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Cash Flow Modeling of Amortizing Structured Pools: Assets

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Synthesizing the Cash Flow Model with Data In structured finance, by far the most difficult task is to model the cash flow generating capacity of the assets in the pool. This is true for an amortizing pool. It is even truer of a revolving portfolio. Although in practice, much of a structuror’s time is spent on liability side modeling, attention to the asset side brings the biggest bang for the buck in terms of value discovery. In this class we will devote equal time to the asset and the liability sides. We begin by identifying the basic financial parameters of an asset pool, from which we will derive its cash flow quantities. Under normal circumstances, these parameters will be given to the analyst by the issuer by the banker, who requests them from the corporation doing the financing. Perhaps the most important data set is the static pool data, preferably on losses and recoveries, going back in time as far as possible. Quantitative borrower- or collateral-level data that does not translate immediately into a default number but that is indicative of relative credit quality may also be included as part of the package. The more this type of data is available on the real pool to be securitized, the better the prediction of performance will be. Starting the Cash Flow Model It is a cash flow model, so we begin with time. In this simplified version, time t will be divided into discrete collection periods and will thus take on integer values only; in other words ].....,,2,1,0[ Tt� . This nomenclature also implies the condition � �Ttt ,0,1 ���� .1 Since time steps are numerically always equal to one, we omit them throughout. The following basic data are required: Number of Loans in the Pool: 0)0(),( NNtN �

Weighted Average Asset Coupon (periodic): r WAC

� 12

Here we implicitly assume monthly payment frequency, which is the case in most consumer asset classes. Weighted Average Maturity (months): TWAM � Initial Pool Balance: 0)0( BB � Expected Loss as a % of 0B : )(LE Organization of the Cash Flow Model Please remember that in structured analysis, assets and liabilities are completely independent and can be treated separately at all times. After computing cash flows below, we will feed them through the liabilities later on. Our spreadsheet-based cash flow model will be organized in the following way:

1. Each collection period (i.e. one month) will represent one row of the spreadsheet.

1 Note: this expression reads “for all t from 0 to termination.”

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2. Each row will then be divided into two columnar sets. The first set of columns will be devoted to the computation of the cash flows generated by the assets, while the second will allocate these to the two classes of bondholders in the two-tranche liability structure just indicated.

3. In this handout, we discuss only the structure of the first set: assets. First Principles: The Distribution of Expected Losses By definition, the expected loss is the first moment of the credit loss density function or credit loss distribution. Although we will not develop this concept further here, the loss density function is required to assign credit ratings to structured securities. The distribution is not visible to the naked eye; it relates the uncertainty known to exist (but in unknown quantities) to the vagaries of pool performance within stable economic environments. Normally, the selection of distribution should be the subject of intense discussion, because it will have an immediate impact on the capital required for the ratings. The market practice for commodity assets like consumer loans is to choose either the normal or the lognormal distribution. Any loss density function )(Lf must in general satisfy the basic boundary condition 1,0)( �� LLf , where the loss L is expressed as a fraction of the initial pool’s outstanding principal balance. This means that losses in excess of 100% of the pool cannot occur. In practice, this condition is routinely violated by the assumption that all losses above 100% have negligible probability. Nevertheless, analysts should be aware that they are making an approximation. In addition, negative credit losses are usually neglected although they are possible in structured as well as corporate finance. An example of a lognormal credit loss distribution is given in Figure 1 below. In this example, )(LE would be ca. 1.5%.

