On the Dynamical Risk Properties

of a Bond Portfolio

Christ Church

University of Oxford

MSc in Mathematical Finance

October 31, 2009

Abstract

In this thesis a study of a portfolio of defaultable bonds and swaps isperformed.

The focus is on the modelling and determination of risk premia associ-ated with different types of risk linked to the portfolio. This comprisesthe market and credit risk and a default contingent market risk associ-ated with the swaps.

Different models are used to investigate the various risks. The mar-ket risk is analysed via Value-at-Risk. The credit risk is described bythe portfolio’s loss distribution. Its calculation is based on a correla-tion expansion technique, which enables fast analytical computationof relevant expectation values. The default contingent market risk ismodelled via EPE-profiles familiar from counterparty risk. The theo-retical predictions are compared with results from a dynamical portfoliosimulation.

The results of this dissertation are twofold. Firstly, it is shown how thedefault contingent market risk can be modelled and consistently inte-grated into the full risk framework. The importance of the modellingissue of the default contingent market risk is highlighted. Secondly, itis shown that the analytical results regarding the correlation expansionof a Gaussian copula can be extended to other copulas as well. Animplementation of the correlation expansion method is provided.

MSc in Mathematical Finance

Christ Church

OXFORD UNIVERSITY

Authenticity Statement

The author confirms:

• This Dissertation does not contain material previously submittedfor another degree or academic award, and

• the work presented here is the author’s own, except when other-wise stated.

Name: Dr. Andre Miemiec

Address: Hasenheide 9210967 BerlinGermany

Signed:

Date: February 12, 2010

Contents

Preface 1

1 Preliminaries 31.1 Simple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Market Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Calibration of a Spread Curve . . . . . . . . . . . . . . . . . 131.3.2 Prices of defaultable Bonds . . . . . . . . . . . . . . . . . . 16

1.4 Default-contingent Market Risk . . . . . . . . . . . . . . . . . . . . 17

2 Individual Approach 182.1 Asset Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Hedging the Market Risk of Coupon bonds . . . . . . . . . . 192.1.2 Risk Properties . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Calculation of the price of risk . . . . . . . . . . . . . . . . . . . . . 222.2.1 Estimating the expected loss . . . . . . . . . . . . . . . . . . 222.2.2 Estimating the unexpected loss . . . . . . . . . . . . . . . . 24

3 Portfolio Approach 273.1 Loss distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Loss distribution of independent obligors . . . . . . . . . . . 273.1.2 Loss distribution of dependent obligors . . . . . . . . . . . . 28

3.2 Multivariate Gaussian distribution . . . . . . . . . . . . . . . . . . 283.2.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 Correlation expansions . . . . . . . . . . . . . . . . . . . . . 303.2.3 Glasserman’s example revisited . . . . . . . . . . . . . . . . 35

3.3 Multivariate Student Distribution . . . . . . . . . . . . . . . . . . . 363.3.1 Example continued . . . . . . . . . . . . . . . . . . . . . . . 38

4 Portfolio Dynamics 414.1 System Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Portfolio Generation . . . . . . . . . . . . . . . . . . . . . . 434.1.2 Bond Specification . . . . . . . . . . . . . . . . . . . . . . . 454.1.3 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . 45

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4.2 Compatibility of the two Approaches . . . . . . . . . . . . . . . . . 464.2.1 Individual Approach . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Portfolio Approach . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Portfolio Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 494.3.2 Discussion of default process assumptions . . . . . . . . . . . 524.3.3 Backtesting the Predictions of Replacement Costs . . . . . . 52

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A Quantiles of Positive Exposure 57A.1 Statistics of the Positive Exposure . . . . . . . . . . . . . . . . . . . 57

A.1.1 Variance of PE . . . . . . . . . . . . . . . . . . . . . . . . . 57A.1.2 Quantile of the distribution of positive exposures . . . . . . 58

B Covariance 60B.1 Calculation of Volatilities and Correlation . . . . . . . . . . . . . . 60

C Code 62C.1 Loss Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.2 EV.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.3 HermiteB.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63C.4 PDtwiddle.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63C.5 V.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64C.6 GenerateAllJ.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65C.7 GenerateAllD.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66C.8 MyFunction.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66C.9 CDOTest.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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Fede e sustanza di cose speratee argomento de le non parventi;e questa pare a me sua quiditate.

Paradiso, Canto XXIV

Preface

The present work is concerned with the analysis of a portfolio of defaultable bonds.I was introduced to the topic in early 2008 by a former colleague of mine fromd-fine GmbH, Peter Gloßner, who was mainly interested in the phenomenon of thedifference between the time at which a default of a loan occurred and the timeat which the P&L was realised. In this thesis we will study a slightly simplerproblem. We will build a portfolio of bonds which grows according to a specifiedfunction and investigate its properties. This includes the hedging with matchingswaps and subsequently the risk properties of the portfolio consisting of bonds andswaps. In a slightly different shape the problem became even more interesting dueto the financial crisis, which displayed a dramatic increase in the number of nonperforming loans and raised a lot of subtle questions with the pricing of simplebond contracts, which might be considered as the tradeable cousins of loans. Sincewe are not much interested in this difference we will not further distinguish betweenthem. But the lessons learned from the dramatic shift in the market opinion aboutthe price of bonds renewed the interest in the understanding of the loss dynamicsof a bond or loan portfolio which is obviously of high importance for properlysupervising and regulating its performance.

The work is organised into five chapters. In chapter 1 we introduce the financialinstruments we are going to use in this thesis. This includes bonds, floating ratenotes, swaps and swaptions. Then we are going to explain in detail the RiskMetricsmethodology used to determine the market Value-at-Risk (market VaR). Finallywe present the default model, which is used to simulate the default process andwhich is based on the rating transition matrix. Referring to the continuous timeMarkov chain underlying the rating transition matrix we derive a term structureof interest rates of defaultable bonds.

In chapter 2 we introduce the structure of an asset swap and it is explained howit may be employed to take the interplay of credit and market risk of a defaultablebond together with its hedging swap into account. In particular we try to estimatethe replacement costs due to the market risk of the swap in the case the default ofthe matching bond has happened by reinvestigating the methodology of expectedpositive exposure profiles. A modification of the way the exposure at default (EAD)is computed is made, which serves our needs.

In chapter 3 we first review an analytical approach to compute expectationvalues of functions of the loss variable in a Gaussian copula model developed byGlasserman & Suchintabandid. A Matlab implementation of the proposed algo-rithm is given. Furthermore it is shown how this methodology can be extended toother copulas as well. Later this method shall be applied to the computation of

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the loss distribution of a portfolio of correlated bonds.In chapter 4 the results of a simulation of the dynamical properties of the bond

portfolio are presented. After introducing the simulation engine in section 4.1 wediscuss in section 4.2 the compatibility of the results obtained in the chapters 2and 3 and show how the seemingly different approaches perfectly fit into a unifiedframework. Then we present the results of the simulation in section 4.3. A criticaldiscussion of the default model is made in section 4.3.2. The results of the modifiedEAD computation are backtested and discussed in section 4.3.3.

At the end of this preface I would like to take the opportunity to express mysincere thanks to Sam Howison, Peter Gloßner and Monika Hejjas, whose kindsupport and patience made this work possible. Furthermore I want to thank IgorSchnakenburg, Matthias Horn and C.A.T. Schulze for valuable discussions. Lastbut not least I would like to thank d-fine GmbH and BHF-Bank AG for the op-portunity to work on their behalf and in particular Thomas Siegl for his support.

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Chapter 1

Preliminaries

1.1 Simple Products

Definition 1 (Coupon Bond). A bond is a debt security, in which the issuerowes the bond holder a debt and is obliged to pay interest (the coupon) and hasto repay the notional at a later date, termed maturity. The time over which theinterest accrues is called the term or the frequency of the bond. Basically, a bondis a loan: the issuer is the borrower, the bond holder is the lender.

The purpose of a bond is to provide the borrower with external funds in order tofinance long-term investments. The cash flow structure of the bond is shown inFig. 1.1.

Coupon

Notional

ttttt0 1 2 n−1 n

Fig. 1.1. Coupon Bond cashflows to the lender.

Assuming the following notation

n number of cash flows of fixed paymentst0 valuation datetj time of the jth cash flow exchange (j = 1 . . . n)τj time between tj−1 and tjNj notional amount for cash flow at tj (non-negative)Rfix coupon of the bondDti,tj discount factor between time ti and time tjT maturity date of the bond (T = tn)

3

the price of a bond is given by

P (t0, T ) = Rfix

n∑

j=1

Nj τj Dt0,tj + Nn Dt0,T (1.1)

Later we will need to compute coupons of special bonds, so called par bonds, whichare defined below.

Definition 2 (Par Bond). A bond trading at its face value is a par bond.

The point here is that a coupon determined that way reflects the view from todayabout a fair interest income in exchange for the bond’s notional.

Definition 3 (Floating Rate Note). Floating rate notes are similar to bonds.FRNs have coupon rates that are reset periodically over the life of the note. Thecoupon rate effective for the payment at the end of one period is usually set twodays before the beginning of the period at the current market interest rate for thatperiod.

A FRN is a benchmark for the current value of money. The zig zag arrows inFig. 1.2 indicate that all coupons (possibly with the exception of the first fixedone) are unknown at initiation. Because they are stochastic they may differ.

t t tt t0 1 2 n−1 n

Fig. 1.2. Floating Rate Note cash flows.

Assuming the following notation

n number of cash flows of fixed paymentst0 valuation datetj time of the jth cash flow exchange (j = 1 . . . n)τj time between tj−1 and tjNj notional amount for cash flow at tj (non-negative)Dti,tj discount factor between time ti and time tjT maturity date of the FRN (T = tn)

the price of a FRN is given by

FRN(t0, T ) =

n∑

j=1

Nj τj

(Dt0,tj−1

− Dt0,tj

)+ Nn Dt0,T (1.2)

4

Definition 4 (Interest Rate Swap). An interest rate payer (receiver) swap is aportfolio consisting of a short (long) position in a Coupon Bond and a long (short)position in a Floating Rate Note with identical issue and maturity dates but notnecessarily equal terms.

An example of a five year receiver swap with annual coupon payments and semian-nual floating payments is given in Fig. 1.3. One purpose of a swap is the maturitytransformation. Thus it constitutes a vehicle to short term refunding1.

1Y 2Y 3Y 4Y 5Y

6M 3.5Y 4.5Y1.5Y 2.5Y

t0 T

Fig. 1.3. Cash flows of a receiver swap.

The price of a receiver swap (receive fix, pay float) is given by

NPVSwap(t0, T ) = P (t0, T ) − FRN(t0, T ) . (1.3)

The NPV of a payer swap has the opposite sign.

Definition 5 (Swaption cf. [1]). A swaption is an option to enter into a swap.We consider only European swaptions. As usual there are two versions:

• a receiver swaption, which gives the right but not the obligation to enter intoa receiver swap and

• a payer swaption, which gives the right but not the obligation to enter into apayer swap.

We will price the swaptions within the Black-76 model [2]. The situation we willlater encounter is the one displayed in Fig. 1.4. It shows the example of a receiverswap starting at the future date T . The payment frequencies of fixed leg and float-ing leg might differ. We will consider the situation with annual fixed paymentsversus semiannual or quarterly floating payments. The reason that the swap startsat time T is that it forms the underlying of a swaption.

1A typical example are Eonia swaps, which are used for overnight refunding.

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1Y 2Y 3Y 4Y 5Y

6M 3.5Y 4.5Y1.5Y 2.5Y

t0 TT + L

Fig. 1.4. Cash flows of a receiver swap underlying a european swaption.

In the generic setting we assume the following notation:

n number of cash flows of fixed paymentsn number of cash flows of floating paymentst0 valuation datet1 swap start datetj time of the jth cash flow exchange (j = 2 . . . n)τj time between tj−1 and tjNj notional amount for cash flow at tj (non-negative)Rfix fixed rate of the swap and thus strike of the swaptionDti,tj discount factor between time ti and time tjT expiry date of the swaption (T = t1 for simplicity)L swap lengthω 1 for a payer swap and −1 for a receiver swap

Then the price of the underlying swap is expressed in the following form

NPVSwap = ω ( Vfloat − Vfix ) (1.4)

where

Vfix = Rfix

n∑

j=2

Nj τj Dt0,tj = Rfix · δ (1.5)

with tjj=1...n a partition of the interval [T, T + L] into (e.g. annual) intervals τj

and

Vfloat =

n∑

j=2

Nj τj

(Dt0,tj−1

− Dt0,tj

)(1.6)

with tjj=1...n a partition of the interval [T, T +L] into (e.g. semi annual) intervalsτj. The swap rate, i.e. the rate which allows to write the expression of the floatingleg in the same form as the expression of the fixed leg, is defined by

Vfloat = RSwap · δ . (1.7)

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The standard approach to swaption pricing is to use the Black formula. In orderto apply this Black-Scholes-like analytical expression one needs a strike and avolatility. The payoff of a swaption reads

VSwaption = δ [ω (RSwap − Rfix ) ]+ = PE , (1.8)

where PE is also called the positive exposure of the underlying swap. Assumingthe swap rate satisfies a log normal process the value of the swaption is given by

EuropeanSwaption = e−r T δ [ ω RSwap Φ(ω d1) − Rfix ω Φ(ω d2) ] . (1.9)

Here d1 and d2 are defined below:

d1 =ln

RSwap

Rfix+ 0.5 σ2 T

σ√

T, d2 =

lnRSwap

Rfix− 0.5 σ2 T

σ√

T. (1.10)

The swaption volatility σ is obtained from market data.

After having introduced the instruments we are going to use we now want to payattention to the basic market risk management techniques related with them.

1.2 Market Risk

The definition of a proper risk measure is a subtle problem. Several risk measureshave been proposed [3, 4]. The measure we are going to use is the Value-at-Risk2

(VaR), which - despite its many deficiencies [5] - is still the most commonly usedone in practice. A survey of the historical development of the Value-at-Risk can befound in [6]. The basic assumption underlying the VaR-methodology is that thechanges of the market value (returns) of a portfolio of instruments over a chosentime horizon (e.g. daily) are normally distributed. The concept of VaR is actuallya very simple example of test theory. Let us denote by the random variable xthe stochastic portfolio returns over the chosen time horizon. If the distributionalassumption we implicitly made is correct than the returns, i.e. concrete realisationsof the random variable x, are with a probability of α (the level of confidence) largerthan the critical value, −VaR, defined by

P [−VaR ≤ x ] = α . (1.11)

In terms of P&L this equation states that with a probability of α losses are smallerthan VaR. In summary this means the following: VaR has the two ingredients, thelevel of confidence and the horizon. We will work with a horizon of one day and alevel of confidence of α = 99%.

