An Integrated Market and Credit Risk Portfolio Model

8/2/2019 An Integrated Market and Credit Risk Portfolio Model

1/18

21ALGO RESEARCH QUARTERLY VOL. 2, NO. 3 SEPTEMBER 1999

Credit risk modeling is one of the most importanttopics in risk management and finance today.The last decade has seen the development ofmodels for pricing credit risky instruments andderivatives, for assessing the credit worthiness of

obligors, for managing exposures of derivativesand for computing portfolio credit losses forbonds and loan portfolios. In light of thesefinancial innovations and modeling advances theBasle Committee on Banking Supervision hastaken the first steps to amend current regulationand is reviewing the applicability of internalcredit risk models for regulatory capital (BasleCommittee on Banking Supervision, 1999a,1999b).

However, common practice still treats marketand credit risk separately. When measuringmarket risk, credit risk is commonly not takeninto account; when measuring portfolio creditrisk, the market is assumed to be constant. Thetwo risks are then added in ad hoc ways,resulting in an incomplete picture of risk.

There are two categories of credit riskmeasurement models: Counterparty Credit

Exposure models and Portfolio Credit Riskmodels.

Derivative desks traditionally manage credit riskby monitoring and placing limits on counterparty

credit exposures. Counterparty exposure is theeconomic loss that will be incurred on alloutstanding transactions if a counterpartydefaults, unadjusted by possible future recoveries.Counterparty exposure models measure andaggregate the exposures of all transactions with agiven counterparty. In the BIS regulatory model,potential exposures are given by an add-on factormultiplying the notional of each transaction(Basle Committee on Banking Supervision,1988). Although simple to implement, the modelhas been widely criticized because it does notaccurately account for future exposures. Sinceexposures of derivatives such as swaps depend onthe level of the market when default occurs,models must capture not only the actualexposure to a counterparty at the time of theanalysis but also its potential future changes.Recently, more advanced methods based onMonte Carlo simulation (Aziz and Charupat1998) have been implemented by financial

An Integrated Market and

Credit Risk Portfolio ModelIan Iscoe, Alex Kreinin and Dan Rosen

We present a multi-step model to measure portfolio credit risk that integrates

exposure simulation and portfolio credit risk techniques. Thus, it overcomes the

major limitation currently shared by portfolio models with derivatives.

Specifically, the model is an improvement over current portfolio credit risk models

in three main aspects. First, it defines explicitly the joint evolution of market

factors and credit drivers over time. Second, it models directly stochastic

exposures through simulation, as in counterparty credit exposure models. Finally,

it extends the Merton model of default to multiple steps. The model is

computationally efficient because it combines a Mark-to-Future framework of

counterparty exposures and a conditional default probability framework.


2/18

22ALGO RESEARCH QUARTERLY SEPTEMBER 1999

Market and credit model

institutions. By simulating counterpartyportfolios through time over a wide range ofscenarios, these models explicitly capture thecontingency of the market on derivativeportfolios and credit risk. Furthermore, they canaccurately model natural offsets, netting,

collateral and various mitigation techniques usedin practice.

Since their main focus is on risk at thecounterparty level, counterparty credit riskmodels do not generally attempt to captureportfolio effects such as the correlation betweencounterparty defaults. In contrast, PortfolioCredit Risk (PCR) models measure creditcapital and are specifically designed to captureportfolio effects, specifically obligor correlations.

They include CreditMetrics (JP Morgan 1997),CreditRisk+ (Credit Suisse Financial Products1997), Credit Portfolio View (Wilson 1997a and1997b) and KMVs Portfolio Manager (Kealhofer1996). Although superficially they appear quitedifferentthe models differ in theirdistributional assumptions, restrictions,calibration and solutionGordy (1998) andKoyluoglu and Hickman (1998) show anunderlying mathematical equivalence amongthese models. However, empirical work showsgenerally that all PCR models yield similar results

if the input data is consistent (Crouhy and Mark1998; Gordy 1998).

A major limitation of all current PCR models isthe assumption that market risk factors, such asinterest rates, are deterministic. Hence, they donot account for stochastic exposures. While thisassumption has less consequence for portfolios ofloans or floating rate instruments, it has greatimpact on derivatives such as swaps and options.Ultimately, a comprehensive framework requiresthe full integration of market and credit risk.