Figure 1 – Lognormal Credit Loss Distribution

Credit Loss Distribution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 2 4 6 8 10

Credit Loss (%of Initial)

P ro

ba bi

lit y

of L

os s

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Modeling the Credit Loss Curve: Before embarking on this task, it is important to remember that credit loss means loss of principal, not interest or interest plus principal, and that the loss curve should never be confused with the loss distribution shown above. You can think of the loss distribution as a scatter plot of many endpoints on loss curves, which have been constructed from many static pools of loans with relatively homogeneous performance expectations. (IE, the borrowers have similar ranges of FICO scores, similar debt-equity ratios, and the assets are somewhat homogeneous in value.) Modeling the loss curve a time-dependent process may be the most difficult computational challenge of the cash flow model. In reality, you only get to see one static pool from a given issuer in a given quarter – if that. But, YOU COULD IMAGINE seeing many such pools. You can create them in an excel file using a simulation method discussed on pages 170-171 in our book. For now, we will limit ourselves to the task of constructing a loss curve. A loss curve is one scenario of how losses may or have occurred. If the pool has amortized completely (principal balance outstanding is 0%) then the end point of that loss curve is the historical loss. If it has yet to amortize fully, the end point is still a matter of conjecture, and it has a statistical dimension. We call it the “expected loss.” We will model the loss curve through its functional form, which we input into our cash flow model using equation (1) below: Although there are many functional forms available to model credit losses, any chosen form must all be able to simulate fairly accurately the empirical behavior of losses encountered in structured pools. Of these, the logistic curve2 is by far the simplest to use and manipulate. Many practitioners have chosen it as the most appropriate to model the time-dependent cumulative credit loss function )(tF of many asset classes in structured finance. Thus, the credit loss is usually expressed via:

0( ) ( )

1 c t t a

F t be

� �

(1)

The first time derivative of the CDF, the marginal loss distribution, is defined as )(tf and is given by:

2)(

)(

]1[ )(

)( 0

0

ttc

ttc

eb ecba

t tF

tf

� �

� �

� (2)

Although readers should verify this, the analytical behavior of Equation (1) can be surmised intuitively by inspection of its form. For instance, when the exponential term in the denominator is negligible, the quantity )(tF clearly becomes equal to the constant a , hence (1) is asymptotic to a as �t since the denominator tends to 1 at �t . Since the exponent of “e” is negative for 0tt � and positive for 0tt , we can readily see that )(tf must rise progressively from zero until

0tt � and then fall progressively back to zero for 0tt . Finally, parameter b simply dictates the region over which the exponential factor will be effective. For example,

2 Note: The logistic curve describes the full cycle of birth and death of a population, from inception to

maturity. Here, the population in question is that of losses.

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for 0�b , atF �)( holds for all t from 0 to T . In fact, via appropriate choices for its four parameters, Equation (1) can replicate the credit loss behavior of most asset pools. By definition, the following continuous-time relationship holds between the cumulative distribution function (CDF) and the probability density function (PDF):

dttftdF )()( � The above equation relates the monthly defaults on the right to the change in cumulative loss curve values on the left. Thus, in our spreadsheet approximation, we may define marginal or monthly losses via: Monthly losses during time t: )1()()( � tFtFtl With time measured in months, examples of marginal and cumulative loss curves are given below in Figures 2 and 3 respectively for the following typical values of the parameters

cba ,, and 0t :

10�a 1�b

1.0�c 550 �t 120�T

Figure 2 – Curve Reflecting Total Cumulative Losses

Cum ulative Los s Curve

0 2 4 6 8

10

0 24 48 72 96 120

Tim e (m onths)

C u

m . L

o ss

R at

e (%

)

Figure 3 – Curve Reflecting Periodic Losses

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Marginal (Periodic) Loss Curve

0.000

0.050

0.100

0.150

0.200

0.250

0 24 48 72 96 120

Time (months)

M ar

gi na

l L os

s R

at e

(% )