For a Gaussian random variable the determination of the critical value, VaR,is actually very easy to perform. Due to the symmetry of the distribution the VaRis identical to the α-quantile of the Gaussian distribution, which can be obtained

2VaR itself shall be a positive number.

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from its standard deviation and the corresponding quantile of the standard normaldistribution. This is a consequence of the scaling property of the quantiles ofa Gaussian distribution: The α-quantile of a Gaussian distribution is equal tothe product of its standard deviation and the α-quantile of the standard normaldistribution. Therefore it is sufficient to restrict the computation of the VaR tothe determination of the portfolio’s standard deviation only. A crucial point is theapproximation which will be used to calculate it. For so called linear instrumentsthe analytic Delta-Normal method is sufficient, which we are going to explain insection 1.2.1. For the nonlinear complements a full Monte Carlo simulation mustbe performed.

But before describing the computation of the VaR for the different instrumentsour portfolio consists of, we just want to focus on the set of risk factors responsiblefor the randomness.

1.2.1 Value at Risk

Risk Factors

The instruments we are going to consider are simple interest rate products such asbonds, FRNs and swaps. The risk of their market value is due to the uncertaintyof the future term structure of discount factors3. In Fig. 1.5 a schematic pictureof the term structure of discount factors is drawn. It assigns to each future date afixed discount factor (black solid line).

S15 Time

Dis

coun

t fac

tor

R030 S01R180 S02

Fig. 1.5. RiskMetrics Tweaks of the Term Structure.

The future uncertainty of the price of an interest rate product depends on the waythis term structure moves. In order to quantify this risk a finite set of maturities t1, t2, . . . is selected on the time axis. For example these maturities could cor-respond to the RiskMetrics nodes, which are listed in Tab. 1.1.

Term 1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 15Y

Node R030 R090 R180 S01 S02 S03 S04 S05 S07 S10 S15

Tab. 1.1. RiskMetrics Nodes

3The term structure of discount factors is often called the term structure of interest rates, too.

8

The term structure is reduced to exactly this set of maturities as displayed inFig. 1.5 (red circles). For each of these nodes one has to provide the volatilities σi

and the correlation matrix ρij or equivalently the covariance matrix Σij as basicingredients for the VaR calculation described below.

The sensitivities of the present value, PV , of an interest rate product with respectto a local change (tweak) of the interest rate at a RiskMetrics node ti is definedby the following formula

∆i =PVtweaked − PVorig

Dtweaked(0, ti) − Dorig(0, ti)· Dorig(0, ti) . (1.12)

Here D(0, ti) denotes the discount factor at maturity ti belonging to the term struc-ture given in Fig. 1.5. The tweak itself is computed by multiplying the discountfactor at time ti by 1 + ǫ and computing the resulting zero rate4. The so definedelongation of the rate at the RiskMetrics node ti generates a new term structureof interest rates, which is basically identical to the old one except that the ratesbetween the selected node and the two neighbouring nodes are interpolated via ahat function as indicated in Fig. 1.5.

Bonds and Swaps belong to the set of linear instruments. Therefore we are in thecomfortable situation that we do not have to consider a Monte Carlo approach inorder to calculate the VaR5.

Delta Normal Method

The Delta Normal Method is based on an analytical approximation of the standarddeviation of the distribution of changes in the market value (returns) of a portfolioof instruments. Assuming that the distribution of returns is Gaussian, its quantilescan be obtained by scaling the standard deviation with the quantiles of the standardGaussian distribution, i.e.

VaRα = Qα=99% ·√

∆T Σ ∆ , (1.13)

where Qα=99% is the 99%-quantile of the standard Gaussian distribution6, Σ is thecovariance matrix of the risk factors and ∆ a vector of sensitivities with respect tothe chosen set of risk factors (RiskMetrics nodes). The covariance matrix and thecorrelation matrix are related via

Σij = σi σj ρij . (1.14)

Since we have reduced the number of nodes on which the term structure is actuallyknown to the RiskMetrics nodes for the purpose of risk calculations, we have tocompute the volatilities and correlations of the zero rates at these nodes, only. The

4The value of ǫ is set to 0.0001.5The actual implementation of the sensitivity calculation was done using the QuantLib [7].6 Qα=99% = Qα=99%(mean = 0, stddev = 1) = 2.32634 . . .

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details of its construction are given in appendix B.1. The set of volatilities7 foreach of the RiskMetrics nodes is displayed in Tab. 1.2

R030 R090 R180 S01 S02 S03 S04 S05 S07 S10 S15

0.0136 0.0107 0.0197 0.0345 0.129 0.1935 0.3081 0.3939 0.5201 0.6962 0.9897

Tab. 1.2. Volatilities of RiskMetrics Nodes8

and the corresponding correlation matrix is given in Tab. 1.3

R030 R090 R180 S01 S02 S03 S04 S05 S07 S10 S15

R030 1 0.621 0.653 0.602 0.227 0.138 0.113 0.079 0.065 0.058 0.061R090 0.621 1 0.8 0.641 0.313 0.227 0.204 0.181 0.161 0.146 0.134R180 0.653 0.8 1 0.887 0.51 0.419 0.334 0.301 0.286 0.286 0.284S01 0.602 0.641 0.887 1 0.597 0.528 0.447 0.415 0.391 0.377 0.366S02 0.227 0.313 0.51 0.597 1 0.979 0.95 0.93 0.918 0.899 0.869S03 0.138 0.227 0.419 0.528 0.979 1 0.963 0.952 0.941 0.925 0.896S04 0.113 0.204 0.334 0.447 0.95 0.963 1 0.997 0.991 0.976 0.954S05 0.079 0.181 0.301 0.415 0.93 0.952 0.997 1 0.996 0.983 0.963S07 0.065 0.161 0.286 0.391 0.918 0.941 0.991 0.996 1 0.994 0.98S10 0.058 0.146 0.286 0.377 0.899 0.925 0.976 0.983 0.994 1 0.994S15 0.061 0.134 0.284 0.366 0.869 0.896 0.954 0.963 0.98 0.994 1

Tab. 1.3. Correlation Matrix of RiskMetrics Nodes

Remark (Historical Covariance). For the purpose of this work we have assumedthat the covariance matrix is constant over time. This is just to simplify the calcu-lations and not a real restriction. The “true” covariance matrix can be estimatedfrom historical data. It can be easily derived from the historical quotes belongingto the benchmark instruments used to generate the zero curve, e.g. Euribor depositrates and IRS quotes [8].

Value at Risk of a Bond

With these preparations it is possible to compute the sensitivities of a typical bond9

with a notional of 100 EUR. The sensitivities with respect to the RiskMetrics nodesof Tab. 1.1 follow from eq. (1.12) and are displayed in Fig. 1.6. In the following wewill refer to these sensitivities as VaR-Maps because they describe the contributionsto the total VaR of effective cash flows mapped to a specific maturity.

7The reference day is 02.01.2008. Quoted are the price volatilities following from the yieldvolatilities actually observed.

8For historical reasons the volatilities displayed here are actually the volatilities multiplied bythe 95%-quantile of the standard normal distribution. This hidden factor has to be taken intoaccount, when applying the prescription of the Delta-Normal-Method to calculate a VaR.

9Spot = 02.02.2009, Start = 01.01.2008, End = 01.01.2013, Annual Coupon 4.5%

10

Fig. 1.6. VaR-Maps of a five year bond.

The pattern of Fig. 1.6 is typical for bonds. The largest VaR-Map comes from therepayment of the bond notional at the maturity of the bond. Compared with therepayment of the notional the coupon payments produce only small VaR-Maps.With these sensitivities and the covariance matrix of section 1.2.1 one obtains dueto eq. (1.13) a one day VaR at a confidence level of 99% of

VaR99% = 0.46 EUR (1.15)

at the spot date.

A technical remark is in place here. When we later study a portfolio of bonds whichmainly differ by their coupons, a technique to reduce the amount of computationsrequired is to write the bond as

Bond = Coupon[%] · Synthetic Bondno redemption + Zero Bond. (1.16)

Due to this decomposition one obtains the sensitivities for all coupons at once byreferring to the sensitivities of two coupon independent building blocks, a syntheticcoupon bond with coupon 100 EUR and no redemption and a zero coupon bondwith redemption 100 EUR.

Value at Risk of a FRN

A FRN has a different behaviour. Opposing a bond whose VaR-Maps peak atmaturity the VaR-Maps of a FRN are large for small terms and usually small atmaturity. Basically a FRN is equivalent to a Zero Bond with maturity equal to theend of the current period. This is due to the following property. Since the couponpayable after a future period [ti, ti+1] is computed from the forward rate for thisperiod as derived from the current term structure, there is no difference betweenobtaining the current coupon, all future coupons and the notional at maturity orobtaining just the current coupon and the notional at the end of the current period.This is because the coupons of future periods just compensate the time value ofmoney. This “pullback” property of an FRN is shown in Fig. 1.7.

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n−10 1 2 nt tttt

Fig. 1.7. Pullback property of a FRN.

Therefore a FRN can be seen as a zero bond with a notional equal to ( 1 +Last Fixing ) times 100 EUR [3].

Fig. 1.8. VaR-Maps of a five year FRN.

The VaR-Maps for a typical FRN10 are shown in Fig. 1.8. With these sensitivitiesand the covariance matrix of section 1.2.1 one obtains a one day VaR of

VaR99% = 0.005 EUR. (1.17)

Due to the fact that the coupon of a FRN is regularly updated to the currentlevel of the term structure it carries significantly less risk with respect to themovement of the term structure when compared with the example of a fixed ratebond considered before.

Value at Risk of a Swap

Since we have defined the swap as a portfolio of a bond and a FRN the VaRin this case can be computed using the results of the two examples (bond, FRN)considered before. Yet, this cannot be done by simply summing up the VaRs of thetwo constituting instruments since the VaR is not additive. Only the VaR-Mapsof the swap are linear superposition of the VaR-Maps of the bond and the FRNinvolved (at least in the Delta-Normal approximation). The VaR itself thereforecontains an additional cross terms since its calculation is based on the bilinearmap 〈, 〉Σ : R

2 7→ R appearing under the root in eq. (1.13). This example isprototypical for the VaR calculation of a portfolio of linear instruments.

10Spot = 02.02.2009, Start = 01.01.2008, End = 01.01.2013, 6M-Euribor

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1.3 Credit Risk

Definition 6 (Credit Risk). Credit risk is the risk of a loss due to a debtor’s non-payment of a loan or other line of credit (either the notional or interest (coupon)or both)

The natural approach to model credit risk is to weight all future cash flows withthe probability they occur. This is the so called survival probability. It is closelyrelated to the default probability, which possesses a whole term structure. It isthe main topic of this section to discuss the method we want to utilise in order todetermine it. But before delving into details we just want to summarise the existingmodels (cf. [9]). There are two main approaches to model credit risk: structuralmodels and reduced form models. Structural models are based on option pricingmethodologies and originate from the seminal paper [10]. The most influentialcommercial version of structural models is the KMV model. Reduced form modelson the other hand may be classified into three categories [11]:

• Default models, which use stochastic processes to model the default directly(intensity models),

• spread models, which decompose risk into credit and recovery risk but solelymodel the spread without reference to its components and

• Credit rating models depicting the evolution of a firm as changes in its creditrating.

We are going to work with the last approach to credit risk, i.e. credit rating mod-els. They are based on discrete or continuous-time Markov chains with transitionmatrices equal to historical averages of rating transition frequencies as publishedby the rating agencies [12]. Despite their simplicity credit rating models have sev-eral drawbacks. For instance, the historical transition matrix overstates the truedefault rate. Therefore there is a difference between the historical transition ma-trices published by the agencies and the credit spreads observed in the market [13].Another problem is due to the timing of the publication of a rating change. Inempirical studies it was shown that especially rating downgrades are followed byan increase in credit spreads [14]. This might lead rating agencies to delay thepublication of a rating adjustment11.

1.3.1 Calibration of a Spread Curve

In this section we are going to explain how a term structure of default probabilitiescan be obtained from the rating transition matrix given in Tab. 1.4 [15, 16].

11A prominent example is the stalled rating downgrade of Sal. Oppenheim & Cie. from A toBBB by Fitch. Instead a downgrade from A to A- was announced.

13

AAA AA A BBB BB B CCC D

AAA 93.06 6.29 0.45 0.14 0.06 0.00 0.00 0.01AA 0.59 91.00 7.59 0.61 0.06 0.11 0.02 0.02A 0.05 2.11 91.43 5.63 0.47 0.19 0.04 0.08BBB 0.03 0.23 4.44 89.01 4.70 0.96 0.28 0.36BB 0.04 0.09 0.44 6.07 82.70 7.89 1.22 1.55B 0.00 0.08 0.29 0.41 5.33 82.23 4.91 6.75CCC 0.10 0.00 0.32 0.65 1.62 10.30 57.65 29.35D 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00

Tab. 1.4. Rating Transition Matrix

The rating transition matrix is also called a stochastic (migration) matrix andcharacterised by the property that for each row the sum over all its elementsequals one. For the product of two stochastic (migration) matrices A = (aij) andB = (bij) we obtain

(A · B)ij =∑

k

aik bkj . (1.18)

Computing for the product the sum over the row i, we obtain

∑

j

(A · B)ij =∑

j

∑

k

aik bkj =∑

k

aik

︸ ︷︷ ︸

1

∑

j

bkj

︸ ︷︷ ︸

1

= 1 . (1.19)

Thus the set of stochastic (migration) matrices is closed under multiplication. Theunit matrix is also a stochastic (migration) matrix. It turns out that the set ofstochastic (migration) matrices carries the structure of a semi group.

Returning to the rating transition matrix of Tab. 1.4 the multiplication justdefined allows us to define a stochastic (migration) matrix RTMn for any integernumber of years, n, via

RTMn = RTMn−1 · RTM1 , (1.20)

where RTM1 is the one year matrix given in Tab. 1.4. We even go one step furtherby assuming that RTM1 is the element of the continuous one parameter semigroup12,

RTMt = exp ( t · Q ) , t ≥ 0 , (1.21)

for the special choice t = 1, i.e. the time, t, is measured in years. The generator,Q, can be defined by the matrix logarithm, which is ensured to exist, if the matrixRTM1 = (aij) is diagonal dominant, i.e. |aii| >

∑

j 6=i |aji|. For the matrix RTM1

displayed in Tab. 1.4 the property of diagonal dominance is satisfied.

Introducing

p(R)t =

(etQ)

i(R),8, (1.22)

12The exponential map has the unique property of mapping an additive group law onto anmultiplicative group law.

14

where R denotes a rating and i(R) is the corresponding row of the matrix displayedin Tab. 1.4, we obtain the term structure of the default probability for the ratingclass R, i.e.

t 7→ p(R)t = P

(R) [ τ < t ] R ∈ AAA, . . . , CCC . (1.23)

In Fig. 1.9 the corresponding probability curves are displayed.

