Credit Risk, Economic Capital and the Rating of Credit Portfolios KajNystr¨om * Department of Mathematics Ume˚ a University, S-90187 Ume˚ a, Sweden Jimmy Skoglund † Swedbank, Group Risk Control S-105 34 Stockholm, Sweden July 26, 2004 Abstract In this paper we describe a multi-period and multi-state portfolio credit risk model and its applications. The applications include economic capital, credit risk pricing, the division of the credit portfolio into tranches reflecting a certain credit quality and the assignment of appropriate ratings to the tranches. In particular we explain how the approach used by rating agencies can be implemented in our framework. The model consists of a methodology for estimation and simulation of systematic tran- sition risk, a methodology for the modelling of recoveries in the case of stochastic col- laterals as well as an approach to dimension reduction of the portfolio. Though general, the model is in particular applicable to retail and mortgage portfolios. Our approach to systematic transition risk extends, to transition matrices, the well-known systematic default risk model of Wilson (1997). One motivation for the extension is to develop a model, where non-traded credit risk can be marked to market. Another is that pure de- fault risk models, like Wilson’s model, tend to give too small weights to large transitions and thereby potentially under-estimating credit risk. Keywords and phrases: credit risk, economic capital, systematic transition risk, recov- ery, capital allocation, conditional value at risk, pricing, rating, collateralization, credit default swap. * email: [email protected]† email: [email protected]1
Transcript
Credit Risk, Economic Capitaland the Rating of Credit Portfolios
Kaj Nystrom∗
Department of MathematicsUmea University, S-90187 Umea, Sweden
Jimmy Skoglund†
Swedbank, Group Risk ControlS-105 34 Stockholm, Sweden
July 26, 2004
Abstract
In this paper we describe a multi-period and multi-state portfolio credit risk modeland its applications. The applications include economic capital, credit risk pricing, thedivision of the credit portfolio into tranches reflecting a certain credit quality and theassignment of appropriate ratings to the tranches. In particular we explain how theapproach used by rating agencies can be implemented in our framework.The model consists of a methodology for estimation and simulation of systematic tran-sition risk, a methodology for the modelling of recoveries in the case of stochastic col-laterals as well as an approach to dimension reduction of the portfolio. Though general,the model is in particular applicable to retail and mortgage portfolios. Our approachto systematic transition risk extends, to transition matrices, the well-known systematicdefault risk model of Wilson (1997). One motivation for the extension is to develop amodel, where non-traded credit risk can be marked to market. Another is that pure de-fault risk models, like Wilson’s model, tend to give too small weights to large transitionsand thereby potentially under-estimating credit risk.
Keywords and phrases: credit risk, economic capital, systematic transition risk, recov-ery, capital allocation, conditional value at risk, pricing, rating, collateralization, creditdefault swap.
Modelling credit risk is a challenge for any financial institution and every bank needs a struc-tured approach to the management and quantification of credit risk. This includes managingthe risk at portfolio level as well as at the counterparty or transaction level. From a portfo-lio perspective management wants to understand and shape the risk profile of the portfolioand in particular know how different (macroeconomic) scenarios affect the overall risk. Atthe counterparty or transaction level managers want to make sure that sufficient marginalsare charged in order to absorb many of the potentially adverse scenarios. In this paper wedevelop a multi-period and multi-state portfolio credit risk model which give guidance for themanagement problems described above. The model includes a methodology for estimationand simulation of systematic transition risk, a methodology for the modelling of recoveries inthe case of stochastic collaterals well as an approach to dimension reduction of the portfolio.Though general, the model is in particular applicable to retail and mortgage portfolios.
From the perspective of dimension reduction we divide the set of counterparties in theportfolio into a number of homogenous segments. Homogeneity is defined in terms of commonrating and systematic transition risk, common features of recoveries as well as with respectto the structure of the cash flows of the counterparties. Homogeneity should not be takenliterally, rather it is a subjective matter, to define the ’degree of homogeneity’, required in ahomogenous segment. For each segment an appropriate model for the evolution of transitionrisk, recoveries and cash flows is specified.
In the case of systematic transition risk our approach extends, to transition matrices, thewell-known systematic default risk model of Wilson (1997). One motivation for the extensionis to develop a model where non-traded credit risk can be marked to market. Another is thatpure default risk models, like Wilson’s model, tend to give too small weights to large transitionsand thereby potentially under-estimating credit risk1. We will furthermore assume that aninternal classification system, based on counterparty rating, is available. More specifically wewill assume, in case of a limited rating history, that expertise knowledge is available for thecalibration of parameters2.
In the modelling of recoveries we map the evolution of collaterals to a set of collateralevolution models. Each such model is driven by an index and the idea is to use, in theconstruction of the index, officially published price indices (e.g., in case of mortgages usingsegmentation on region and type of property). To come closer to reality our model alsoincludes the notion of haircut, introduced here as an estimate of the fraction of the marketvalue retained when the property is sold at (default) auction. Finally we define the exposureat default value as well as the legal aspects of recovery. In the modelling of collateral recoverieswe will also assume that an internal (”going concern”) collateral valuation tool is available.
1Wilson’s model is obtained as a special case of the model proposed here by allocating non-current states(excl. default state) transition probabilities to current state.
2Best practice in rating classification systems consists of logistic regression scorecards for the retail andSME segments and an expert model with score judgements for banks and large corporates. The expert modelis often complemented with a benchmark rating (PD) from a structural model of Merton-type. For the logisticscorecards a rating classification is generated by a partitioning of the interval [0, 1], i.e., the range of the model.For the expert model score judgements are, due to the scarcity of internal data for calibration, usually mappedto the rating classification of rating agencies.
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Having historical data from that tool it can of course be used to construct relevant priceindices in case of a lack of official data. However, official price indices have the benefit ofobjectivity and public scrutiny.
Having defined the components of the credit risk model properly we focus on certain keyapplications of the portfolio model. In particular we focus on the calculation of economiccapital and subsequently on
• the allocation of economic capital,
• the pricing of credit products based on economic capital.
To quantify the amount of economic capital needed to support the portfolio we argue asfollows. We consider a fixed time horizon T and we denote by PV out (0, ω) and PV in (0, ω)the present value under future world ω, as seen today at t = 0 from the perspective of thebank, of liabilities respectively of incoming cash flows during the interval (0, T ]. We define
ec = −minρα
(PV in (0, ω)− PV out (0, ω)
), 0
as the amount of economic capital needed, at t = 0, to support the portfolio during the interval(0, T ]. Here the risk buffer is quantified using the risk measure ρα and ρα should be thoughtof as either V aRα (Value at Risk at the confidence level α) or CV aRα (Conditional Valueat Risk at the confidence level α). In the case of V aRα, economic capital can be interpretedas the amount of equity capital needed, in order to ensure that, the probability of the eventthat the present value of our assets is smaller than the present value of our liabilities, is lessthan some very small pre-defined number. Having computed the total amount of economiccapital we also compute risk contributions with respect to our exposures. This is done atthe level of the groups constructed in the division of the portfolio into homogenous segments.Subsequently the economic capital is allocated to the groups depending on their contributionto the overall risk. However, it is important to emphasize that the economic capital numbersobtained from the model are, in the capital planning process within the bank, only one ofseveral inputs. The analysis must be complemented with information concerning other riskslike operational risk and in particular the statistical scenarios should be supplemented withstress scenarios as well as scenarios based on expertise. Furthermore, strategic plans, i.e.,expectations on future flows should be taken into account.
