valuing cdos of bespoke portfolios with implied multi ... · portfolio, such as the itraxx or cdx...

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Research Paper Implied Multi-Factor Models

Valuing CDOs of Bespoke Portfolios with

Implied Multi-Factor Models 1, 2

Dan Rosen 3 and David Saunders 4

First version: October 4th, 2006

This Version: April 5th, 2007

The authors gratefully acknowledge financial support from a research grant by Algorithmics, Inc. Additional financial support from R2 Financial Technologies (DR, DS), The Fields Institute (DR), NSERC (DS) and the University of Waterloo (DS) is gratefully acknowledged.

2 The authors would like to thank Ben De Prisco, Alex Kreinin Philippe Rouanet, Roger Stein, and the Fitch Academic Advisory Board for many insightful comments and suggestions, and Zhuqing Jin for research assistance. Further thanks are extended to participants of the Fitch academic Advisory Board meeting (October 2006), RiskUSA (November 2006), ICBI Risk Conference (December 2006), Fitch Ratings Seminar (January 2007), Carnegie Mellon Probability and Math. Finance Seminar (March 2007), Moody’s Seminar (March 2007), and The Fields Institute Quantitative Finance Seminars (March 2007), where this paper has been presented.

3 Corresponding author. The Fields Institute for Research in Mathematical Sciences and R2 Financial Technologies, Toronto Canada. [email protected]

4 University of Waterloo, Canada. [email protected].

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Abstract

This paper presents a robust and practical CDO valuation framework based on the application of multi-factor credit models in conjunction with weighted Monte Carlo techniques used in options pricing. The framework produces arbitrage-free prices and can be seen as an extension to the implied copula methodology of Hull and White (2006). We demonstrate the practical advantages of working through multi-factor models, rather than directly on a common hazard rate (or a set of them), to value consistently CDOs of bespoke portfolios, CDO-squared and cash CDOs.

The multi-factor credit models which determine the codependence of obligor defaults are defined generally within the mathematical construction of Generalized Linear Mixed Models (GLMMs). The implied copula approach can be seen as a special case of a GLMM, as are other common credit portfolio models. For a given model, the quoted prices of various credit portfolio instruments, such as CDO tranches, are used to imply the “risk-neutral” distributions (or processes) for the underlying systematic risk factors, which drive joint obligor defaults. We solve numerically the inverse problem of implying the factors’ joint distribution, by creating first discrete scenarios on the factors. Although standard quadrature points may be used for low dimensions, more realistic problems require Monte Carlo simulation. We describe various numerical techniques for effectively sampling factor scenarios and obtaining well behaved factor distributions as the solutions from the optimization problem. While this paper focuses on a static version of the model (by defining the codependence of default times), the framework is general and can be potentially extended to a dynamic setting.

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1 Introduction

The overwhelming growth of the CDO market is driving significant innovations in the modelling and pricing of portfolio credit derivatives, as can be witnessed by the large literature on the subject in recent years. The current standard model for pricing synthetic CDOs is commonly referred to as the single-factor Gaussian copula model (Li, 2001). As the model with a single parameter does not simultaneously match the prices of all quoted tranches, it is common practice to quote an implied “correlation skew” – a different correlation which matches the price of each tranche. Implied tranche correlations not only suffer from interpretation problems, but they might not be unique and cannot be interpolated to price other tranches with different attachment points. An alternative to tranche correlations is to use base correlations, the implied correlations of various equity tranches with different detachment points (which coincide with the quoted tranche prices). A mezzanine or senior tranche can then be simply expressed as the difference between two equity tranches (with different detachment points). Base correlations resolve calibration and interpolation issues encountered with tranche correlations. However, they are not guaranteed to yield arbitrage free prices and they can they can be even more difficult to interpret than tranche correlations. Alternative approaches to the single-factor Gaussian model include more sophisticated (multi-parameter) copula models and loss processes (e.g. Burtschell et al. 2005, Hull and White 2005, Joshi & Stacy 2006), implied distribution static models (Hull and White 2006, Walker 2006), and dynamic (top-down) models (e.g. Sidenius et al. 2006, Schönbucher 2006).

Once a model has been calibrated to observed tranche prices of a given reference market portfolio, such as the iTraxx or CDX indices (and produced an implied correlation skew), the model is used to price new “bespoke” CDO tranches, whose prices are unobserved in the market. We talk about bespoke tranches when

1. The underlying portfolio and maturity is the same as the reference, but the tranche attachment and/or detachment points are different

2. The underlying portfolio is the same but the maturity of the portfolio is different from that of observed references

3. The underlying portfolio differs from the reference

Commonly, cases (1) and (2) are treated through some standard interpolation or extrapolation methods. In the base correlation framework, case (3) is commonly treated by “mapping” the new CDO to a reference CDO with “similar risk” (e.g. St. Pierre et al. 2004). In this sense, the idea is to scale the correlation skew of the model to fit the “riskiness” of the new portfolio. A common mapping methodology is based on expected tranche losses of the underlying and reference portfolios. While mathematically consistent, the bespoke portfolio mapping methodologies are ad-hoc and often produce unsatisfactory results. Furthermore, they generally do not account for systematic (sector) risk concentrations and further compound the drawbacks of base correlation approaches.

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The difficulty of extending a model to price bespoke portfolios is not exclusive to copula methods. Top-down approaches are not designed to price bespoke portfolios, since they directly model the loss distribution of a given index. Applications of other models to price bespoke portfolios are generally ad-hoc in nature as well, as explicitly acknowledged by their authors (e.g. Hull and White 2006, and Joshi & Stacy 2006).

This paper presents a robust and practical CDO valuation framework based on the application of multi-factor credit models in conjunction with weighted Monte Carlo techniques used in options pricing. The framework produces arbitrage-free prices and can be seen as an extension to the implied copula methodology of Hull and White (2006), where sector concentrations of bespoke portfolios are explicitly modelled using a multifactor credit model. We demonstrate the practical advantages of working through multifactor models, rather than directly on a common hazard rate scenario (or a set of them), to value consistently CDOs of bespoke portfolios, CDO-squared and cash CDOs.

We use the mathematical construction of Generalized Linear Mixed Models (GLMMs) to define the underlying multi-factor credit models which determine the codependence of obligor defaults (e.g. McNeil and Wendin, 2006). The implied copula approach can be seen as a special case of a GLMM, as are extensions of common credit portfolio models such as CreditMetrics (Gaussian copula) or Credit Risk+ (mixed Poisson). In this sense, the proposed approach also permits more general calibration of factor models for risk purposes (and not only CDO valuation).

For a given multi-factor model, we can think of the quoted prices of given credit portfolio instruments, such as CDO tranches, as providing information on the implied “risk-neutral” distributions (or processes) for the underlying systematic risk factors, which drive joint obligor defaults. We can solve numerically the inverse problem of implying the factors’ joint distribution from observed market prices, by creating first discrete scenarios on the joint outcome of the factors. Although standard quadrature points may be used for low dimensions, more realistic high-dimensional problems require Monte Carlo simulation. We describe various numerical techniques for effectively sampling systematic factor scenarios (discretizing the problem), computing the convolution on each scenario, and obtaining well behaved factor distributions as the solutions from the optimization problem.

In this paper, we focus on a static version of the model (by defining the codependence of default times), and present a numerical example using a multi-factor Gaussian copula model as a staring point. However, more general GLMMs can be used and the model can be formulated also in a dynamic setting. The latter case results in a substantial increase in computational effort and is the subject of future work.

The rest of the paper is organized as follows. The second section provides background on CDO pricing, implied and base correlations, and algorithms for pricing bespoke portfolio CDO tranches (EL mapping and the Hull-White implied copula method). The third section reviews factor models of credit risk, presents the implied factor model method for pricing bespoke portfolio CDOs, and discusses numerical and modelling issues in using

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the method. The fourth section presents examples using implied factor models to price bespoke portfolio CDOs based on market data. The fifth section concludes and discusses directions for future work.

2 Background – pricing bespoke CDOs

This section briefly reviews the background material necessary for understanding the pricing of bespoke CDO tranches. We begin by reviewing the basic structure of CDOs, and presenting the general pricing equations. This is followed by a brief discussion of standard tranches, and bespoke tranches of CDOs. We then review the Gaussian copula model, which is the market standard against which other methods are compared. We also discuss the market practice of inverting the model to produce “base correlations” for standard CDO tranches implied from market data. Finally, we review the most popular methods for determining prices of bespoke CDO tranches; the “EL Mapping” approach that serves as a benchmark for other methods, and the Hull-White (2006) “Implied Copula” method.

2.1 Synthetic CDOs

A synthetic CDO is constructed on an underlying pool of credit default swaps. The pool is divided into “tranches”, which are separated into a hierarchy according to the order in which losses from the CDS pool are assigned. Tranches are defined by their attachment point and their detachment point (sometimes also called the lower and upper attachment points respectively). Table 1 presents an example of attachment and detachment points of a sample Synthetic CDO:

Tranche Name Attachment Point Detachment Point Equity 0% 3% First Mezzanine 3% 7% Second Mezzanine 7% 10% Senior 10% 15% Super Senior 15% 30%

Table 1. Example of CDO Tranches

The size of the tranche is its total notional value, given by the equation

S = N U − N A = N ⋅ (U − A )n n n n n

where N is the total portfolio notional, and An and Un are the tranche attachment and detachment points respectively.

Credit protection is bought and sold on each tranche separately in the following way.