0 10 20 30 40 50 60 70 80 90 100Time in Years

0

0.2

0.4

0.6

0.8

1P

rob

ab

ility

of

De

fau

lt

Default Probabilities from Rating Transition Matrix

AAA

CCC

Fig. 1.9. Term structure of default probabilities p(R)t .

The probability density function for a given rating class (R) is defined by

f(R)(t) =d

dtp

(R)t . (1.24)

With this function an expectation value can be defined by

Et [ h(t) ] =

∞∫

0

h(t) · f(R)(t) dt . (1.25)

Choosing a discretisation of the time axis ti i=1...∞ with step size h = ti+1 − tiindependent of i the integral may well be approximated by the Riemann sum

Et [ h(t) ] ∼∞∑

i=0

h(ti) ·[P

(R)( τ < ti+1 ) − P(R)( τ < ti )

](1.26)

with ti ∈ [ti, ti+1) a suitable mean value. A special situation occurs if one hasto compute expectation values of the form E [ h(t) · 1lτ < T ]. The symbol 1lτ < T is

15

a default indicator, with τ the default time and T the maturity. In this case theinfinite sum in eq. (1.25) is truncated to become finite ti i=1...N . Defining the set

of discrete probabilities p(R)i by

p(R)i =

P(R) ( τ < ti+1 ) − P

(R) ( τ < ti )

P(R) ( τ < T ),

N∑

i=1

p(R)i = 1 (1.27)

and a matching expectation value

E [ h(t) ] =

N∑

i=0

p(R)i · h(ti) (1.28)

the expectation value of eq. (1.25) is approximated by

Et [ h(t) · 1lτ < T ] ∼ E [ h(t) ] · P(R) ( τ < T ) . (1.29)

1.3.2 Prices of defaultable Bonds

The term structure of default probabilities derived in subsection 1.3.1 implicitlyassumes that the credit spreads are time homogeneous and solely dependent onthe credit class [13]. Assuming the following notation

n number of cash flows of fixed paymentst0 valuation datetj time of the jth cash flow exchange (j = 1 . . . n)τj time between tj−1 and tjNj notional amount for cash flow at tj (non-negative)Rfix coupon of the bondDti,tj discount factor between time ti and time tjT maturity date of the bond

p(R)t default probability at time t of a name with rating R

the price of a defaultable bond with zero recovery (restriction not essential) is givenby

P (R)(t0, T ) = Rfix

n∑

j=1

Nj τj Dt0,tj ·(

1 − p(R)tj−t0

)

+ Nn Dt0,T ·(

1 − p(R)T−t0

)

(1.30)

Hence, the price of the risky bond is given by the sum of discounted cashflowsweighted with the probability that these cashflows actually occur (survival proba-bility).The probability curves can be absorbed in the corresponding discount factorsthereby defining a credit quality adjusted discount curve (risky curve). Since thediscount factors are bootstrapped on the basis of deposit rates and IRS quotes (all

16

having a credit quality of AA) the credit quality adjusted discount curve is givenby

D(R)t0,t = Dt0,t

1 − p(R)t−t0

1 − p(AA)t−t0

. (1.31)

While this method is a conceptually sound treatment of credit risk it is not alwaysapplicable. In particular if the presence of credit risk is evident but due to lack ofdata there is no direct way to obtain a customised PD. A detailed discussion of thelast sort of problem and possibilities to solve it for the case of a private customerportfolio of exceedingly high credit quality can be found in [17, 18].

1.4 Default-contingent Market Risk

The distinction between market and credit risk is not always as sharp as suggestedin sections 1.2 and 1.3. One form of mixture of these risks are replacement costs.

Definition 7 (Replacement costs). Replacement costs are the estimated ex-penses of buying or building an asset to replace another asset.

This type of risk becomes relevant in a synthetic structure where part of its com-ponents might be exposed to cancellation events such as default or callabilities.We are interested in default events only. If such a default event occurs, a syntheticinstrument might develop a market risk because a previously closed position - withrespect to market risk - opens up due to the default of one of its components13.

13For the sake of completeness it should be also mentioned that the other mixture also exists,i.e. a market risk contingent default risk. For example interest rate products with knockoutfeatures have the property that a discrete event depends on the development of the market data.

17

Chapter 2

Individual Approach

In this chapter we are going to consider a synthetic structure consisting of a bondand a swap (asset swap) which is basically immune against the daily movementsof the term structure of interest rates. In the following we want to develop amethodology to quantify the market risk connected with the swap, which mightappear if the matching bond defaults.

2.1 Asset Swap

Definition 8 (Asset Swap). An asset swap consists of a fixed rate coupon bondand a matching interest rate swap that has a maturity equal to that of the bond.The fixed leg is exactly opposite to coupon payments of the bond, so that the couponreceived from the bond is used to pay the fixed interest on the swap. The swapcounter party pays a floating rate of interest in turn. Therefore the fixed rate bondhas been converted into a FRN.

Financial institutions are major investors in asset swaps. The main reason forcreating the synthetic structure is to reduce interest rate risk as we are going toshow in subsection 2.1.1. It also suits the funding profile of banks, since it ischeaper if banks fund themselves short term [19]1.

1This statement must be taken with the appropriate care. A bank has two main tasks toperform: credit intermediation and maturity transformation. The latter is the vehicle to refinancelong term engagements (loans) on the active side via short term engagements (debts) on thepassive side. The refinancing of loans is made via customers’ deposits (e.g. call money) or theemission of bank notes. The latter was the dominating method of refinancing in the case ofNorthern Rock. Due to subprime mortgage crisis the request of market participants for emissionsof bank notes of Northern Rock dropped to an all time low. The only alternative of the bankto refinance its business was to borrow money at the inter bank money market. Here the liquidvolume of money was so small, that the interest for short term borrowing increased sharply andan economically sensible refinancing became impossible. Therefore the bank had to face a seriousliquidity crisis.

18

2.1.1 Hedging the Market Risk of Coupon bonds

Definition 9 (Hedging). Practice in which a risky trading position and its hedgeare treated as one item so that the gains in one automatically offset the losses inthe other2.

The hedging of coupon bonds against market risk can be realised by constructinga synthetic asset swap thereby transforming the coupon bond into a FRN. Thispractice is known in banks as “hedge accounting” and reviewed in [20]. To see theeffect of market risk reduction we have displayed a histogram of the daily changesin the mark-to-market value of a bond and the associated asset swap by assumingthat the credit quality of the bond does not change in Fig. 2.13.

Clean P&L

Absolute Price Changes

Freque

ncy

050

100150

200

−1 −0.8 −0.4 0 0.2 0.4 0.6 0.8 1

Bond onlyBond + Swap

Fig. 2.1. Daily changes for all business days in 2008.

The width of the distribution of daily changes belonging to the asset swap (FRN)(blue bar) compared to width of the distribution of daily changes belonging to thebond (red bars) is negligible. Consequently while the value of the bond movessignificantly as time goes by the value of the associated asset swap is quite stable.The time dependence of the dirty and of the clean price of an asset swap are dis-played in Fig. 2.2.

Fig. 2.2. Stability of MtM-values during 2008.

2The word “hedge” can be traced back to the German root “umbibiheggen” (about AD 800)used in the meaning of “to protect” and was later used by the Anglo-Saxons when planting rowsof hawthorn which they called “hege”. Hedging a bet comes from the same linguistic roots ashedging a field. The idea of a hedge as a defence or enclosure making something secure is thestarting point for the image of hedging bets.

3Bond: Start = 01.01.2008, End = 01.01.2013, Annual Coupon 4.5%FRN: Start = 01.01.2008, End = 01.01.2013, 3M-Euribor, Spread 0.07%

19

The sawtooth pattern in the dirty MtM value is due to the accrued interest andthe downside spikes in the clean MtM values are artifacts of the refixing events.

Comments

These observations are consistent with the predictions of the market risks (VaR)as done for a specific spot date in section 1.2.1. The detailed comparison of timeseries of precomputed VaR-values with realised clean P&L changes forms the basisof back testing a VaR model which we are not going to discuss here [21].

−3 −2 −1 0 1 2 3

−0.

50.

00.

51.

0

QQ−Plot

Theoretical Quantiles

Sam

ple

Qua

ntile

s

Fig. 2.3. Quantiles of the daily changes of the Bond-MtM.

The only aspect we want to comment on is the validation of the model assump-tion regarding the distribution underlying the daily changes of the mark-to-marketvalue. We focus on the example of the bond whose empirical distribution was al-ready presented in Fig. 2.1. In order to apply the VaR methodology we assumedthis distribution to be normal. In Fig. 2.3 a QQ-Plot of the sample quantiles againstthe quantiles of a standard normal distribution are displayed. Any deviation fromthe straight line points to a violation of the model assumption. Obviously theempirical quantiles are in good agreement with the model assumption.The three rare events which are out of line correspond to the reductions of theinterest rates by the European Central Bank (ECB) in the aftermath of the LehmanBrothers event.

Date Funding Rate

02-Jan-2008 4.00%03-Jul-2008 4.25%08-Oct-2008 3.75%06-Nov-2008 3.25%04-Dec-2008 2.50%

Tab. 2.1 Movement of the ECB Funding Rate within 2008.

20

In table 2.1 the dates of changes in the funding rate and the corresponding ratesare shown. In October, November and December the level of the funding rate wasreduced by at least 50bp. Therefore the corresponding events in the QQ-Plot canbe explained. They are inconsistent with the assumption of a Gaussian distributionbut do not question the whole procedure.

2.1.2 Risk Properties

The most important thing to understand about an asset swap is that the assetswap buyer takes on the credit risk of the bond. If the bond defaults, the assetswap buyer has to continue paying the fixed side on the interest rate swap thatcan no longer be funded with the coupons from the bond. The asset swap buyeralso loses the redemption of the bond that was due to be paid at maturity and iscompensated with whatever recovery rate is paid by the issuer.Marking-to-market the bond and the swap will not avoid the credit risk but itwill highlight any deterioration in the creditworthiness of the bond. The mark-to-market of the asset swap package comprises two items:

1. The market price of the bond

2. The net present value (NPV) of the associated swap

Any change in value of the bond due to changing interest rates will be reflectedby an equal and opposite change in the value of the swap4. Any change in thecredit spread of the bond will remain unhedged. If the credit spread of the bondimproves there will be a net profit, if it deteriorates there will be a net loss [19].As a result, the asset swap buyer has a rating-contingent exposure to the mark-to-market value of the interest rate swap and to the redemption on the asset and theasset swap can be considered as an improper credit derivative [24, 25].

Fig. 2.4. Change of the Value of an Asset Swap due to Rating Transitions.

4The key point here is that the sensitivity of the bond price to parallel movements in the yieldcurve will be less than the sensitivity of the fixed side of the swap to parallel shifts in the Liborcurve. This is only true provided the issuer curve is above the Libor curve. The asset swap buyertherefore has a residual exposure to interest rate movements. In practice this sensitivity is smalland only becomes apparent when Libor spreads widen significantly [22]. This is exactly whathappened in the recent market turmoil starting in August 2007. A study of the behaviour of theyield curves constructed with respect to different Libor spreads can be found in [23].

21

In Fig. 2.4 an example of a five year asset swap5 is displayed. Its initial rating was Aand there are three rating changes A 7→ BB 7→ B 7→ CCC. At the correspondingdates the value of the asset swap changes step like.

2.2 Calculation of the price of risk

As explained in section 1.3 the credit risk of the bond is captured by the use ofspreads which account for the probability of not receiving the promised cash flows.In addition there is a default-contingent market risk due to the interest rate swap.This potential market risk must also be calculated and taken into account. Thecorresponding margin can be computed by separating the potential loss into thesum of expected and unexpected losses

margin = EL + UL . (2.1)

Some clarifications must be made. The loss we are referring to is not the loss causedby the defaulted bond. It is the loss due to a possible negative market value ofthe remaining payer swap (hedge). To close out this position after a default hasoccurred, we have to buy the matching receiver swap, which cancels all remainingcash flows of the payer swap. Therefore the margin are the replacement costs ofdefinition 7. This costs are given by

PE = max ( NPVReceiver, 0 ) , (2.2)

where PE denotes the positive exposure of the matching receiver swap at the timeof default τ < T . Here T refers to the bond’s maturity. In order to formulate thedependency on the default of the bond, we also use the bond’s default indicator,1lτ < T , in the following. In the next sections we are going to describe, how the ELand UL can be obtained from exposure profiles [26, 27] and how other moments ofthe loss distribution can be constructed [28, 29].

2.2.1 Estimating the expected loss

The expected loss (EL) can be obtained by the standard ansatz

EL = E [ PE · 1lτ < T ] = Et [ E [ PE ] · 1lτ < T ] . (2.3)

The inner expectation value is related to the distribution of the par rate, whichdetermines the NPV of the corresponding receiver swap. The outer expectation isthe time average with respect to the time dependent probability density functionconstructed in section 1.3.Only if the stochastic positive exposure (swap) and the stochastic loss indicator(bond) can be assumed to be independent, the above formula can be factorisedand one obtains the regulatory expression

EL = E [ PE ]︸ ︷︷ ︸

EAD

·Et [ 1lτ < T ]︸ ︷︷ ︸

PD

= EAD · PD , (2.4)

5Start date = 02.01.2008, Maturity 02.01.2013, yearly fixed payments (coupon 4.542%) againstsemiannual Euribor.

22

i.e. the expected loss is the product of the exposure at default (EAD) and theprobability of default (PD) for the time horizon T 6. Here this property is notsatisfied but using eq. (1.25) the expectation value7 of eq. (2.3) reads

EL =

T∫

0

E [ PE ] (t) · fR(t) dt . (2.5)

Using eq. (1.25) and eq. (1.29) the expected loss can be approximated by

EL ∼ E [ E [ PE ] ] · PD (2.6)

and has precisely the regulatory form again. The corresponding EAD reads

EAD = E [ E [ PE ] ] . (2.7)

This definition of the EAD is also called the mean exposure of the interest rateswap. The function E [ PE ](τ) of the default time τ forms the so called exposureprofile and corresponds to the compounded price of a European swaption:

E [ PE ] (τ) = erτ ·

e−rτ · E [max(NPVSwap, 0) ]︸ ︷︷ ︸

price of a european swaption

, (2.8)

The compounding/discounting is expressed in continuous compounding and r isthe applicable interest rate.

Interpreted correctly one recognises that the profile can be considered as for-ward prices of receiver swaptions with exercise dates τ , written on the remainingcash flows of a common underlying receiver swap. In the special case where theunderlying risk factor (swap rate) is assumed to be log-normally distributed theswaption price is given by eq. (1.9) so that the EPE-profile8 is determined in termsof a Black Scholes formula.