Concerning the pricing of credit products based on economic capital we know that, fortraded bonds, the credit quality of the underlying counterparty is reflected in an appropriatelydefined credit spread with respect to a curve of reference. The discounting curves for thesebonds can be constructed using market data and standard interpolation techniques. For non-traded bonds, such as mortages, we use the economic capital model to internally generate,for each homogenous group, a credit discount curve. In particular for each time horizon T wecompute the expected loss on the segment as a fraction of the total notional of the segment.This results in a term structure reflecting the expected loss of notional . Similar for each timehorizon T we compute the economic capital contribution of the segment. This results in aterm structure reflecting the economic capital contributions. Based on these curves the minimalpremia that the bank must charge the counterparty in order to, on average, break-even can becalculated. Furthermore, strategic decisions concerning potential expansions in, on average,
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profitable segments and contraction in, on average, non-profitable or low-profitable segmentscan be made and the term-structures obtained can also be used to mark the banking bookto market. Combining this with e.g., a RAROC risk-adjustment and a fixed hurdle rate forRAROC yields a credit risk pricing tool. Such a credit risk pricer can be used for instance as alower limit on required premia, a guideline or recommendation or an automatic pricer. Fromthe perspective of rating agencies an evaluation of management risk is important. Concerningmanagement risk, this risk can, at least in principle be reduced by a credible commitment toa pricing methodology. One such commitment is to achieve certain minimum RAROC levelson the flow into the portfolio.
Finally, as a second application of our portfolio model, we show how to divide the creditportfolio, assuming an additional collateralization in terms of a stochastic cash/reserve ac-count, into tranches reflecting certain credit qualities and how to assign ratings to the tranches.In particular we explain how the approach used by rating agencies can be implemented in ourframework. The rating agencies assign a certain rating to a tranche j based on an upperlimit on the expected loss on the tranche. As explained in the paper for each scenario ω ∈ Ω,generated from the underlying model, we get in the case of a time horizon of T , one realizationof the ratio
πj(T , ω) =Aj − TAj(T , ω)− Ej(T , ω)
Aj
.
Here Aj is the notional of the tranche, TAj(T , ω) is the accumulated amortizations including
recoveries and Ej(T , ω) is the outstanding notional, under the scenario ω, on the tranche at
t = T . πj(T , ω) is therefore the total loss, as a fraction of the notional, incurred on tranche j,
up to time T . Tranche j is assigned, by the rating agency, the rating R at horizon (0, T ] if
πj(T ) = E[πj(T , ·)] =∑
ω
pωπj(T , ω) ≤ βR(T ),
where βR(T ) is a constant associated to the rating R and pω is the probability that scenario ωoccurs. βR, as a function of rating R, is increasing as the rating decreases. We note that therating depends heavily on the scenario set used as well as on the probabilities attached to eachscenario. In practice, in any form of securitization, the division of the portfolio into tranchesas well as the assigment of ratings to the tranches is done by the rating agencies. Still, it isimportant to have an internal view on the appropriate ratings and an understanding of thediscrepancies between the internal model and the model used by the rating agency. Moreover,in case the internal portfolio model is audited and approved by the rating agencies, the ratingagencies may choose to use the internal portfolio model directly in the rating process. Hence,in this case there is no discrepancy between the internal model and the model used by theagency, and the remaining potential disagreement is on the relevance (probability weights) ofthe scenarios underlying the model. For an insight into the structured finance rating process werefer to e.g., Moody’s investor service The Binomial Expansion Method Applied to CBO/CLOAnalysis (1996), The Lognormal Method Applied to ABS Analysis (2000) and for servicingquality judgement Residential Services Quality (”SQ”) Ratings in EMEA (2003).
The rest of the paper is organized as follows. In section 2 the portfolio model and itscomponents are described in detail. Section 3 is devoted to the application to economiccapital, its allocation and pricing based on economic capital. Section 4 describes the details
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of the tranching as well as the rating of the tranches described above. In Section 5, the finalsection of the paper, an illustration of the model is given in terms of an application to amortgage-type portfolio.
2 The credit portfolio model
In order to develop a model for the computation of potential future cash flows there areessentially three factors to consider. First, the transition probabilities between different ratingclasses have to be modelled. In particular it is crucial to understand how these transitionprobabilities vary, across different segments of the portfolio as well as over time, as functionsof so-called systematic variables or credit drivers. Throughout this paper we will assume thereare K ratingclasses (e.g., the master rating scale for the bank) and that the counterpartiesin the portfolio are divided into a number, m, of homogenous segments (in a sense madeclear below). The number of segments is less than the total number of counterparties in theportfolio. To each segment, i.e., for j ∈ 1, ..., m we assign a matrix of transition probabilities,Aj. This matrix describes the probabilities of transition, for a counterparty in the segment,from its current rating to the different rating classes. Secondly, the vector of recovery rates(δ1, . . . , δn) has to be modelled and thirdly a model for the cash flows (C1, . . . , Cn) is needed.In the most general cases both
δk
and
Ck
are stochastic. One such example is a mortgage
loan with a floating interest rate.3
2.1 The model for systematic transition risk
As described above the portfolio is divided into m homogenous segments, indexed by j =1, . . . ,m. An example of a segmentation is private counterparties, small and medium sizedcorporates (SME) and large corporates. In practice finer segmentation based on industryclasses and geographic regions may be used. For each segment we then need a model for theK-dimensional transition matrix Aj (t), describing the probabilities of transitions, at time t,for each counterparty in segment j.
We let Z =(Z1, . . . , Zh
)denote a vector of systematic variables like interest rates and
output gaps and we assume that these are observable at discrete time intervals, say monthlyor quarterly. In the following we consider an explicit functional form linking Z to transitionprobabilities. In particular we let
Ajls (t) = f j
ls
(Z, βl
).
Here Ajls denotes the entry of the matrix Aj on row l and on column s. Note that since
transition probabilities must satisfy f jls
(Z, βl
)> 0,
∑s f j
ls
(Z, βl
)= 1, the choice of functional
form for f jls is somewhat limited. We will choose the multinomial logit formulation introduced
by Theil (1969) although other choices are certainly feasible. Moreover, we restrict ourselves
3In the case of marking the cash flowsCk (t)
to market there are K − 1 degrees of freedom in the sense
that, each cash flow should be discounted using the credit spread curve. The spread reflects the credit qualityof the counterparty paying the flow and in particular there are K − 1 non-default states.