In this paper, we present a multi-step, stochasticmodel to measure portfolio credit risk thatintegrates exposure simulation and portfoliocredit risk methods. Through the explicitmodeling of stochastic exposures, the modelovercomes the major limitation currently sharedby portfolio models in accounting for theexposure caused by instruments with embedded

derivatives. By combining a Mark-to-Futureframework of counterparty exposures (see Azizand Charupat 1998) and a conditional defaultprobability framework (see Gordy 1998;Koyluoglu and Hickman 1998; Finger 1999), weminimize the number of scenarios where

expensive portfolio valuations are calculated, andcan apply advanced Monte Carlo or analyticaltechniques that take advantage of the problemstructure.

We restrict this paper to a default mode model;that is, the model measures credit losses arisingexclusively from the event of default. However,default mode models cannot account for dealsthat have direct contingency on migrations (e.g.,credit trigger features) without furthermodifications. Although perhaps

computationally intensive, it is not difficult toextend the model to account for migration losses.Note, however, that since credit migrations areactually changes in expectations of futuredefaults, a multi-step model captures migrationlosses indirectly.

Specifically, the model presented in this paper isan improvement over current portfolio models inthree main aspects:

First, it defines explicitly the joint evolutionof market risk factors and credit drivers.Market factors drive the prices of securitiesand credit drivers are non-idiosyncraticfactors that drive the credit worthiness ofobligors in the portfolio. Factors are generaland can be microeconomic, macroeconomic,economic and financial.

Second, it models directly stochasticexposures through simulation, as do theCounterparty Credit Exposure models. In

this sense, it constitutes an integration ofcounterparty exposure and Portfolio CreditRisk models.

Finally, it extends the Merton model ofdefault (1974), as used, for example, inCreditMetrics, to multiple steps. It explicitlysolves for multi-step thresholds andconditional default probabilities in a generalsimulation setting.


3/18



The rest of the paper is organized as follows. Webegin by introducing a general framework forPortfolio Credit Risk Models. The framework isfirst illustrated through the commonly knownsingle-step model with deterministic exposures.Next, we present the multi-step, stochastic

model in two stages. First, we extend the singlestep model with deterministic exposures toaccount for stochastic exposures, and second, weextend that model to a multi-step version. Thepaper closes with some concluding remarks andoutline of future work.

Framework for Portfolio Credit Risk models

Current portfolio models fit within a generalizedunderlying modeling framework. Gordy (1998)and Koyluoglu and Hickman (1998) first

introduced the framework to facilitate thecomparison between the various models. Finger(1999) further points out that formulating themodels in this framework permits the use ofpowerful numerical tools known in probabilitythat can improve computational performance bydramatically reducing the number of scenariosrequired. The main idea behind the framework isthat conditional on a scenario all defaults andrating changes are independent. A state-of-the-world is a complete specification at a point in

time of the relevant economic and financialcredit drivers and market factors(macroeconomic, microeconomic, financial,industrial, etc.) that drive the model. A scenariois defined by a set of states-of-the-world overtime. In a single-period model there is a directcorrespondence between a state-of-the-worldand a scenario; in a multi-period model ascenario corresponds to a path of states-of-the-world over time.

In this section, we introduce the basic

components of the framework, which wesubsequently use to present various models. Wemake several steps explicit in the framework,which were previously implicit in the originalpresentations. This further specification permitsus to present the models in a manner that betterexplains the assumptions made and allows us toaddress the generalizations of the model.

The frameworkconsists of five parts:

Part 1: Risk factors and scenarios. This is amodel of the evolution of the relevantsystemic risk factors over the analysis period.These factors may include both credit drivers

and market factors.

Part 2: Joint default model. Default andmigration probabilities vary as a result ofchanging economic conditions. An obligorsprobabilities are conditioned on the scenarioat each point in time. The relationshipbetween its conditional probabilities and thescenario is obtained through an intermediatevariable, called the obligors creditworthiness index. Correlations amongobligors are determined by the joint variation

of conditional probabilities across scenarios.