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Payments on the premium leg of the CDO are made at a pre-defined set of times, which we denote by tj, j = 1,...,T. At each time, payments are made by the protection buyer to the protection seller of a fixed percentage (the spread) multiplied by the remaining tranche notional (i.e. the original tranche notional less the cumulative losses that have been assigned to the tranche up to the payment date). In return, any losses on the tranche are paid by the protection seller to the protection buyer. Thus, the payment made by the protection buyer to the protection seller at time tj is sn(tj-tj-1)·(Sn - Ln,j) , where sn is the tranche spread, and Ln,j is the cumulative loss on the nth tranche up to time tj. As mentioned above, losses are assigned to different tranches according to their seniority. In particular, if we denote by Lj the cumulative loss of the entire portfolio up to time tj then:

Ln, j = min( S ,max( Lj − Aj )) 0, n

That is, losses start to be assigned to the nth tranche once portfolio losses exceed the tranche’s attachment point, and continue to be assigned (exclusively) to that tranche until the total losses on the portfolio exceed the tranche’s detachment point, at which time the tranche is exhausted and all payments on the CDO cease. The protection seller pays to the protection buyer at time tj the losses that have been assigned to the tranche during the time interval (tj-1,tj), which are given by the difference in the cumulative tranche losses between time j and time j -1 , Ln,j – Ln,j-1.

The price of each set of payments is calculated by computing the present value of its expectation under the risk-neutral probability measure (specific methods for computing these expectations are discussed below). To simplify the expressions for these expectations, denote by Dj the discount factor for present valuing cash flows at time ti. The price of all the payments made by the protection buyer (the premium leg) is:

T T

PV Buy = ∑ s (t j − t j−1) E D Q [S − L ] = ∑ s (t j − t j−1)Dj ⋅ (S − EQ [L ]) n j n n, j n n n, j j=1 j=1

While the price of all payments made by the protection seller (protection leg) is:

T

PV Sell = ∑Dj (EQ [L ] − EQ [L ]) n, j n, j−1 j=1

The value of the CDO tranche to the protection buyer is PVSell – PVBuy , while the value of the CDO the protection seller is PVBuy – PVSell. When the CDO is initiated, the value of the tranche spread sn is set in order to make the price of the tranche equal to zero, i.e. in order to make PVBuy = PVSell. A bit of simple algebra leads to the expression:

T

∑Dj (EQ [L ] − EQ [L ]) n, j n, j −1 j =1 s = n T

∑ (t j − t j −1)D ⋅ (S − EQ [L ]) j n n, j j=1

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The equity (0-3%) tranche bears the first losses incurred by the portfolio, and thus would have a very high spread if quoted as above. To make the equity tranche more attractive (and reduce the chance of default on the part of the protection seller, whose losses are expected to occur relatively quickly), the structure of the protection buyer’s payments is modified. The spread is fixed at 500 basis points (5%), and the protection buyer makes an upfront payment of a fixed percentage of the tranche notional, referred to as the upfront percent. The structure of the payments made by the protection buyer thus becomes:

T

PV Buy = UFP ⋅ N ⋅ (U − A ) + ∑ .0 05 ⋅ (t j − t j−1) E D Q [S − L ]n n j n n, j j=1

where UFP is the upfront percent. Since the spread is fixed at 500 basis points, it cannot be adjusted to make PVBuy = PVSell. For equity tranches, it is instead the upfront percent that is adjusted to make the initial tranche price zero (so market prices of equity tranches are quoted in terms of the upfront percent, and we will do so below). Again, solving for the value of the upfront percent that makes PVBuy = PVSell gives

T

∑Dj (EQ [L ] − EQ [L ] − .0 ( 05 t j − t j−1)EQ [Sn − Ln, j ]) n, j n, j−1

UFP = j=1

N ⋅ (U − A )n n

2.2 Gaussian copula model and base correlations

In this section, we review the standard Gaussian copula model and the notion of base correlations, which are implied from market quotes of CDO tranches. The Gaussian copula model provides a method for computing the expected tranche losses E[Ln,j] needed in order to compute CDO tranche prices.

We assume that we are given the marginal distributions (under the risk-neutral measure) of the default times of each of the names in the portfolio underlying the CDO. We denote by τi the default time of the i-th name in the portfolio, and by Fi we denote its cumulative probability distribution function. Thus:

) ( = Q(τ ≤ t)t F ii

In practice Fi is obtained by calibrating Fi (tj) to market prices of credit default swaps for a fixed set of times tj and then interpolating to find Fi (t).

In order to specify the full portfolio loss distribution, we need the codependence structure (copula) between the different default times. We obtain this by assigning a Gaussian copula to the default times in the following way. For each name i, we define a creditworthiness index:

K K 2Yi = ∑β Z + 1− ∑βik ⋅εik k i

k =1 k =1

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where the random variables Zk and εi have standard normal distributions and are mutually independent so that Yi has a standard normal distribution. The random variables Zk are referred to as systematic factors, and the coefficients βik are the factor loadings. The systematic factors often can be interpreted as representing the health of the economy as a whole, as well as the performance of specific sectors. The default time of name i is then defined to be

− 1(τ = Fi Φ (Y )) i i

This ensures that τi has the correct marginal default distribution. Defaults between two names are more highly correlated if their loadings are large (and positive). The cumulative portfolio loss at time tj is:

Lj = ∑wi1{Y Φ ≤ − 1( Fij )} i i

where wi is the (loss-given-default adjusted) exposure to name i, and we have used the notational simplification Fij = Fi(tj). The model described above is referred to as the Gaussian copula model for portfolio credit risk. When K > 1, it is called a multi-factor model, while when K=1, it is called a single-factor model.

As shown later, it is important to compute the conditional distribution of losses given the values of the factors (Z1,...,ZK) = Z. The conditional distribution of Y given Z is normal:

K y − ∑β Z

k = 1 ik k

Y Q i ≤ y | Z ] Φ = K[

2 1− ∑β ik k = 1

This distribution is particularly easy to work with in the single factor case, where many analytic results are available.

Methods for computing expected tranche losses often follow a two step procedure (see, for example, Andersen et al. 2003, 2004 ; De Prisco et al., 2005; Gregory and Laurent, 2002; Li and Liang, 2005; Okunev, 2005):

1. Computation of the conditional expected tranche loss given a particularrealization of the systematic factor scenario, E[Ln,j | Z ].

2. Computation of the unconditional expectation by integrating the conditional expected losses over the marginal distribution of the systematic factors.

2.2.1 Base correlations

Observed market prices of CDO tranches cannot be reproduced with a basic single-factor Gaussian copula model. Nonetheless, since this model is so easy to work with, it forms a basis of comparison for the prices produced by other models. The situation is analogous to the use of Black-Scholes implied volatilities to quote equity option prices. Tranche

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prices are often given in terms of their implied correlations. The implied correlation of a CDO tranche is the value of the factor correlation in the single factor Gaussian copula model (with all names assumed to have the same factor correlation) that makes the model reproduce the market price. Typically, different tranches will have different implied correlations, producing a correlation smile much like the volatility smile observed in equity options markets.

However, there are some difficulties with implied correlations which do not arise for Black-Scholes implied volatilities. In particular, since CDO prices for non-equity tranches are not monotonic functions of the correlation, one can sometimes find that there are multiple solutions to the equation defining the implied correlation. In such a situation, there is no obvious method for choosing which of the multiple implied correlations is the correct one. Also there is no obvious interpolation method for other tranches with different attachment and detachment points. A resolution to these difficulties was proposed in McGinty and Ahluwalia (2004). Rather than calculating the implied correlation of each market tranche, one instead calculates the correlation of each base tranche. The nth base tranche is the (fictitious) equity tranche with attachment point 0 and detachment point Un. That is, its detachment point is the same as the nth detachment point of the CDO, but its attachment point is 0 (the base).

Cumulative tranche losses can be calculated from cumulative base tranche losses in an obvious manner. If we denote the cumulative loss of the nth base tranche until time tj by LB then: n, j

BL ,1 j = L ,1 j

L B B n, j = Ln, j − Ln− ,1 j , n ≥ 2

The base correlation curve is the function mapping the upper attachment point U to its base correlation ρ(U). All other values on the base correlation curve need to be computed by interpolation. Examples of base correlation curves are given in the section on Index CDOs below.

2.2.2 Model extensions to Gaussian copula

Additional CDO model enhancements to the standard single-factor Gaussian copula model include:

• Single-factor copula models – these models basically attempt to fit quoted tranche prices by using a more sophisticated copula than the Gaussian, perhaps with more parameters (though not necessarily factors), as well. An example of this is the double t-copula (Hull and White 2005).

• Multi-parameter and multi-factor copula models – these models provide a more flexible correlation structure by allowing for different correlation parameters for each obligor within the single-factor model, or even using a multi-factor copula model to drive credit correlations.

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• Advanced static models – still within a static framework (i.e. no description of the underlying processes driving the portfolio loss distributions), these models attempt to fit the price of observed tranches using (generally, non-parametric) descriptions of the portfolio loss distributions, without an explicit functional definition of the underlying copula. Examples of these include the “implied copula model” (Hull and White, 2006a) and the term structure model of Walker (2006).

• Dynamic and term structure models – the copula models can be described as “static” models, since they do not describe the dynamic evolution of the underlying credit events. More advanced term structure and dynamic models directly model the underlying processes driving the portfolio losses over time. These models can be classified generally into:

o Top-down approaches – where a process for the portfolio loss distribution is modelled directly, without using any information on the underlying credits (Schönbucher 2006, Sidenius et al. 2006, Hull and White 2006b). This is akin to modelling, for example, an equity or bond index process directly from observable quotes on the index level and various options traded. For a given reference portfolio, these models guarantee consistent prices and hedge ratios of all tranches and maturities, but do not provide direct information on their relationship to the underlying securities or to other underlying portfolios.

o Bottom-up approaches – these models start with the joint processes describing the evolution of credit events of the underlying names in the portfolio and build the portfolio loss distribution over time from these. Typically these models require MC simulation. Examples of this type of model are given in Duffie and Garlenau, 2001 (for a reduced form model) or Hull, Predescu and White, 2006 (for a structural type model).