There are two main effects that determine the expected positive exposure overtime: diffusion and amortisation. As time passes, the “diffusion effect” tends toincrease the exposure; the “amortisation effect”, in contrast, tends to decrease theexposure over time, because it reduces the remaining cash flows. These two effectsact in opposite directions. For products with multiple cash flows, such as interest-rate swaps, the exposure usually peaks at one-third to one-half of the way into thelife of the transaction [30].

6The loss given default (LGD) is 100%.7The “R” in the probability density function, fR(τ), denotes the rating of the bond.8Note that what we call E(xpected)P(ositive)E(xposure)-profile is usually called

E(xpected)E(xposure)-profile.

23

Fig. 2.5. EAD determined from an EPE-Profile.

Example:

In Fig. 2.5 we display an example of an EPE-profile for a receiver swap9. The EADcomputed by eq. (2.7) for the two limiting credit ratings AAA (blue line) and CCC(red line) are also displayed. Both are significantly smaller than the peak exposureof the swap.

2.2.2 Estimating the unexpected loss

Now we turn the attention to the determination of the unexpected loss (UL) givenby

UL = Et [ Qα − EL ] (2.9)

with Qα the α-quantile of the distribution of positive exposures implicitly definedby the identity P [ PE < Qα ] = α, sketched in Fig. 2.6 and determined in ap-pendix A.1.2, eq. (A.14).

α

αQ PE

Fig. 2.6. α-Quantile

9Evaluation date = Start 11.02.08, End 11.02.13, annual coupon of 3.9%, against 6M-Euriborwith additional spread 0.07%

24

Consequently the UL describes the loss due to the swap, which is not exceededwith probability α.For the computation of the UL, i.e. the loss that exceeds EL, the tails of thejoint distribution of the swap’s positive exposure and the default time of the bondbecome important. Due to this tail dependence it turns out that the EAD asdetermined in the last subsection is not an appropriate estimate. An approach totackle this difficulty and to obtain the EAD from the point of view of the tailsof the joint distribution is to determine the EAD of a loan in such a way thatthe variance10 of its one dimensional loss distribution equals the variance of thejoint loss distribution of the swap’s positive exposure and the bonds default time.The EAD of this loan is called loan equivalent (LEQ). Using again eq. (1.25) andeq. (1.29) and the identity 1l2τ < T = 1lτ < T we obtain the variance of the joint lossdistribution by straightforward algebra and it is equal to

V [ PE · 1lτ < T ] = E[(PE · 1lτ < T )2

]− E [ PE · 1lτ < T ]2

= Et [ V [ PE ] · 1lτ < T ] + Vt [ E [ PE ] · 1lτ < T ] (2.10)

Starting with the equivalent loan, which has the constant exposure LEQ, the vari-ance of its one dimensional loss distribution reads

V [ PE · 1lτ <T ] = PD(1 − PD) · LEQ2 . (2.11)

By matching the moments, i.e by equating the variance of the joint loss distributionwith the variance of an equivalent loan, we conclude that the constant EAD is givenby the loan equivalent [31]

LEQ =

√

Et [ V [ PE ] · 1lτ < T ] + Vt [ E [ PE ] · 1lτ < T ]

PD · (1 − PD). (2.12)

The result is that the replacement of EAD in the UL part of eq. (2.1) is the meanexposure corrected by the timely variance of the exposure distribution. Henceforththe replacement costs read

replacement costs = ELEL[EAD= EPE] + ULEL[EAD= LEQ] . (2.13)

Example continued:

Continuing with the example already studied in Fig. 2.5 we are now going to presentthe details necessary for obtaining the unexpected loss as well. We consider thecase of a swap used to hedge a CCC rated bond.The quantile of the positive exposure as a function of the default time τ is derivedin appendix A.1.2 and replacing the variable y in eq. (A.14) with the par rate of theswap RSwap and the strike K with the fixed coupon Rfix, the quantile of a receiverswap (ω = −1) reads

Q99%(τ) = − δ(τ)[

RSwap e−Q99%(0,1) σ√

τ − 12

σ2 τ − Rfix

]+

. (2.14)

10The variance is the most elementary measure for the width of a distribution.

25

Here Q99%(0, 1) denotes the 99%-quantile of the standard normal distribution,whose value is given on page 9 and δ(τ) denotes the annuity or level of the swap.In Fig. 2.7 we display the EPE-profile and the 99%-quantiles of the positive expo-sure. The red line denotes the average of the quantile according to eq. (1.28)

Q99% = E [ Q99% ] . (2.15)

Fig. 2.7. Unexpected loss computation.

In order to compute the LEQ of eq. (2.12) we must evaluate the variance of thejoint distribution of the positive exposure and the default process. Using therepresentation

V [ PE · 1lτ < T ] = E[E[PE2

] ]· PD − E [ E [ PE ] ]2 · PD2 (2.16)

this can be computed by averaging eq. (2.8) and eq. (A.6) according to eq. (1.28).The resulting LEQ is the solid blue line displayed. In addition the broken blue linedenotes the EAD as used to determine the EL as determined previously.The unexpected loss of eq. (2.9) reads

UL =(Q99% − LEQ

)· PD ∼ 2 EUR · 67% . (2.17)

26

Chapter 3

Portfolio Approach

In contrast to the last chapter this chapter is going to study the credit risk of aportfolio of correlated bonds. We adopt to the rules and study the correspondingloss distribution. To avoid Monte Carlo simulations we use a method utilisingcorrelation expansions.

3.1 Loss distributions

3.1.1 Loss distribution of independent obligors

In modelling credit risk of n independent names one is mainly interested in theconstruction of the so called loss distribution, i.e. the set of discrete loss probabil-ities with which a specific loss may occur. Each name i ∈ Ω = 1, . . . , n mayhave a different exposure ci. We assume each exposure to be an integer, which isa reasonable assumption. Defining the minimal loss unit, u, as the greatest com-mon divisor of all exposures ci, then each exposure may be written in terms of aweight wi, implicitly defined by ci = u · wi. From now we restrict our attentionto normalised losses, which are the integers obtained as the quotients of the actuallosses and the loss unit. The true values can be retrieved by multiplying the nor-malised values with the loss unit. For j = 1, . . . , n, define the random variablesof (normalised) losses

Lj = w1 · X1 + . . . + wj · Xj (3.1)

with Xi ∈ 0, 1 the default indicator for the i-th obligor. The joint probabilityof experiencing a loss of amount k was constructed in [32] using a recursion inj. The recursive relation1 builds the loss distribution of the portfolio of n namesincrementally from the relation

Lj+1 = Lj + wj+1 · Xj+1 (3.2)

via

P (Lj+1 = k ) = P (Xj+1 = 0 ) · P (Lj = k ) +

P (Xj+1 = 1 ) · P (Lj = k − wj+1 ) (3.3)

1implemented in LossDistribution.m (cf. appendix C.1)

27

with initial value P (L0 = k ) = 1lk=0.

Expectation values of functions of the loss variable, L, can now be obtained viathe standard formula2

E [ f(L) ] =Lmax∑

l=0

f(l) · P( Ln = l ) . (3.4)

3.1.2 Loss distribution of dependent obligors

In order to construct the loss distribution of a dependent obligor model one has tospecify a correlation structure via a copula [33]. In principle one may obtain the lossdistribution afterwards by computing the expectation values in the correspondingcopula model, i.e.

Pρ ( Ln = k ) = Eρ [ 1lLn=k ] , (3.5)

where ρ indicates that here a correlation structure was taken into account. Inthe case of elliptical copulas the symbols ρ and Σ are actually used to denote thecorrelation and covariance matrix, respectively.

3.2 Multivariate Gaussian distribution

A commonly used copula model is based on the multivariate Gaussian distribution.In order to apply this model in practice one is often limited to use it in conjunctionwith Monte Carlo techniques. Another approach to the construction of the multi-variate Gaussian distribution was taken in [34] and recently used to obtain analyticapproximations to interesting quantities of a portfolio of defaultable names, suchas expectation values of functions of the loss variable. This section is intended toreview the corresponding results and to prepare the ground for the extension of themethodology to other copulas such as the student copula further investigated insection 3.3. The methodology we are going to review basically relies on two facts:

• the Rodriguez formula of the set of orthogonal polynomials associated withthe marginal probability density function and

• the factorisation of the multivariate density distribution function of iid ran-dom variables into the product of the corresponding univariate density dis-tribution functions.

Another multivariate distribution function which for the aforementioned reasonscould also be treated with the methodology of [35] is the multivariate Gammadistribution. The necessary identities can be found in [36].

In the next subsections we collect basic facts on Hermite polynomials and reviewgenerating functions for different products of Hermite polynomials.

2implemented in EV.m (cf. appendix C.2)

28

3.2.1 Basic facts

Hermitian Polynomials

The set of orthogonal polynomials [37] belonging to the normal density distributionfunction,

f(x) =1√2π

e−x2/2 , (3.6)

i.e. being orthogonal with respect to the scalar product

〈 g, h 〉 =

∞∫

−∞

g(x) · h(x) · f(x) dx (3.7)

are the Hermite polynomials of degree n, Hen(x), defined by the Rodriguez formula

Hen(x) = (−1)n 1

f(x)

dn

dxnf(x) = (−1)n ex2/2 dn

dxn

(

e−x2/2)

. (3.8)

The generating function of this set of polynomials is

etx−t2/2 =∞∑

k=0

Hek(x) · tk

k!(3.9)

from which the property

Hen(−x) = (−1)n · Hen(x) (3.10)

and the most important recursion formulae follow easily. The recursion used forthe generation of the polynomials is given by

Hen(x) = xHen−1(x) − (n − 1) Hen−2(x) (3.11)

with He0(x) = 1 and He1(x) = x.

For n > 0 the definite integral of a Hermite polynomial multiplied by a Gaussiandensity is easy to compute:

∞∫

x

f(y) Hen(y) dy =

∞∫

x

(−1)n dn

dynf(y) dy = (−1)n dn−1

dyn−1f(y)

∣∣∣∣

∞

x

= f(x) Hen−1(x) . (3.12)

If n = 0 the integral becomes the definition of the marginal probability

p = P [ y > x ] =

∞∫

x

f(y) dy . (3.13)

29

Both identities are unifed by introducing the function3 Bn(x), which is definedbelow:

Bn(x) =

∞∫

x

f(y) Hen(y) dy =

P [ y > x ] if n = 0

f(x) Hen−1(x) if n > 0 .(3.14)

3.2.2 Correlation expansions

Correlation expansions are based on identities, which express the multivariate prob-ability density function in terms of the univariate probability density function anda complete set of orthogonal polynomials. As the simplest version of this typeof identities we can consider eq. (3.9), i.e. the generating function of the set oforthogonal polynomial itself. In the next two paragraphs we review the results forthe multivariate Gaussian distributions.

Mehler’s Theorem

It was shown by Mehler [38], that the density, ρ, of the two-dimensional Gaussiandistribution, i.e.

ρ(x1, x2) =1

2π√

1 − ρ212

exp

− x21 − 2 ρ12 x1 x2 + x2

2

2 (1 − ρ212)

,

possesses an expansion in powers of the only correlation parameter ρ12 of a twodimensional correlation matrix

ρ =

(1 ρ12

ρ12 1

)

and Hermite polynomials given by:

ρ(x1, x2) =∞∑

k=0

ρk12

k!f(x1) Hek(x1) f(x2) Hek(x2) .

Due to eq. (3.8) the products f(xi) Hek(xi) can be expressed as a total derivativeand the integration is easy to perform:

P (x1 > ν1, x2 > ν2 ) =

∞∫

ν1

∞∫

ν2

ρ(x1, x2) dx1 dx2 ,

P (x1 > ν1, x2 > ν2 ) = p1 · p2 +

∞∑

k=1

ρk12

k!f(ν1) Hek−1(ν1) f(ν2) Hek−1(ν2) .

3implemented in HermiteB.m (cf. appendix C.3)

30

Here pi = P(xi > νi) is the marginal probability of the i-th obligor. Using thefunctions Br(x) defined in eq. (3.14) the above expansion can be compactly sum-marised as

P (x1 > ν1, x2 > ν2 ) =∞∑

k=0

ρk12

k!Bk(ν1) Bk(ν2) .

Kibble’s Theorem

When dealing with more than two correlated normally distributed variables, let ussay n, which are all rescaled to have variance one4, then the probability density,ρ(x1, . . . , xn), is given by

ρ(x1, . . . , xn) =1

(2π)n/2 ·√

det ρ· exp

(

−xT ρ−1 x

2

)

, (3.15)

where the correlation is specified by the positive definite symmetric matrix, ρ,which reads

ρ = (ρij) =

1 t · ρ12 . . . t · ρ1n

t · ρ21. . .

. . ....

.... . .

. . . t · ρn−1n

t · ρn1 . . . t · ρnn−1 1

. (3.16)

The parameter t ∈ [ 0, 1 ] is included to match the notation used in [35]. There itis used to deform the model in a continuous manner from an uncorrelated model tothe correlated one by letting t vary from 0 to 1. The n-dimensional generalisationof Mehler’s result was given in [34]. There a proof of the following identity isspelled out and based on a simple combination of basic properties of the Laplacetransform (differential theorem) and integrals of Gaussian type. The result reads

exp(

−xT ρ−1 x2

)

(2π)n/2 ·√

det ρ= f(x1) · . . . · f(xn) ·

∞∑

r=0

tr

r!·

∑

k1<l1,...,kr<lr

ρk1l1 . . . ρkrlr︸ ︷︷ ︸

r−times

Heh1(x1) . . .Hehn(xn)

and the summation in the last equation has to be understood as follows:

• The summation is over all different indices k1, l1 . . . kr, lr. from 1 to n, subjectonly to the conditions ki < li and

• by ht we denote the total number of times the index t occurs among thesuffixes k1, l1 . . . kr, lr.

4Thus the marginal distribution of each of the n variables is standard normal.

31

A more compact derivation of this expansion was given in [39] by solving theCauchy problem of a heat equation. If we let D denote the covector

D =

(∂

∂x1

, . . .,∂

∂xn

)

(3.17)

and choose a decomposition of the correlation matrix, ρ, into a diagonal and anoff-diagonal part

ρ = 1l + t B (3.18)

then up to a factor of (2π)−n/2 the expansion above is equivalent to the followingoperator identity:

1√det ρ

exp

− xT ρ−1x

2

= exp

tDBDT

2

exp

−‖x‖2

2

. (3.19)

From the correlation expansion the following expression for the joint cumulativeprobability function follows easily by paying attention to the fact that due toeq. (3.8) the integration in each variable is trivial and leads to expressions given ineq. (3.14) evaluated at the lower boundary of the domain of integration

P (x1 > ν1, . . . , xn > νn ) =

∞∫

ν1

dx1 . . .