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to first-order Markov models4.The multinomial logit formulation expresses the log of a ratio of probabilities as a function
of exogenous variables. If the model has K sets of probabilities, one for each row of thetransition matrix, there will be K sets of ratios. Each such ratio will be defined using thetransition probability from the last column as a denominator,
ln
(f j
ls
(Z, βl
)
f jlK (Z, βl)
)= f
j
ls
(Z, βl
), s = 1, . . . , K − 1, l = 1, . . . , K. (1)
By definition the functions introduced comprise a transformation from the exogenous variablesto the transition probabilities such that all elements of Aj (t) are non-negative and such thatthe elements on each row sum to unity. Expressing the transition probabilities for row lseparately, as a function of exogenous variables and coefficients, yields
f jls
(Z, βl
)=
exp[f
j
ls
(Z, βl
)]
1 +∑
s exp[f
j
ls (Z, βl)] , s = 1, . . . , K − 1 (2)
f jlK
(Z, βl
)=
1
1 +∑
s exp[f
j
ls (Z, βl)] .
We will, for convenience, let fj
ls be a linear function. The degrees of freedom, i.e., the vectorsβl, have to be calibrated using empirical data or alternatively, in case of a lack of sufficienthistorical data, calibrated to reproduce an expert view on future transition probabilities. Foreach j we let Ej (t) be the empirical transition matrix to be used for calibration. For anarbitrary K ×K matrix M we introduce the norm || · || of M , ||M ||, as ||M ||2 =
∑i
∑j M2
ij.
We will use this L2-norm to measure distances but we could also have used other norms. Inthe following we assume, for the moment, that the matrix of parameters, β, is known andthat prediction errors are zero for every j, i.e.,
T∑t=1
∣∣∣∣Aj (t)− Ej (t)∣∣∣∣ = 0 (3)
Based on this calibration we can simulate future transition matrices as follows.
1. Simulate the vector of systematic variables Z at future time steps T + 1, . . . , T using amultivariate model e.g., a Gaussian Markov process, dZ (t) = C (v − Z (t)) dt+DdW (t),where C and D are d-dimensional matrices, v is a d-dimensional vector and W is astandard d-dimensional Wiener process.
2. Transform to transition matrices, Aj (t), for t = T + 1, . . . , T and j = 1, . . . , m usingequation (2).
4The present model is reduced form in the sense that systematic variables are not structurally related tocounterparties financial health as in e.g., Merton models. In case of a structural model we obtain any row ofAj (t) through the assignment of rating trigger points on the domain of the structural index, being for examplea proxy for asset value. In practice the probabilities of these trigger points are obtained from unconditionaltransition matrices, CreditMetrics being an example of this approach.
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We can conclude that, conditional on Z, the transition of any counterparty is a realization froma multinomial distribution. For a large pool of homogenous counterparties (i.e., homogenouswith respect to rating at t = 0 and transition matrix j = 1, . . . , m) we want to understandthe frequency with which the counterparties have migrated, after the time steps T + 1, . . . , T ,to the different rating classes. This can be done by using the following iteration
y (t) = A (t)y (t− 1) (4)
where y (0) is a K-dimensional vector having entry at position k equal to 1 and all otherentries equal to 0. In case all counterparties in the pool can be considered as homogenouswith respect to recoveries and cash flows a large pool can be treated as a single (fractional)counterparty, hence reducing the computational burden considerably5. In particular, as wewill discuss further below, for retail and SME portfolios the construction of homogenous poolsis a sensible approach to portfolio credit risk 6.
The matrix of coefficients in the regression, β, can be estimated in several ways. Forinstance we can estimate β by minimizing (3) row-wise. This estimator is clearly not the mostefficient one since it does not take heteroscedasticity and potential cross-equation correlationbetween rows into account. A linearized version of the estimator is obtained by applyingSeemingly Unrelated Regression (SUR) techniques to the K − 1 log-odds ratios in (1) for agiven row l. Sacrificing efficiency even further, one can apply ordinary least square techniquesequation by equation, ignoring the cross-equation correlation altogether. This is though notrecommended in practice.
In the application of section 5 we consider calibration when only two distinct rating periodsare available. In that case the constant parameter matrices of the multinomial logit regressionsare chosen such that current transition matrices are replicated. To further calibrate theparameter matrices linking the exogenous variables we proceed qualitatively. In particularparameters are set so that realized transition matrices under high, middle and low values ofthe systematic variables are consistent with our prior beliefs.
2.2 The recovery process
For counterparties, supplying stochastic collateral, we also need a model for recovery takingthe evolution of collateral values into account. This is particularly relevant in the case ofmortgages. In the following we will assume that counterparty i, i = 1, . . . , n, has Li ≥ 1 in-struments and Si ≥ 1 stochastic collaterals. We will furthermore assume that counterparty ieither defaults on all or none of the obligations and hence that the random nature of promisedcash flows (possibly conditional on market factors as in e.g., derivatives) arise from the un-certainty of the single default time, τ i, and from the uncertainty of collateral values at thedefault time.
5However if the pool is not large enough for an application of the Law of Large Numbers (LLN) we haveto use multinomial sampling for each member of the pool.
6Most internal rating classification systems use ’90 days due on payments’ as the definition of default.Still this technical default does not necessarily correspond to a ”true default”, i.e., there is a priori only aprobability, x%, that a default in the definition of the rating classification system turns out to be a truedefault. To account for this discrepancy one can proceed as follows: If there is a transition to default usebinomial sampling to decide wether the default is a ”true” default or not. In case of a purely technical defaultthe counterparty will be assumed to remain in the ratingclass it was in pre-default.
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For large (mortgage) portfolios it is neither feasible nor desirable to have collateral specificmodels for the evolution of collateral value. It is therefore natural to map the evolution ofcollateral to a set of collateral evolution models. We index these models by j = 1, . . . , M, M <∑n
i=1 Si, and for each counterparty i the value process of each of its collaterals s ∈ 1, . . . , Siis modelled by one of the models indexed by j. We assume that for each j there is a processZj, such that Zj(0) = 1 and such that the evolution of the value of the collateral is
V is (t, ω) = V i
s (0)× Zj (t, ω)
ω ∈ Ω the scenario set.
Here V is (0) is the value of the collateral today. In the case of mortgage loans, with property as
collateral, it is natural to use officially published price indices (using segmentation on region
and type of property) to construct the processes Zj. Similarly, having historical data from aninternal collateral valuation tool one can use this to construct price indices.