Account-Space Calculations The market convention is to model dollars rather than accounts. This model starts with account- or loan-space estimates because loans are the units that are known to default, not dollars. This starting point gives the modeler maximum flexibility in analyzing the impact of different account-wise behavioral assumption without loss of generality or having to rebuild the model for each new scenario. Transforming account-space defaults to dollar-space defaults is not difficult and is shown below. Normalization of Loss Curves Loss curves of the type of Equation (1) normally require normalization in order to properly reflect the expected loss known to be applicable to the pool under study. We, now show how to normalize defaults so that cumulative defaults at Tt � are indeed equal to the chosen value of )(LE adjusted as required for account-space utilization. For instance, with an assumed 2,000 accounts initially in the pool, assume we wish to simulate an expected loss value of 10% of the account base, or 200 accounts. In effect, we are saying that over the life of the pool (120 months), 200 accounts will default at one point or another according to the loss curve shown in Equation (1). Using the parameters above, the following results would emerge:

814.0)0( �F 899.0)1( �F 993.0)2( �F

This means that account-space losses during 1�t are 085.0)0()1()1( � � FFl , and that for 2�t , they are 094.0)1()2( � FF . There are no account-space losses during t=0, because t=0 is a point in time (“the close”) rather than a duration. Also, we note that 7.199)120( �F instead of the required 200. The table below shows percentage of completion of losses (i.e., the loss curve) and corresponding number of defaulted accounts in the last two time periods (N). Figure 4 – The End of the Curve, Normalized and Non-Normalized

N E(L) Accts. Normalized

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119 9.983% 199.668 199.968 120 9.985% 199.700 200.000 In the fourth column of Figure 4, the number of defaults is normalized by a process that is known as “normalization.” In a nutshell, it is equivalent to multiplying the elements of the

non-normalized curve by this ratio ���������� � ��� � �� � ��� � ��� � �� �

� � � �

.

Formally, that process is described thus: to correct this small error, simply normalize each monthly entry as follows: With )(tl as the marginal non-normalized losses in loan-space, )(tnD as the equivalent normalized loan-space losses, 0N as the number of initial accounts and rD as the cumulative total default rate, we have by definition: rDNLE 0)( � . Thus, )(tnD can be defined from )(tl as:

� �)0()( )(

)(

)( )( 0

1

0

FTF DNtl

tl

DNtl tn r

T

t

r D

��

� �

.

Ordinarily we would think of the denominator as being equal to )(TF , not )0()( FTF . The reason for the adjustment is simply that )0(F does not begin at 0, hence the distance from )0(F to )(TF is less than the actual unit length. For this reason, using the formula 0 0

1

( ) ( ) ( )

( ) ( )

r r D T

t

l t N D l t N D n t

F T l t

� �

� still would not bring cumulative

defaults to the target 10%. Account-Space Adjustment What if, on a real portfolio, the percentage of accounts in default is not equivalent to the percentage of defaulted dollars? We can assume they are the same for a hypothetical case, but in real portfolios we cannot simply assume this problem away. If defaults turn out to be concentrated in the accounts that have higher balances, then to assume equivalency between the account-space and the loan-space percentages could materially underestimate total losses when it comes to simulating asset performance. To adjust the loss curve for discrepancies, first note the corresponding dollar-space loss amount and multiply the account-space cumulative default rate by the ratio of discrepant dollar-space defaults to loan-space defaults. For instance, if the dollar-space default rate is 10% but the loan-space cumulative default rate is only 6.5%, you would need to multiply

f(t) at each point in time (see above) by the ratio 54.1 5.6

10 � in order to project the loss

curve leading to an a (in the formula) of 10%. Thus we would be able to simulate the given dollar-space loss distribution by using loan-space values multiplied by the ratio 1.54. Please note that the ability to so easily switch from loan- to dollar-space values stems from the linear way parameter a appears in Equation (1). Note, too, that this discussion