∞∫

νn

dxn ρ(x1, . . . , xn) =

∞∑

r=0

br ·tr

r!.

The expansion coefficients, br, are actually functions of ν1, . . . , νn. For brevity wesuppress these arguments. The expansion coefficients read:

br =∑

k1<l1,...,kr<lr

ρk1l1 · . . . · ρkrlr︸ ︷︷ ︸

r−times

Bh1(ν1) · . . . · Bhn(νn) . (3.20)

y-dependence of the expansion coefficients

Later we have to consider a y-dependence of the expansion coefficients just intro-duced. For later convenience we record the related formulas here. The correspond-ing integrals are given by

P (x1 > y · ν1, . . . , xn > y · νn ) =

∞∫

y·ν1

dx1 . . .

∞∫

y·νn

dxn

exp(

− xT ρ−1 x2

)

( 2 π )n2

√det ρ

=∞∑

r=0

br(y) · tr

r!(3.21)

with

br(y) =∑

k1<l1,...,kr<lr

ρk1l1 · . . . · ρkrlr︸ ︷︷ ︸

r−times

Bh1(y · ν1) · . . . · Bhn(y · νn) (3.22)

Again the expansion coefficients, br(y), are also functions of ν1, . . . , νn. Thesearguments are suppressed for brevity.

32

The approach of Glasserman & Suchintabandid

In [35] a different form of eq. (3.20) was derived by assuming the following represen-tation of the correlation matrix in terms of a d-dimensional factor decomposition5:

ρij =

d∑

k=1

λk · aik ajk , i 6= j . (3.23)

It was shown in proposition 1 of [35] that under this assumption about the corre-lation matrix eq. (3.20) reads

br =1

2r

∑

J∈Ir[d]

λJ∂

∂s21 . . . ∂s2

r

(

p(J)1 · . . . · p(J)

n

)

. (3.24)

Here I[d] = 1, 2, . . . , d and Ir[d] denotes its Cartesian product and by J we

denote a multi-index J = ( j1, . . . , jr ) ∈ Ir[d]. Furthermore n denotes the number

of obligors and to the i-th obligor belongs the set of perturbed default probabilities,

p(J)i , labeled by J and defined by

p(J)i =

∑

T⊂I[r]

B|T |(νi)∏

ℓ∈T

sℓ ai,jℓ. (3.25)

Here |T | denotes the number of elements in the subset T . In order to put eq. (3.24)into a more appealing shape an extension of the definition of the objects λk andthe factor matrix aij proves useful6:

λ0 = − 2 ( λ1 + . . . + λd ) ai,0 = 0 (3.26)

λ−j = λj ai,−j = − ai,j . (3.27)

Approximating the second derivatives in eq. (3.24) by the expressions for the nu-

merical second derivative, setting s1 = s2 = . . . = sr!= s and using the extensions

of the indices as defined in eq. (3.26) and eq. (3.27) the generated terms can beregrouped and one obtains that

br = lims→0

∑

J∈Dr

wJ · p(J)1 · . . . · p(J)

n (3.28)

where D = 0, ±1, ±2, . . . , ±d and Dr denotes its Cartesian product andwJ = λJ/(2s2)r and λJ = λj1 · . . . · λjr

. Eq. (3.28) was proven in proposition2 of [35].

5Here all λi = 1. Their correct meaning would become clear if we were not considering thecorrelation matrix but the covariance matrix instead. In this case the λi appear as the eigenvaluesin a principal component analysis and represent the resolution of variance.

6The reason behind the extension of the definition of these variables is the intention to obtaincompact expressions afterwards.

33

Each term in the sum can be interpreted as the probability that a group of indepen-

dent obligors, each having a marginal probability7 of p(J)i , default simultaneously.

The p(J)i in eq. (3.25) can also be written as

p(J)i =

∑

T⊂I[r]

B|T |(νi)∏

ℓ∈T

sℓ ai,jℓ=

r∑

k=0

Bk(νi) · v(J)(i, k) . (3.29)

The big advantage is, that the p(J)i can now be computed recursively. To this

purpose we have introduced the function

v(J)(i, k) =∑

T⊂I[r],|T |=k

∏

ℓ∈T

sℓ ai,jℓ. (3.30)

The recursion formula for this function follows easily from its definition and reads8

v(J)(i, k) = v(J ′)(i, k) + sr ai,jrv(J ′)(i, k − 1) . (3.31)

Here J ′ = ( j1, . . . , jr−1, 0 ) denotes a truncation of the multi-index J . The recur-sion boundary is

J ∈ Dr v(J)(i, k) = 0 if k < 0 or k > r

j ∈ D v(j)(i, 0) = 1

j ∈ D v(j)(i, 1) = s1 aij .

The usefulness of the representation of br given in eq. (3.28) lies in the fact that thisequation provides an expansion of the joint probability of default of an n-obligormodel given in eq. (3.20) into a sum of joint probabilities each factorised inton products, thus interpretable as marginal probabilities of artificial independentobligor models.The probability that only the names S = i1, . . . , im ⊂ Ω = 1, . . . , n default while the names in Ω \ S will survive is given by

Pρ ( S, Ω \ S ) =∞∑

r=0

tr

r!lims→0

∑

J∈Dr

wJ · p(J)S · q(J)

Ω\S

Here p(J)S denotes the product p

(J)i1

· . . . · p(J)im and q

(J)Ω\S is understood analogously in

the natural sense.

7implemented in PDtwiddle.m (cf. appendix C.4)8implemented in V.m

34

The expectation value of a function of the loss variable, L, now reads

Eρ [ f(L) ] =

Lmax∑

l=0

f(l) · Pρ [ L = l ]

=Lmax∑

l=0

f(l) ·∑

S,L=l

Pρ(S, Ω \ S)

=

Lmax∑

l=0

f(l) ·∑

S,L=l

∞∑

r=0

tr

r!lims→0

∑

J∈Dr

wJ · p(J)S · q(J)

Ω\S

=∞∑

r=0

tr

r!lims→0

∑

J∈Dr

wJ ·Lmax∑

l=0

f(l) ·∑

S,L=l

p(J)S · q(J)

Ω\S

=∞∑

r=0

tr

r!· lim

s→0

∑

J∈Dr

wJ · E [ f(L) ]

︸ ︷︷ ︸

δr

=∞∑

r=0

tr

r!· δr , (3.32)

i.e. the expectation value in the correlated model is expressed in terms of sums ofexpectation values of uncorrelated models9. Each of these expectations is computedin the discrete model described in section 3.1.1.

3.2.3 Glasserman’s example revisited

In order to check our implementation of the algorithm proposed in [35] and givenin appendix C, we recomputed the first example given there. It is a portfolio ofn = 50 names with exposures ci i=1...n and marginal probabilities of default pi i=1...n given by

ci = i (3.33)

pi = 0.02 . (3.34)

The correlation between the obligors are specified by a factor matrix according toeq. (3.23) given by

A = (~a1, . . . , ~a5 ) =

a1,1 0...

... a9,2 0...

a12,1... a19,3 0

...

0 a22,2... a29,4 0

... 0 a32,3... a39,5

... 0 a42,4...

... 0 a50,5

. (3.35)

9implemented in CDOTest.m

35

All non zero entries, aij, are taken to be 0.2. The quantity we are going to inves-tigate is the expectation value of

max( L − 200, 0 ) . (3.36)

This expectation value possesses an expansion according to eq. (3.32) with coeffi-cients given in Tab. 3.1.

δ0 δ1 δ2 δ3

0.0011 0.0016 0.0022 0.0026

Tab. 3.1. Glasserman Example with Gaussian copula.

We obtain basically the same coefficients as in the original paper. However, thevalue of δ0 published in [35] is 0.0012. This is not in correspondence with ourresult.

3.3 Multivariate Student Distribution

In this section we would like to show how the results of the previous section 3.2can be used in order to derive a similar correlation expansion for the case of theStudent distribution. The student distribution was introduced by W. S. Gos-sett, an employee of the Guinness brewery in Dublin, who published under thepseudonym “Student” [40]. Its n-dimensional multivariate probability densityfunction reads [41]

ρν(x) =Γ(ν + n

2)

Γ(ν2)

1

( πν )n2

√det ρ

( 1 +xT ρ−1 x

ν)− ν+n

2. (3.37)

Here ν denotes the degrees of freedom. Restricting eq. (3.37) to the case of uncorre-lated random variables, the distribution does not possess the property of factorisinginto the product of one dimensional student distributions. This is the reason whythe expansion method as discussed in section 3.2 does not seem to be applicable.But it can be related to the case considered before. The relation of the multivari-ate student distribution to its Gaussian counterpart is based on a simple identityoften used in renormalisation theory and which is given in [42], for instance. Thisidentity reads

( 1 +xT ρ−1 x

ν)− ν+n

2=

22−ν−n

2

Γ(ν+n2

)

∞∫

0

yν+n−1 exp

−y2

2

(

1 +xT ρ−1 x

ν

)

dy

(3.38)

and due to the standard procedure of passing from a probability density functionto the probability itself one obtains for the probability of observing n joint defaults

36

the expression

P ( x1 > ν1, . . . , xn > νn ) =1

2ν−22 Γ(ν

2) ν

n2

∞∫

0

dy yν+n−1 e−y2

2 (3.39)

·∞∫

ν1

dx1 . . .

∞∫

νn

dxn

exp(

−y2

νxT ρ−1 x

2

)

( 2 π )n2

√det ρ

.

Performing a change of variables defined by

ui = y · xi√ν

(3.40)

the joint probability of observing n defaults reads:

P (x1 > ν1, . . . , xn > νn ) =1

2ν−22 Γ(ν

2)

∞∫

0

dy yν−1 e−y2

2 (3.41)

·∞∫

u−

1 (y)

du1 . . .

∞∫

u−

n (y)

dun

exp(

− uT ρ−1 u2

)

( 2 π )n2

√det ρ

.

Here by u−i (y) we denote the transforms of the lower bound of the domain of

integration:

u−i (y) = y · νi√

ν. (3.42)

Inserting the expansion of eq. (3.21) one obtains

P (x1 > ν1, . . . , xn > νn ) =1

2ν−22 Γ(ν

2)

∞∫

0

dy yν−1 e−y2

2

( ∞∑

r=0

br(y) · tr

r!

)

=

∞∑

r=0

tr

r!·

1

2ν−22 Γ(ν

2)

∞∫

0

dy yν−1 e−y2

2 br(y)

(3.43)

from which the resolution of the probability of n joint defaults into (an integralover) a sum of joint default probabilities of independent obligor models followsonce more. For the expression of an expectation value of a function of the lossvariable one obtains likewise:

Eρ [ f(L) ] =

∞∑

r=0

tr

r!· 1

2ν−22 Γ(ν

2)

∞∫

0

dy yν−1 e−y2

2 lims→0

∑

J∈Dr

wJ · E [ f(L) ] (y)

=

∞∑

r=0

tr

r!· 1

2ν−22 Γ(ν

2)

∞∫

0

dy yν−1 e−y2

2 · δr(y) . (3.44)

37

3.3.1 Example continued

Here we continue the example of section 3.2.3. In the last section we explainedhow the correlation expansion with respect to a Student copula can be obtained.Expectation values with respect to a student distribution are determined in terms ofintegrals over y-dependent expectation values calculated with respect to a Gaussiandistribution as shown in eq. (3.44). Therefore the main insight was that we had torecord the y-dependence of the expansion coefficients δr(y) in order to be able toperform the additional integration.In Fig. 3.1 the dependence of the first three expansion coefficients δr(y) r=1,2,3

on the choice of the marginal probabilities pi is displayed because this is the mostgeneric way to precompute the needed quantities.

0 0.1 0.2 0.3 0.4 0.5Marginal default probabilitiy

0

1

2

3

4

Dependence of expansion coefficients

n = 1

n = 2 n = 5

δ i(y

)

δ1(y)δ2(y)δ3(y)

Fig. 3.1. Dependence of the expansion coefficients, δr(y), on the marginalprobability of default for the Glasserman example of section 3.2.3.

The exact relation of this dependence to the y-dependence we are actually in-terested in follows from eq. (3.41) and is summarised in eq. (3.42). The u−

i (y)appearing there correspond to the marginal probabilities displayed on the abscissain Fig. 3.1, i.e. those marginal probabilities with which the calculation of the expec-tation values was done. Obviously the relation between the marginal probabilitieswith respect to the Gaussian copula used in calculating y-dependent expectationvalues and the u−

i (y) is given by

Pmarg

[x > u−

i (y)]

= pi , (3.45)

where Pmarg denotes the univariate Gaussian cumulative distribution function. Theexpansion of the corresponding student distribution can be obtained by integratingthe functions in Fig. 3.1 (seen as functions depending on y) according to eq. (3.44).The coefficients computed this way are given in Tab. 3.2. The expansion is of thesame quality as in the Gaussian case. The fact that δ0 is significantly larger reflectsthe fact that the student distribution has fat tails10.

10The value of δ0 was cross checked with a MC simulation in Gnu R using the fCopulae package

38

δ0 δ1 δ2 δ3

RS 2.4382 0.2037 0.0104 0.0008

Tab. 3.2. Glasserman Example with Student copula11

Fat Tails

This example is a modification of the portfolio considered before. It again containsn = 50 obligors each having a default probability of pi = 0.02, i = 1, . . . , 50. Thecorrelation matrix is also the same and therefore specified by the factor matrix ofeq. (3.35). The only difference to the example before is that this time all nameshave the same exposure, say one for simplicity.

In the sense of section 3.1.2 we would like to construct and visualise the lossdistribution of the dependent obligor model. This is the basic object needed forall computations one might be interested to do. It is similar to a Green’s functionin the theory of partial differential equations. The probability distribution of adependent obligor model can be computed using eq. (3.5). This turns out to bea special expectation value so that the expansion method discussed before can beapplied. In Fig. 3.2 the probability distribution as a function of the loss variable,L, and identical marginal default probabilities of all obligors, PD, is displayed. Togenerate the plot we had to compute a PD at 60x50 mesh points.

Fig. 3.2. Dependence of the loss distribution on the Gaussian marginalprobability of default for the portfolio at hand.

It is now possible to extract from Fig. 3.2 the loss distributions with respect toseveral assumptions of the underlying distribution, e.g. Gaussian marginals corre-lated via a Gaussian copula or Gaussian marginals correlated via a Student copula.The latter one can be obtained by the same procedure as described in subsection3.3.1, i.e. by performing suitable integrations.

[43] based on a sample of size 105. The value so obtained is 2.37 with a standard deviation of0.188 determined by resampling 100 times 10% of the sample.

