CDOs and the Search for Simplicity: Speeding up the valuation of Synthetic CDOs using a grouped recursion method
To download the latest trial version of FINCAD Analytics, contact a FINCAD Representative.
Overview
Once considered a novelty item, Collateralized Debt Obligations (CDOs) have become a serious investment vehicle in the asset-backed securities market and for the credit risk market as a whole. CDOs have not arrived without controversy as some investors have incurred huge losses using these instruments, not to mention the evolving complexity of CDO offerings that continue to appear in the marketplace.
CDO assets are chosen from the full range of credit risk products, but for this article's purposes we will focus on how we might improve the valuation of synthetic CDOs using the grouped recursion method, which improves the results that the recursion method developed by Andersen et al (2003) gives. We will explain the concepts of synthetic CDOs (single tranche CDOs) and one factor copula models. We then proceed to work through the math by giving detailed algorithms for the calculation of portfolio loss distributions and the valuation of synthetic CDOs. Finally, we present examples of a homogeneous CDO and a generic CDO to compare the efficiencies of different calculation methods.
The Search for Simplicity
CDOs come in many colours and flavours based on their reference assets and the reason for the structure. The majority of traditional CDOs are fully cash funded. Some CDOs are partially cash funded and partially synthesized through credit derivatives - hence the name synthetic CDOs. Cash CDOs have complicated waterfall cash flow structures and often require the ownership of the reference assets. The variety and complexity of such traditional CDOs make CDOs look like a complicated maze.
Recently, the creation of a new type of CDO, the single tranche (synthetic) CDO, offers simpler structures that are much easier to understand and manage than traditional CDOs. Stimulated further by the exchange traded standardized single tranche CDO products, e.g., the Dow Jones CDX and ITRAXX series, single tranche CDOs have enjoyed rapid expansion for well over a year now. Single tranche CDOs have almost become a synonym of synthetic CDOs. For the purposes of this discussion by synthetic CDO we mean a single tranche CDO.
The Three Pillars of Single Tranche CDOs
- The separation of the credit risk of a CDO's reference pool from the ownership of the assets in the pool. The Issuer or manager of a single tranche CDO does not own the assets in the pool. The reference pool means nothing more than the credit risk of its entities. This way, CDOs can be constructed from essentially any types of credit-related entities and therefore a CDO manager has the freedom to choose reference entities that may be impossible to acquire in a traditional CDO due to liquidity, legal, or other issues.
- The independent capital structure of tranches. This makes it possible to trade customized tranches, i.e., single tranches, and relieves the pain of dealing with complicated waterfall capital structures.
- The adoption of the one-factor copula model. This significantly simplifies the modeling of CDOs and hence its valuation. Such simplification makes efficient computation possible. This also helps investors, traders and financial engineers speak the same language of CDO pricing and hedging.
One-factor Copula Model to the Rescue
One of the challenging tasks in the management of CDOs is the computational efficiency. With a full correlation structure of the reference entities, Monte Carlo methods are the pre-dominant methods due to their flexibility. The issue with the seemingly all-powerful Monte Carlo simulation is that it moves too slowly. Without a large computer system dedicated to parallel processing, it is unrealistic to use Monte Carlo methods to handle tasks such as correlation calibration and simultaneous valuation of multiple tranches. Under a one-factor copula model the computation of a synthetic CDO can be greatly simplified, and non-Monte Carlo methods can play a role. The most popular non-Monte Carlo methods are the traditional fast Fourier transforms (FFT) (see e.g. Debuysscher and Szego 2003) and the more recently developed recursion methods by Andersen et al (2003) and Hull and White (2004). These methods are often called semi-analytic or quasi-analytic methods.
We will further exploit the power of the recursion method developed in Andersen et al (2003) by approximately grouping the reference entities into homogeneous subgroups and using the recursion method in both the individual subgroups and in the integration of the subgroups. Implementation experience shows that when the accuracy level (described later) is high, the grouped recursion method significantly improves the recursion method of Andersen et al (2003). When the level of accuracy is low, the improvement is less significant.