Part 3: Obligor exposures, recoveries andlosses in a scenario. The amount that will belost if a credit event occurs (default ormigration) as well as potential recoveries arecomputed under each scenario. Based on thelevel of the market factors in a scenario ateach point in time, Mark-to-Future (MtF)exposures for each counterparty are obtainedaccounting for netting, mitigation andcollateral. Similarly, recovery rates in the

event of default can be state dependent.

Part 4: Conditional portfolio lossdistribution in a scenario. Conditional upona scenario, obligor defaults are independent.Various techniques based on the property ofindependence of obligor defaults can beapplied to obtain the conditional portfolioloss distribution.

Part 5: Aggregation of losses in allscenarios. Finally, the unconditional

distribution of portfolio credit losses isobtained by averaging the conditional lossdistributions over all possible scenarios.

We illustrate the framework with a single timestep Portfolio Credit Risk model withdeterministic exposures, PCR_SD. Commonnotation and key concepts are also introduced.


4/18



Next, the model is extended to allow forstochastic exposures in a single-step setting,PCR_SS. Finally, we present a third model,PCR_MS, which allows for multiple time stepsand stochastic exposures.

A set of four tables (Appendix 1) summarize thefeatures of the models and highlight thesimilarities and differences of the modelspresented here. Table A1 presents a summary ofthe features of the three PCR models. Table A2summarizes definitions of the risk factors andscenarios in Part 1 of the framework. Table A3summarizes the components of the joint defaultmodel of Part 2. Table A4 summarizes thecalculations for conditional obligor losses,conditional portfolio losses and unconditionallosses of Parts 3 to 5 of the framework.

PCR_SD - Single-step with deterministicexposures

The first model, PCR_SD, measures single-stepportfolio credit losses with deterministic obligorexposures and recovery rates. This is a two-stateform of the CreditMetrics model. We consider adefault mode model, where default is driven by aMerton model.

Consider a portfolio withN obligors or accounts.

Each obligor belongs to one ofNs < N sectors.We assume that obligors in a sector arestatistically identical. The grouping of obligorsinto sectors facilitates the estimation andsolution of the problem.

Part 1. Risk factors and scenarios

Consider the single period [t0, t] where, generally,t = 1 year. In this single period model a scenariocorresponds to a state-of-the-world. At the endof the horizon, t, the scenario is defined by qc

systemic factors, the credit drivers, whichinfluence the credit worthiness of the obligors inthe portfolio.

Denote byx(t) the vector of factor returns attime t; i.e.,x(t) has componentsxk(t) = ln{rk(t)/rk(t0)}, where rk(t) is the value ofthe k-th factor at time t. Assume that at thehorizon the returns are normally distributed:x(t) ~ N(, Q), where is a vector of mean

returns and Q is a covariance matrix. Denote byZ(t), the vector of normalized factor returns; i.e.,Zk(t) = (xk(t) k) / k. For ease of exposition,and without loss of generality, assume that thefactor returns are independent; independentfactors can always be obtained, for example, by

applying Principal Component Analysis to theoriginal economic factors.

Part 2. Joint default model

The joint default model consists of threecomponents. First, the definition ofunconditional default probabilities. Second, thedefinition of a credit worthiness index for eachobligor and the estimation of a multi-factormodel that links the index to the credit drivers.Finally, a model of obligor default, which links

the credit worthiness index to the probabilities ofdefault, is used to obtain conditional defaultprobabilities. Below, we explain thesecomponents in more detail.

Denote by j the time of default of obligorj, andbypj(t) its unconditional probability of default,the probability of default of an obligor in sectorjby time t:

(1)

Note that all obligors in sectorj have the sameunconditional probability of default. We assumethat unconditional probabilities for each sectorare available from an internal model or from anexternal agency.

The credit worthiness index, Yj, of obligorjdetermines the credit worthiness or financialhealth of that obligor at time t. Whether anobligor is in default can be determined byconsidering the value of its index. We assume

that Yj, a continuous variable, is related to thecredit driversthrough a linear, multi-factormodel:

(2)

where

pj t( ) Pr j t{ }=

Yj

t( ) jk

Zk

t( ) j

j

+

k 1=

qc

=


5/18



is the volatility of the idiosyncratic componentassociated with sectorj, jk is the sensitivity of

the index of obligorj to the k-th factor and j,j = 1, 2,...,N, are independent and identicallydistributed standard normal variables. Thus, thefirst term on the right side of Equation 2 is thesystemic component of the index while thesecond term is the specific, or idiosyncratic,component. Note that the distribution of theindex is standard normal; it has zero mean andunit variance.

