated year

Appendix C

Average annual rating

migration probabilities

The following two matrices represent average annual rating probabilities

over long time horizons for S&P and Moody’s.

71

APPENDIX C. AVERAGE ANNUAL RATING MIGRATION PROBABILITIES72

Figure C.1: S&P average annual rating transitions, 1981 - 2005 (conditional

on no rating withdrawal)

Figure C.2: Moody’s average annual rating transitions, 1920 - 1996 (condi-

tional on no rating withdrawal)

Bibliography

Allen, L., G. DeLong, and A. Saunders (2004), “Issues in the credit risk

modeling of retail markets”, Journal of Banking and Finance, pp. 31-

40.

Altman, E. I., A. Saunders (1998), “Credit risk measurement: developments

over the last 20 years”, Journal of Banking and Finance, 21(1998), pp.

1721-1742.

Berndt, A., R. A. Jarrow and C. Kang (2005), “Restructuring Risk in Credit

Default Swaps: An Empirical Analysis”, Journal of Derivatives, 8(1),

pp. 29-40.

Choudhry, M. (2005), “Credit Derivatives”, Fixed-Income Securities and

Derivatives Handbook: Analysis and Valuation, Bloomber Press, pp.

173-188.

D Amato, E. M. Remolona (2003), “The credit spread puzzle”, BIS Quar-

terly Review, December 2003, pp. 51-63.

Das, S. R., P. Hanounas (2006), “Credit default swap spread”, Journal of

Investment Management, 4(3), pp. 93-105.

73

BIBLIOGRAPHY 74

Elizalde, A. (2005), Credit risk models IV: Understanding and pricing CDOs,

CEMFI and UPNA.

Elton, J. E., M. J. Gruber, D. Agrawal and C. Mann (2001), “Explaining

the rate spread on corporate bonds”, Journal of Finance, 56(1), pp.

247-278.

EU report (2004), “Study on asset-backed securities: impact and use of ABS

on SME finance”, Contract ENTR 03/44.

Fabozzi, F. J. (2004), The handbook of European structured financial prod-

ucts, Wiley Finance.

Fitch report (2001), “European SME CDOs: An investors guide to analysis

and performance”, pp. 1-8.

Gagliardini, P., and C. Gourieroux (2005), “Migration correlation: definition

and efficient estimation”, Journal of Banking and Finance, 29, pp. 865-

894.

Guan, L. K. (2005), “Estimating credit risk premia”, Financial Management

Association Conference 2005, Chicago October 11-14 2005.

Habersaat A. (2005), Die KMU in der Schweiz und in Europa, Staatssekre-

tariat fur Wirtschaft, SECO.

Hull, J., M. Predescu (2004), “The relationship between credit default swap

spreads, bond yields, and credit rating announcements”, Journal of

Credit Risk, 1(2), pp. 53-60.

BIBLIOGRAPHY 75

Israel, R. B., J. S. Rosenthal and J. Z. Wei (2001), “Finding generators for

markov chains via empirical transition matrices, with applications to

credit ratings”, Mathematical Finance, 11(2), pp. 245-265.

Jafry, Y., T. Schurmann (2003), “Metrics for comparing credit migration

matrices”, Centre for Financial Institutions Working Paper, 3(9), pp.

3-43.

Jarrow, R. A., D. Lando, and S. M. Turnbull (1997), “A Markov model for

the term structure of credit risk spreads”, Review of Financial Studies,

10(2), pp. 481-523.

Jarrow, R. and S. M. Turnbull (1995), “Pricing derivatives on financial se-

curities subject to credit risk”, Journal of Finance, 50(1), pp. 53-85.

Jobst, A. (1997), “What is structured finance?”, Working Paper.

Lando, D. (2000), “Some Elements of Rating-Based Credit Risk Modeling”,

in N. Jegadeesh and B. Tuckman (eds), Advanced Fixed-Income Valu-

ation Tools , Wiley.

Longstaff, F. A., S. Mithal, E. Neis (2000), “Corporate yield spreads: De-

fault risk or liquidity? New evidence from the credit-default swap mar-

ket”, Journal of Finance, forthcoming.

McNeil, A. J., R. Frey, P. Embrechts (2005), Quantitative Risk Management:

Concepts, Techniques and Tools, Princeton University Press.

Moody’s report (2000), “RiskCalc for Private Companies”,

www.moodyskmv.com.

BIBLIOGRAPHY 76

Moody’s report (1997), “Moody’s rating migration and credit quality

correlation,1920-1996”, www.moodyskmv.com.

Schonbucher, P. (2003), Credit Derivatives Pricing Models: Models, Pricing,

Implementation, Wiley Finance.

Stuttard, J. (2006), “Uses & Abuses & Some Quirks of Credit Default

Swaps”, Schroders Talking Point library.

Tang, D. Y., H. Yan (2006), “Liquidity, liquidity spillover, and credit default

swap spreads”, Working Paper.

Wolf, U. (2001), “Taking the risk out of retail credit modelling”, ERisk -

July 2001, pp. 1-5.

20070131 portolio credit derivatives based on rating migrations

Documents