But before we delve into the math, let's review some basic concepts and definitions…
Synthetic CDOs (Single Tranche CDOs)
Perhaps a bit of misnomer, a single tranche CDO does not mean a CDO has only a single tranche. It is so named because the cash flow structure of a credit derivative linked to a particular tranche depends only on the loss to this single tranche, not the losses or cash flows to any other tranches. A single tranche CDO is linked to certain ranges of the total possible loss of a pool of entities. The entities can be actual assets, such as bonds, or simply names of companies or countries. A typical single tranche CDO consists of between 100 to 200 entities. After inception of a CDO, some entities in its reference pool may default or suffer other types of credit events. When a credit event occurs to an entity, part of its notional will be lost. A synthetic CDO absorbs the accumulated historical losses of the pool that fall within certain ranges. A loss range is called a tranche. The lower bound of a tranche is called an attachment point of the tranche and the upper bound a detachment point. A tranche with an attachment point of 0 is called an equity tranche. For example, a 5-10% tranche has an attachment point of 5% and a detachment point of 10%. When the accumulated loss of the reference pool is no more than 5% of the total initial notional of the pool, this tranche will not be affected. However, when the pool's accumulated loss exceeds 5%, any further loss will be deducted from the tranche's notional until the detachment point, 10%, is reached, when the tranche is completely wiped out.
The most commonly used credit derivatives in synthetic CDOs are credit default swaps (CDSs on CDO tranches). They can be viewed as extensions of the corresponding single entity credit default swaps. Like a single-entity CDS, a CDS on a CDO tranche has a payoff leg and a premium leg. The buyer of a CDS on a tranche will be compensated by the seller for any loss to the tranche and in return he/she pays periodic premium to the seller. As the loss to the tranche increases, the premium notional of the tranche decreases and hence the premium payment decreases. For some CDO deals, some tranches, mostly the equity tranche, may also charge an upfront fee to protect buyers. The main differences between a CDO credit derivative and a single-entity credit derivative are twofold: (1) a CDO derivative has a variable notional whereas a single entity credit derivative has a constant notional; (2) a CDO derivative has a protection layer (except for an equity tranche derivative) whereas a single-entity derivative does not.
Roughly speaking, there are two classes of single tranche CDOs: standardized CDOs and bespoke CDOs. Standardized CDOs are exchange-traded CDOs. These CDOs are homogeneous; each entity in the reference pool has the same notional and the same expected recovery rate. Bespoke CDOs are customized CDOs. Typically, the notional and the recovery rate of a reference entity vary from entity to entity.
By a homogeneous CDO or a homogeneous reference pool we mean a CDO with all of its reference entities having the same loss given default. Here the loss given default of an entity is its notional minus its expected recovery value from default.
One Factor Copular Models
Credit Indices
In some sense, copula models use asset correlations to infer default correlations. Under a one factor copula model correlations of all the reference entities come from the same source of risk - the single systematic risk. Mathematically, the credit status of entity can be represented as
where ζ, ε1, ε2, ε3,... εn are independent random variables with the same mean of 0 and the same variance of 1, and each αi, called the factor loading of entity i, is a constant between -1 and 1. The random variable Xi is sometimes called the credit index of entity i. Roughly speaking, it is the average asset return of the firm. If all the variables ζ, ε1, ε2, ε3,... εn are Gaussian (normal), the copula model is called a one-factor Gaussian copula model. It is currently the standard correlation model in the credit derivatives industry. If all the Xi's are Student-t variables, the copula is a one-factor Student-t copula model. If all the variables ζ and εiare Student-t variables, the copula model is called a one-factor double-t copula model. Note that a double-t copula model is, in general, not a Student-t copula model. For more detailed discussions on copula models the reader is referred to Laurent and Gregory (2002).
For a one-factor copula model, the asset correlation of entities i and j is αiαj. A special case is that the factor loadings of all the entities are identical:αi ≡ α. In this case all the asset correlations of the entities are the same: ρ=α2. Note that for this case the single correlation is always nonnegative. The common factor loading α can be any number between -1 and 1. Without loss of generality we will always assume that 
to be a positive number.
A very important implication of a one-factor copula model is that the credit indices of all the entities are conditionally independent given the common factor ζ = x for any fixed real number x. This conditional independence allows us to calculate various types of conditional joint default probabilities of the entities conveniently.
Default thresholds
Let ρi(t) be the default probability of entity i by time t. The credit index of a firm is related to its default status through a so-called default threshold hi(t) (t > 0), which is the percentile of the credit index Xi at ρi(t):
ρi(t)=Prob(Xi ≤ hi(t)).
The above indicates that entity defaults i by time t if and only if Xi ≤ hi(t). From this, joint default distributions can be calculated easily. For example, the probability that both entity i and entity j default by time t is simply Prob(Xi ≤ hi(t), Xj ≤ hj(t) ).