11RS = Riemann sum

39

Fig. 3.3. Comparison of tail dependence with respect to different correlationassumptions.

The different results are compared in Fig. 3.3. There three loss distributions areshown. In black we displayed the loss distribution of the independent obligor modelwith marginals pi. In red we plotted the loss distribution computed with respect toGaussian marginals correlated via a Gaussian copula. Last but not least in yellowappears the loss distribution of Gaussian marginals correlated via a Student cop-ula. The different behaviour of the tail dependence of the last version comparedwith the other two counterparts is obvious. So the student copula model predictsmuch larger losses than the analogous Gaussian copula model. Therefore in realportfolios it might be appropriate to choose a suitable copula to match the histori-cally observed defaults better than it can be done by just relying on the correctnessof the distribution implied by the Gaussian copula model.

40

Chapter 4

Portfolio Dynamics

4.1 System Architecture

After the technical preparations detailed in the previous chapters we are now goingto apply and test our ideas by studying the dynamics of a complex bond portfolio.Basically the complexity is due to the following two properties we are going tosimulate: portfolio growths and default events. To this purpose we first explain insection 4.1.1 the design of the bond portfolio, which mimics many of the featuresfound in reality. This includes the market risk due to the changing level of interestrates, the hedging of the market risk via matching interest rate swaps, the creditrisk associated with a bond and the default contingent market risk due the hedgingswaps.

The market risk with respect to the changing level of interest rate can becaptured with relative ease by bootstrapping for each of the simulation dates adiscount curve from real market data, i.e. from the deposit rates and IRS quotesobserved in the market on the simulation date. The bootstrapping of the yieldcurves, the pricing of all instruments (bonds & swaps) and the computation of therisk measures is completely encoded using QuantLib [7].

The default process in the bond portfolio is simulated via the rating transitionmatrix already considered in section 1.3 (continuous time Markovian model), i.e.the credit risk is modelled via a rating transition model. This can be rephrased interms of risky discount curves as shown in eq. (1.31).

In Fig. 4.1 the main components of the simulator are displayed.

Portfolio−Generator

Pricing Engine VaR−Module

VaR−M

apsInstrum

ents

Trades

DB

Fig. 4.1. System architecture

41

These components are realised in terms of an Access database and several Ex-cel spread sheets which combine the computational strength of the QuantLib withVBA’s database functionality. The spread sheets are used to perform the requestedcomputation for each of the simulation date and store the results in the database.

There are six main calculations which must be performed:

• Generation of the bond portfolio,

• calculation of par bonds for each credit quality,

• calculation of par spreads for the corresponding swaps,

• calculation of market prices in the toy model,

• calculation of the swaps’ exposures at default (EAD) and

• calculation of the VaR-Maps.

The results are stored in a database which is built in the spirit of the Front Arenadata model. The main tables are shown in Fig. 4.2.

+ VALUE

Table HEDGES

+ PERDATUM+ INSID

+ SWAP

Table VaR_Maps

+ INSID+ PERDATUM+ RTG+ RISK_FACTOR

Table PRICES

+ INSID+ PERDATUM+ RTG+ PRICE

Table TRADES

+ ACQUIRE_DATE+ QUANTITY

+ INSID+ TRADEID

Table RTG_EVENTS

+ TRADEID

+ CURRENT_RTG+ PERDATUM

+ DEF_OBS_DATE+ DEF_SETTLE_DATE+ EWB

Table INSTRUMENTS

+ INSID+ START+ END+ INITIAL_RTG+ CPN

Fig. 4.2. The design of the main tables used to store results and to aggregatethem.

The portfolio is realised by the table TRADES, which stores possible multipletrades to similar instruments, whose credit quality at start may coincide but maychange independently in the future. These prototypical coupon bonds are storedin table INSTRUMENTS1. For each maturity there exist eight versions of a bondcorresponding to the eight different ratings it might take. They differ by thecoupons making them on the start day to be quoted at par with respect to theappropriate risky discount curve. The table RTG EVENTS stores the ratings ofall bonds at each of the simulation days as they are generated by the simulation ofthe rating transition process and might therefore deviate from the initial ratings

1For simplicity all bonds are five year coupon bonds with annual coupon payments.

42

assigned on the day of the bonds’ inceptions. The table PRICE stores links toprecomputed bond prices. We have precomputed the bond prices for coupon bondswith all possible ratings. The actual bond price is selected by joining this tablewith the table RTG EVENTS, i.e. if during time evolution the rating changes,then the price link is moved to the price of the bond with the matching rating.The table HEDGES contains the swap prices belonging to a coupon bond. Itslinks are not changed if the rating of the underlying bond changed during timeevolution. The remaining tables and its relations are self-explanatory.The simulation time covers the whole year 2008 and the market data used duringthe simulation consist of yield curves and swaption volatilities (needed for EPE-profiles) valid on the simulation day.

4.1.1 Portfolio Generation

The bond portfolio we are going to consider should mimic a real business situation.Often banks take on new loans according to a specific risk profile, i.e. specifiedamounts of the new deals must fall into the standard eight rating classes.Our initial portfolio2 consists of one thousand five-year bonds with a rating dis-tribution as shown in Tab. 4.1, whose start dates all correspond to the simulationstart date.

Rating Initial Nbr.

AAA 500AA 300A 100BBB 40BB 30B 20CCC 10D 0

Sum 1000

Tab. 4.1. Initial Rating Distribution

A rating transition is simulated by generating a sample according to the Markovchain displayed in Fig. 4.3, whose one year rating transition matrix was given inTab. 1.4 and which must be rescaled to the time horizon under consideration (e.g.1 month).

2Simulation start date 2.1.2008, Simulation end date 02.01.2009

43

8

π11

π12

π13

π14

π15

π16

π17

π18

1

2

3

4

5

6

7

Fig. 4.3. Rating transitions

The Markov chain has the peculiar structure that it is possible to get from a givenstate i to each other state j with a certain probability πij with only one excep-tion: the state labelled “8” is a so-called absorbing state and once the system hasreached it, it is forced to stay there forever. In Fig. 4.3 we have sketched onlythe transitions from the state labelled “1” to all the others. Also the absorbingstate is singled out (black) for the convenience of the reader. The simulation worksas follows: at each start day of a month additional five year bonds are added tothe portfolio, whose start dates corresponds to the current simulation date. Thestructure of the added bonds is such that its rating distribution is identical to theoriginal rating distribution (this is the situation mentioned above).In Fig. 4.4 a comparison of the rating distribution of the initial and final bondportfolio is shown (02.01.2008-02.01.2009). Here the final portfolio is the re-sult of a 12 months observation period after portfolio inception. The monthlygrowth rate is 10% and the final number of bonds has more than tripled (3138).

Fig. 4.4. Development of the portfolio’s rating distribution in time.

By construction there is no major change in the percentage composition of thebond portfolio.

44

4.1.2 Bond Specification

The types of bonds contained in the portfolio are very similar. They are all five-year bonds with annual coupon payments. Basically they are characterised by thestart date and the rating. The coupons are determined such that each bond at theday of its inception is quoted at par (cf. definition 2).For the seven rating classes and the twelve inception dates (start dates) this leadsto the par coupons listed in Tab. 4.2.

Sim. date AAA AA A BBB BB B CCC

02-Jan-08 4.374 4.421 4.542 5.079 7.063 12.861 36.51501-Feb-08 3.988 4.034 4.154 4.69 6.665 12.429 35.90103-Mar-08 3.9 3.946 4.067 4.602 6.576 12.333 35.7601-Apr-08 4.169 4.216 4.337 4.873 6.856 12.644 36.24306-May-08 4.329 4.375 4.496 5.033 7.018 12.816 36.45602-Jun-08 4.631 4.678 4.799 5.338 7.331 13.155 36.93801-Jul-08 5.099 5.145 5.266 5.806 7.803 13.649 37.56501-Aug-08 4.818 4.864 4.985 5.524 7.515 13.342 37.17301-Sep-08 4.567 4.613 4.734 5.273 7.263 13.08 36.82901-Oct-08 4.583 4.629 4.75 5.287 7.272 13.074 36.76403-Nov-08 3.954 4 4.121 4.654 6.625 12.371 35.75801-Dec-08 3.302 3.348 3.468 3.999 5.961 11.669 34.82302-Jan-09 3.12 3.165 3.285 3.814 5.767 11.445 34.463

Tab. 4.2. Par coupon per rating class valid on each simulation date.

Obviously the huge coupons for sub-investment-grades reflect the risk contained ina bond with a bad credit rating. A more convenient way of rephrasing basically thesame information is to express the coupons in terms of spreads over e.g. 3-monthsEuribor. This is the spread necessary to set up hedges via swaps, whose NPV areequal to zero. This is a more convenient measure since it expresses the risk interms of the current level of market rates used for short term funding.

Sim. date AAA AA A BBB BB B CCC

02-Jan-2008 -0.047 0 0.115 0.636 2.56 8.18 31.1101-Feb-2008 -0.047 0 0.115 0.635 2.553 8.148 30.93603-Mar-2008 -0.046 0 0.116 0.636 2.555 8.149 30.91401-Apr-2008 -0.046 0 0.117 0.638 2.562 8.181 31.08806-May-2008 -0.046 0 0.116 0.637 2.563 8.189 31.12302-Jun-2008 -0.046 0 0.117 0.639 2.57 8.214 31.25901-Jul-2008 -0.046 0 0.116 0.638 2.569 8.224 31.35701-Aug-2008 -0.047 0 0.115 0.637 2.565 8.208 31.28401-Sep-2008 -0.046 0 0.116 0.639 2.568 8.207 31.22901-Oct-2008 -0.044 0 0.118 0.638 2.562 8.187 31.15203-Nov-2008 -0.048 0 0.114 0.632 2.547 8.131 30.85801-Dec-2008 -0.048 0 0.114 0.632 2.543 8.102 30.65302-Jan-2009 -0.051 0 0.11 0.626 2.528 8.061 30.486

Tab. 4.3. Swap spreads per rating class valid on each simulation date.

4.1.3 Data Generation

The generation of the time series of bond prices, swap NPVs and VaR-Maps followsthe rules detailed in chapter 1. The generation of the predictions of the defaultcontingent market exposures was explained in section 2.2. Results are stored andcan be retrieved from the database whose basic layout is given in Fig. 4.2. Theevaluations of the next section are based on these stored results.

45

4.2 Compatibility of the two Approaches

In this section we are going to apply the two techniques developed in chapter 2 andchapter 3 to the computation of risk premia associated with the different sourcesof risk. We are especially interested in the compatibility of the two approaches, i.e.to which extend a double counting of contributions to risk margins is caused by theuse of the individual and the portfolio approach in conjunction. For definiteness wefocus on the portfolio defined in Tab. 4.1. It consists of thousand bonds distributedover seven rating classes. In Tab. 4.8 we collected the five year PDs belonging toeach of the seven rating classes. Each bond has the same exposure equal to itsnotional which we will assume to be 100 EUR. This is therefore equal to theminimal loss unit and the exposure weights of all bonds are equal to one.

4.2.1 Individual Approach

Bonds

Returning to our synthetic structure consisting of a bond and a swap, we mustconsider two contributions to the cumulative loss. At first the expected loss of aR rated bond which is given by

E [ L ] = Notional · PD . (4.1)

We assumed the notional to be 100 EUR. Than the expected loss corresponds nu-merically to the PD percentage of 100 EUR, where the PD belonging to a givenrating class is given in Tab. 4.8. The unexpected loss of the bond is obtainedfrom the portfolio approach discussed later in section 4.2.2. This is because it isrelated to the properties of the full loss distribution, which in turn depends on thecorrelation between individual names.

Swaps

In addition we had to determine the loss of the hedging swap due to a possible de-fault of the associated bond. Actually these are the replacement costs of definition7 and determined according to eq. (2.13). In section 4.3.3 we will argue in favour ofthe weighting as defined in eq. (1.28) as an appropriate procedure to determine av-eraged quantities such as the exposure at default or quantiles, which are necessaryto determine the replacement costs. The backtesting considered there will showthat the replacement costs defined as the sum of the expected and unexpected lossof the hedging swap behave reasonably well.

In Tab. 4.4 we collected the replacement costs estimated over a horizon of fiveyears for the portfolio at hand anticipating the measure mentioned before. It isnoteworthy that the quantities are only significant for sub-investment-grades.

46

Rating (R) EL+UL in EUR

AAA 0.002AA 0.006A 0.019BBB 0.085BB 0.411B 1.907CCC 8.445

Tab. 4.4. Estimated sum of expected and unexpected lossesfor a five year bond with credit rating R.

4.2.2 Portfolio Approach

As indicated before the limitation of the individual approach is that it cannot takethe correlation effects in a bond portfolio into account. Therefore the main newingredient in the portfolio approach is the correlation between individual names.Our approach is to approximate the correlation matrix with a factor model. As anillustrative example we choose a one-factor-model, i.e. each name, si, reads

si = ai · Z + bi · εi , (4.2)

which corresponds to a decomposition into a systematic (Z) and an idiosyncraticrisk (εi). Z and εi i=1...N are independent random numbers (standard normals).The volatilities of Z and εi are therefore equal to one. The correlation of twonames reads

ρ(si, sj) = aiaj + bibj δij . (4.3)

Here bi is implicitly defined by a2i + b2

i = 1. The correlation matrix reads

(ρij) =

(1l aiaj

aiaj 1l

)

. (4.4)

In our example we assume all ai to be equal to 0.2 and the number of names waschosen to be N = 1000.

Using the procedure developed in chapter 3 one obtains the correlation expan-sion of the expected loss EN [ L ]. Removing a bond with rating R from the portfolioand recomputing the expected loss EN−1 [ L ] one can consider the difference,

∆(R) = EN [L ] − EN−1 [ L ] , (4.5)

which is the contribution to the total expected loss assigned to a specific bond,the so called marginal risk. For the seven rating classes we obtained the results3

collected in Tab. 4.5.3To speed up the calculation we merged always ten bonds with the same rating into one with

an exposure ten times the exposure of one of the original bonds. Then we applied the procedureas described. The marginal risk contribution of the bond we are actually interested in is than10% of the marginal risk contribution of the computed bond.

47

N 100 99 99 99 99 99 99 99

RTG AAA AA A BBB BB B CCC

δ0 2.089 2.089 2.086 2.080 2.055 1.966 1.763 1.391δ1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0δ2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0δ3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Scale 1000 1000 1000 1000 1000 1000 1000 1000

Sum 2089.40 2088.5 2086.2 2080.30 2054.80 1966.40 1763.40 1391.40

∆(R) 0.09 0.32 0.91 3.46 12.30 32.60 69.80

Tab. 4.5. The marginal risk weights, ∆(R), of bonds with rating R.