Since all obligors in a sector are statisticallyidentical, obligors in a given sector share the

same multi-factor model. However, while allobligors in a sector share the same jk and j,each has its own idiosyncratic, uncorrelatedcomponent, j.

The conditional probability of default of anobligor in sectorj, pj(t; Z), is the probability thatan obligor in sectorj defaults at time t,conditional on scenario Z:

(3)

The estimation of conditional probabilitiesrequires a conditional default model whichdescribes the functional relationship between thecredit worthiness index Yj (and hence thesystemic factors) and the default probabilitiespj.

We assume that default is driven by a Mertonmodel (Merton 1974). In the Merton model(Figure 1), default occurs when the assets of thefirm fall below a given boundary or threshold,generally given by its liabilities. We consider thatan obligor defaults when its credit worthiness

index, Yj, falls below a pre-specified thresholdestimated from historical data. In this setting, anobligors credit worthiness index, Yj, can beinterpreted as the standardized return of its assetlevels. Default occurs when this index falls belowj, the unconditional default threshold.

From an econometrics perspective, the Mertonmodel is referred to as a probit model. It isconceptually straightforward to substitute a

different default model, such as a logit model, as

presented in Wilson (1997a, 1997b).

Figure 1: Merton model of default

The first step in the model defines the

unconditional default threshold, j, for eachobligor. The second step calculates the

conditional default probabilities in each scenario.

The unconditional default probability of obligorj

is given by

(4)

where denotes the normal cumulative densityfunction. For simplicity, we have dropped the

dependence on time, t, from the notation.Thus,

the unconditional threshold, j, is obtained bythe inverse of Equation 4:

(5)

The conditional probability of default is then the

probability that the credit worthiness index fallsbelow the threshold in a given scenario:

(6)

j 1 jk2

k 1=

qc

=

pj

t Z;( ) Pr j

t Z t( ){ }=

j

pj

pj

Pr Yj

j


6/18



The conditional threshold, , is thethreshold that the idiosyncratic component of

obligorj, j, must fall below for default to occur inscenario Z.

Note that obligor credit worthiness indexcorrelations are uniquely determined by thedefault model and the multi-factor model, whichlinks the index to the credit driver returns. Thecorrelations between obligor defaults are thenobtained from the functional relationshipbetween the index and the event of default, asdetermined by the Merton model. For example,the indices of obligors that belong to the same

sector are perfectly correlated if theiridiosyncratic component is zero.

Part 3. Obligor exposures and recoveries in ascenario

Define the exposure to an obligorj at time t, Vj,as the amount that will be lost due tooutstanding transactions with that obligor ifdefault occurs, unadjustedfor future recoveries.An important property of PCR_SD is theassumption that obligor exposure isdeterministic, not scenario dependent:

.

The economic loss if obligorj defaults in anyscenario is

(7)

where j is the recovery rate, expressed as afraction of the obligor exposure. Recovery, in theevent of default, is also assumed to be

deterministic. (Expressing the recovery amountas a fraction of the exposure value at default doesnot necessarily imply instantaneous recovery of afraction of the exposure when default occurs.)

The distribution of conditional losses for each

obligor is given by

(8)

Part 4. Conditional portfolio loss distribution in ascenario

Conditional on a scenario, Z, obligor defaults areindependent. This follows from Equation 6 andthe assumption that the idiosyncratic

components of the indices are independent. Todetermine whether an obligor default occurs in ascenario, all that remains to be sampled is itsidiosyncratic component.

In practice, the computation of conditional lossescan be onerous. In the most general case, aMonte Carlo simulation can be applied todetermine portfolio conditional losses. However,the observation that obligor defaults areindependent permits the application of moreeffective computational tools. Some of these

techniques are described in Credit Suisse (1997),Finger (1999) and Nagpal and Bahar (1999).

For the purpose of exposition only, consider aportfolio with a very large number of obligors,each with a small marginal contribution. In thiscase, we can use the Law of Large Numbers(LLN) to estimate conditional portfolio losses.As the number of obligors approaches infinity,the conditional loss distribution converges to themean loss over that scenario; the conditionalvariance and higher moments become negligible.