It is worth mentioning that, while their theoretical appeal and potential are high, the application and calibration of dynamic models in practice is still fairly recent and exploratory, as even acknowledged by some of the authors. At this point, we do not know of general examples of these models with practical calibrations.

2.3 The implied copula approach

Hull and White (2006a) introduced an alternative method to price CDOs. Although the method’s name has the term “copula”, it is very different in nature from the copula methods. The basic idea is similar to that used in methods for implying risk-neutral distributions of stock prices based on observed prices of derivative securities. Specifically, it assumes a set of scenarios λi for the hazard rates of all the names in the portfolio. Given a particular value for the hazard rates, the distribution of the default of each name is given by:

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i (t F | λ ) = Q(τ ≤ t | λ ) = 1 − exp( −λ t)k k k

If we denote by Vn(λk) the value of the nth tranche given that the hazard rate λ = λk, and by pk the probability that λ = λk, then to match current market prices, we should have

k ∑

∑

V p (λ ) = 0k n k

p = 1 (1) k k

p ≥ 0 for all kk

The authors are able to match quoted index prices, assuming that losses are given by the default mechanism described above, and assuming a random recovery rate of the form estimated by Hamilton et al. (2005):

(Q R ) = max( .0 52 − 9.6 Q ) 0 ), 1(

The values of λk are selected to produce evenly spaced values of the sum of the tranche prices between the minimum defined by λ=0, and the maximum defined at λ=λmax for some pre-chosen value λmax. In general, there are many possible solutions p to the above system. The final formulation chooses the “smoothest” set of implied probabilities by minimizing the discrete approximation to the integral of the squared second derivative:

−

∑ L 1

C = ( p − 2 p + pk −1)k +1 k 2

(λ − λk −1)k +1k 2=

The implied copula method is developed originally when a homogeneous portfolio is assumed and thus a single hazard rate describes all names. The authors present some extensions for the case when the portfolio is inhomogeneous and for a bespoke portfolio (see section 2.4.2. below).

2.4 Methods for pricing bespoke tranches

The growing popularity of the CDO market has led to the trading of standard synthetic single tranche CDOs on standard credit default swap indices. These liquid, standardized tranches have greatly increased the activity and liquidity in the CDO market. The CDS indices pool together liquid CDS contracts to form the reference portfolio underlying the CDO tranches. These indices include the Dow Jones CDX indices on North American and emerging market names, as well as the iTraxx European and Asian indices. For each index, there is a standard set of CDO tranches that are traded in the market. These tranches are defined by a set of attachment and detachment points, which may differ by index. For example, Table 1 presents the standard tranches on the CDX North American 5 Year Investment Grade Index. For more information on the standard CDS indices, and standard CDO tranches on them, see Bluhm and Overbeck (2006), www.djindexes.com and www.intindexco.com .

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http://www.djindexes.com/

http://www.intindexco.com/


Any CDO tranche that is not one of the standard tranches on one of the standard index portfolios is referred to as a bespoke tranche. If the tranche is on one of the standard credit default swap indices, and only differs due to the attachment points (or times to maturity), then it can usually be priced simply by interpolating base correlations. The more challenging case is when the portfolio underlying the CDO is different from the standard CDS indices.

In this section, we briefly review the most popular method for pricing bespoke tranches of CDOs: the EL Mapping developed by researchers at Bear Stearns. In addition, we briefly discuss the treatment of bespoke portfolios in the Hull-White (2006a) “Implied Copula” method.

2.4.1 EL Mapping

The EL (Expected Loss) mapping was introduced by St. Pierre et al. (2004) to provide a simple method for pricing bespoke CDO tranches. It takes a relatively simple approach, which has become the market standard and can be used as a benchmark for other methods. The basic idea is to transform the base correlation curve of a standard portfolio into a base correlation curve for the bespoke portfolio. The algorithm involves preserving the same correlation levels, but calculating a new set of upper attachment points. This is shown in Figure 1.

ρ(U) Index Base Correlation Curve

Bespoke Base Correlation Curve

3% 7% 10% 15% 30% U

Figure 1. Base correlation mapping for bespoke portfolios

In the market, only the base correlations corresponding to the standard upper attachment points are observed directly. All others are calculated through interpolation. For example, market data for the CDXIG provides the points (Un, ρ(Un)), for Un=3%, 7%, 10%, 15%, 30% for the base index portfolio. The EL mapping transforms these into points (Un*, ρ(Un*)=ρ(Un)) on the base correlation curve for the bespoke portfolio. Notice that the correlation remains the same, but the tranche detachment point is changed. Once the new

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detachment points have been calculated, base correlations for any other detachment points are computed by interpolation. The new values for the upper attachment points corresponding to the market base correlations for the bespoke portfolio are calculated by solving an equation of the form:

( *, U , ρ (U )) = U P S , ρ(U )) P S * ( ,n n n n

where S is a given “risk statistic”, and calculations are done under the single-factor Gaussian copula model, with all names having correlation ρ(Un) with the one systematic factor Z. Here, the index portfolio is denoted by P, the bespoke portfolio by P*, and the superscript * denotes values for the bespoke portfolio. Notice that only the tranche detachment point, and not the correlation differs on both sides of the equation. The right-hand side of the above equation then gives the value of the risk statistic for the index portfolio, calculated using a single-factor Gaussian copula with all factor correlations equal to the tranche base correlation. The left-hand side is the same risk statistic again, computed for the bespoke portfolio using a single-factor Gaussian copula with all factor correlations equal to the index tranche base correlation. The upper attachment point Un * is varied to produce equality, with the idea being that tranches with similar risk should have equal base correlations. The most commonly used risk statistic is the tranche expected loss divided by the expected portfolio loss (hence the name ‘EL Mapping’), however other risk statistics have been considered, including probability of tranche exhaustion, and simple tranche size.

By using only the first moment of portfolio losses, the EL mapping does not account for concentration risk (or the correlation between all the underlying names in the portfolio). Two portfolios with the same EL characteristics have the same “implied correlation skew” regardless of their composition (for example one could be fully concentrated in one sector while the other one might be diversified in various sectors, and thus intuitively have a lower correlation). Rosen and Saunders (2006) present a simple adjustment to this mapping, which accounts for the differences in concentration risk of the index and the portfolio. While this “concentration-adjusted mapping” explicitly incorporates systematic risk into the pricing of bespoke portfolios, it suffers from the same limitations as the models it extends, and is more effectively used as a communication (and perhaps didactic) tool.

2.4.2 Bespoke portfolios in the Implied Copula method

In this section, we review the treatment of bespoke portfolios in Hull and White (2006a).

The authors first consider the situation where the bespoke portfolio is “as well diversified” as the portfolio underlying the reference index (iTraxx or CDX). Assuming a homogeneous approximation for both the index and bespoke portfolios, one can calibrate the hazard rate probabilities of the index portfolio to the prices of standard tranches using the algorithm described in section 2.3. This yields a probability πk for the hazard rate λk. Hazard rates are then modified for the bespoke portfolio as follows:

*λ = βλkk

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where β is chosen so that the average CDS spread for names in the bespoke portfolio is matched (the probabilities πk for the set of bespoke hazard rate scenarios are taken to be the same as the implied probabilities for the corresponding index hazard rate scenarios). A more sophisticated approach where CDS spreads for all names in the bespoke portfolio are matched when implied probabilities are calculated is also considered.

According to the authors “dealing with portfolios that are less (or more) well diversified than the index requires some judgment and, whether the Gaussian copula/base correlation or implied copula approach is used, is inevitably somewhat ad hoc.” In this case, the authors suggest the following as one possible approach. First compute the average pairwise equity correlations between names in the index and bespoke portfolios, denoted by y and y* respectively. Then, compute the correlation ρ of an “equivalent Gaussian copula” consistent with the particular implied copula for the index. Finally, increase the dispersion of hazard rate paths for each obligor (while maintaining CDS spreads), in such a way that the results are consistent when ρ is replaced by ρ y/y*.

3 Implied multi-factor models with weighted Monte Carlo

In this section we introduce a CDO valuation framework based on the application of multi-factor credit models in conjunction with weighted Monte Carlo techniques. The framework produces arbitrage-free prices and can be seen as an extension to the implied copula methodology of Hull and White (2006). Working through multi-factor models rather than directly on a common hazard rate (or a set of them) allows a straightforward and consistent valuation of CDOs of bespoke portfolios, CDO-squared and cash CDOs.

Within a given multi-factor model, the quoted prices of various credit portfolio instruments, such as CDO tranches, are used to imply the “risk-neutral” distributions (or processes) for the underlying systematic risk factors, which drive joint obligor defaults. We formulate an optimization problem to find the “best” implied risk-neutral probabilities for a set of simulated factor scenarios, subject to the constraint that the probabilities reproduce market prices of liquid credit derivatives (perhaps within a given level of accuracy). The method can be applied to any credit risk model that has a “factor structure”, and we focus on the class of Generalized Linear Mixed Models (GLMMs) (c.f. McNeil and Wendin, 2006). We present the algorithm and then discuss some technical details of the implementation.

3.1 Generalized Linear Mixed Models for credit risk

We begin by reviewing the mathematical structure of Generalized Linear Mixed Models (GLMMs), and discussing their application to credit risk.5 We consider a set of systematic factors, which can in general be time varying and which we denote by:

Z t t= (Z t ,..., ZK )1

5 Note that McNeil and Wendin (2006) focus on the statistical estimation of the parameters of GLMMs for credit risk. Here, we take the parameters as given and focus on the mathematical structure of the models, and the resulting loss distributions once parameters are known.