To capture correlation between the variables driving transition risk and the price indicies
a joint scenario model for Z (t) =(Z (t) , Z (t)
)can be specified. An example of such a model
is a combined multivariate Geometric Brownian motion for price indices and multivariateGaussian process for systematic variables. For more on multivariate scenario models andtheir simulation we refer the interested reader to Glasserman (2003).
Starting from the current internal valuation of collaterals, V is (0), i = 1, . . . , n, s =
1, . . . , Si, we model the evolution of the value of collaterals as above. Still an additionalcomponent is needed in order to properly model reality. The fact is that in many cases thedefault value of the collateral is lower than its ”going concern” value. A common way ofadjusting for this is to introduce an estimate of the fraction of the market value retainedwhen the property is sold at (default) auction. We denote this fraction by γi
s. Using thisnotation, the (default) value of collateral of counterparty i is described, in scenario ω ∈ Ω, by
V i (t, ω) =∑Si
s V is (t, ω) γi
s.To be able to compute recoveries and losses in a given scenario it remains to define the
exposure at default as well as the legal aspects of recovery. We denote by Ei (t) the exposureto counterparty i at time t. Then the value recovered at t is
min[Ei(t), Ri (t)
]. (5)
Here Ri (t) is a function of V i (t) which captures the fact that only part of the collateral valuemay be legally secured by the financial institution7. In many cases this legal function is quitesimple. However, an example of a more complicated case is when only different ”tranches”of the collateral is legally secured by the financial institution. In such cases Ri (t) can beexpressed as
Ris (t) =
fM∑
em=1
max[(
V is (t) γi
s + min[(
Bus,em − V i
s (t) γis
), 0
]−Bls,em
), 0
]
Ri (t) =Si∑
s=1
Ris (t)
7An expected non-zero blanco portion of recovery can simply be added to Ri (t) in (5).
9
Here Bus,em and Bl
s,em are, for collateral s, the upper respectively lower threshold of collateral
tranche (i.e., interval, segment) m = 1, . . . , M . Having obtained the recovery value it isconvenient to introduce the recovery rate
δi (t) =1
Ei(t)min
[Ei(t), Ri (t)
].
Using this notation we can define the counterparty loss (of notional) process up to T as
Li(T ) = Ei(τ i)(1− δi
(τ i
))χ(0,T ]
(τ i
)
Here τ i is the stochastic default time of counterparty i and χ(0,T ](t) is the indicator function
for the interval (0, T ]. For a portfolio with n counterparties the total loss process is obtainedby summation of individual counterparty loss processes,
L(T ) =n∑
i=1
Ei(τ i)(1− δi
(τ i
))χ(0,T ]
(τ i
)(6)
Finally we point out that in analogy with (internal) validation requirements of ratingclassification models we also have to validate the recovery model. In the case of rating systemsthis is often done using so-called cumulative accuracy profiles or accuracy ratio measures, seeSobehart et al. (2001). In our model for recoveries the validation must include:
• a validation of the accuracy of the current methodology or tool for going concern valu-ation of collateral,
• a validation of the accuracy of haircuts employed,
• a calculation of the accuracy of the employed blanco portions of recovery.
In addition to this one must ensure that a relevant model for the evolution of collateral valuesis specified. Although this might seem as an overwhelming check list, there are benefits fromusing this model compared to using simple portfolio mean recovery rates. One such benefitis that it can be used for collateral-specific pricing. Another is that the model includes anexplicit specification of the dependence between the variables driving transition probabilitiesand the variables driving collateral values.
2.3 Cash flows
In practice cash flow based risk measurement is important because it allows for managementof both ALM and the present value at risk of portfolios. For instance, in economic capitalcomputations we ultimately like to ascertain that we have enough equity capital, in order toensure that the probability, that the present value of our assets is smaller than the presentvalue of our liabilities, is less than some very small pre-defined number.
In the following we will focus on instruments with cash flows that depend on the term-structure of interest rates e.g., variable-rate mortgages. For reference our apprach is principallysimilar to Fabozzi (1997, p. 362-366). We introduce the following notation.
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• r(Tk, ω), denotes the spot rate used for discounting cash flows due at Tk.
• C (Tk, ω) = C (Tk, f(δ, Tj, ω)), denotes a cash flow occurring at t = Tk where f(δ, Tj, ω)is the forward term-structure, δ indexing the term, at Tj ≤ Tk used to determine cashflows at Tk
8.
• Each counterparty i has a set of Li instruments and for l = 1, . . . , Li, T li denotes the
maturity of instrument l. C li(Tk, ω) denotes the cash flow on instrument l at t = Tk.
We will evaluate credit events on a time horizon of T , i.e., we will consider credit events occur-ring in the interval [0, T ]. We denote, in scenario ω ∈ Ω, by PV i
NonDef (0, ω) and PV iDef (0, ω),
the present values of the cash flows which are default free during the interval [0, T ] respectivelyexposed to default. The explicit definitions are given below. The present value of future cashflows can therefore be expressed, in scenario ω ∈ Ω, as
PV i(0, ω) =(PV i
Def (0, ω)− PV iNonDef (0, ω)
)+ PV i
NonDef (0, ω).
That is, we have induced a partitioning of the cash flows based on wether or not it is exposedto default.
PV iNonDef (0, ω) =
Li∑
l=1
∑
Tk≤T
C li (Tk, ω) exp(−r(Tk, ω)Tk)χ(0,T i
l ](Tk)
PV iDef (0, ω) = χ(0,T ](τ
i)Li∑
l=1
∑
Tk≤τ i
C li(Tk, ω) exp(−r(Tk, ω)Tk)χ(0,T l
i ](Tk)
+χ(0,T ](τi) min
[Ei(t), Ri
(τ i
)]exp(−r(τ i, ω)τ i)
+χ(T ,∞](τi)PV i
NonDef (T , ω)
Again, τ i is the stochastic default time of counterparty i and χ(0,T ](t) is the indicator function
for the interval (0, T ]. In summary the evaluation of PV iDef (0, ω) for any ω ∈ Ω require the
joint scenario set for the forward term-structure, the systematic variables and the collateralevolution indices. A joint scenario set ω and the assignment of the associated probability canbe achieved either by Monte-Carlo models, expertise scenarios or a combination of both.
2.4 Portfolio dimension reduction
In this section we describe our approach to portfolio dimension reduction. This is important inmortgage and more general retail applications as well as to some extent in SME applications.We introduce the following definition.
8An example of this is a mortgage with re-setting of the rate every 2:nd period. Assuming that re-settinghas occurred at T0 and applied to cash flows at T0 we further have
C (T1, ω) = C (T1, f (T2 − T0, T0, ω)) ,
C (T2, ω) = C (T2, f (T2 − T0, T2, ω))
etc.