Hence, the conditional portfolio losses, L(Z), aregiven by sum of the expected losses of eachobligor:

(9)

Assuming that the LLN is appropriate simplifiesthe presentation which permits us to focus this

Pr j

j

jk

Zk

k 1=

qc

j

----------------------------------------=

Yj

tn 1

( ) j n 1

Yj

tn

( ) ji

}

pj

t1

( ) j1

{ } v( ) vd

j1

= =

v( )e

v2

2

2--------------------=

j1

1

pj

t1

( ){ }=

pj

t2

( ) Pr t2

={ } Pr Yj

t1

( ) j1

> Yj

t2

( ) j2

Yj

t1

( ) Yj t2( )+

2--------------------------------------

j2 j1

j2+

j2 1u +v < j2


11/18



where . Default does not occur in thefirst time step, thus the limits of integration are

; default must occur in the second timestep, thus the limits of integration are .

The unconditional threshold for the second timestep, j2, is defined implicitly in Equation 24from the probabilitypj(t2) and the t1-thresholdj1.

More generally, for any time stepn, the thresholdjn is determined implicitly from the defaultprobabilities and the thresholds at all previoustime steps:

(25)

where .

Given the cumulative default probability curvefor each sector, thresholds can be computed

using a Monte Carlo method which solvesrecursively for the limits of the integrals inEquation 25.

We define the conditional probability ofdefault,pj(tn;x

c), asthe probability that default ofan obligor in sectorj occurs in then-th time stepconditional on the realization of the creditdrivers up to time tn:

(26)

The computation of the conditional defaultprobabilities is as follows: for the first time step,the conditional probability of default is, as in theprevious models, given by

(27)

The threshold, adjusted by the drift and volatilityof the index returns, is .Note that Equation 27 is equivalent toEquation 6.

For the second time step, the conditionalprobability is given by

Then, in general, for time stepn

(28)

where

Equation 28 can be restated as

u( ) v u( ) ud vdj1

j2

=

j2 j2 2=

[j1 ),( j2 ],

pj tn( ) Pr tn={ }=

vn vn 1( ) vn 1 vn 2( )j1

j n 1

jn

=

v2 v1( ) v1( ) v1d vn 1d vnd

ji ji i=

pj tn xc

;( ) Pr j tn xc

ti( ) i 1 n, ,=,=

=

pj

t1

xc

; Pr Y

jt1

( ) j1

xc

t1

( )<

=

Pr

jk

xkc

t1

( ) j

t 1

+

k 1=

qc

j t1( )

j

t1

( )-------------------------------------------------------------------------------------------------

j1 x

ct1

( ),>

=

pj

tn x

c;

Pr Y

jti

( ) ji

>[ ]

i 1=

n 1

Yj tn( ) jn xc

ti

( ),i 1 n, ,=


12/18



(29)

The right side of Equation 29 can be computedusing numerical integration. Details of thecomputation of the multi-step conditionaldefault probabilities are presented in Appendix 3.

Part 3. Obligor exposures and recoveries in ascenario

As in the single-step stochastic model, PCR_SS,obligor exposures are stochastic. However, in thismodel, the exposure to obligorj is dependent onthe path of the market risk factors up to time ti,

. Sinceexposures at various times are summed, each is

already discounted to today. Thus, discountedexposures express the capital that must be heldtoday to cover future defaults (unadjusted forrecoveries).

Exposures for each obligor are obtained througha multi-step simulation of all outstandingtransactions, accounting for all nettingagreements, mitigation and collateral. Aziz andCharupat (1998) present examples of thecomputation of these exposures. The table ofobligor exposures over every scenario, Vj(x

m, ti) isreferred to as the Multi-step Exposure MtF Table.

The economic loss if obligorj defaults in timestep i, is the exposure of obligor j at time step i,net of recoveries, where it is also assumed thatrecoveries in the event of default, ji, arestochastic:

(30)

The probability of this default ispj(ti,xc). Thus,

for every time step i,the distribution of

conditional obligor losses is given by

(31)

The table of conditional obligor losses, Lji(x), isreferred to as the Multi-step MtF Table ofObligor Losses.