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as in the above exposition of the Gaussian copula model (although here we do not necessarily assume normally distributed factors). For each obligor i we denote its credit variable for time t by Uit. A GLMM is specified by:

1) A set of parameters with constant terms ait, and factor loadings:

t tbi = (bt ,..., biK )i1

for each obligor.

2) A probability distribution from the exponential family, and a link function h, such that, conditional on Zt , Uit has the given distribution, with mean:

U E it | Z ] = a h Z b k ) (2) [ ( it ik + ∑k

K

=1

t t

with the variables U*t being conditionally independent given Zt .

Losses on a credit portfolio at time t are then defined to be:

U w it t iL = ∑i

N

=1

where wi is the (loss given default adjusted) exposure to obligor i.

Some comments on the general model framework are in order. First, it is most common for the distribution from the exponential family6 to be the Bernoulli distribution, in which case Uit can be thought of as an indicator variable taking the value one if obligor i defaults before time t and zero otherwise (this is the case in the Gaussian copula model for credit risk, see below). In this case, the conditional expectation in (2) is the conditional default probability of obligor i given the systematic factor Zt and we write:

K t t[ (PDit (Zt ) = U E | Z t ] = a h + ∑k =1

Z b k )it it ik

However, the specification of a Bernoulli distribution is not strictly necessary, and other distributions are indeed used in practice. For example, it is sometimes assumed that the conditional distribution of Uit given Z is Poisson (this is the case with the CreditRisk+ model, see below).

Examples of models that fit into the GLMM framework include the standard Gaussian copula model. In this case, we take the factor vector Z and the coefficient vector b to be time independent, and assume that the conditional distribution of U given Z is Bernoulli (with Uit = 1 indicating default of the ith name before time t). Then the specification h = Φ and

6 For a discussion of the exponential family, see P. McCullagh and J.A. Nelder, “Generalized Linear Models”, Chapman & Hall, 1989. The family includes the Bernoulli, Binomial, Poisson, Normal, Inverse Gaussian and Gamma distributions, among others.

15


Φ −1 (PDit ) β kait = , =bik K 2 2βk1− ∑k =11− ∑k

K

=1 β k

leads to the Gaussian copula model (here PDit denotes the probability that obligor i will default by time t).

The same specification for the parameters, with a different choice for the link function:

1) ( =x h

1( + exp( −x))

leads to the LOGIT model.

Finally, we briefly demonstrate that the CreditRisk+ model, which is a Poisson mixture model, can also be expressed as a GLMM. This is a single period model, and so we suppress the time variable. In this case the factors Zk are independent random variables with the Gamma distribution. Conditional on Z the variables Ui are independent and have the Poisson distribution, with parameter:

λ (Z ) = U E | Z ] = ci ∑k

K

=1 β Zi [ i ik k

where ci > 0 and

βik = ,1 βik ≥ 0∑k

K

=1

This specification clearly fits the form of a GLMM with a linear link function h. We have the conditional default probability:

KPD (Z ) = U P > 1| Z ] = 1 − exp (− λ (Z )) = 1 − exp (− ci ∑ β Zk )i [ i i k =1 ik

We note that in theory there is a positive probability that a given obligor will default more than once in the CreditRisk+ model. However, given the small size of default probabilities for most obligors, this probability is usually negligible in practice.

3.2 Weighted Monte Carlo – general formulation

In this section, we formulate an application of the weighted Monte Carlo method to calculate implied factor models (GLMMs) for pricing portfolio credit derivatives. We take an approach similar to the Implied Copula method (Hull and White, 2006). Rather than working directly with hazard rates and hazard rate scenarios, the weighted Monte Carlo method finds an implied distributions of the systematic factors, assuming as a starting point a GLMM credit risk model with a specified set of parameter values. We can think of this distribution as a “risk neutral” distribution for the systematic factors. An intuitive justification for this approach follows a standard CAPM type argument, where market prices only account for systematic risk, coupled with the application of a

16


conditional independent credit framework (where conditional on a given scenario for the systematic credit factors, obligor defaults are independent).

In this case, we simulate a large number of possible values of the systematic factors Z1,...,ZK according to the prior distribution specified by the GLMM. We note that for any specific set of values for the factors, the expected portfolio loss (and tranche value) conditional on the factor scenario can be computed (often semi-analytically, in the common GLMMs, such as the Gaussian copula, LOGIT, and CreditRisk+ models discussed above). In particular, we find the following:

TnPV Buy (Z ) = EQ [PVBuy | Z ] = ∑ s (t j − t j− 1)Dj ⋅ (S − EQ [L ]) n n n, j

j= 1

TnPV Sell ) ( = EQ [PV | Z ] = ∑Dj (EQ [L | Z ] − EQ [LZ Sell n, j n, j− 1 | Z ])

j= 1

where

EQ [Ln, j | Z ] = ∑wi PDi, j (Z ) i

K − 1Φ (PDi, j ) − ∑ β ,

PD k i Zk

i, j (Z ) = EQ [name i defaults by time t j | Z ] Φ = k = 1 K

2 1 − ∑ β , k i k = 1

We seek a set of implied probabilities qm for the M simulated factors scenarios. We restrict the probabilities to price all CDO index tranches correctly, and also match cumulative default probabilities of individual names (i.e. match CDS prices). In order to be a set of probabilities, the q’s must satisfy:

M

∑q = 1m (3) m= 1

q ≥ ,0 m = 1,..., Mm

The constraints that current market prices of standard CDO tranches are matched are that for each n:

M M n n m )∑q ⋅ PVBuy (Z

m ) = ∑q ⋅ PVSell (Z (4) m m m= 1 m= 1

where Zm represents the vector of simulated values of the systematic risk factors in the mth scenario. Finally, matching default probabilities gives:

M m∑qmPDi, j (Z ) = Fi, j (5)

m= 1

17


where, as before Fi,j is the cumulative (risk-neutral) probability that name i will default before time tj. It is possible that there are many choices of the vector q that satisfy all of the above constraints, and so we seek one that maximizes a given measure of fitness. This leads to the optimization problem:

max q G ) (

subject to: M M

n n∑q PVBuy = ∑q PV for all nm m Sell m= 1 m= 1

M m∑qmPDi, j (Z ) = Fi, j for all i, j

(6)

m= 1

M

∑qm = 1 m= 1

q ≥ 0 for all mm

Different choices for the fitness measure for the probabilities q are possible including:

1. Closest to a prior distribution. Minimizing ║q-p║r for some choice of a prior distribution p and vector norm (e.g. r=1, 2, ∞).

2. Maximum smoothness probability distribution. As in Hull and White (2006), minimize the discretized integral of the squared second derivative

M − 1 2C = ∑ (q − 2q + qm− 1)m+ 1 m

m= 2

M

) ( − = ∑q ln( q )3. Maximum entropy. q G m m m= 1

We observe that in general (8) is a nonlinear (convex) optimization problem with linear constraints, and therefore can be solved efficiently using standard solvers. The constraints on CDO tranche prices should hold for all traded index tranches, while the constraints on default probabilities should hold for all names in both the bespoke and index portfolios. In practice, we have found that variations of the model often result in superior performance; for example dropping some of the constraints corresponding to cumulative probabilities at times before expiration, or turning the “hard constraints” on matching prices and default probabilities into penalized terms in the objective function. See below for discussion of these issues and further implementation details.

While the methodology is, in principle, applicable in a dynamic setting, throughout the rest of this paper we focus on a single maturity CDO and static model for the factors. More specifically, throughout the rest of the paper, we work with a specific GLMM, the standard multi-factor Gaussian copula model.

18


3.3 Computational issues

We briefly discuss some computational considerations in the implementation of the weighted MC method.

3.3.1 Scenario generation – prior distribution and sample points

The first issue is with the selection of the method for generating the prior scenarios. A standard set of scenarios could be generated by transforming a pseudo-random sequence of uniformly distributed random numbers into normally distributed random numbers using standard simulation techniques (e.g. by applying the inverse of the cumulative distribution function). It is well known that this leads to scenario sets where some scenarios “cluster” and does not provide optimal coverage of the probability space (c.f. Glasserman, 2004). This is particularly relevant for the weighted Monte-Carlo algorithm, as its efficiency is dependent on the number of scenarios (see the section on coefficient generation below). An alternative method is to replace the sequence of pseudo-random numbers with a quasi-random sequence, for example Sobol points of the appropriate dimensionality.

Both of above methods generate prior factor scenarios according to a normal distribution, and place equal probabilities on each of the simulated factor scenarios. The weakness of these methods is that they provide the optimizer with insufficiently few scenarios in the tail of the factor distributions to match observed market prices (particularly for senior tranches). It has been consistently observed that implied distributions from the weighted MC algorithm have “fat tails”, with significant excess kurtosis (see examples below). In order to get a good match to market prices, one needs to give the optimizer more freedom in the tail of the factor distributions. This can be accomplished by using Sobol points that are scaled to be uniformly distributed on the cube [-5, 5]K. When normal prior probabilities are required for the objective function of the optimization, they can be calculated using the normal cumulative distribution function (in this case the prior probabilities would not be equal for each scenario).

3.3.2 Computing coefficients of the optimization problem

The bulk of computation time taken by the method is not in solving the optimization problem itself, but rather in computing the coefficients in the linear constraints. The constraints on default probabilities have the form

M m∑qmPDi, j (Z ) = Fi, j

m=1

There is one constraint for each name and each time (frequently, the constraints are only applied at the final time, or a selection of times, see the discussion of the tradeoff between constraint matching and distributional properties below). In the Gaussian copula model, each evaluation of PDi,j(Z

m) requires an evaluation of the cumulative normal distribution function. Generating the coefficients of for these constraints therefore

19


requires NumTimes * NumScenarios * NumObligors calls to this function. The cost of this will depend on the efficiency of the implementation of Φ.

