Differences in Tranching Methods-Some Results and Implications

Professor Roger Stein

Differences in Tranching Methods: Some Results and Implications

Ashish Das and Roger M. Stein

Moody’s Research Labs

New York

First draft: May 20, 2009

Current Draft: January 7, 2011

We review the mathematics of evaluating the credit risk of

tranches of structured transactions with simple loss-priority structure for

two common tranching approaches: PD-based tranching where the

probability of default of a tranche is the quantity of interest; and EL-

based tranching where the expected loss on a tranche is the quantity of

interest. While comparing the attributes of these two tranching

approaches, we examine their relative (level of credit enhancement)

conservatism. While the mathematics is simple some implications of the

results are interesting. We discuss the impact of the collateral LGD

assumption on attachment points; show that, all else equal, lowering

attachment point or the detachment point necessarily increases the

tranche EL; and provide upper-bounds on senior-most tranche LGD

under reasonable distributional assumptions. One implication of the low

LGDs associated with the senior tranches is that under some EL

definitions, it may be impossible to create a tranche with a given EL

under the EL-based approach, even though they are always possible

under the PD-based approach.

1 Introduction1

We review the mathematics of evaluating the credit risk of tranches of structured transactions with simple loss-priority structure for two common tranching approaches: PD-based tranching where the probability of default of a tranche is the quantity of

interest; and EL-based tranching where the expected loss on a tranche is the quantity of interest.2 These approaches are used to either evaluate the credit quality of an

exogenously defined tranche or to determine the theoretically appropriate attachment and detachment points for a tranche to meet an exogenously defined target PD or EL. We compare the attributes of different tranching approaches. 1 We are grateful to John Hull, Alan White, David Lando, Jordan Mann, Yufeng Ding, and Aziz Lookman for

extensive comments on earlier drafts of this article. We also thank all of the members of Moody’s Academic Research and Advisory Committee for useful discussions during presentations of this work at MAARC meetings. We wish to thank Shirish Chinchalkar for extensive programming work to generate the numerical examples and for his comments. All errors are of course, our own. Finally, the views presented here are those of the authors and do not represent the views of our current or former employers.

2 Please note that we abuse the notation a bit to use probability of default (PD) to denote the probability of an economic loss on the tranche since default for a tranche as a legal or contractual term may not be very well-defined.

Professor Roger Stein

We also examine the relative conservatism of different approaches to tranching. We define tranching framework X as more “conservative” than tranching framework Y in a specific setting if, in that setting, tranche credit enhancement (CE) levels are higher when using framework X than when using framework Y. For our discussions, we assume that the “target” or idealized levels of EL and PD are exogenously defined.

We collect a series of sometimes disparate observations that follow simply from the mathematics of calculating PD and EL for a given tranche. Though these

observations follow logically, a number of results could seem to be counter-intuitive. To illustrate, we use stylized collateral loss distributions, though the results hold for more realistic loss distributions as well. Our goal is to provide more transparency into the credit risk analytics of structured transactions as well as to provide some clear

mathematical results and implications that market participants can use in thinking about these risks.

We show that (a) neither approach results in uniformly more or less conservative tranche levels;3 (b) for a fixed detachment (attachment) point of a tranche, lowering the attachment (detachment) point necessarily increases the EL of the tranche, even if the tranche is made thicker by this move and regardless of the probability distribution of the underlying collateral; (c) because different organizations use different target PD and EL levels for determining risk grades and use different assumptions in generating these targets, it is typically unclear (without reference to the underlying loss distribution) which tranching method will imply higher CE levels in any given instance; (d) there is generally no constant LGD assumption (including 100% LGD) that can be made on collateral to translate a PD-based tranche into an EL-based tranche; (e) assuming a

monotonically decreasing density of the loss distribution in the tail, the upper bound on the LGD of the senior-most tranche is 50% under EL-based tranching (in practice, for loss distributions often used by market participants, the LGD of the senior-most tranche may be much lower); (f) in the EL-based approach, it may be impossible to create a

tranche with a target EL, given a specific capital structure and loss distribution; this will occur whenever the PD of the tranche above is higher than the target EL of the given tranche.

The remainder of this paper is organized as follows: Section 2 outlines notation and terminology. Section 3 briefly discusses the basic mathematics of tranching and some implications, including the observation that EL-based and PD-based tranching approaches do not produce similar attachment and detachment points for any tranche in the capital structure. Section 4 shows that, for a fixed detachment (attachment) point of a tranche, lowering the attachment (detachment) point necessarily increases the EL of the tranche regardless of the probability distribution of the collateral. Section 5 presents the upper-bound for the LGD on the senior-most tranche. Section 6 illustrates why some tranches with a target EL are unattainable under the EL-based tranching approach, even though they can exist under the PD-based approach. 3 This does not mean that one approach is “better” than the other. Rather, analyses that focus on EL vs. PD

answer different questions, both of which are of interest to investors. Provided the analyses are used in the manner defined (i.e., as either an EL or a PD), there is no inconsistency.

Professor Roger Stein

2 Defining a tranche

A tranche is specified by attachment and detachment points of a tranche to

collateral losses. These points partition losses on the underlying collateral such that losses below the attachment point do not affect the tranche, whereas losses above the attachment point are absorbed entirely by holders of the tranche until these losses exceed the detachment point, when the tranche has lost everything.