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Definition 1 (Credit homogeneity tree) A credit homogeneity tree is a tree with threelayers. The first layer is a k1-dimensional vector, V = (v1, ...., vk1), such that each componentrepresents a set of counterparties homogenous with respect to rating at t = 0 and homogenouswith respect to the (systematic) transition matrix. For each component of V , vj, a second layeris constructed and this layer is a k2-dimensional vector, Uj = (uj1, ...., ujk2), such that eachcomponent represents a set of counterparties homogenous with respect to common featuresof the recovery process. Finally for each component of Uj, uji, a third layer is constructedand this layer is a k3-dimensional vector, Wji = (wji1, ...., wjik3), such that each componentrepresents a set of counterparties with homogenous cash flow structures.
In total, phrasing our credit portfolio in terms of a credit homogeneity tree implies that wedivide all the counterparties into k1 × k2 × k3 homogenous segments, where homogeneity isdefined in the definition. The homogeneity requirement should not be taken literally. Rather itis a subjective matter, to define the ’degree of homogeneity’, required in a homogenous group.In practice one can proceed with the ”bucketing” as follows. First the k = 1, . . . , K − 1rating levels together with transition matrix j ∈ 1, 2, . . . , m are used to define the k1-dimensional vector of counterparties homogenous with respect to initial rating and transitionmatrix. Second, recovery classes are defined in a similar fashion by using existing modelsegmentation j ∈
1, . . . , M, legal tranches and value of collateral, Blanco. Finally cash
flows are segmented. In case of bonds this cash flow segmentation amounts to constructionof e.g., notional and amortization grids. To ensure that the (no default) cash flows of theoriginal and compressed portfolio match any leaf in the tree is assigned the leaf (weighted)conditional mean of the bucketing characteristic.
Portfolio bucketing in accordance with the credit homogeneity tree provides a nice par-titioning of the portfolio and this partitioning is especially important, as we will see below,for the purpose of capital allocation and for the pricing. Another important aspect is that,if the number of counterparties in any three-layer homogenous subgroup is large (≥ 300),then we can apply a conditional law of large numbers to reduce the amount of time neededfor computations. This makes it possible to treat any leaf in the tree as effectively a singlecounterparty.
3 Economic capital
Having described the credit portfolio model in the previous section we can for any timehorizon, T , quantify the cash flows on the interval [0, T ] and calculate the relevant presentvalue. Under every scenario, ω generated from the underlying model, we get present valuesand we can therefore talk about present value distributions. In economic capital computationswe ultimately like to ascertain that we have enough equity capital, in order to ensure thatthe probability, that the present value of our assets is smaller than the present value of ourliabilities, is less than some very small pre-defined number. To quantify economic capital wefocus, in this section, on the application of measures of risk to the present value distributions.
In general terms a risk measure is a mapping ρ : V → R ∪ ∞ where V is a non-emptyset of F -measurable real-valued random variables defined on an underlying probability space
12
(Ω, F, P ). If α ∈ (0, 1] and if Z is a random variable defined on (Ω, F, P ) then
V aRα(Z) = supλ ∈ R : P [Z ≤ λ] ≥ αis called the Value at Risk (VaR) at the confidence level α of Z. The simplest and stillintuitively clear coherent risk measure is that of Conditional Value at Risk, CV aRα. Usingthe notation above,
CV aRα(Z) = E[Z|Z ≤ V aRα(Z)]
for α ∈ (0, 1]. I.e., CV aRα(Z) is the expectation of the random variable Z conditional on theevent that Z ≤ V aRα(Z). Consider a portfolio, P , and let h = (h1, . . . , hn) denote the unitposition vector of the instruments defining the portfolio. I.e., P = P (h1, . . . , hn). If ρ is arisk measure and if ρ(P ), as a function of (h1, . . . , hn), is C1 (i.e., continously differentiable)then by the Euler formula
ρ(P ) =n∑
i=1
hi∂ρ
∂hi
.
The derivatives of ρ(P ) with respect to hi are refered to as risk contributions. In the simula-tionbased case an interesting question is how to compute the risk contributions. That is, howdo we allocate capital based on a set of simulations. In the case of Value at Risk, given a setof scenarios and a confidence level α, there will be a scenario which realize the VaR-number.Following Mausser and Rosen (1988) we may, in the simulationbased case, take that scenarioas our estimator of the derivative of VaR. As
CV aRα =1
1− α
∫ 1
1−α
V aRudu
we can similarly estimate marginal CV aRα by averaging over estimated VaR derivatives.To quantify the amount of economic capital needed to support the portfolio we argue as
follows. We consider a fixed time horizon T and we denote by PV out (0, ω) and PV in (0, ω)the present value under scenario ω, as seen today at t = 0 from the perspective of the bank,of liabilities respectively of incoming cash flows during the interval (0, T ]. We define
ec = −minρα
(PV in (0, ω)− PV out (0, ω)
), 0
as the amount of economic capital needed, at t = 0, to support the portfolio during the interval(0, T ]. Here the risk buffer is quantified using the risk measure ρα and ρα should be thoughtof as either V aRα (Value at Risk at the confidence level α) or CV aRα (Conditional Value atRisk at the confidence level α). In the case of V aRα, economic capital can be thought of asthe amount of equity capital needed, in order to ensure that, the probability of the event thatthe present value of our assets is smaller than the present value of our liabilities, is less thansome very small pre-defined number.
3.1 Allocation of economic capital
The question of capital allocation is here taken as synonomous with allocation of risk contri-butions to counterparties or homogenous pools of counterparties, homogenous in the sense ofsection 2.4.
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Consider a large retail portfolio with e.g., 1 million counterparties. It is clear that eventhough we can model every counterparty in the computation of the total present value distri-bution it is neither feasible nor desirable to compute risk contributions at counterparty level.Instead we divide the portfolio into homogenous groups and calculate the risk contributionscoming from each group. Subsequently this generates a three-dimensional cube of allocatedcapital. As any single deal does not have a large impact on a retail portfolio, the capitalallocation cube will be stable over time and can be used as an input, as we will discuss below,into a risk-adjusted pricing tool. In contrast, for a large-corporate portfolio, where any newdeal will contribute significantly to the risk, the estimation of the derivatives described abovemay not be very useful and should be reconsidered.Clearly, the economic capital contributions from a homogenous segment, computed from ρα (·),can be either positive or negative. This is natural as, when focusing on cash flows, someportfolio segments may be economic capital hedges due to excess (credit) spreads. Hence,such segments allow for the reduction of total economic capital.
3.2 Pricing based on economic capital
In this section we will describe a simple tool which computes the minimal premia the bankmust charge the counterparty in order to, on average, break-even. Combining this tool withe.g., a RAROC risk-adjustment and a fixed hurdle rate for RAROC yields a credit risk pricingtool. Such a credit risk pricer can be used for instance as a lower limit on required premia, aguideline or recommendation or an automatic pricer. Taking expectations in the formula forPVDef derived in section 2.3, we get by standard deductions that
E [PVDef (0)] ≈∑
Tk≤T
C (Tk) exp [−r(Tk)Tk − EL (Tk, Tk−1)] .