Computation of the coefficients corresponding to matching prices of liquid CDO tranches is more complicated. These constraints have the form:

M M n n m )∑q ⋅ PVBuy (Z

m ) =∑q ⋅ PVSell (Zm m m=1 m=1

The conditional expectation of the present values of the payments corresponding to the buy and sell legs must therefore be calculated for each of the simulated factor scenarios. This requires (at each simulation time) the calculation of expected tranche losses conditional on the realized systematic factors. A number of algorithms exist for performing this calculation, including a recursion developed by Andersen et al. (2003), approximating conditional portfolio losses with a normal distribution (e.g. Li and Liang, 2005) or with a Poisson distribution (e.g. De Prisco et al. 2005), and using full Monte Carlo Simulation.

Determining the trade-off between speed and accuracy for these algorithms, and assessing the impact on the implied risk-neutral factor distributions produced by the weighted Monte-Carlo optimization are important practical considerations. For example, the conditional Normal approximation is fast, but can have problems with accuracy, particularly when valuing equity tranches (see Li and Liang, 2005).

3.4 Price matching and distribution “quality” trade-off

This section discusses various modifications that can be made to the standard form of the weighted MC optimization problem (6).

• Omitting/modifying Constraints. The first modification is to omit some constraints entirely. In particular, in the examples calculated below only the cumulative implied default probability to the end of the lifetime of the CDO is matched. The advantage of dropping constraints is that it loosens the conditions required of the probabilities q, thus enabling a higher value of the objective function G(q) (and consequently a distribution with “superior” qualities). Occasionally, it is even found that attempting to enforce all constraints may lead to an infeasible system. Dropping constraints also reduces the number of coefficients to be calculated, resulting in a substantial reduction in computation time.

• Penalties. Another modification applied in practice is to replace the “hard constraints” with penalized terms in the objective function. This may reflect the fact that market prices cannot be considered perfectly accurate. Particularly for the senior tranches of index STCDOs, the relative illiquidity of the tranche means that price quotes are of limited reliability. Hence, we replace the above optimization problem with the following “penalized form”:

20

2


2M M M − ∑

−Fi, j

N max q G ) ( −∑ ∑qmPV n

Buy −∑ ∑n Sell

m )(C qmPV K qmPD i, j Zi jnn= 1 i j ,, = 1 = 1 m= 1m m

subject to: (7) M

∑q = 1m m= 1

q ≥ 0 for all mm

The additional terms in the objective function replace the default probability and tranche pricing constraints in the original optimization problem. Deviations from equality in these constraints are penalized, since their squares are subtracted from the objective function that is being maximized. We preserve the constraints requiring that q be a probability vector in order to avoid potential arbitrage problems when using the probabilities to price other securities. The coefficients Cn ≥ 0 and Ki,j ≥ 0 give the relative importance of matching the prices of the traded index tranches and the cumulative default probabilities for each time and name respectively. Taking larger values of these coefficients means matching prices and default probabilities more closely, at the expense of having an implied distribution that has a lower value of the fitness measure G(q). Taking lower values emphasizes G(q), while sacrificing the matching of market prices and implied default probabilities. An example showing the trade-off between the “fitness” of the distribution G(q) and the degree of matching of tranche prices is shown below in Figure 10.

Note that alternatively (or in addition) one could consider matching prices within the bid-ask spread (either using hard constraints to bound prices produced by the model, or again using penalization for prices that fall outside the bid-ask spread).

3.5 Interpreting implied distributions

When analyzing the results of an implied factor model, it is important to keep in mind that implied distributions of systematic factors should be considered relative to the prior factor model, and not in isolation. In particular, the mean and standard deviation of implied factor distributions can be strongly influenced by the assumptions on the prior factor loadings βi,k.

To illustrate this point, consider the following simple example. Imagine a completely homogeneous portfolio, with all names having the same default probability PD. Suppose that market prices actually agree with those from a single factor Gaussian copula model with correlation ρM. Percentage portfolio losses conditional on the systematic factor Z will be:

Mρ 1

Z L E ][ | Φ =

,− ρ MPD Z PD Φ = (Y ), Y ~ N

1 , 1− ρ Mρ M ρ M− −

21


If one considers a prior factor model that is the single factor Gaussian copula model with correlation ρP ≠ ρM then one gets the expression for the percentage loss conditional on the systematic factor Z* to be

Pρ 1

L E | Z*] Φ = [

−PD Z *

− ρ P

A simple calculation shows that one will get identical distributions to those given by the above expression using the “true” market correlation and a standard normal Z, if one takes the implied distribution to be normal, but with a different mean and variance:

− 1

P

P

ρ ρ

Z* ~NΦ − 1

⋅

1

Pρ −

, ρ 1( − ρ )M P( )PD

ρ M ρ P ρ M− −1( ) 1( )

4 Example

We demonstrate the valuation techniques on index data as of May 31st 2006. CDS and CDO data are sourced from ValueSpread, and the risk-free yield curve is obtained from Bloomberg. We first present the Index CDO quotes and corresponding base correlations for the CDX and iTraxx indices, and discuss the concentration characteristics of the underlying index portfolios. Thereafter we construct two bespoke portfolios and present the prices and corresponding base correlations using EL mapping as well as the concentration-adjusted mapping (Rosen and Saunders, 2007). Finally we present the results and discuss the calibration of the implied factor model using the weighted Monte Carlo technique.

4.1 CDO Indices

Figures 1 and 2 show the quotes and base correlations implied for the CDX and iTraxx indices at the analysis date. Spreads are quoted for the index as well as for five tranches, across three different maturities (5, 7 and 10 years). Base correlations are obtained using the Poisson approximation in De Prsico et al. (2005). In both cases, a homogeneous portfolio approximation is used. Base correlations are generally increasing with detachment points ranging from around 10% to 60%. In addition, the base correlations obtained by relaxing the portfolio homogeneity assumption (and using each individual CDS spread curve) are presented for the 5y iTraxx index. In this case, the homogeneous portfolio assumption results in slightly higher base correlations.

22

Base Correlation (Poisson)

00.20.4

0.60.8

0 - 3% 0 - 7% 0 - 10% 0 - 15% 0 - 30%

CDXIG5YR CDXIG7YR CDXIG10YR

IndexName Index Price Up FrontSpread (%) 0 - 3% 3 - 7% 7 - 10% 10 - 15% 15 - 30% 0 - 3%

CDXIG5YR 42.04 500 99 21 10.5 5.5 31.81CDXIG7YR 52.00 500 246 46.5 21 7.5 48.69CDXIG10YR 65.00 500 595 118.5 55.5 16 55.63

Tranch spreads

Market Quotes - CDX


00.20.4

0.60.8

0 - 3% 0 - 7% 0 - 10% 0 - 15% 0 - 30%



0

0.1

0.2

0.3

0.4

0.5

0.6

0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22%

ITRAXXEUR5YR ITRAXXEUR7YR ITRAXXEUR10YR

Spread (%) 0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22% 0 -3%ITRAXXEUR5YR 32.32 500.00 70.28 20.00 8.50 4.25 24.03ITRAXXEUR7YR 42.21 500.00 189.13 50.00 27.00 9.00 42.25ITRAXXEUR10YR 53.56 500.00 536.98 124.61 57.83 20.97 51.23

Market Quotes ITRAXX


IndexName Index Price Up Front Spread (%) 0 - 3% 3 - 7% 7 - 10% 10 - 15% 15 - 30% 0 - 3%

CDXIG5YR 42.04 500 99 21 10.5 5.5 31.81 CDXIG7YR 52.00 500 246 46.5 21 7.5 48.69 CDXIG10YR 65.00 500 595 118.5 55.5 16 55.63

Tranch spreads

Market Quotes - CDX

Ba se Correlation (Poisson)

0 0.2 0.4

0.6 0.8

0 - 3% 0 - 7% 0 - 10% 0 - 15% 0 - 30%


Figure 1. CDX index: CDO quotes and base correlations

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22%

ITRAXXEUR5YR (Homo) ITRAXXEUR5YR (Hete)

Base Correlations iTraxxEUR5YR

IndexName Index Price Up Front FeesTranch Prices


0

0.1

0.2

0.3

0.4

0.5

0.6

0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22%

ITRAXXEUR5YR ITRAXXEUR7YR ITRAXXEUR10YR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22%



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22%



IndexName Index Price Up Front FeesSpread (%) 0 -3% 3 - 6% 6 - 9% 9 -12% 12 -22% 0 -3%

ITRAXXEUR5YR 32.32 500.00 70.28 20.00 8.50 4.25 24.03 ITRAXXEUR7YR 42.21 500.00 189.13 50.00 27.00 9.00 42.25 ITRAXXEUR10YR 53.56 500.00 536.98 124.61 57.83 20.97 51.23

Tranch Prices

Market Quotes ITRAXX

Figure 2. iTraxx index: CDO quotes and base correlations

4.2 Concentrations of index portfolios

Figure 3 presents a summary of industry sector concentrations for five quoted indices (two in North America, two in Europe and one in Asia). The Fitch industry sector classification is used (with 25 sectors), and concentrations are expressed in terms of notional exposures. The Herfindahl index (and its inverse) provides a simple summary industry concentration measure. The CDX IG and iTraxx Europe portfolios contain 125 equally-weighted, names in US and Europe, respectively. These portfolios are well diversified across industry sectors (with 14 and 10 effective sectors, respectively).