Tranching is the structured finance analog to the use by a corporation of multiple classes of liabilities. In both cases, the mechanism seeks to provide different liability holders with different liens on the assets supporting the liabilities should the issuer

default. The function of tranching is to apportion losses in the underlying collateral loss distribution among tranche holders in a manner that provides more or less risk of loss to the holders of the different tranches. The most common tranching methods used target a specific PD (PD-based tranching) or a specific EL (EL-based tranching) in which the attachment and detachment points are set to achieve a target expected default frequency (PD) or expected loss (EL) for the tranche in order to appeal to investors with a specific risk preference. Clearly, in the settings above, tranche attachment and detachment points depend critically on the assumptions of the collateral loss distribution.4

Conversely, given a set of exogenously defined tranches and a loss distribution

for the collateral, one can use the same analytic machinery to determine the PD or EL for a tranche. This is often done in the rating process where the capital structure of a transaction is defined by bankers or other structurers and ratings are then determined by raters, based on the credit enhancement implicit in the capital structure of the

transaction. (e.g., Lucas, Goodman, and Fabozzi (2006)).

For purposes of our discussion here, we consider tranching in a very simple

setting, similar to that which is done for synthetic transactions, by assuming no cash

flow waterfall and a simple loss-priority approach. Assuming no waterfall simplifies the analysis but it ignores some features of cashflow timing that can affect tranching for cashflow transactions. Despite this limitation, many of our results carry over to the waterfall case. (Mahadevan, et. al (2006), for example, provides an overview of some typical securitization structures.)

The mathematics of tranching is relatively simple and involves only basic

calculus and probability theory that can be found in any undergraduate textbook on probability. (e.g., Feller, W. (1978)). However, the subtlety lies primarily in the interpretation and the logical consequences of these results. In the next section we review the mathematical machinery in detail before going on to explore these

consequences. 4 The discussion of which distributional assumptions are appropriate for which assets and of how to

parameterize these distributions is a widely debated topic that is beyond the scope of this article. For our purposes, we assume a collateral loss distribution that is given and may have been calculated analytically, through simulation, empirically or otherwise.

Professor Roger Stein

3 The mathematics of tranching

We examine two tranching approaches used to determine the attachment point,

A, for a tranche with a given detachment point of D. Under both approaches, when the collateral pool underlying a transaction experiences a loss (L) that is less than A, the

tranche experiences zero loss, whereas the tranche is wiped out if the pool losses exceed

D. For simplicity, we assume that the pdf of the collateral loss distribution is given.

3.1 PD-based tranching

Tranche PD is the probability of collateral losses exceeding the attachment point.

A tranche meets a given PD target if the tranche PD is less than an exogenously defined value, PDT:

TranchePD P(L A)

f(L) dLA1 (1.1)

PDT

where A

L

f(·) ≡ the tranche attachment point ≡ the percentage loss on the portfolio ≡ the pdf of the percentage loss on the portfolio, and PDT ≡ predefined target default rate for a tranche.

PD-based credit enhancement (CE or subordination) is equal to the collateral

portfolio credit VaR5 with α = PDT .

Note that the tranche width, (D-A), does not appear anywhere in this expression, underscoring the insensitivity of the PD-based approach to the width of the tranche (or the severity of losses on it). Thus a very thick tranche and a very thin tranche, with the same attachment point, would have the same credit quality under the PD-based

tranching approach. Thus, the required subordination to achieve a given target PD is the same for all variations on the size of the tranche and configuration capital structure of the transaction.6 5 e.g., Bohn and Stein (2009).

Recently, there has been much attention focused on so called “thin tranches” that span only a small region of 6

the loss distribution. These tranches are viewed by some as being more risky that traditional tranches since when they default, they can be completely wiped out very quickly. The PD approach treats thin tranches and thicker tranches the same with respect to subordination levels.

Professor Roger Stein

In the PD-based tranching approach, tranche credit enhancements can be

calculated simply by determining the 1-PDT th quantile of the loss distribution.7

3.2 EL-based tranching

For EL-based tranching the EL of the tranche is calculated. This calculation is

more involved as it depends not only on the underlying loss distribution but also on the width of the tranche. Consider two tranches with identical attachment points but

different detachment points. These tranches will experience different percentage losses for each dollar lost, since the loss will represent a different percentage of the total size of the respective tranches.

As in the case of PD-based tranching, whenever the pool experiences losses less than A, the tranche is unaffected. If losses are greater than D, the tranche is completely wiped out, i.e., it experiences its maximum loss, D – A. However, when the losses are between A and D, the measure is affected differentially: in absolute terms, the tranche loss increases linearly from 0 (at A) to D – A (at D), or, in percentage terms, relative to the tranche par, from 0% at A to 100% at D. This can be expressed as

L A Percentage TrancheLoss min 1,max ,0 D A

The expected loss of this tranche, relative to the tranche width (D – A), can be

expressed (cf., Pykhtin (2004)), using the pdf of the collateral loss distribution, as

1

TrancheEL Percentage TrancheLoss fL(L)dL

0 (1.2)

L A min 1,max ,0 fL(L)dL D A 01

where the terms are defined as above and

Note that the tranche width, (D-A), is explicit in this expression, underscoring the sensitivity of the EL-based approach to the width of the tranche (and the severity of losses on it). It can be shown that, for the same attachment point, A, the smaller the width of the tranche (the closer we make D to A), the higher the tranche EL. (We provide more detail on this point in Section 5.) 7L ≡ percentage loss on the portfolio fL( ) ≡ pdf of the loss on the portfolio For some distributions, this can be done analytically. For empirical distributions, it can be done non-

parametrically by simply choosing the quantile.