cash flow
calculate the loss curve, cash flow.
I gave excel, and two notes.
should read notes first then calculated it
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-��$�$#��� G��� Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 9
Synthesizing the Cash Flow Model with Data In structured finance, by far the most difficult task is to model the cash flow generating We begin by identifying the basic financial parameters of an asset pool, from which we will .
Starting the Cash Flow Model It is a cash flow model, so we begin with time. In this simplified version, time t will be 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 �
1 2
Here we implicitly assume monthly payment frequency, which is the case in most consumer 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 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.”
Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 1 0
2. Each row will then be divided into two columnar sets. The first set of columns will be 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 Any loss density function )(Lf must in general satisfy the basic boundary An example of a lognormal credit loss distribution is given in Figure 1 below. In this
Figure 1 – Lognormal Credit Loss Distribution Credit Loss Distribution
0 1
0.2 5
0.6 0 2 4 6 8 10
Credit Loss (%of Initial )
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ba li t
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of os Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 11
Modeling the Credit Loss Curve: Before embarking on this task, it is important to remember that credit loss means loss of Modeling the loss curve a time-dependent process may be the most difficult computational For now, we will limit ourselves to the task of constructing a loss curve. A loss curve is one Although there are many functional forms available to model credit losses, any chosen form 0 ( ) 1 c t t F t � (1)
The first time derivative of the CDF, the marginal loss distribution, is defined as )(tf and is 2 )(
)( ]1[ )( 0 ttc
ttc eb t tf � � (2)
Although readers should verify this, the analytical behavior of Equation (1) can be surmised For instance, when the exponential term in the denominator is negligible, the quantity )(tF 0tt � and then fall progressively back to zero for 0tt . Finally, parameter b simply
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. Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 12
for 0�b , atF �)( holds for all t from 0 to T . In fact, via appropriate choices for its four dttftdF )()( � The above equation relates the monthly defaults on the right to the change in cumulative Monthly losses during time t: )1()()( With time measured in months, examples of marginal and cumulative loss curves are given cba ,, and 0t : 10�a 1.0�c
Figure 2 – Curve Reflecting Total Cumulative Losses Cum ulative Los s Curve
0 10 0 24 48 72 96 1 20
Tim e (m onths)
C m o R e ) Figure 3 – Curve Reflecting Periodic Losses
Linna Dai Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 13
Marginal (Periodic) Loss Curve
0.0 00
0.050
0.100
0.150
0.200
0.250
0 24 48 72 96 120
Time (months)
M gi l L s at (% Account-Space Calculations The market convention is to model dollars rather than accounts. This model starts with Transforming account-space defaults to dollar-space defaults is not difficult and is shown Normalization of Loss Curves Loss curves of the type of Equation (1) normally require normalization in order to properly For instance, with an assumed 2,000 accounts initially in the pool, assume we wish to 814.0)0( �F
This means that account-space losses during 1�t are 085.0)0()1()1( � Also, we note that 7.199)120( �F instead of the required 200. The table below shows Figure 4 – The End of the Curve, Normalized and Non-Normalized N E(L) Accts. Normalized Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 14
119 9.983% 199.668 199.968 In the fourth column of Figure 4, the number of defaults is normalized by a process that is non-normalized curve by this ratio ��
� ��� � �� � �
� �
.
Formally, that process is described thus: to correct this small error, simply normalize each With )(tl as the marginal non-normalized losses in loan-space, )(tnD as the equivalent � �)0()( )( )( 1 FTF tl
tl DNtl T
t r �� .