In the following we will only consider Gaussian copulas. For this case h(it) = (ρi(t)), where Φ-1is the cumulative normal distribution function with a mean of 0 and a standard deviation of 1.
Portfolio Loss Distributions
To value a CDO, the key is to calculate the loss distribution of the reference portfolio. Let τi be the default time of entity i and Mi and αi its notional and recovery rate, respectively. Set Li = Mi x (1 - αi), which is the loss given default of entity i. Then the total loss of the reference pool up to a given time horizon t is
where N is the number of entities in the pool and 1(A) is the indicator function of A. Under the one-factor Gaussian copula model
where
If the conditional loss distributions
are known, the loss distribution of L(T) can be obtained by use of the Gaussian quadrature.
When the reference entities are homogeneous, the calculation of the conditional loss distribution is relatively simple: it can be determined by the number of defaults. However, for more general cases, to determine the loss amount at a default time one has to trace down the defaulted entity. Hence this simple counting method no longer works, and a more sophisticated method is needed.
The conditional probability of an entity given the common factor ζ = x
With the one-factor Gaussian copula model the basic building blocks of the loss distributions of a reference pool are the entity's default probabilities conditional on the common factor. Set
ρi(t, x) = Prob(Entity i defaults by time t | ζ = x). Then by equation (1)
The conditional default probability of exact number of defaults
When a reference pool is homogeneous, to calculate its loss distribution is equivalent to finding out the probabilities of specific numbers of defaults. Due to the conditional independence of the entities the conditional default probability of a specific number of defaults given the common factor ζ can be calculated through a simple recursion.
Define q(t,x,l) = Prob(The reference pool has exactly l defaults by time t | ζ = x) and set
q(0)(t,x,0) = 1 and q(0)(t,x,0) = 0 for l ≥ 1. For each i = 1,… ,N we consider iteratively the probability of exactly number of defaults when only the first entities are to be used in the pool. Then the probability that we are looking for is the one we will get from the last iteration, i.e., when i = N. Set for l ≤ i
q(i)(t,x,l) = Prob(The portfolio of the first i entities has exactly l defaults by time t | ζ = x)
At the first iteration, i = 1, there are only two possibilities: entity 1 doesn't default by time t with probability 1 - ρi(t, x), or it defaults by t with probability ρi(t, x). Mathematically
In general, at the iteration, a new entity, entity i is added to the basket considered in the previous iteration. For each i = 0,1,… ,i, the probability of exactly l defaults consists of two parts; part 1: entity i doesn't default by time t, in which case the number of defaults keeps the same as that of the (i - 1)th iteration, and part 2: entity i defaults, in which case the number of defaults increases by 1. Mathematically, for i = 2,… ,N
Therefore the conditional probability that the reference pool has exactly l defaults is
Loss distributions when the reference pool is homogeneous
When all the entities have an identical loss given default L ≡ Ni(1 - α,i), i = 1, 2, …, N, the possible loss points are , and the conditional loss distribution can be easily calculated with the recursion algorithm given above. The result is
Loss distributions when the reference pool consists of homogeneous groups
In this section we will extend the recursion method given above to derive the conditional loss probabilities of the whole reference pool when the pool consists of homogeneous subgroups. We also assume that losses from default are non-negative integers. Define
Suppose the reference pool can be divided into J subgroups. For j = 1, …, J let 0, 1,…,Kj be the loss points of subgroup j and
For convenience set qj(t, x, l) = 0 all for l < 0. Suppose in the above setup each of the J subgroups has an identical loss given default. Set
Then the conditional loss distribution of the reference pool given the common factor ζ = x is
Loss distributions when the reference pool is not homogeneous
When the reference pool is not homogeneous, we can approximate the pool with homogeneous subgroups. To achieve this use the following steps
Find out a greatest common divisor d of the loss points {Li, i = 1,…,N} such that
where the 
's are integers. One of such algorithms to search for a proper greatest common divisor is given in Andersen et al (2003). Another obvious method is to represent first the loss given default of each entity by an integer and then find the greatest common divisor of all such integers. Our implementation experience shows that combining the two methods gives the best balanced result in terms of efficiency and accuracy.
Sort the integer loss points in ascending order. Let 
be the minimum and the maximum of the loss points.
Divide the loss range into intervals of equal length and for each interval form a subgroup that consists of the entities whose integer loss points fall into the interval.