Here C(Tk) and r(Tk) are the, by today’s term-structure, implied cash flow and discount factorrespectively. Further EL (Tk, Tk−1) is the, by the underlying model, generated expected lossof notional between time periods Tk, Tk−1. Hence the expected cash flows are discountedwith a spread factor reflecting the credit quality of the underlying counterparty. For tradedbonds these spreads can, at least in principle, be constructed using market data and standardinterpolation techniques. For non-traded bonds, such as mortgages, we have to generateinternally, for each homogenous group, a credit spread curve. In particular for each timehorizon T we compute the expected loss on the segment as a fraction of the total notional ofthe segment. This results in a term structure reflecting the expected loss of notional . Similar,for each time horizon T we compute the economic capital contribution of the segment. Thisresults in a term structure reflecting the economic capital contributions. To explain this furtherwe decompose the exponential discount term above as
r (u) = (1− g (u)) r (u) + α (u) g (u)
Here r (u) is, as a function of u, the spot rate curve faced by the institution, for the financingof debt and α (u) is the function describing the term-structure of demanded (expected) returnon equity capital. Further g(u) is the internally computed economic capital contribution of theinstrument/segment. The decomposition therefore captures the fact that loans are financed
14
by a combination of traditional debt and equity capital and, in particular, that the fractionof debt vs. equity is determined by the risk contribution of the loan. As we in general haveα (u) > r (u), ∀u, ”good” credits get cheaper financing9.
Focusing on a minimal premia calculator we can interpret the exponential discount termabove as the minimal premia which, on average, generates non-negative profits (after equityholders have been compensated). Rates above the minimum premium give, on average, anadded economic value and can be compared on a risk-adjusted basis e.g., RAROC. Wethersuch rates can be achieved depends on the market situation.If we for a counterparty, condition on λ (rating), and assuming a fixed structure for the loan,focus is shifted to δ, i.e., to the collateral supplied by the counterparty. If for some reasonoffer prices are independent of rating classes, the most important flexibility for the bank is tohave different policies concerning collateral requirements for the different rating classes.
4 Rating the credit portfolio
In this section we consider the problem of dividing the credit portfolio into tranches as wellas the rating of the tranches. We assume that the portfolio of reference initially consists of nnon-defaulted counterparties. The reference portfolio is divided into L tranches denoted byM1, . . . , ML having notionals A1, . . . , AL. The tranches are ordered in the sense that creditlosses start to consume the notional on the bottom tranche before the other tranches aretaken into consideration. We denote this bottom-up ordering between the tranches by ≺,i.e., if A ≺ B then A is subordinated to B in the sense that credit losses will consume thenotional on A before the notional of B is effected. In the following we will assume thatM1 ≺ M2 ≺ · · · ≺ ML. We will also assume an additional collateralization of the portfolio interms of a cash/reserve account construction in the sense that the first losses will be coveredby the money in the reserve. We can consider the cash account as an additional tranche, M0,consisting of a stochastic amount of cash Z(Tk, ω) available in the account at time Tk, i.e.,Z(Tk, ω) represents the balance of the account.
Before continuing we need to introduce some more notation. We will assume that the timeinterval [0, T ] is split by a set of reference timepoints Tk and that any kind of cash flows areexchanged only at these time points.
• L(Tk, ω) denotes the losses on the portfolio occuring during the interval (Tk−1, Tk].
• TL(Tk, ω) denotes the total accumulated losses on the portfolio up to t = Tk.
• TA(Tk, ω) denotes the total accumulated amortizations incl. realized recoveries on theportfolio up to t = Tk.
• E(Tk, ω) denotes the total outstanding notional of the portfolio at t = Tk.
9We previously remarked that the economic capital contribution may very well be negative for a poolof instruments (i.e., positive number). In that case we may take g = 0 for that pool and re-distribute theeconomic capital reduction to pools with positive economic capital contribution (i.e., negative number). Hence,this portfolio effect may be viewed as enhancing our competitiveness.
15
• E(Tk, ω) denotes the losses, occurring during the interval (Tk−1, Tk] and not covered bythe cash account. As such the process is referred to as the excess loss process at t = Tk.
• TE(Tk, ω) denotes the cumulative excess losses up to time Tk.
We will assume that amortizations on the tranches are given as top-down amortizations. Inthis case ML will be fully amortized before amortizations start on ML−1 etc. We note therelation
E(Tk, ω) = E (0)− TA(Tk, ω)− TL(Tk, ω)
where E (0) is the initial notional. We also note that
TE(Tk, ω) =∑t≤Tk
E (t, ω)
and if TE(T , ω) = 0 we have not been forced to reduce the notional on any of the tranches.We will furthermore assume that there is an instrument of credit transfer attached to eachtranche10. We denote these instruments of credit transfer by I1, . . . , IL. The notionals onthese instruments match the notionals on the tranches i.e., A1, . . . , AL and we will assumethat the instruments are Credit Default Swaps (CDSs). In case of a CDS there is a premiumXj associated to the premium paying leg of instrument Ij, in return the seller agrees to deliverthe losses incurred on tranche Mj at the time they arise. The obligation of the seller is referredto as the default paying leg. Other natural instruments attached to the tranches are CreditLinked Notes (CLNs).
The reserve account can be constructed in many ways. Here we will, for convience, assumethat the construction is such that the amount of cash will vary in the interval [0, B] and we willassume that the amount of cash paid into the account, during the time period (Tk−1, Tk], isX0(Tk−Tk−1)E(Tk−1, ω) unless there is already an amount B in the account. In the latter caseno payment is needed. Here X0 is a fix premium determined on yearly basis. To understandthe dynamics of the account we argue as follows. At t = Tk−1 there is an amount Z(Tk−1, ω)in the account. If there are no credit losses during the period (Tk−1, Tk] we have
assuming that interest is earned on the amount already in the account. Similarly if creditlosses of a magnitude L(Tk, ω) occur during the period (Tk−1, Tk] we have, as the amount onthe account never can become negative,
Starting with Z(0) ∈ [0, B], the specified dynamics will ensure that Z(Tk, ω) ∈ [0, B] for allfuture time points Tk. In particular, if no or very few losses occur the account will grow to itsmaximum amount B and remain at that level. Relating the process Z(Tk, ω) to the processes
10In a so-called synthetic construction where there is transfer of credit risk to an Special Purpose Vehicle(SPV) there is an instrument of credit transfer attached to each tranche. These credit transfer instrumentsare typically credit linked notes, matching the notes issued to investors, or credit default swaps.