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concerns defaults and not to losses net of recoveries. Recoveries on defaults may operate according to a different mechanism, which we will discuss below. Modeling Prepayments Serious prepayment modeling is difficult and lacks an all-purpose functional form like the S- curve for defaults. Although many variables affect the level of prepayment in structured pools, most important by far is the prevailing interest rate environment. Prepayment modeling thus amounts to interest rate prediction, a notoriously inaccurate science. Even if many reliable prepayment models exist, all are conditional prepayment models. This means that if interest rates are assumed known, the corresponding pool prepayment rates can be determined fairly precisely. Unfortunately, this is equivalent to assuming the problem away. However, in most ABS asset classes (mortgages being the great exception) we can treat prepayment as a secondary credit issue since very few borrowers have the necessary incentive to prepay solely based on interest rate variations. Whereas a mortgage borrower may save $50,000 or more by refinancing his or her home at a lower rate, an automobile borrower barely breaks even by doing so. As a result, the variability of prepayment behavior within the relatively short-maturity asset pools of most ABS asset classes is largely driven by factors other than interest rates (none any more predictable than interest rates). To avoid false precision, for non-mortgage asset classes, we usually restrict prepayment models to simple functional forms like the one shown in Figures 4 and 5 below, giving the marginal and cumulative curves, respectively, for a typical prepayment model. The marginal prepayment curve says that the monthly amount of prepayments (g) begins at 0 and rises at a constant rate of increase until reaching its characteristic steady-state rate, at t00. For this example, we assume that 20% (= PN ) of the pool’s initial account inventory will prepay at one point or another over the life of the 10-year transaction. Figure 4 – Marginal Prepayment Curve for a 10-Year Transaction

Prepayment Density Function

0.00

0.07

0.14

0.21

0 24 48 72 96 120

Time (months)

M ar

gi na

l P re

pa ym

en t R

at e

(% )

To compute marginal prepayments, calculations identical to the calculation of periodic defaults are required. So, for a prepayment cumulative distribution function )(tG , we simply

define monthly account-space prepayments )(tnP as )1()()( � tGtGtnP . From Figure 4 above, we can easily define )(tG as:

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2

0

2 0

0 0 0

, 0 2( )

( ) , 2

P

P P P P

a t t t

G t at

t t at t t

� � ���

� � � � � ���

In the above relation, parameter a is the slope of the straight line from 0 to 0t months and point t0P

3 is the time when prepayments reach a steady-state condition. Note that t0P is

generally not the same time as the inflection point on the default curve, t0. The derivative of these two equations over the range of t = 0 to t = T is the line g(t) = {at, 0�t�t0P ; g(t) = at0P, t0P�t�T. Geometrically, in the formula for G(t), the area under the curve from t=0 to 48 is triangular and corresponds to a triangle. The second term defines G(t) after the inflection point. It corresponds to a rectangle with a base of (T- t0P) and a height a t0P . The graph of G(t) is depicted in Figure 5, below.

Cumulative Prepayment Distribution

0

5

10

15

20

0 24 48 72 96 120

Time (months)

C um

. P re

pa ym

en t R

at e

(% )

Figure 5 – Prepayment Distribution Function for a 10-Year Transaction

To normalize this loan-space cumulative distribution function, we simply use the fact that at Tt � , the following boundary condition holds:

Ppp

t

NtatTdtta p

� �� 00 0

)(][ 0

Solving fora , we find:

pp p

P

ttT t

N a

00

2 0 )( 2

� �

3 Here, ��� is 48 months; in the assignment, it is 45 months; and in the conventions of CPR notation, it is 30 months.

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Here again, note the linearity of )(tG with respect to parameter a . We are now ready to compute cash flows from a pool of consumer assets with the above default and prepayment characteristics. Asset Cash Flow Analysis Assume pool assets consist of 0N level-pay loans, each with a weighted average periodic coupon r , a weighted average maturity of T months and a time-dependent principal balance )(tV with 0)0( VV � . This allows us to describe the entire cash flow process from closing until the assets pay off. Ending Loan Count Logically, we have at the end of the month:

)()()1()( tntntNtN PD � Before we jump into the detailed computation of asset cash flows, a comment on the nature of level pay loans is in order, because the typical payment profile for consumer assets going into the SPE is that of a level pay loan. The Level Pay Loan The standard level pay loan is a series of cash flows with a principal and an interest portion. The total size of the cash flow does not vary, but the relative proportions of interest and principal in each time step do change. Amortization in A Discrete Framework The principal amount at every time step always equals the present value of the remaining principal balance outstanding at par. The interest amount is equal to the difference between the level pay amount and the principal amount. As time passes, the principal amount outstanding decreases, which lowers the interest cost. Hence, with the passage of time, interest decreases as a percentage of the total payment amount, and principal increases. Amortization in A Continuous Framework A level-pay loan is defined by the following first-order linear differential equation:

dBdtBrdtM � (3) We do not use equation (3) in the cash flow model, but it shows us that the value of a level-pay loan (M) is a function of time and the level of principal balance. As in the discrete case, M has two components:

i. Interest (r), which is a product of the (continuous) interest rate and the remaining principal balance outstanding that is changing in time, B(dt); and

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ii. Decrements of the principal balance outstanding, dB. In Equation (3) M is the total payment rate, a constant number by definition. Equations like (3) are all solvable and the solution can be looked up in any calculus textbook. The minus sign in Equation (3) stems from the fact that the loan’s principal balance is decreasing as time increases. We do not go through the derivation of equation (4) from equation (3). But with (4), we are back to the periodic process described at the beginning of this sidebar, which are more descriptive of real asset cash flows. Skipping the details, then, we have:

� �Ttr r M

tB � � )1(1)( (4a)

This formula gives us the principal balance outstanding at the end of time period (t). Note that the principal balance outstanding at the beginning of the period is B(t-1) and the corresponding equation is

� �1( ) 1 (1 )t TMB t r r

� � (4b)

We use equations (4a and 4b) in the cash flow analysis, as you will see below. The two boundary conditions 0)0( BB � and 0)( �TB can be satisfied by setting:

0

1 (1 ) p T

B r M

r �

� (5)

This formula is used to calculate MP, the aggregate of level pay amounts. Dividing MP by N0, we arrive at the level pay amount for a single loan, M. Since our analysis is on the “average loan,” we use M in our cash flows. Interest Collections To calculate the interest collected during the month, we can use Equation (4) above and the definition � � )1()()1()( � tBrtntNtI D , to yield:

� � � �TtD rMtntNtI � � 1)1(1)()1()( The right-hand side of this equation can be broken into three components: the number of performing loans ( 1) ( )DN t n t � �� � ; the periodic payment on a single loan, M; and the

inverse of the discount factor, 1[1 (1 ) ]t Tr � . Multiplied together, it means “the number of performing loans times the interest component of the single-loan, level pay amount.” Principal Collections

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Monthly principal received consists of a) regular or scheduled principal )(tPR and b) prepayments )(tPP .

a. Regular principal is computed via the scheduled monthly amortization given by Equation (4):

� � � �TtTtDR rrr M

tntNtP � � � 1)1()1()()1()(

Here, the right-hand side of the equation can be broken down into three components: the number of performing loans (as above); the future value of the

sum of all the single-loan level pay amounts, M r

; and amortization of the discount of

principal balance at par, which is 1[(1 ) (1 ) ]t T t Tr r � � .

b. Prepayments are equal to the number of prepaid loans multiplied by the ending balance at time step t . We use the ending balance amount as the “prepayment” amount since the monthly amortization is technically a regular principal payment.

� �TtP rr M

tntPP � � )1(1)()(

Total principal )(tP actually received during this collection period is computed as:

)()()( tPtPPtP R�� , but this does not go into the cash flow model. Defaults Defaulted balances )(tD are equal to the number of defaulted loans multiplied by the outstanding principal balance at time 1 t :

� �TtD rr M

tntD � � 1)1(1)()(

Principal Due Each Collection Period, the principal due )(tPD to bondholders that allows them to remain fully collateralized is thus:

)()()()( tDtPPtPtP RD ��� Clearly, only the first two components were actually received while defaults must be covered from excess spread.