23

tainment

res

rtainment

sores

tainment

res

rtainment

sores

r

tt

Exposure Per Name 10MM 8MM 8MM 20MM 1BNumber of Names 100 125 125 50 50

Aerospace & Defence 3.00% 4.00% 2.40%Automobiles 9.00% 7.20% 4.00% 4.00%Banking & Finance 3.00% 17.60% 20.00% 20.00% 24.00%Broadcasting/Media/Cable 7.00% 8.00% 7.20%Business Services 4.00% 1.60% 10.00%Building & Materials 4.00% 3.20% 4.00% 6.00%Chemicals 6.00% 3.20% 4.80% 2.00% 2.00%Computers & Electronics 10.00% 4.00% 1.60% 8.00% 6.00%Consumer Products 5.00% 3.20% 3.20% 4.00%Energy 10.00% 4.80% 1.60% 10.00%Food, Beverage & Tobacco 4.00% 5.60% 7.20% 4.00%Gaming, Leisure & Entertainment 3.00% 1.60% 0.00% 2.00%Health Care & Pharmaceuticals 3.00% 5.60% 0.80% 0.00%Industrial/Manufacturing 2.00% 3.20% 2.40% 8.00% 6.00%Lodging & Restaurants 3.00% 3.20% 0.80%Metals & Mining 2.00% 1.60% 0.80% 12.00% 4.00%Packaging & Containers 2.00%Paper & Forest Products 5.00% 3.20% 1.60%Real Estate 1.60% 0.00% 4.00%Retail (general) 3.00% 6.40% 4.00% 4.00% 8.00%Supermarkets & Drugstores 1.00% 4.80% 4.80%Telecommunications 4.00% 4.80% 8.80% 2.00% 12.00%Transportation 3.00% 4.80% 0.80% 8.00% 4.00%Utilities 4.00% 5.60% 14.40% 4.00% 4.00%Sovereign 14.00%

Herfindahl 5.7% 7.0% 9.5% 9.8% 12.2%Effective number of sectors 17.5 14.2 10.5 10.2 8.2

Industry concentration by Notional

r

tt


The fact that, by construction, CDO indices are well diversified across industrial sectors is an important point worth emphasizing when considering the valuation of bespoke portfolio CDOs. Index prices contain virtually no pricing information on specific industry concentrations. To the degree that a bespoke portfolio is not as well diversified as the index, we can possibly expect measurable pricing differences.

ITRAXX EUR

Aerospace & Defence Automobiles Banking & Finance Broadcasting/Media/Cable Business Services Building & Materials Chemicals Computers & Electronics Consumer Products Energy Food, Beverage & Tobacco Gaming, Leisure & Ente Health Care & Pharmaceuticals Industrial/Manufacturing Lodging & Restaurants Metals & Mining Packaging & Containers Paper & Forest Products Real Estate Retail (general) Supermarkets & Drugsto Telecommunications Transportation Utilities Sovereign

CDX IG

Aerospace & Defence Automobiles Banking & Finance Broadcasting/Media/Cable Business Services Building & Materials Chemicals Computers & Electronics Consumer Products Energy Food, Beverage & Tobacco Gaming, Leisure & Ente Health Care & Pharmaceuticals Industrial/Manufacturing Lodging & Restaurants Metals & Mining Packaging & Containers Paper & Forest Produc Real Estate Retail (general) Supermarkets & Drugs Telecommunications Transportation Utilities Sovereign

CDX HY CDX IG ITRAXX EUR ITRAXX CJ ITRAXX ASIA

Industry (Fitch)

ITRAXX EUR

Aerospace & Defence Automobiles Banking & FinanceBroadcasting/Media/Cable Business Services Building & MaterialsChemicals Computers & Electronics Consumer ProductsEnergy Food, Beverage & Tobacco Gaming, Leisure & EnteHealth Care & Pharmaceuticals Industrial/Manufacturing Lodging & RestaurantsMetals & Mining Packaging & Containers Paper & Forest ProductsReal Estate Retail (general) Supermarkets & DrugstoTelecommunications Transportation UtilitiesSovereign

CDX IG

Aerospace & Defence Automobiles Banking & FinanceBroadcasting/Media/Cable Business Services Building & MaterialsChemicals Computers & Electronics Consumer ProductsEnergy Food, Beverage & Tobacco Gaming, Leisure & EnteHealth Care & Pharmaceuticals Industrial/Manufacturing Lodging & RestaurantsMetals & Mining Packaging & Containers Paper & Forest ProducReal Estate Retail (general) Supermarkets & DrugsTelecommunications Transportation UtilitiesSovereign

CDX HY CDX IG ITRAXX EUR ITRAXX CJ ITRAXX ASIAExposure Per Name 10MM 8MM 8MM 20MM 1B Number of Names 100 125 125 50 50

Industry (Fitch)Aerospace & Defence 3.00% 4.00% 2.40% Automobiles 9.00% 7.20% 4.00% 4.00% Banking & Finance 3.00% 17.60% 20.00% 20.00% 24.00% Broadcasting/Media/Cable 7.00% 8.00% 7.20% Business Services 4.00% 1.60% 10.00% Building & Materials 4.00% 3.20% 4.00% 6.00% Chemicals 6.00% 3.20% 4.80% 2.00% 2.00% Computers & Electronics 10.00% 4.00% 1.60% 8.00% 6.00% Consumer Products 5.00% 3.20% 3.20% 4.00% Energy 10.00% 4.80% 1.60% 10.00% Food, Beverage & Tobacco 4.00% 5.60% 7.20% 4.00% Gaming, Leisure & Entertainment 3.00% 1.60% 0.00% 2.00% Health Care & Pharmaceuticals 3.00% 5.60% 0.80% 0.00% Industrial/Manufacturing 2.00% 3.20% 2.40% 8.00% 6.00% Lodging & Restaurants 3.00% 3.20% 0.80% Metals & Mining 2.00% 1.60% 0.80% 12.00% 4.00% Packaging & Containers 2.00% Paper & Forest Products 5.00% 3.20% 1.60% Real Estate 1.60% 0.00% 4.00% Retail (general) 3.00% 6.40% 4.00% 4.00% 8.00% Supermarkets & Drugstores 1.00% 4.80% 4.80% Telecommunications 4.00% 4.80% 8.80% 2.00% 12.00% Transportation 3.00% 4.80% 0.80% 8.00% 4.00% Utilities 4.00% 5.60% 14.40% 4.00% 4.00% Sovereign 14.00%

Herfindahl 5.7% 7.0% 9.5% 9.8% 12.2% Effective number of sectors 17.5 14.2 10.5 10.2 8.2

Industry concentration by Notional

Figure 3. Industry sector concentrations for indices

Figure 4 presents the concentrations by ratings for the CDO indices. The CDX IG and iTraxx Europe indices are constructed from investment-grade names (with below A ratings for about two thirds of these firms).

24


Concentration by Expected Loss Rating CDX HY CDX IG ITRAXX EUR ITRAXX CJ ITRAXX ASIA AAA AA+ AA AAA+ A A

BBB+ BBB BBBBB+ BB BBB+ B B

CCC+ CCC CCCCC

1.49% 0.28% 0.38% 1.49% 0.51%

1.10% 0.24% 1.10% 0.39% 5.28% 2.13% 0.39% 4.70% 7.53% 4.70%

12.56% 6.70% 9.13% 12.56% 11.16% 11.93% 19.25% 11.16% 16.36% 19.54% 5.22% 16.36% 25.49% 35.00% 13.63% 25.49%

0.47% 19.61% 12.98% 14.00% 19.61% 4.20% 4.40% 10.77% 4.40% 9.14% 10.10% 6.85% 15.39%

12.12% 2.74% 2.74% 8.50% 9.79% 4.22%

34.57% 7.34% 2.81%

Figure 4. Ratings concentrations for indices

4.3 Multi-factor model

To illustrate the effect of sector concentrations in CDO prices and asses the pricing methodologies introduced earlier in this paper, we define a multi-factor model with seven industrial sectors, consolidated from the 25 Fitch sectors. Each name in the portfolio is mapped to one industry sector, as given in Table 2. The industry concentrations for the CDX index are given in the right side of Figure 5 (both weighted by notional and by expected losses).

We define a multi-factor Gaussian copula model as follows. First, risk-neutral default probabilities for each name are implied from individual name credit default swap spreads. The correlation model consists of 8 factors: the first factor, Z1, is a global factor which is interpreted as representing economy-wide effects and the following seven factors represent specific industry systematic factors (see for example Garcia et al. 2006). As we show later, this results in a simple model definition which is general and parsimonious in terms of the number of parameters, and further leads to results which are interpretable.

25


Consolidated Sector Corresponding Fitch Sectors TECH Computers & Electronics,

Broadcasting/Media/Cable Telecommunications Aerospace & Defence

SERVICE Gaming & Leisure & Entertainment Lodging & Restaurants Transportation

PHARMA Health Care & Pharmaceuticals RETAIL Consumer Products

Food & Beverage & Tobacco Retail (general) Supermarkets & Drugstores

FINANCIAL AND REAL ESTATE Banking & Finance Real Estate

INDUSTRIAL Building & Materials Chemicals Industrial/Manufacturing

NATURAL RESOURCES & ENERGY Energy Metals & Mining Paper & Forest Products Utilities

Table 2. Sector model classification

We further assume constant intra-sector and cross-sector asset correlations. Let k(i) denote the sector in which the i-th name resides. The correlation of the creditworthiness Yi for all names to the global factor Z1 is the same:

Error! Objects cannot be created from editing field codes.

Each name has the same factor loading for its own sector, and a zero factor loading for all other sectors:

Error! Objects cannot be created from editing field codes.