16
introduced in the previous section we note that the excess loss during the interval (Tk−1, Tk],
E(Tk, ω), registered at Tk, and the total excess loss up to time Tk are given as
E(Tk, ω) = maxL(Tk, ω)− Z(Tk, ω), 0TE(Tk, ω) =
∑Tj≤Tk
maxL(Tj, ω)− Z(Tj, ω), 0.
In order to understand the consumption on the tranches we introduce the intervals
I1 := (A0, A1]
I2 := (A1, A1 + A2]
...... ....
IL := (A1 + .... + AL−1, A1 + .... + AL].
with A0 = 0. Recall that the total initial notional on the reference portfolio is E(0) =A1 + .... + AL. Hence the intervals I1, ...., IL represent a partitioning of the interval (0, E(0)].We also note that if TE(Tk, ω) ∈ Ij then the cumulative excess loss processes is, at timet = Tk, consuming the notional of tranche j.
4.1 Collateralization and rating
To assign a rating to the tranches the rating agency considers, for j ∈ 1, ..., L, the ratios
πj(Tk, ω) =Aj − TAj(Tk, ω)− Ej(Tk, ω)
Aj
.
Aj is the notional of the tranche, TAj(Tk, ω) is the accumulated amortizations includingrecoveries and Ej(Tk, ω) is the outstanding notional at t = Tk. I.e., πj(Tk, ω) is, under scenarioω ∈ Ω generated from the underlying model, the total loss expressed as a fraction of thenotional incurred on tranche j up to time Tk. The rating assigned by the rating agency totranche j reflects an upper limit on expected losses on the tranche over a time horizon T . I.e.,
πj(T ) = E[πj(T , ·)] =∑
ω
pωπj(T , ω) ≤ βR(T ),
where βR(T ) is a constant associated to the rating R. We note that, from this perspective, therating depends on the scenario set used as well as the probabilities attached to each scenario.In practice, in any form of securitization, the division of the portfolio into tranches as well asthe assigment of ratings to the tranches is done by the rating agencies. Still, it is importantto have an internal view on the appropriate ratings and an understanding of the discrepanciesbetween the internal model and the model used by the rating agency. Moreover, in case theinternal portfolio model is audited and approved by the rating agencies, the rating agenciesmay choose to use the internal portfolio model directly in the rating process. Hence, in thiscase there is no discrepancy between the internal model and the model used by the agency,and the remaining potential disagreement is on the relevance (probability weights) of thescenarios.
As the specific features of the reserve account collateralization is of crucial importance inall of this we consider its degrees of freedom.
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• The initial balance of the account, Z(0),
• the yearly premium for the account, X0,
• the maximum amount, B.
Increasing all of these increases the credit protection for all of the tranches and hence theirrating. From the perspective of the manager of the reference portfolio there is an obvioustrade-off since increasing the credit protection may be costly while a too low protection mayresult in a low rating of the tranches creating other disadvantages when selling off the piecesin a securitization.
Assuming that the degrees of freedom of the reserve account are fixed we can solve for themaximal fraction αR1 ∈ [0, 1], of the notional of the portfolio of reference, which has a ratingof at least R1. Then proceeding iteratively we can similarly solve for αR2 , . . . , αRL
. Hence,conditioned on the current reserve account construction, we can divide the portfolio into Ltranches having rating R1,....,RL.
The construction can also be turned around in the following way. Assume that we have arating target for the portfolio in the sense that we want a fraction α1 of the notional to haverating R1, a fraction α2 of the notional to have rating R2 and so on. Based on this we canconstruct the degrees of freedoms, of the reserve account, so that the targets are achieved.
4.2 Pricing credit default swaps
For each scenario ω ∈ Ω, generated from the underlying model, we can quantify the premium,Xj(ω), on a credit default swap linked to tranche j. The net present value of the premiumpaying leg is quantified as
Xj (w)∑
Tk≤T
Ej (Tk−1, ω) (Tk − Tk−1) exp (−r (Tk−1, ω) Tk−1)
where Ej (Tk−1, ω) is the outstanding notional on tranche j at t = Tk−1. Here we have madethe natural assumption that insurance is paid in advance. The discounted flow for the sellerof the protection becomes
Denoting by pω the probability of scenario ω. One way to define the premium, Xj, is then
Xj = E[Xj (w)] =∑
ω
Xj(ω)pω.
This implies that the two parties will have different views on the ”fair premium” dependingon what probabilities pω they attach to the scenarios.
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5 Application
To illustrate the model described in this paper we will in the following consider a smallmortgage portfolio consisting of 32605 borrowers. The portfolio is, for convenience, assumedto be financed by interest-rate risk matched liabilities. The smallest exposure in the portfoliois 200.000 SEK and the largest is 26 MSEK (corresponding to a commercial property). Theportfolio has been divided into 10 homogenous groups in analogy with the portfolio dimensionreduction method described in section 2.4. The features of these groups are summarized inTable 1 below. The following notation is used in the table.
• N denotes the number of counterparties in each pool,
• VModel is a label for the collateral value evolution model. Groups with common valuemodel have the same property type and are located in the same region,
• Rating is the common rating of the members in the group at t = 0 (we have 8 ratingclasses, rating class 8 being the default class),
• LTV is the current Loan To Value ratios for members of the groups (without taking anydefault value discounts into account). In terms of legal recovery structure the mortgagepools are all prime ranking, except for the last pool which has second ranking mortgageswith another institution holding the first ranking mortgage up to 1 MSEK,
• Notional is the current notional of the members of the group in MSEK,
• A is the amortization rates as % of current notional,
• M is the marginal over financing costs, in %, for the loans in the group,
• R is the reset time, in years, of the variable mortgage rate.
We furthermore assume a uniform default value haircut equal to 25% and that there are nopre-payments. Any recoveries in the event of default are invested in Treasury.To simulate future cash flows from the portfolio we also need the following input.
• Today’s term-structure of mortgage financing interest rates and scenarios for forwardterm-structures to obtain pool specific mortgage rates (pool-specific due to pool-specificreset time and marginals) and discount factors, hence realized discounted cash flowsconditional on no default11.
• Scenarios for systematic risk factors and the collateral value multipliers.
• Parameter matrices linking a ”creditindex” to transition probabilities, hence transform-ing scenarios for the creditindex into scenarios for the transition matrix.
11We define the forward mortgage rate at reset time as the relevant mortgage financing rate on the simulatedforward term-structure plus the marginal. The marginals are assumed fixed here but in practice we mightemploy expertise judgement on the expected future evolution of marginals (e.g., as multiplicative factors) anduse these as an input to the simulation.
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We define a creditindex for a pool as the standardized weighted sum of the pool-specificmortgage rate and a single systematic risk factor12. The systematic risk factor is common to allpools and can be interpreted as an output gap measure. Both the nodes on the term-structureand the systematic variable are assumed to follow mean-reverting processes. Consideringcollateral value evolution each of the value growth models is described by a log-normal processstarting at 1. Univariate model parameters as well as correlations are calibrated on historicaldata covering two business cycles and scenarios are obtained.