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Ending Pool Balance4 The ending pool balance at time t is the initial pool balance at the beginning of time t less regular principal, prepayments and defaults viz.:

)()()()1()( tPtPPtDtVtV R � Recoveries (or “Loss Given Default” in the corporate world) Defaulted balances are usually subject to recoveries )(tRC , for instance if the collateral is sold or auctioned off after a time delay rt and the proceeds fed back through the trust. We would then have:

� �)(1)()( rrC ttLGDttDtR � (6) In Equation (6) above, )( rttLGD is the loss given default as a percentage of the loan’s outstanding principal balance at the time of default, i.e. rtt . The value of LGD is time- dependent since recoveries will depend on the coincident values of the collateral and the loan’s outstanding principal balance. The time-dependent phenomenon is displayed in Figure 6 below where we show a typical loan amortization curve next to an asset depreciation curve, normally a durable good, like a car. Figure 6 – Loan Amortization and Asset Depreciation Curves

Loan Amortization vs. Asset Depreciation

0.00

0.20

0.40

0.60

0.80

1.00

0 24 48 72 96 120

Time (months)

A ss

et V

al ue

o r

P ri

nc ip

al

B al

an ce Asset

Loan

In the middle of the range (two-headed arrow) the recovery amount as a percentage of the loan’s outstanding principal balance would tend to be the smallest since the gap between the loan’s principal balance and the asset’s depreciated value is relatively larger than either before or after that. But, when both curves intersect (~time step 96), recoveries would theoretically be 100%, i.e. LGD = 0, since the value of the asset is now equal to the loan balance outstanding. Beyond that point, there would be a credit “profit” instead of a credit ”loss”, although certain jurisdictions do not allow lenders to make money from defaults.

4 In this relationship, )(tV means the principal balance outstanding of the asset pool while )(tB refers to an individual loan’s balance.

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So, in a realistic simulation, LGD would start out at 0�t with a certain, relatively small value corresponding to the difference between the retail and wholesale value of the asset, rise monotonically until approximately month 40 and then decline monotonically from month 40 to month 120, including the region 12096 �� t for which LGD < 0. In practice, obligors who have reached time step 96 without a default would most likely not default after that since they could simply sell the asset in the open market and pay off the loan balance in full. In reality though, very few obligors are savvy enough to accomplish this and many defaults are in fact seen after the point at which credit losses are theoretically negative. However, for purposes of your cash flow model, we will ignore this clear time-dependence of

LGD and assume LGD is a constant value )](1[)()( tLGDttDtR rC � . You can choose our own lag value rt such that 72 �� rt . Collections Putting it all together, we would compute the very important quantity Available Funds

)(0 tAF for the current collection period as:

)()()()(0 tRtPtItAF C��� Here, as we will describe in the Liabilities Analysis, the subscript attaching to AF signifies the level of the waterfall, so )(0 tAF is the top of the waterfall, before any disbursements. Other Assets of the Trust In this course, we will not discuss other potential sources of cash in the form of reserve accounts, insurance policies and the like. The discussion of some of these sources will be postponed to more advanced classes. Homework: Please use the following data to estimate the cash flow generated from a pool of these parameters: Collateral:

000,000,30$)0( �V 20000 �N

120��TWAM %12�WAC

5.0)( �tLGD Defaults: Use the logistic curve for )(tF with the following parameters:

ra D� 1�b

1.0�c 550 �t

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1.0�rD (10% expected loss)

[Remember to normalize loan-space defaults] Prepayments: Use the model given in class with the following parameters:

400�PN (20% expected prepayments) 0 45Pt �

[Remember that you have to compute the parameter a before you can begin] Liabilities (separate from the Assets, but an integral part of the cash flow model):

8.0�� %7�Ar %9�Br %1�fs %20�rs