The marginal risks computed by applying this procedure are exactly equal to theexpected losses of individual bonds as obtained in subsection 4.2.1 from the no-tional of the bond and the PDs in Tab. 4.8. This is the expected behaviour, i.e.the expected portfolio loss does not depend on the correlation4. To elaborate thispoint further we generated for three different correlations determined by ai = 0,ai = 0.2 and ai = 0.5 the loss distributions with Gnu-R using the fCopulae pack-age again and display the results in Fig. 4.5

Fig. 4.5. Effect of the correlation parameter onto the shape of the lossdistribution.

Thus correlation affects the shape but not the mean of the loss distribution. Theusefulness of this property is that it makes the subsequent approach to the determi-nation of the unexpected loss compatible with the individual approach consideredbefore because both methods agree on the size of the expected loss.

Unexpected loss

Now we ought to pay attention to the computation of the marginal contribution ofa bond with rating R to the 99% quantile of the (red) loss distribution of Fig. 4.5.

4This is due to the identity E [ L ] =N∑

i=1

ci E [ 1li ] =N∑

i=1

ci pi with ci the exposure, 1li the

default indicator and pi = E [ 1li ] the marginal probability of the i-th name.

48

Obviously there is no objection for us to perform this computation. Yet thereis a problem with the example. The discrete loss distribution at hand possessesmarginal contributions of only two types (zero or a constant non zero value). Thisis due to the concentration of the interesting quantile onto basically two buckets ofthe discrete distribution. The resolution of the marginal contributions computedthis way is simply not good enough to assign a rating specific contribution to eachof the seven rating classes. Therefore we have to work around and use the standarddeviation of the loss distribution as a proxy instead even though we are aware ofthe fact, that it is not that ideal measure as in the case of a Gaussian distribution.At least the standard deviation provides us with a suitable estimate for the magni-tude of the unexpected loss. To obtain the standard deviation we have to computea higher moment of the loss distribution (e.g. E [ L2 ]). The corresponding resultsare displayed in Tab. 4.6.

N 100 99 99 99 99 99 99 99

RTG AAA AA A BBB BB B CCC

δ0 5.704 5.699 5.687 5.657 5.527 5.097 4.228 3.063δ1 0.265 0.264 0.263 0.260 0.249 0.225 0.198 0.200δ2 0.016 0.016 0.016 0.016 0.015 0.014 0.015 0.018δ3 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

Scale 106 106 106 106 106 106 106 106

Sum 5977051 5971686 5958289 5924578 5783634 5328917 4433796 3272765

Tab. 4.6. Computation of EN

[L2]

for bonds with rating R.

From Tab. 4.5 and Tab. 4.6 we obtain now the standard deviations of portfolio lossdistributions with specified bonds removed. From these quantities we build themarginal contributions by the same type of differencing already used in eq. (4.5).The results are displayed in Tab. 4.7.

Rating (R) UL in EUR

AAA 0.063AA 0.213A 0.574BBB 1.986BB 6.022B 11.868CCC 11.324

Tab. 4.7. Estimated contribution of a bond with credit rating Rto the width of the loss distribution.

We come back to the results of this computation in the discussion, which is thesubject of section 4.4. Now we turn our attention to the results of the simulation.

4.3 Portfolio Dynamics

4.3.1 Simulation Results

The specific sort of problems we are experiencing with the bond portfolio describedin section 4.1.1 stems from the simple fact that is is growing. This innocent prop-

49

erty causes a lot of troubles in determining its actual value. Here we do not refer toits market value which is easily determined but instead we refer to the losses, whichare slowly piling up due to defaults of contracts, which have started in the past.The huge volume of new deals entering each month might spoil the monitoring ofthe credit quality of the old deals.In Fig. 4.6 we display the growth of the portfolio over time. As mentioned beforethe volume has tripled during one year.

Fig. 4.6. Growth of Portfolio Value

In Fig. 4.7 we show the accumulated losses occurring during the simulation time.Obviously the losses are about two orders of magnitude smaller than the monthlyincrease of the portfolio value.

Fig. 4.7. Accumulated Losses

50

The proper measure for the intrinsic quality of the portfolio is the loss ratio whichwe define as the quotient of the accumulated losses to the current portfolio value5.In Fig. 4.8 the development of the loss ratio is shown and compared to Fig. 4.7it behaves more like a constant. This is also the expected result because we havemodelled the default process as being time homogeneous.

Fig. 4.8. Percentage Loss

In fact from the approach we have chosen to model the default process we shouldbe able to determine the loss ratio from first principles as follows: In table 4.8 wecollected the percentage composition of the portfolio and the five year probabilityof default. A rough estimate for the percentage of losses can be obtained by mul-tiplying the fraction belonging to a certain rating class by its five year probabilityof default. The result is shown in the last row below.

Weight 5Y-PD

50% 0.09%30% 0.32%10% 0.91%4% 3.46%3% 12.3%2% 32.6%1% 69.8%

Sum: 2.1%

Tab. 4.8 Computation of the Loan Loss Ratio

5To be precise, the loss ratio should be defined as the left sided limit of this quotient.

51

Per year this roughly corresponds to a loss ratio of 0.4% which is actually veryclose to the value observed in Fig. 4.8. This quantity can now be multiplied bythe volume (3138 deals of equal volume) which leads to a prediction of 13 defaults.Actually we observed 11 defaults.

4.3.2 Discussion of default process assumptions

In order to make the problem tractable we had to make a choice of the underlyingdefault process. The Markov chain approach of rating transitions has an importantdeficiency, which is related to its time homogeneity. In Fig. 4.9 the VaR of thehedged bond portfolio belonging to the pure market risk of changing level of in-terest rates is shown for the complete year 2008. Here the solid blue line indicatesthe Value at Risk for a horizon of one day, i.e. the estimated 99% percent quantileof the distribution of daily P&L changes of the portfolio at hand.

Fig. 4.9. Portfolio VaR during 2008 in EUR

It can be observed that several outliers suddenly occurred in October 2008. On the15th of September 2008 the bank “Lehman Brothers” went bust and the centralbanks started with drastic measures to provide liquidity in the markets. Thesehad their effects on the volatilities and correlations of the RiskMetrics nodes witha small time delay. But the original event on the 15th of September which wasmuch more severe is not visible in Fig. 4.9. The reason for this is that the spreadsas determined by the time homogeneous Markov chain are idealised. In real bondportfolios the spreads reflect the current, i.e. changing opinion on the default riskand a premium for the illiquidity of a bond (cf. credit spread models in section 1.3).This latter spread component was very important about that time, too. The com-plexity of the spread dynamics is not captured by the time homogeneous Markovchain model. A practical approach to overcome these problems by consideringhistorical time series of bond spreads was presented in [44].

4.3.3 Backtesting the Predictions of Replacement Costs

In this section we want to discuss the issue of applying the EPE-methodology asa tool to estimate the future replacement cost of a swap due to a default of theassociated bond. The most conservative estimate is based on the so called peakquantile. The peak quantile is defined as the maximum of the quantile profile overthe lifetime of a swap as determined by eq. (A.14). The quantile profile has asimilar shape to the EPE-profile discussed in subsection 2.2. Both are displayed in

52

Fig. 2.7. In Fig. 4.10 we compare the actual swap market exposures as observedat the default times of the associated bonds with the sum of this market exposureand the peak of the 99% quantile-profile of the EAD. While the relevant swapexposure is always lower or equal to zero its sum with its peak quantile used as aproxy for the swap’s market exposures should be distributed around zero (as closeas possible). Yet, that is not the case if we use the peak quantile. Actually the sumis very positive. This is not an inconsistency of the model and a premium basedon this quantity is certainly on the safe side. But at the same time the estimatedreplacement costs are grossly overpriced.

Fig. 4.10. Backtesting Predictions of Replacement costs (Quantile)

An additional problem can be illustrated by refering to the situation displayedin Fig. 4.10. For a bond with a bad credit quality the peak exposure can be reallylarge. Otherwise it is more likely to default soon and the swap had not enoughtime to deviate much from par (cf. the first two bars in Fig. 4.10). So it is notadvisable to use the peak quantile for bonds with bad credit quality, since it isattained only after a significant amount of time of its ageing (cf. comments inthe example of section 2.2). This was one of the reasons that the concept of theeffective (positive) exposure was introduced, which tries to take take the ageingeffect into account [27]. There a time horizon of one year was selected. In oursituation this concept is no improvement, yet. This is because we consider thewhole lifetime of the swap and not a specific time horizon. Considering the wholelifetime eliminates the distinction of peak and effective (positive) exposure.

A not quite original bon mot states that a prognosis becomes more difficult assoon as it tries to predict the future. This is summarising the fact that our approachis using an additional implicit assumption. We assume that the market behavesmore or less time homogeneous. Otherwise today’s predictions of the future haveno reason to become true on average. This is certainly not the case under stressedconditions like after the Lehman Brothers event where this approach becomes

53

dubious. Thus the assumption of time homogeneity comprises the behaviour ofmarket data and the behaviour of the default process. We could chose the defaultprocess to be time homogeneous. Therefore the model is at least self consistentin this respect. But the behaviour of the market data are beyond our designcapabilities.

We developed in section 2.2 our modification of the standard approach to thedetermination of the rating weighted EAD. Comparing now the observed swapmarket exposures as they occurred with the predictions from the modified model,we obtain a much better behaviour. This is displayed in Fig. 4.11.

Fig. 4.11. Backtesting Predictions of Replacement costs (Weighted Exposure)

From the data plotted in Fig. 4.11 we derive the following numerical example:summing up the observed losses due to open market positions of hedging swapsdue to defaulted bonds, one obtains a total market exposure of -75 EUR. Addingto this value the estimated risk premiums the sum is equal to 11 EUR. This is“close” to zero and a good result for such a simple model.

The distribution of the green bars confirms the message of the numerical ex-ample, too.

Stability of the EPE-profile

In order to judge the stability of the prediction for the EAD in the future weconsider its variation over a whole month. We have selected the time intervalstarting on the 1st of September and ending on the 6th of October. This choicewas made due to the following two reasons:

• First, according to table 4.2 the par coupons during that period are reason-ably constant and

• second, this period includes the Lehman Brothers event and this could havehad an effect on the view of the market about future level of interest rates.

54

For the example we use the mean of the two coupons at the beginning of Septemberand October, i.e. a coupon of 4.621%. The rating of the bond is AA and thereforethe spread vanishes according to table 4.3. We compare the EPE-profiles at thefollowing dates: 01.09.2008, 15.09.2008, 01.10.2008 and 06.10.2008. In Fig. 4.12the corresponding EPE-profiles are displayed.

Fig. 4.12. Development of EPE-Profile with Time

It is interesting to see that during the interval starting on Sept., 1st and endingon Oct., 1st the EPE-profile first falls to a minimum on Sept., 15th and then soonrecovers to nearly the old level on Oct., 1st. The variation itself is not large enoughto be genuinely worried about the stability of the predicted average of the profile.On the other hand, there is a huge difference between the two profiles belongingto Oct, 1st and Oct, 6th (the business day after Oct, 1st). This is due to a massiveshift in the level of interest rates and an increased level of swaption volatilities6. Sothe market opinion about the future exposure may change fast and sharply. Thisunderlines once more the importance of the fact that modelling of the swaps’ futuremarket exposure must be taken seriously into account in order to make meaningfulpredictions about the future replacement costs.

4.4 Summary

From the three sorts of risks encountered so far the market risk is by far the mostimportant one. This is mainly due to the fact that it is purely due to exogenousreasons and therefore independent of the idiosyncratic properties of the contractsforming the portfolio. Thus all contracts are exposed to the market risk in an equalway. Opposite to that are the risks, which are related to the idiosyncratic properties

6On Oct., 8th the ECB decided to cut interest rates by 50bp. This was the first step. Otherfollowed in each of the remaining months of 2008. The market always anticipated the measuressome days in advance (cf. Tab. 2.1).

55

of names. They were the subject of the two chapters on the individual and theportfolio approach. Due to the compatibility of the individual and the portfolioapproach which was based on the fact, that the expected loss of the portfolio lossdistribution did not depend on the correlation itself, the individual and the portfolioapproach focused onto two separated aspects of the idiosyncratic risk associatedwith a portfolio of bonds and hedging swaps. The correlation of bonds affectsthe shape of the loss distribution but not its mean. This implies a correlationdependence of the unexpected loss caused by defaulting bonds, only. We haveestimated the size of these effects in section 4.2 and it was shown by an examplethat both are of the same size. The example made use of the decomposition ofthe correlation matrix into factors as advocated in section 3.2.2. The results ofthe estimations of the credit VaR for a time horizon of five years was shown inTab. 4.7. It turned out to be more severe than the matching replacement costs inthe following sense: for bonds with a good credit quality the credit risk dominatesthe replacement costs.

We were especially interested in the inclusion of the replacement costs intothe full picture of determining the risk in the bond portfolio. Independent of theopinion one might develop with respect to the proposed method of determiningthe replacement costs one should realise that they form a significant fraction of theP&L and deserve a dedicated study. In particular the discussion of Fig. 4.12 pointsinto a direction that jump diffusion could play a role in more advanced models ofreplacement costs.

A more advanced investigation should also address the problem of default re-covery. By this we mean the problem, that a technical default triggers certainP&L effects (e.g. loan loss provision). Yet, the defaulted entity often recovers. Sothe true value of the defaulted quantity is only known within a stochastic timelag. Analysing the performance of a portfolio under inclusion of these effects isstill more complicated [45]. Indeed the test system is designed to include these ef-fects into the simulation, too. But the intriguing additional problems in separatingthe effects from each other is beyond the time schedule of this work and will beaddressed in a future publication [46].

56

Appendix A

Quantiles of Positive Exposure

A.1 Statistics of the Positive Exposure

A.1.1 Variance of PE

We assume that y(T ) is log-normally distributed, i.e.

y(T ) = y · exp

(

−1

2σ2 T + σ

√T x

)

, (A.1)

with x ∈ N(0, 1).

The variance of the positive exposure is defined as usually and reads

V ( PE ) = E[PE2

]− E [ PE ]2 . (A.2)

For definiteness we consider the exposure profile of a payer swap. Since the expec-tation value of the positive exposure is given by the corresponding compoundedeuropean swaption value, the part which still must be computed is

E[PE2

]=

∞∫

−∞

δ(T )2 · max ( y(T ) − K, 0 )2 1√2π

e−x2

2 dx . (A.3)

The support of the integral is determined by

δ(T ) · ( y(T ) − K ) = 0 ⇔ x = − ln yK

− 12σ2 T

σ√

T= − d2. (A.4)

Therefore we are left with the integral

E[PE2

]=

δ2(T )√2π

∞∫

−d2

( y(T ) − K )2 e−x2

2 dx . (A.5)

57

which turns out to be equal to

E[PE2

]= δ2(T )

[

y2 eσ2 T Φ( ω [ d1 + σ√

T ] ) − 2 K y Φ(ω d1) + K2 Φ(ω d2)]

.