This ensures that the correlation between the creditworthiness indices for names i and i* is equal to 0.23 if the names are in the same sector (intra-sector correlation) and 0.17 if they are not (inter-sector correlation). These values for the intra-sector correlations and inter-sector correlations were selected consistent with the general empirical data as described in Akhavein et al, 2005.

Note that the specific model used in this exercise, while simple and realistic, is only used to illustrate the methodology. A production implementation would likely use a more sophisticated version of this model. As discussed above (section 3.5), the precise “absolute” values of the multi-factor model may not have a substantial impact on the pricing, but rather the “relative” value of these correlations.

26

16.9%17.5%13.5%15.2%ENERGY

0.600.500.180.160.190.16

1.672.005.636.305.406.07

15%

10%

17.5%

5%

15%

20%

Weight(Notional)

Bespoke Portfolio(40)

14.2%

6.6%

27.2%

3.9%

16.7%

14.5%

Weight(EL)

Bespoke Portfolio(20)CDX Index

9.6%

19.2%

20.0%

5.6%

9.6%

20.8%

Weight(Notional)

9.9%

11.8%

29.0%

3.6%

10.9%

21.4%

Weight(EL)

50%

50%

Weight(Notional)

INDUSTRY

27.7%FINANCE

RETAIL

PHARMA

SERVICE

72.3%TECH

Weight(EL)

Sector(aggregate)


4.4 Bespoke portfolios and EL mapping

We construct two bespoke portfolios with subsets of 40 names and 20 names of the names in the CDX index. A summary of the industry concentrations of both portfolios is given in figure 5 (and compared with the index itself). The 40-name portfolio is well diversified across sectors and hence can be expected to have similar systematic risk as the index (although higher name concentration). In contrast the 20-name portfolio, is concentrated in two sectors – finance and technology. The average 5-year PD of the index is 3.65% , and those of the 40-name and 20-name portfolios are 3.92% and 3.36%, respectively.

HI

No. Eff. sectors

16.9%17.5%13.5%15.2%ENERGY

0.600.500.180.160.190.16HI

1.672.005.636.305.406.07 No. Eff.sectors

15%

10%

17.5%

5%

15%

20%

Weight (Notional)

Bespoke Portfolio (40)

14.2%

6.6%

27.2%

3.9%

16.7%

14.5%

Weight (EL)

Bespoke Portfolio (20)CDX Index

9.6%

19.2%

20.0%

5.6%

9.6%

20.8%

Weight (Notional)

9.9%

11.8%

29.0%

3.6%

10.9%

21.4%

Weight (EL)

50%

50%

Weight (Notional)

INDUSTRY

27.7%FINANCE

RETAIL

PHARMA

SERVICE

72.3%TECH

Weight (EL)

Sector (aggregate)

Figure 5. Concentration of index and bespoke portfolios in seven-factor model

Figure 6 presents the EL-mapping results for the 40-name portfolio. The left side presents the base correlation for the index and the bespoke portfolio; it further provides the bespoke detachment points obtained from the mapping. The right side presents the prices for the standard tranches of the CDX as well as for tranches with same attachments and detachment points for the bespoke portfolio.

27

27.77%30%

14.01%15%

9.75%10%

7.01%7%

3.45%3%

EL MappedPoint

CDXDetach Point

27.77%30%

14.01%15%

9.75%10%

7.01%7%

3.45%3%

EL MappedPoint

CDXDetach Point


0

18

43

139

34.04%

Bespoke (EL)

5.5

9.9

21

99

31.81%

Index

15-30%

10-15%

7-10%

3 - 7%

0 - 3%

Tranche

0

18

43

139

34.04%

Bespoke (EL)

5.5

9.9

21

99

31.81%

Index

15-30%

10-15%

7-10%

3 - 7%

0 - 3%

TranchePRICES

27.77%30%

14.01%15%

9.75%10%

7.01%7%

3.45%3%

EL Mapped Point

CDX Detach Point

27.77%30%

14.01%15%

9.75%10%

7.01%7%

3.45%3%

EL MappedPoint

CDXDetach Point

0

18

43

139

34.04%

Bespoke (EL)

5.5

9.9

21

99

31.81%

Index

15-30%

10-15%

7-10%

3 - 7%

0 - 3%

Tranche

0

18

43

139

34.04%

Bespoke (EL)

5.5

9.9

21

99

31.81%

Index

15-30%

10-15%

7-10%

3 - 7%

0 - 3%

TranchePRICES

Figure 6. EL mapping – bespoke portfolio (40 names)

Given that the bespoke portfolio has similar sector concentration characteristics as the index, we can expect the EL mapping to provide reasonable results (in general), which adjust for the different portfolio PDs and specific risk (name concentrations). For example, given its slightly higher PD, the equity tranche of the bespoke portfolio is a little more expensive. The three intermediate tranches are substantially more expensive than the index, but the most senior tranche results in zero spread (indeed slightly negative). In the end, the methodology faces the standard problems encountered with base correlations, which make it hard to explain some of the results and can result in “noarbitrage” violations.

Figure 7 presents the results also for the 20-name bespoke portfolio, and compares them with the index and the 40-name bespoke portfolio. The EL methodology assumes that the 20 name concentration has similar concentration characteristics and does not explicitly factor in the fact that the portfolio is concentrated in two sectors. Thus, the pricing differences are largely attributable to the differences in average PD and the higher idiosyncratic risk it has (which should result in bigger tail default losses). The slightly better credit quality (lower PD) results in slightly cheaper first loss protection premiums than the index. While the tranche prices remain below those for the 40-name portfolio, they are higher than the index for the rest of the tranches, except for the most senior tranche, which shows the same problems as for the 40-name portfolio.

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IndexIndex BespokeBespoke BespokeBespoke(40)(40) (20)(20)

31.81%31.81% 34.04%34.04% 28.26%28.26%

9999 139139 113113

2121 4343 2929

9.99.9 1818 1111

5.55.5 00 00

Figure 7. EL mapping –bespoke portfolio (20 names)

4.5 Weighted Monte Carlo results

We now illustrate the application of the systematic weighted MC technique on the CDX data, and its application to price bespoke portfolios. As a first exercise, we perform a “raw” implementation and calibration of the model. We assume Normal (uncorrelated) prior scenarios for each of the eight factors. We sample 300 Quasi MC scenarios using Sobol points. We also assume constant LGDs for each name. For computational simplicity we assume a conditional Normal approximation when constructing the convolution of portfolio losses. Finally the optimization problem uses “soft” matching constraints (penalty function) and a minimum distance to prior objective function. The results of the calibration are shown in Figure 8.

27.77%

14.01%

9.75%

7.01%

3.45%

EL Mapped Point (40)

24.65%30%

11.55%15%

7.98%10%

5.89%7%

3.17%3%

EL Mapped Point (20)

CDX Point

27.77%

14.01%

9.75%

7.01%

3.45%

EL MappedPoint (40)

24.65%30%

11.55%15%

7.98%10%

5.89%7%

3.17%3%

EL MappedPoint (20)

CDXPoint

15-30%

10-15%

7-10%

3 - 7%

0 - 3%

Tranche

15-30%

10-15%

7-10%

3 - 7%

0 - 3%

TranchePRICES

5.5

10.5

21

99

31.81%

Index (Market)

15-30%

10-15%

7-10%

3-7%

0-3%

Tranche

0

9.2

24

102

31.93%

Index (WMC)

5.5

10.5

21

99

31.81%

Index(Market)

15-30%

10-15%

7-10%

3-7%

0-3%

Tranche

0

9.2

24

102

31.93%

Index(WMC)

PRICES

5.5

10.5

21

99

31.81%

Index(Market)

15-30%

10-15%

7-10%

3-7%

0-3%

Tranche

0

9.2

24

102

31.93%

Index(WMC)

5.5

10.5

21

99

31.81%

Index(Market)

15-30%

10-15%

7-10%

3-7%

0-3%

Tranche

0

9.2

24

102

31.93%

Index(WMC)

PRICES

Figure 8. Weighted MC calibration (raw calibration)

While the first four tranches are matched reasonably well, the model cannot match the price of the most senior tranche. This is expected, since we have used Normal prior scenarios, where all samples lie within three standard deviations. Scenarios which result

29


in extreme events are necessary in order to hit the senior tranche and thus result in a premium.

Next, we analyze the implied factor distributions which we obtain from the calibration. Figure 9 gives the moments of the implied factors as well as the histogram representing the implied distribution of the systematic factor. The implied distribution of the global factor has the biggest changes from the prior Normal. In general the sector factors show moments which remain close those of a standard normal distribution. The global factor, in contrast shows high kurtosis and skew. These results are indeed a consequence of the concentration risk properties of the index and choice of the multi-factor model, and an important property of CDO prices and multi-factor models:

Since the index is well diversified across industries, the CDO prices in the market generally contain information only on “global” risk premiums and not specifically on

any particular industry concentration.

In addition to the problem of matching super senior tranches, a “naïve” weighted Monte Carlo model may result in sectors with badly behaved implied distributions, as shown in the histogram of the systematic global factor in Figure 9. These can be interpreted as some form of over-fitting of the problem (as we have a large number of unknowns).