Part 4. Conditional portfolio loss distribution in ascenario

At each time step, obligor defaults areindependent, conditional on a scenario. Thelosses in a given scenario are simply the sum ofthe losses at each time step in that scenario.

If the portfolio contains a very large number ofobligors, each with a small marginal contribution,the LLN dictates that conditional portfolio lossesat each time step converge to the sum of theexpected losses of each obligor:

(32)

Expected portfolio losses in a given scenario are

the sum of the expected losses in each time step:

(33)

Part 5. Aggregation of losses in all scenarios

Unconditional portfolio losses are obtained byaveraging the conditional losses over allscenarios:

(34)

where is the probability distribution in thescenario space. This integral is generallycomputed using Monte Carlo techniques.

The third column of Table A1 (Appendix 1)summarizes the features of the PCR_MS model.In the third columns of Tables A2 to A4 wesummarize the components of the PCR_MSmodel associated with the framework.

Concluding remarks

We have presented a new multi-step PortfolioCredit Risk model that integrates exposuresimulation and advanced portfolio credit riskmethods. The integrated model, PCR_MS,overcomes a major limitation currently shared byportfolio models in accounting for the credit riskof portfolios whose exposures depend on the levelof the market.

pj

tn

xc

; Pr

luji

>

l 1=

i

i 1=

n 1

l ujn

l 1=

i

i 1=

n 1

=

uji ji jk xkc

ik 1=

qc

js

=


16/18

36

ALGO

RESEARCHQUARTERLY

SEPTEMBER

1999

PCR_SD PCR_SS

Obligor exposure

Obligor recovery

Obligor losses in

the event of

default

Equation 7 Equation 11 Equation 30

Conditional

portfolio losses

(LLN)


Equation 33

Unconditional

portfolio credit

losses


Table A4: Parts 3, 4 and 5 Obligor exposure, recovery and losses

VjV

jxm

( ) Vji

xm

( )

jj x

mZ,( ) ji x

mx

c,( )

LjZ( ) V

j1

j( )= Lj x

mZ,( ) Vj x

m( ) 1 j x

mZ,( )( )= L

jix( ) V

j=

L Z( ) E Lj

Z( ){ }

j 1=

N

Vj 1 j( ) pj Z( ) j 1=

N

= = L xm

Z,( ) Vj

xm

( )

j 1=

N

1 j xm

Z,( ) p

jZ( ) = L t

ix,( )

j

=

L x( )

i =

M

=

Pr LP


17/18



Appendix 2. The canonical creditworthiness process

Consider a single obligor. The canonical creditworthiness process, Y(t), is described by

where wt is a one-dimensional Wiener process. Inthe continuous time Merton model, the time ofdefault is the first time when the process fallsbelow a boundary :

Let the function P(t), with for all

t > 0 and , represent the probability of

default before time T; i.e.,

This continuous time problem is depicted inFigure A1.

Figure A1: Continuous time default model

Appendix 3. Computation of multi-stepconditional default probabilities

In general, for time stepn, the conditionaldefault probabilities of a given obligor are givenby Equations 28 and 29:

where

For simplicity, the indexj, denoting a given

obligor, is removed from the notation.

Denote by

and

Then it follows that

The probability is a function ofn

variables, Gn(u1,..., un). The function

Gn(u1,..., un) satisfies the relation

(A1)

where . The

integrals in Equation A1 are evaluated using

numerical integration techniques.

Y t( )w

t

t-----=

t( )

in ft

Y t( ) t( )

l 1=

i

i 1=

n 1

l ujn

i 1=

n 1

ll 1=

n

un

i 1=

n

=

Pr Bn 1( ) Pr An( ) Pr Bn( )+=

Pr Bn

( )

G

n 1+

u

1

, u

n 1+

,( ) G

n

u

1

, u

n

,( ) u

n 1+

u

n

( )=

Gn

u1

u2

,, un 1 u,,( ) un 1+ u( ) ud

un

+

t( ) t( ) Pr n 1+

t>{ }= =


18/18


Date post:	06-Apr-2018
Category:	Documents
Upload:	runawayyy
View:	217 times
Download:	0 times

An Integrated Market and Credit Risk Portfolio Model

Documents