20070131 portolio credit derivatives based on rating migrations
TRANSCRIPT
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
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)
CHAPTER 2. CREDIT RISK TRANSFER 10
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
CHAPTER 2. CREDIT RISK TRANSFER 11
Figure 2.2: Unfunded - Credit Default Swap (CDS)
Figure 2.3: Funded - Credit Linked Note (CLN)
CHAPTER 2. CREDIT RISK TRANSFER 12
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
CHAPTER 2. CREDIT RISK TRANSFER 13
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.
CHAPTER 2. CREDIT RISK TRANSFER 14
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
CHAPTER 2. CREDIT RISK TRANSFER 15
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
CHAPTER 2. CREDIT RISK TRANSFER 16
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).
CHAPTER 2. CREDIT RISK TRANSFER 17
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
CHAPTER 2. CREDIT RISK TRANSFER 18
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
CHAPTER 2. CREDIT RISK TRANSFER 19
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
CHAPTER 2. CREDIT RISK TRANSFER 20
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.
CHAPTER 2. CREDIT RISK TRANSFER 21
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
CHAPTER 2. CREDIT RISK TRANSFER 22
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
CHAPTER 3. RATING MIGRATION 25
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.
CHAPTER 3. RATING MIGRATION 26
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.
CHAPTER 3. RATING MIGRATION 27
• 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.
CHAPTER 3. RATING MIGRATION 28
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.
CHAPTER 3. RATING MIGRATION 29
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).
CHAPTER 3. RATING MIGRATION 30
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
CHAPTER 3. RATING MIGRATION 31
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-
CHAPTER 3. RATING MIGRATION 32
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.
CHAPTER 3. RATING MIGRATION 33
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
CHAPTER 3. RATING MIGRATION 34
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.
CHAPTER 3. RATING MIGRATION 35
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
CHAPTER 3. RATING MIGRATION 36
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.
CHAPTER 3. RATING MIGRATION 37
• 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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 40
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
CHAPTER 4. RATING MIGRATION DERIVATIVE 41
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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 42
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
CHAPTER 4. RATING MIGRATION DERIVATIVE 43
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
CHAPTER 4. RATING MIGRATION DERIVATIVE 44
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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 45
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
CHAPTER 4. RATING MIGRATION DERIVATIVE 46
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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 47
Figure 4.7: Payoff on rating drift
Figure 4.8: Payoff on rating activity
CHAPTER 4. RATING MIGRATION DERIVATIVE 48
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
CHAPTER 4. RATING MIGRATION DERIVATIVE 49
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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 50
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).
CHAPTER 4. RATING MIGRATION DERIVATIVE 51
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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 52
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
CHAPTER 4. RATING MIGRATION DERIVATIVE 53
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.
CHAPTER 4. RATING MIGRATION DERIVATIVE 54
• 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
CHAPTER 5. MODELLING AND PRICING 57
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.
CHAPTER 5. MODELLING AND PRICING 58
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
CHAPTER 5. MODELLING AND PRICING 59
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
CHAPTER 5. MODELLING AND PRICING 60
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-
CHAPTER 5. MODELLING AND PRICING 61
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.
CHAPTER 5. MODELLING AND PRICING 62
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.
CHAPTER 5. MODELLING AND PRICING 63
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
CHAPTER 5. MODELLING AND PRICING 64
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)
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