Ordinarily we would think of the denominator as being equal to )(TF , 1 ( ) ( ) ( ) r r t l t N D l t N D F T � � 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 To adjust the loss curve for discrepancies, first note the corresponding dollar-space loss f(t) at each point in time (see above) by the ratio 54.1 10 curve leading to an a (in the formula) of 10%. Thus we would be able to simulate the given Please note that the ability to so easily switch from loan- to dollar-space values stems from Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 15
concerns defaults and not to losses net of recoveries. Recoveries on defaults may operate Modeling Prepayments Serious prepayment modeling is difficult and lacks an all-purpose functional form like the S- However, in most ABS asset classes (mortgages being the great exception) we can treat As a result, the variability of prepayment behavior within the relatively short-maturity asset The marginal prepayment curve says that the monthly amount of prepayments (g) begins at 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 l P pa en at To compute marginal prepayments, calculations identical to the calculation of periodic define monthly account-space prepayments )(tnP as )1()()( From Figure 4 above, we can easily define )(tG as: Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 16
� 2 0 0 0
, 0 ( ) , P P a t G t t t at t t
� � �
In the above relation, parameter a is the slope of the straight line from 0 to 0t months and 3 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, Cumulative Prepayment Distribution
0 C . P pa
Figure 5 – Prepayment Distribution Function for a 10-Year Transaction
To normalize this loan-space cumulative distribution function, we simply use the fact that Ppp
t NtatTdtta � )(][
Solving fora , we find: pp P ttT N 00 2
�
3 Here, ��� is 48 months; in the assignment, it is 45 months; and in the conventions of CPR notation, it Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 17
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 Asset Cash Flow Analysis Assume pool assets consist of 0N level-pay loans, each with a weighted average periodic 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 � The Level Pay Loan The standard level pay loan is a series of cash flows with a principal and Amortization in A Discrete Framework The principal amount at every time step always equals the present value Amortization in A Continuous Framework A level-pay loan is defined by the following first-order linear differential dBdtBrdtM We do not use equation (3) in the cash flow model, but it shows us that i. Interest (r), which is a product of the (continuous) interest rate Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 18
ii. Decrements of the principal balance outstanding, dB. In Equation (3) M is the total payment rate, a constant number by We do not go through the derivation of equation (4) from equation (3). � �Ttr tB
This formula gives us the principal balance outstanding at the end of time � �1( ) 1 (1 )t TMB t r �
We use equations (4a and 4b) in the cash flow analysis, as you The two boundary conditions 0)0( BB � and 0)( �TB can be satisfied by 0 1 (1 ) B r r
�
This formula is used to calculate MP, the aggregate of level pay amounts. Interest Collections To calculate the interest collected during the month, we can use Equation (4) above and the � 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 � �� � ; 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 Principal Collections
Linna Dai Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 19 a. Regular principal is computed via the scheduled monthly amortization given by � � � �TtTtDR rrr � Here, the right-hand side of the equation can be broken down into three � b. Prepayments are equal to the number of prepaid loans multiplied by the ending � �TtP rr 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 � �TtD rr Principal Due Each Collection Period, the principal due )(tPD to bondholders that allows them to remain )()()()( tDtPPtPtP RD ��� Clearly, only the first two components were actually received while defaults must be Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 20 The ending pool balance at time t is the initial pool balance at the beginning of time t less )()()()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 � �)(1)()( rrC ttLGDttDtR � (6) In Equation (6) above, )( rttLGD Figure 6 – Loan Amortization and Asset Depreciation Curves Loan Amortization vs. Asset Depreciation B In the middle of the range (two-headed arrow) the recovery amount as a percentage of the 4 In this relationship, )(tV means the principal balance outstanding of the asset pool Cash Flow Modeling of Amortizing Structured Pools: Assets R&R Consulting � Copyright 2003 � All Rights Reserved 21 So, in a realistic simulation, LGD would start out at 0�t with a certain, relatively smalla
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S&P Structured Ratings
Moody’s Structured Ratings
Def Rate
Letter Grade
DIRR
0
A
Aaa
0.000
1
A
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0.0
6
Aa1
0.000
2
5
0.6
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B1
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B2
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23
B3
100
CC
3
11
Ca
CC
2
50
Deal
Variable Name
Value
Pool Initial Balance
Loss Curve
Parms
Prepayments
Recovery
Number of Loans
t0
Average Loan Balance
ERROR:#DIV/0!
Monthly Payment
Servicing Fees
Current Loan
Defaults
Expected Loan Prepay
Loss Given Default
Asset
Liability
WAC (
12
Losses
Collections
Preliminaries
Level 1 of the Waterfall
Level 2 of the Waterfall
Level 3 of the Waterfall
Level 4 of the Waterfall
Level 5 of the Waterfall
Reserve Account Mechanics
Level 6 of the Waterfall
Reduction of Yield Calculations
WAM (T)
Time
Trial Defaults
Def Loans
PP Curve
PP Loans
End Loans
Interest Coll
Reg Principal
Recoveries
Principal Coll
End Pool Bal
Amortization
Avail. Funds
Beg Cl A Bal
Beg Cl B Bal
Tot Prin Due
Cum Prin Due
Svc Fee Due
Svc Fee Paid
Svc Fee Shrt
Cl A Int Due
Cl A Int Paid
Cl A Int Shrt
Cl B Int Due
Cl B Int Paid
Cl B Int Shrt
Cl A Prin Due
Cl A Prin Paid
Cl A Prin Shrt
Cl B Prin Due
Cl B Prin Paid
Cl B Prin Shrt
Reserve Draw
Avail. Collect.