To obtain corresponding scenarios for the transition matrices we filter the creditindex sce-narios through the multinomial logit transition matrix model. This filtering process requirescalibrated parameter matrices for the transition matrix models. Specifically, denote by B1
the K − 1 × K − 1 constant parameter matrix. Further, denote by B1 the correspondingK− 1×K− 1 parameter matrix linking the values of the creditindices to transition probabil-ities. As previously described we can, if sufficient rating classification history is available, usequantitative methods to calibrate these parameter matrices. Otherwise we need a combinationof quantitative and qualitative methods. Here we consider the case when we only have twodistinct rating periods available. In this case the constant parameters of the multinomial logitmodel i.e., B1 is defined so that the current transition matrix for the groups are replicated(i.e., we solve (3) for T = 1 using only the constant parameters). Table 2 displays the currenttransition matrix used.
The sensitivity parameter matrix B1 is qualitatively calibrated in the sense that the outputin terms of actual transition matrices under high, middle and low values of the creditindex isconsistent with our prior beliefs. This approach to calibration of sensitivity matrices is thesame as the methodology used to calibrate large-scale econometric systems, where parametersare set so that responses to shocks in the system, agrees with our a priori beliefs. Table 3displays the sensitivity parameter matrix.
After the scenarios have been defined we calculate the present value distribution of assets,the net present value distribution of assets and liabilities and the loss of notional distributionof the portfolio. We furthermore apply performance and risk measures to the obtained dis-tributions. This is done for several risk horizons, specifically, 1 year, 5 years, 10 years andmaturity of the portfolio. In the calculations we apply, due to their sizes, the law of largenumbers to group 2, 3, 4, 6, 9 and 10. Tables 4, 5, 6 and 7 contains the results. All values inthe tables are in BnSEK and the notation used in the tables is explained below.
• M PV VaR is the assets present value risk contribution using VaR as the measure ofrisk (see section 3 for the computation of this) at the 99 percent confidence level,
• M PV CVaR is the assets present value risk contribution using CVaR as the measureof risk (see section 3 for the computation of this) at the 99 percent confidence level,
• Mean PV is the assets mean present value of the cash flows under the set of scenarios(corresponding to Fair Value of the portfolio),
12More specifically, we define the creditindex for a multivariate normal vector y with (unconditional) meanµ as
w′Σ−12 (y − µ)
where w is a vector obtained by taking the element by element square root of an index weight vector and Σ−12
is the Cholesky decomposition of the inverse (unconditional) covariance matrix of y.
20
• M Net PV VaR is the net (assets - liabilities) present value risk contribution usingVaR as the measure of risk at the 99 percent confidence level,
• M Net PV CVaR is the net (assets - liabilities) present value risk contribution usingCVaR as the measure of risk at the 99 percent confidence level,
• Mean Net PV is the net (assets - liabilities) mean present value of the cash flowsunder the set of scenarios (corresponding to a net Fair Value of the portfolio),
• M CumLoss VaR is the assets cumulative loss of notional risk contribution using VaRas the measure of risk at the 99 percent confidence level,
• M CumLoss CVaR is the assets cumulative loss of notional risk contribution usingCVaR as the measure of risk at the 99 percent confidence level,
• Mean CumLoss is the assets cumulative mean loss of notional.
From the tables the reader notice that from the perspective of a negative net value the portfoliois of exceptional quality under the current scenarios. In fact the only risky pool is number 10and pool 10 is the only pool which has a negative risk contribution to net value. This negativerisk contribution occurs at risk horizon maturity and with CVaR as risk measure. Consideringthis risk horizon and that risk measure, pool 10 therefore requires economic capital. However,as is clear from an inspection of Table 7, although pool 10 requires economic capital the otherpools are sufficiently overpriced in order to hedge the economic capital required by the pool.That is, their economic capital (M Net PV CVaR) contribution is negative (positive numbers).Therefore in total the amount of economic capital required for the portfolio at all horizons,given the CVaR risk measure and the current scenarios, is 0. Still the historically calibratedMonte-Carlo scenarios have to be complemented with expertise and stress scenarios.
To obtain the minimum fixed (theoretical) marginals discussed in section 3.2 we proceedas follows. First we decompose the cumulative expected losses into incremental losses in orderto get the approximate annual incremental expected loss. Secondly we decompose the choiceof net risk contributions i.e., M Net PV VaR or M Net PV CVaR into annual incremental netrisk contributions. The term-structure of (break-even) spreads obtained should then, in orderto find out if the pools are properly priced or not, be compared with the actual marginals forthe pools (see Table 1). Also from this perspective it is clear that the portfolio is, under thecurrent scenarios, of exceptional quality, and that the pools are being overpriced rather thanunderpriced e.g., Mean Net PV is positive for all pools at all risk horizons.
Finally, Table 8 shows the portfolio cumulative expected loss of notional in percentage ofinitial notional vs. Moody’s idealized cumulative expected loss rates for rating class Baa2and Baa3. Here Moody’s idealized cumulative expected loss rates are obtained from Moody’sspecial report The Lognormal Method Applied to ABS Analysis (2000) and are used heremerely as an illustration. From the table the reader notices that the total portfolio qualifiesfor at least a Moody’s Baa3 rating at the horizons considered. In general the horizon considereddepends on the ”average life” of the tranche under consideration. If a stronger rating is desiredfor the total portfolio one might make use of some of the excess spreads (i.e., the marginals)in a first-loss piece reserve account. An alternative approach seeks to upgrade the rating of a
21
subset of the portfolio by construction of tranches. Still, to upgrade the bottom tranches inthe hierarchy a first-loss piece reserve account is necessary.
6 References
• Fabozzi, (1997), ’Fixed Income Mathematics ’, McGraw-Hill.
• Glasserman, (2003), ’Monte Carlo Methods in Financial Engineering ’, Springer Verlag.
• Mausser and Rosen, (1998), ’Beyond VaR: From Measuring Risk to Managing Risk ’,Algo Research Quarterly Vol. 1, No. 2.
• Moody’s special report, (1996), ’The Binomial Expansion Method Applied to CBO/CLOAnalysis ’.
• Moody’s special report, (2000), ’The Lognormal Method Applied to ABS Analysis ’.
Table 8: Portfolio cumulative expected loss vs. Moody’s idealized cumulative expected lossrates
Horizon Cumulative expected loss rate Moody’s treshold Baa2 Moody’s treshold Baa31 year 0.206% 0.0935% 0.231%5 years 0.961% 0.869% 1.6775%10 years 1.499% 1.980% 3.355%Maturity 1.667% NA NA