(A.6)

In the final result we have inserted the parameter ω which toggles between theresults for payer (+1) and receiver (-1) swaptions.

A.1.2 Quantile of the distribution of positive exposures

As we said in subsection A.1.1 the swap rate is assumed to follow a log-normaldistribution (cf. eq. (A.1) ). The α-quantile, Qα, of the distribution of positiveexposures is determined by the defining equation α = P [ PE(T) < Qα ]. Thisequation reads

α =1√2π

∞∫

−∞

1lPE(T) < Qα e−x2

2 dx . (A.7)

The domain in which the indicator function is non-vanishing corresponds to thosex which satisfy

PE(T) < Qα ⇔ δ(T ) · ( y(T ) − K )+ < Qα . (A.8)

What seems to make the evaluation difficult is how to treat the positive partfunction. So let us first compute the quantile, Qα, of the exposure distribution byskipping the positive part function.

α = P

[

E(T) < Qα

]

=1√2π

∞∫

−∞

1lE(T) < Qα e−x2

2 dx . (A.9)

The indicator function is non-vanishing, iff

E(T) < Qα ⇔ x <ln K δ + Qα

y δ+ 1

2σ2 T

σ√

T

!= x+ . (A.10)

Therefore eq. (A.9) reads

α = P [ E(T) < Qα ] =1√2π

x+∫

−∞

e−x2

2 dx . (A.11)

From this representation one concludes that x+ is the α-quantile of the standardnormal distribution:

x+ = Qα(0, 1) . (A.12)

58

Combining the last formula (A.12) with the definition of x+ in eq. (A.10) gives thefollowing result:

Qα = δ(T )(

y · e− 12

σ2 T +Qα(0,1) σ√

T − K)

. (A.13)

Returning now to the original problem given in eq. (A.7) we just note that thequantile we are looking for is given by taking the positive part of the quantilecomputed in eq. (A.13)

Qα = Q+α . (A.14)

If one thinks of this equation in terms of a histogram, then by taking the positivepart all buckets on the negative line are merged into one bucket centred aroundzero. But then the quantiles of both distributions coincide as soon as they arelarger than zero.

59

Appendix B

Covariance

B.1 Calculation of Volatilities and Correlation

The volatilities and correlations of the RiskMetrics nodes are the volatilities andcorrelations computed with respect to time series of historically observed log-retunsof discount factors or zero bond prices belonging to the RiskMetrics nodes. Thediscount factor associated with an annual compounded zero rate r(ℓ) belonging tothe maturity t(ℓ) is given by

Dt(ℓ) =1

( 1 + r(ℓ) )t(ℓ)

. (B.1)

By ℓ we are going to label the ℓ-th RiskMetrics node. The discount factors areused to build the statistics of the daily changes of logarithmic returns. The use oflog-returns has the additional advantage that a possible drift contained in the timeseries is automatically removed1. For a time series xii=0...N−1 the exponentiallyweighted average is defined by

E [x ] =1 − λ

1 − λN

N−1∑

i=0

λi xi . (B.2)

We will work with a λ of 0.94.The computation of the (price-)volatilities can be slightly simplified by observ-

ing that from eq. (B.1) one obtains

d log Dt(ℓ) =−t(ℓ) r(ℓ)

1 + r(ℓ)d log r(ℓ) (B.3)

and therefore

σ(ℓ)Price Vola =

t(ℓ) r(ℓ)

1 + r(ℓ)σ

(ℓ)Yield Vola . (B.4)

The yield volatility is more directly accessible from the values of the rates of the

zero curve r(ℓ)i i=0...N−1, which is usually the output of the bootstrapping of a zero

1Compare chapter 19 of [47].

60

curve from benchmark quotes2. The yield vola in turn is now obtained from the

time series of log returns x(ℓ)i = ln

r(ℓ)i

r(ℓ)i+1

i=0...N−1 and since the drift is already

removed we may set in the formula for the volatility the expectation value of thelog returns to zero. Thus the volatility of the ℓ-th RiskMetrics node is given by

σ(ℓ) =

√√√√ 1 − λ

1 − λN

N−1∑

i=0

λi x(ℓ)i x

(ℓ)i . (B.5)

Using eq. (B.4) one obtains the price volatility from the yield volatility computedabove.

The correlation of the discount factors is obtained analogously. But this timethe log returns must be actually computed on the time series of discount factors.Denoting by x(ℓ) the corresponding time series, the correlation is obtained by

ρℓℓ =E

[

x(ℓ) x(ℓ)]

σ(ℓ) σ(ℓ). (B.6)

In this expression we again neglected the terms based on the product of expectationvalues of discount factors by the same argument used before.

2The bootstrapping was performed using the QuantLib library.

61

Appendix C

Code

C.1 Loss Distributionfunction [ p ] = LossDistribution(pdmarg,weights)

%

% Author: A. Miemiec

%

% Description:

%

% Builds the loss distribution, p, of an independent obligor model according

% to the algorithm in Anderson, Sidenius, Basu "All your hedges in one

% basket" (2003).

% The input are the marginal probabilities, pdmarg, of each obligor, all of which

% are assumed to be independent and the exposure weight, weights, of each

% engagement.

%

NBR_ITEMS = length(pdmarg);

MAX_LOSS = sum(weights);

p = zeros(MAX_LOSS+1,1);

p(1) = 1; % 0 obligors

% add next defaultable item

for newItem=1:NBR_ITEMS

MAX_LOSS = sum(weights(1:newItem));

for k=MAX_LOSS:-1:0

if( k >= weights(newItem) )

p(k+1) = p(k+1)*(1-pdmarg(newItem))+ p(k-weights(newItem)+1)*pdmarg(newItem);

else

p(k+1) = p(k+1)*(1-pdmarg(newItem));

end

end

end

end

C.2 EV.mfunction [ res ] = EV( func, pd )

%

% Author: A. Miemiec

%

% Description:

%

% Computes for a given function and a given discrete

% probability distribution the expectation value

%

res = sum(func.*pd’);

end

62

C.3 HermiteB.mfunction [y] = HermiteB(x,n)

%

% Author: A. Miemiec

%

% Description:

%

% Computes the function exp(-x*x/2)/sqrt(2*pi)*He_n-1(x),

% where He_k(x) = (-1)^n*exp(x*x/2)*d^k/dx^k exp(-x*x/2)

% is the Hermite Polynomial with coefficient of the leading

% power equal to 1, which satisfies the following recursion

%

% He_k+1 = x*He_k - k*He_k-1;

%

y = zeros(1+n,1);

% a special case

y(1+0) = (1-erf(x/sqrt(2)))/2;

if(n==0) return; end

% generic case

Bn_2 = 1;

y(1+1) = Bn_2*exp(-x*x/2)/sqrt(2*pi);

if(n==1) return; end

Bn_1 = x;

y(1+2) = Bn_1*exp(-x*x/2)/sqrt(2*pi);

if(n==2) return; end

for i=3:n

Bn = x*Bn_1 - ((i-1)-1)*Bn_2;

Bn_2 = Bn_1;

Bn_1 = Bn;

y(1+i) = Bn*exp(-x*x/2)/sqrt(2*pi);

end

end

C.4 PDtwiddle.mfunction [ pdtwiddle ] = PDtwiddle(p,J,A,s)

%

% Author: A. Miemiec

%

% Description:

%

% Function, which computes for a point J in a dim-dimensional

% cone from

%

% - the given probabilities p of an independend obligor

% model and

% - the correlation expressed through the loading matrix A and

% - the value of the parameter s1=..=sn ! \over = s

%

% a value of the perturbed probabilities ptwiddle, which is then

% used to compute expectation values in a modified independent

% obligor model.

%

% \beginrem should be computed only once

dim = length(J);

Obligors = length(p);

B = zeros(Obligors,dim+1);

63

for i=1:Obligors

% determine default boundaries

nu = fzero((@(x)(0.5-0.5*erf(x/sqrt(2))-p(i))),0.02);

% determine value of HermiteB-functions at the boundaries

B(i,:) = HermiteB(nu,dim)’;

end

% \endrem

pdtwiddle = B(:,1);

% \beginrem computation of perturbed probabilities

for i=1:Obligors

for k=1:dim

v_J_i_k = V(J,A,i,k,s);

pdtwiddle(i) = pdtwiddle(i) + B(i,1+k)*v_J_i_k;

end

end

% \endrem

end

C.5 V.mfunction [ y ] = V(J,A,i,k,s)

%

% Author: A. Miemiec

%

% Description:

%

% Recursion, which determines the contribution the correlation

% to the expression of perturbed probabilities as given in the

% paper Glasserman, Suchintabandid: "Correlation expansion ..."

%

v_J_i_k = 0; %default

dim = length(J);

j = J(dim);

% Loading matrix

if( j < 0 )

a_i_j = -A(i,-j);

elseif( j == 0 )

a_i_j = 0;

else

a_i_j = A(i,j);

end

% recursion

if( dim > 1 )

if( ( k < 0 ) || ( k > dim ) )

v_J_i_k = 0;

else

v_J_i_k = V(J(1:(dim-1)),A,i,k,s)+s*a_i_j*V(J(1:(dim-1)),A,i,k-1,s);

end

else

%recursion boundary

v_J_i_k = 0; %default

if( k==0 )

v_J_i_k = 1;

end

if( k==1 )

v_J_i_k = s*a_i_j;

end

end

y = v_J_i_k;

end

64

C.6 GenerateAllJ.mfunction [ J, dim ] = GenerateAllJ( n,d )

%

% Author: A. Miemiec

%

%

% Description:

%

% Given a number ob oligors, n, and a number of factors, d,

% generates all possible (j_1<=...<=j_n) in J

%

dim = factorial(n+2*d)/factorial(2*d)/factorial(n);

J = zeros(dim,n);

%set initial values vor (j_1,...,j_n) in the sense

%of the given order j_1<=...<=j_n

J(1,:) = 0;

%increment previous value (j_1,...,j_n) by one in the sense

%of the given order j_1<=...<=j_n

for i=2:dim

J(i,:) = increment(J(i-1,:),d);

end

J = J - d;

end

function [J_i] = increment(J_i_1,d)

%

%

% Description:

%

% Increments the multi index (j_1<=...<=j_n) by 1

% while preserving the order j_1<=...<=j_n

%

n =length(J_i_1);

J_i = zeros(n,1);

number = 0;

% encode

for j=n:-1:1

number = number + J_i_1(j)*(2*d+1)^(n-j);

end

while true

% increment

number = number + 1;

% decode

difference = 0;

for j=n:-1:1

J_i(j) = rem((number-difference)/(2*d+1)^(n-j),2*d+1);

difference = difference + J_i(j)*(2*d+1)^(n-j);

end

% check if index order is valid

if( valid(J_i) )

break;

end

end

end

function [flag] = valid(J_i)

%

%

% Description:

%

% Tests for the order j_1<=...<=j_n

%

flag = true;

n = length(J_i);

for k=2:n

if ( (J_i(k-1) > J_i(k)) )

flag = false;

return;

end

end

end

65

C.7 GenerateAllD.mfunction [ D, dim ] = GenerateAllD( n,d )

%

% Author: A. Miemiec

%

%

% Description:

%

% Given a number ob oligors, n, and a number of factors, d,

% generates all points (j_1,..,j_n) of D^n

%

dim = (2*d+1)^n;

D = zeros(dim,n);

%set initial values vor (j_1,...,j_n)

D(1,:) = 0;

%increment previous value (j_1,...,j_n) by one

for i=2:dim

D(i,:) = increment(D(i-1,:),d);

end

D = D - d;

end

function [D_i] = increment(D_i_1,d)

%

%

% Description:

%

% Increments the multi index (j_1,...,j_n) by 1

%

n =length(D_i_1);

D_i = zeros(n,1);

number = 0;

% encode

for j=n:-1:1

number = number + D_i_1(j)*(2*d+1)^(n-j);

end

% increment

number = number + 1;

% decode

difference = 0;

for j=n:-1:1

D_i(j) = rem((number-difference)/(2*d+1)^(n-j),2*d+1);

difference = difference + D_i(j)*(2*d+1)^(n-j);

end

end

C.8 MyFunction.mfunction [ y ] = MyFunction( x )

y = max(x-200,0);

%y=x;

end

66

C.9 CDOTest.mfunction [coeff] = CDOTest()

%

% Author: A. Miemiec

%

%

%

clear all;

format long;

NBR_OBLIGORS = 50 % number of obligors

NBR_FACTORS = 5 % number of factors in the approx of the correlation

DIM_EXPANSION = 2 % order of the correlation expansion

EPSILON = 0.1 % the expansion parameter s

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %

% Definition of exposures and marginal pd’s %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Exposures=zeros(NBR_OBLIGORS,1);

pd = zeros(NBR_OBLIGORS,1);

for i=1:NBR_OBLIGORS

Exposures(i)=i;

pd(i)=0.02;

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %

% Definition of the correlation structure %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A=LoadingMatrix(NBR_OBLIGORS,NBR_FACTORS);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %

% Definition of possible losses %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Loss=0:sum(Exposures);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %

% Minimal set of expextation values, which %

% must be computed i order to perform the %

% correlation expansion up to the specified %

% order %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[J,NbrOfJ] = GenerateAllJ(DIM_EXPANSION,NBR_FACTORS);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %

% Precompute all expectation values up to the %

% given order %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

ExpectationValues = zeros(NbrOfJ,1);

for j=1:NbrOfJ

pd_twiddle = PDtwiddle(pd,J(j,:),A,EPSILON);

PD_Loss = LossDistribution(pd_twiddle,Exposures);

ExpectationValues(j) = EV(MyFunction(Loss),PD_Loss);

end

67

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %

% Computes all coefficients in the %

% correlation expansion up to the %

% given order %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

coeff = zeros(DIM_EXPANSION,1);

% order 0

PD_Loss = LossDistribution(pd,Exposures);

coeff(1) = EV(MyFunction(Loss),PD_Loss);

% orders > 0

for order=1:DIM_EXPANSION

% all D^l for the correct order of expansion

[D,NbrOfD] = GenerateAllD(order,NBR_FACTORS);

coeff_order = 0;

for j=1:NbrOfD

index = MapDToJ(D(j,:),J,DIM_EXPANSION,NBR_FACTORS);

lambda = 1; %default

for k=1:order

if( D(j,k) == 0 )

lambda = lambda * (-2*NBR_FACTORS);

end

end

coeff_order = coeff_order + lambda*ExpectationValues(index)/(2*EPSILON*EPSILON)^order;

end

coeff(1+order) = coeff_order;

end

% exit

format short;

end

68

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