Target Reserve
Res. Contrib.
End Res Acct
Residual Pmt
Tot Pmt Cl A
Tot Pmt Cl B
Advance Rate (alfa)
Class A Balance
$0
Class A Rate
Class B Balance
Class B Rate
Total Residuals
Delta IRR Class A
ERROR:#NUM!
Delta IRR Class B
Monte Carlo Runs
Standard Deviation
Moodys Cl A Rating
Moodys Cl B Rating
Moodys Cl A DIRR
Moodys Cl B DIRR
Priority of Payment
please select
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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.
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
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
WAC
asset classes.
independent and can be treated separately at all times. After computing cash flows below,
we will feed them through the liabilities later on.
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.
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.
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.
example, )(LE would be ca. 1.5%.
0.
0.3
0.4
0.
0.7
bi
L
s
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.)
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.
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:
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:
( )
a
be
�
given by:
)(
0
ecba
tF
�
�
�
intuitively by inspection of its form.
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
dictates the region over which the exponential factor will be effective. For example,
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):
loss curve values on the left. Thus, in our spreadsheet approximation, we may define
marginal or monthly losses via:
� tFtFtl
below in Figures 2 and 3 respectively for the following typical values of the parameters
1�b
550 �t
120�T
2
4
6
8
u
. L
ss
at
(%
ar
na
os
R
e
)
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.
below.
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.
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:
899.0)1( �F
993.0)2( �F
� 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.
percentage of completion of losses (i.e., the loss curve) and corresponding number of
defaulted accounts in the last two time periods (N).
120 9.985% 199.700 200.000
known as “normalization.” In a nutshell, it is equivalent to multiplying the elements of the
���������� �
��� �
���
�
�
monthly entry as follows:
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
DN
tn r
D
�
�
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
( )
( )
D T
n t
l t
� �
still would not bring cumulative
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.
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
5.6
� in order to project the loss
dollar-space loss distribution by using loan-space values multiplied by the ratio 1.54.
the linear way parameter a appears in Equation (1). Note, too, that this discussion
according to a different mechanism, which we will discuss below.
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.
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.
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.
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.
Time (months)
M
ar
gi
na
re
ym
t R
e
(%
)
defaults are required. So, for a prepayment cumulative distribution function )(tG , we simply
� tGtGtnP .
2
0
0
2( )
2
P P P
t t
at
� ���
� �
� ���
point t0P
is the time when prepayments reach a steady-state condition. Note that t0P is
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.
5
10
15
20
0 24 48 72 96 120
Time (months)
um
re
ym
en
t R
at
e
(%
)
at Tt � , the following boundary condition holds:
p
�� 00
0
0
p
t
a
0 )(
2
�
is 30 months.
default and prepayment characteristics.
coupon r , a weighted average maturity of T months and a time-dependent principal
balance )(tV with 0)0( VV � .
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.
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.
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.
equation:
� (3)
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:
and the remaining principal balance outstanding that is changing
in time, B(dt); and
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.
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:
r
M
�
� )1(1)( (4a)
period (t). Note that the principal balance outstanding at the beginning of
the period is B(t-1) and the corresponding equation is
r
� (4b)
will see below.
setting:
p T
M
�
(5)
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.
definition � � )1()()1()(
�
performing loans ( 1) ( )DN t n t
performing loans times the interest component of the single-loan, level pay amount.”
Monthly principal received consists of a) regular or scheduled principal )(tPR and b)
prepayments )(tPP .
Equation (4):
M
tntNtP
�
� 1)1()1()()1()(
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
� .
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.
M
tntPP
�
� )1(1)()(
outstanding principal balance at time 1
t :
M
tntD
�
� 1)1(1)()(
fully collateralized is thus:
covered from excess spread.
Ending Pool Balance4
regular principal, prepayments and defaults viz.:
sold or auctioned off after a time delay rt and the proceeds fed back through the trust. We
would then have:
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.
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
al
an
ce Asset
Loan
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.
while )(tB refers to an individual loan’s balance.
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
Linna Dai
Linna Dai
Cash Flow Modeling of Amortizing Structured Pools: Assets
R&R Consulting � Copyright 2003 � All Rights Reserved 22
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