-0.220.041.00-0.01ENERGY

-0.04-0.011.04-0.01INDUSTRY

-0.230.030.97-0.02FINANCE

-0.390.011.00-0.03RETAIL

-0.20-0.130.99-0.01PHARMA

-0.120.041.00-0.01SERVICE

-0.050.271.04-0.07TECH

2.76-0.480.63-0.15Global

Ex. Kurt.SkewStdMeanFactor

-0.220.041.00-0.01ENERGY

-0.04-0.011.04-0.01INDUSTRY

-0.230.030.97-0.02FINANCE

-0.390.011.00-0.03RETAIL

-0.20-0.130.99-0.01PHARMA

-0.120.041.00-0.01SERVICE

-0.050.271.04-0.07TECH

2.76-0.480.63-0.15Global

Ex. Kurt.SkewStdMeanFactor

Global Systemic Factor

Figure 9. Implied factor moments (raw calibration)

In order to understand the properties of the calibration and the implied distributions, we explore in more detail the single-factor model calibration. Based on the observations in the previous example, we may expect a single-factor model, properly defined, to match the prices of all tranches for a given index.7 As a prior for the systematic factor, we sample 500 Quasi MC Sobol points on (-5,5) and assign prior weights to each point to match a normal distribution. This generates extreme scenarios which will allow us to

Note that the implied copula approach (which is a single-factor model) matches the prices of traded securities. The authors highlight the need to include extreme scenarios as well as correlated LGDs.

30

7

MEAN STD SKEW EX.KURT 0-3% 3-7% 7-10% 10-15% 15-30%None 0.00 0.99 -0.11 -0.27 28.47% 229.06 45.97 9.81 1.12level 1 0.37 1.90 1.49 1.02level 2 0.30 1.81 1.68 2.05 32.22% 111.58 34.8 18.67 10.37level 3 0.22 1.69 1.90 3.33level 4 0.14 1.54 2.23 5.01 31.89% 101.58 24.46 12.47 7.5

Final 0.12 1.47 2.42 5.98

MARKET 31.81% 99 21 10.5 5.5

Global factor statisticsPenalties Tranche Prices


match the prices of senior tranches. We further assume constant LGDs and again, for computational simplicity, we use a conditional Normal approximation. Figure 10 illustrates the behaviour of the implied factor distribution as the prices are matched more closely (by increasing the penalties in (7)). A multimodal distribution is required to match tranche prices.

Figure 10. Evolution of implied factor distribution (single-factor)

Figure 11 presents the results for various “levels” of increasing penalties. It gives the prices of each tranche, the first four moments of the implied factor distribution as well as the final implied factor distribution and base correlations for the final solution and the index. Prices of all tranches can be matched with great accuracy. This results however in multimodal distributions with fatter tails and higher skewness. The example further demonstrates how the framework can be used to trade-off price matching and “quality” of the implied factor distribution.

Penalties MEAN STD SKEW

Global factor statistics EX.KURT 0-3% 3-7% 7-10% 10-15%

Tranche Prices 15-30%

None 0.00 0.99 -0.11 -0.27 28.47% 229.06 45.97 9.81 1.12 level 1 0.37 1.90 1.49 1.02 32.31%32.31% 118.39118.39 32.8532.85 23.2723.27 9.479.47level 2 0.30 1.81 1.68 2.05 32.22% 111.58 34.8 18.67 10.37 level 3 0.22 1.69 1.90 3.33 32.04%32.04% 106.54106.54 30.7830.78 16.1216.12 10.7710.77level 4 0.14 1.54 2.23 5.01 31.89% 101.58 24.46 12.47 7.5

Final 0.12 1.47 2.42 5.98 31.82%31.82% 99.3599.35 21.4721.47 10.7710.77 5.775.77

MARKET 31.81% 99 21 10.5 5.5

Figure 11. Calibration results (single-factor)

31


As an example of the sensitivity of the implied distribution to the matching of various tranches, Figure 12 compares the implied factor distribution when prices are matched for all tranches as well as when the most senior tranche is removed. Removing the senior tranche results in a simpler distribution, with one less mode.

Figure 12. Global factor distribution – calibration without senior tranche

Since the 40-name bespoke portfolio has essentially the same concentration characteristics as the index, we use the single-factor model to price various tranches. Figure 13 gives the weighted MC prices for the bespoke portfolio tranches, for various levels of calibration (as the index is matched more closely). It also compares the prices and implied base correlations to those obtained using EL mapping. The EL mapping prices are quite different from those obtained using MC. In particular, the super-senior tranche has a spread of 6 bps (compared to zero for EL mapping). Also the 3-7% tranche goes from 139 to over 200 bps. Note that the MC prices are arbitrage-free.8

In addition to the Weighted MC and EL mapping, the figure plots the base correlations resulting from a “concentration adjusted” mapping as described in Rosen and Saunders (2007).

32

8

0-3% 3-7% 7-10% 10-15% 15-30%

None 29.75% 302.4 77.78 19.53 1.83level 1level 2 31.93% 232.1 39.48 22.28 10.85level 3level 4 32.48% 207.92 35.2 15.41 7.78

Final

EL Mapping 34.04% 139 43 18 0

WMC Index 31.82% 99.35 21.47 10.77 5.77Market Index 31.81% 99 21 10.5 5.5

PenaltiesTranche Prices


Penalties 0-3% 3-7% 7-10%

Tranche Prices 10-15% 15-30%

None 29.75% 302.4 77.78 19.53 1.83 level 1 31.58%31.58% 244.48244.48 39.239.2 25.1625.16 10.6510.65level 2 31.93% 232.1 39.48 22.28 10.85 level 3 32.22%32.22% 219.38219.38 38.4838.48 19.6719.67 10.9910.99level 4 32.48% 207.92 35.2 15.41 7.78

Final 32.57%32.57% 203.51203.51 33.0733.07 13.3713.37 6.096.09

EL Mapping 34.04% 139 43 18 0

WMC Index 31.82% 99.35 21.47 10.77 5.77 Market Index 31.81% 99 21 10.5 5.5

Figure 13. Weighted MC pricing of bespoke portfolio (40 names) – single factor

The results for the 40-name bespoke portfolio process using a full multi-factor model are presented in Figure 14. Figure 15 further gives the moments for the implied factor distributions after calibration. As expected the prices are very close to the single-factor model.

0-3% 3-7% 7-10% 10-15% 15-30%

MARKET 31.81 99 21 10.5 5.5

WMC INDEX 31.81 99.53 19.96 10.86 5.48

BESPOKE (WMC MF) 33.13 214.54 24.66 11.18 5.53 BESPOKE (WMC SF) 32.57 203.51 33.07 13.37 6.09 BESPOKE EL Mapping 34.04 139 43 18 0

Figure 14. SWMC pricing of bespoke portfolio (40 names) – multi-factor model

33


Factor MEAN STD SKEW KURT Global 0.66 1.26 1.35 2.89 TECH 0.15 2.91 0.14 - 1.13 -

SERVICE 0.13 2.86 0.07 - 1.09 -PHARMA 0.31 2.59 0.09 - 1.14 -RETAIL 0.36 - 2.73 0.13 1.01 -

FINANCE 0.13 2.41 0.06 0.72 -INDUSTRY 0.39 2.78 0.21 - 1.13 -ENERGY 0.37 2.83 0.16 - 1.10 -

Figure 15. Implied factor distributions (moments)

The multi-factor model further allows us to price more accurately portfolios with higher concentrations. Figure 16 shows the differences of the weighted MC compared to the EL mapping methods for the 20-name bespoke portfolio (expressed as both prices and implied base correlations).

8

i 0

0-3% 3-7% 7-10% 10-15% 15-30%

MARKET 31.81 99 21 10.5 5.5

WMC INDEX 31.81 99.53 19.96 10.86 5.48

BESPOKE (40 name) WMC 32.57 203.51 33.07 13.37 6.09

BESPOKE (20 Name) WMC 20.98 266 81 30 BESPOKE EL Mapp ng 28.26 113 29 11

Figure 16. Weighted MC pricing of bespoke portfolio (20 names)

34


5 Concluding remarks

We have presented a robust and practical CDO valuation framework based on the application of multi-factor credit models in conjunction with weighted Monte Carlo techniques used in options pricing. The framework produces arbitrage-free prices and can be used to value consistently CDOs of bespoke portfolios, CDO-squared and cash CDOs. We demonstrate the practical advantages of working through multi-factor models, rather than directly on the portfolio hazard rates, as in the implied copula methodology. The mathematical framework of Generalized Linear Mixed Models provides a general basis for defining the underlying multi-factor models, and understanding their impact.

We have described the numerical solution of the inverse problem of implying the factors’ joint distribution from observed market prices, by creating first discrete scenarios on the factors. Although standard quadrature points may be used for low dimensions, more realistic high-dimensional problems require Monte Carlo simulation. In particular, for this purpose, we have demonstrated the application of Quasi-MC techniques. We further discussed how to obtain well behaved factor distributions as the solutions from the optimization problem. Although not discussed in this paper, the approach can also produce pricing sensitivities via standard MC Greek techniques.

At the heart of the methodology lies the explicit definition of an underlying multi-factor credit model. While this might seem like a limitation to some practitioners, we believe that a multi-factor credit model is a fundamental requirement of any model that explicitly models systematic concentrations in credit portfolios. These models are also used widely in the industry for measuring credit risk and calculating economic capital. As explained in the paper, for pricing bespoke CDOs, we have found that the “absolute values” of the factor weights are less important than the “relative” differences in concentration they define. Note that, recently, other models attempting to price bespoke portfolios have also, incorporated fundamental information contained in a multi-factor model, either implicitly or explicitly, (e.g. Hull and White 2006a use “average correlations” for the index and the bespoke portfolio).

Finally, while we have focused on a static version of the model (by defining the codependence of default times), the framework is general and allows the use of dynamic models. Extensions to dynamic models can be done via hazard rate processes (Cox processes) or multi-step structural models (e.g. Kreinin et al. 1999, Hull and White 2001) This of course results in an increase in computational complexity. Future work will explore the numerical implementation of a dynamic model.

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http://www.defaultrisk.com/

http://www.schonbucher.de/

http://www.defaultrisk.com/

valuing cdos of bespoke portfolios with implied multi ... · portfolio, such as the itraxx or cdx...

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