Form homogeneous subgroups by setting each of the integer loss points in a subgroup to the median of the integer loss points of this group.
Valuation of Synthetic CDOs
Consider a CDO tranche with attachment and detachment amounts (the actual dollar amounts), i and u, (0 <=l < u), respectively. Let L(t) denote the accumulated loss of the pool at time t (the current time is 0) and Nl,u(t) the notional of the tranche at time t. At inception of the CDO, the notional of the tranche equals the size of the tanche: Nl,u(t) = u - l . As time goes on, some entities in the reference pool may default. At any future time the tranche's notional will be reduced by the loss to the tranche, if any. The relationship between the tranche's notional and the accumulated loss of the reference pool at any future time t is
In the following we will derive the formulas for the values and break-even spread of a protection seller credit default swap on the above tranche, where the holder of the swap receives periodic premium payments.
Let Bl,u(t) be the time t value of the premium leg. Let C be the annual premium coupon rate and Ti, i = 1,… ,m the premium coupon payment time points. Then
where E stands for the expectation under a risk-neutral probability measure, {D(s),(s >=0)} is a risk-free discount factor curve and Li-1,i = Nl,u(Ti) - Nl,u (Ti+1) is the total loss in the time period (Ti-1, Ti) due to default.
Let be the time tvalue of the payoff leg (protection leg). Then
If we assume that compensation is made at the next premium coupon payment date when a default occurs, then the above formula becomes
Remark 1: In the above the default correlation is modeled with a one-factor Gaussian copula model. The most important case is that the asset correlations of all entities are a constant. For this case, one would hope that the single correlation can be calibrated from market quotes of synthetic CDOs. Unfortunately, due to the imperfect model, only for the equity tranche can the implied correlation be identified uniquely for a given par CDS spread of the tranche. Because of this we have to rely on the equity tranches in order to use CDO market quotes for calibration. This leads to the concept of so-called base tranches and base correlations. The base tranche of a CDO tranche is an equity tranche with its detachment equal to the detachment of the CDO tranche. The implied correlation of a base tranche is called a base correlation. Given market quotes of CDO tranches, base correlations of the tranches can be calculated from market quotes through a so-called bootstrapping procedure. When base correlations are known, a non-base tranche can be valued through the following relationships between the values of a non-base tranche and its related base tranches;
The topic of base correlations is beyond the scope of this paper. Interested readers are referred to the paper of McGinty and Ahluwalia (2004) and that of Willemann (2005).
Remark 2: Some tranches, particularly equity tranches, charge upfront fees. For this case, the par spreads of the tranches should be adjusted. This can be easily done by calculating the present value of the upfront fee and subtract this amount from the payoff leg in the calculation of a par spread.
An Example
Consider a synthetic CDO with a reference pool of 119 entities, which is a slightly modified version of a market traded CDO. The pool has 18 homogeneous subgroups and some single entities. The details of the subgroups are shown below.
Data of the homogeneous groups
| group | number of entities | notional |
| 1 | 9 | 175 |
| 2 | 6 | 200 |
| 3 | 2 | 210 |
| 4 | 5 | 225 |
| 5 | 2 | 235 |
| 6 | 6 | 250 |
| 7 | 2 | 275 |
| 8 | 10 | 300 |
| 9 | 5 | 325 |
| 10 | 8 | 350 |
| 11 | 5 | 375 |
| 12 | 11 | 400 |
| 13 | 2 | 410 |
| 14 | 9 | 425 |
| 15 | 3 | 428 |
| 16 | 5 | 450 |
| 17 | 2 | 475 |
| 18 | 1 | 500 |
The remaining 5 individual entities have notionals 238, 525, 528, 550 and 625, respectively. The expected recovery rate of all entities in the reference pool is 40% flat, and their par CDS spread is 0.5% flat (for bootstrapping a default probability curve from par CDS spreads see Luo 2005). The CDO has three tranches whose details are given in the following table:
Tranche information
| attachment | detachment | base correlation | premium rate | position | upfront fee |
| 0 | 5% | 9% | 2% | sell protection | 5% |
| 5% | 8% | 12% | 1.35% | sell protection | 0 |
| 8% | 11% | 18% | 3.5% | buy protection | 0 |
The CDO has a time to maturity of 5 years. The premium leg of all three tranches has a quarterly payment frequency. The fair values, the values of the two legs and the par spreads of the three tranches are shown below:
Valuation of an inhomogeneous synthetic CDO
| Value\Tranche | 0-5% | 5-8% | 8-11% |
| Fair value | -728.3362 | 31.0649 | -199.5527 |
| Protection leg value | -877.3320 | -46.5611 | 3.7146 |
| Premium leg value | 148.9957 | 77.6260 | -203.2673 |
| Par spread | 0.1323 | 0.0081 | 0.0006 |
Computation is carried out for four different levels of accuracy. Each level corresponds to a different tolerance level in the approximation of the pool losses from default, and a different number of points used in the Gaussian quadrature for integration. The fair values in the above table are the results corresponding to the highest accuracy level. The following table gives the maximum relative errors of the values of the payoff and the premium legs:
Maximum relative errors*
| Tranche\accuracy level | Level 1 | Level 2 | Level 3 | Level 4 |
| 0-5% | 0.422% | 0.135% | 0.010% | 0.000% |
| 5-8% | 0.167% | 0.657% | 0.005% | 0.000% |
| 8-11% | 8.081% | 1.550% | 0.055% | 0.000% |
* Same for all three methods
The three quasi-analytic methods (the grouped recursion method introduced in this article, the recursion method of Andersen et al (2003) and the FFT) are now compared for speed. The following table gives the CPU time used in calculation on an Intel Pentium D processor in the Excel VBA debug mode (used for prototyping at FINCAD) for the three methods.
CPU time (in minutes)
| Calculation method\accuracy level | Level 1 | Level 2 | Level 3 | Level 4 |
| Grouped recursion | 0.2 | 1.37 | 10.32 | 61.72 |
| Straight recursion | 0.22 | 1.88 | 16.45 | 111.87 |
| FFT | 1.33 | 20.87 | 139.68 | 555.85 |
From the above table we see that at the highest level of accuracy, the grouped recursion is about twice as fast as the straight recursion method and is about 9 times as fast as the FFT. For the lowest level of accuracy, the grouped recursion and the straight recursion method are almost indistinguishable, and at this accuracy level the recursion method is about 6 times as fast as the FFT.
Note that for the lowest level of accuracy the efficiency ratio between the recursion method and the FFT is similar to that of a homogeneous CDO. Our implementation experience shows that for a homogeneous CDO with 120 entities the CPU times for its valuation are about 0.32 and 1.47 minutes for the recursion method and the FFT, respectively. The recursion method is nearly 5 times as fast as the FFT.
Conclusions
The grouped recursion method introduced in this paper enhances the power of the recursion method developed by Andersen et al (2003) on efficient pricing of synthetic CDOs. The grouped recursion method uses the recursion method in two stages. In stage 1, the reference pool is approximated by homogeneous subgroups and the recursion method is then applied to each subgroup. In stage 2, the recursion method is applied to integrate the results of the subgroups obtained in stage 1. Detailed algorithms are given for the calculation of portfolio loss distributions as well as the valuation of synthetic CDOs. An example is given to show that for a typical bespoke inhomogeneous CDO the grouped recursion method is about twice as fast as the straight recursion method and about 9 times as fast as the FFT. For a typical homogeneous CDO the recursion method is about 5 times as fast as the FFT.
References
Andersen L., J. Sidenius and S. Basu (2003), All your hedges in one basket, Risk, November, 67-72.
Debuysscher, A. and Szego, M., (2003), The Fourier Transform Method - Technical document, Moody's Investors Service.
Hull, J. and A. White (2004), Valuation of a CDO and an nth to default CDS without Monte Carlo simulation, Journal of Derivatives, 12, No. 2, 8-23.
Laurent J-P and J. Gregory (2002), Basket default swaps, CDOs and factor copulas, working paper
Luo L. (2005), Bootstrapping default probability curves, Journal of Credit Risk, 1 (4), p. 169 - 179.
McGinty, L., and R. Ahluwalia (2004), A model for base correlation calculation, Research paper, JP Morgan.
Willemann S. (2005), An evaluation of the base correlation framework for synthetic CDOs, Journal of Credit risk, 1, 180 - 190.
Disclaimer
Your use of the information in this article is at your own risk. The information in this article is provided on an "as is" basis and without any representation, obligation, or warranty from FINCAD of any kind, whether express or implied. We hope that such information will assist you, but it should not be used or relied upon as a substitute for your own independent research.
For more information or a customized demonstration of the software, contact a FINCAD Representative.