October 22, 1999
How the
subsidy and its components
are derived
from cash flow observations
Direct loans and loan guarantees have a common set of
subsidy components:
Interest
Fee
Default,
net of recoveries
All other
For each component, the subsidy percentage is calculated by
dividing the present value of certain cash flow observations (described below)
by loan disbursements. The quotient is
multiplied by 100 and rounded to two decimal places.
The total subsidy percentage is the sum of the four components.
A few definitions
“Cash flow
observations” refers to a stream of cash transactions between a Federal
credit program and the public. For
direct loan programs, cash flow observations might refer to loan disbursements,
contractual principal or interest payments by borrowers, adjustments for
defaults or recoveries, prepayments, or fees received. In each case, the cash flow observations are
one or more cash inflows or outflows of a particular type.
“Present value” or
“net present value” is the value
today of a dollar in some future period, adjusted for the time value of money.
When used in regard to a series of payments, “present value” is the sum of the
present value of each payment. It is
derived by multiplying the payment in each period by a present value factor for
that period. “Net present value” is
sometimes used in regard to a series that includes both payments and receipts.
The “present value
factors” are factors, based on assumptions about interest rates, that, when
multiplied by the dollar amount for that period, yield the value in today’s
dollars. See “Description of the “basket-of-zeros” discounting method and the
derivation of present value factors from the yield curve” for details on
the derivation of present value factors.
Interested readers might also consult “How the CSC selects a basis for computing present value factors.”
The “basket-of-zeros”
discounting method defines the present value of a series of payments as the
value today of a collection of zero-coupon bonds that, at maturity, exactly
match the cash flow observations. The
derivation of present value factors from the yield curve is described in a
related paper, “Description of the
“basket-of-zeros” discounting method and the derivation of present value
factors from the yield curve.”
Derivation of the
subsidy components from cash flow observations
The subsidy components, in dollar terms, are calculated as
described below. To convert them into
percentage terms, the dollar amounts are divided by loan disbursements. Then,
the result is multiplied by 100 and rounded to two decimal places.
The total subsidy cost is the sum of the four
components.
Financing/interest subsidy costs
are defined as the portion of the subsidy attributable to subsidizing the
borrower’s interest costs by charging lower rates than the discount rate (in
certain direct loan programs) or by direct interest subsidy payments (in
certain loan guarantee programs). For
direct loans, this is calculated as the excess of the amount of the loans
disbursed over the present value of the interest and principal payments
required by the loan contracts. For
loan guarantees, this is calculated as the present value of estimated interest
supplement payments, before adjustment for defaults.
Defaults, net of recoveries,
subsidy cost, defined as the portion of the subsidy attributable to
defaults, net of recoveries. It is
calculated as the sum of discounted cash flow observations for defaults and
recoveries.
Fee subsidy cost, defined as
the portion of the subsidy percentage attributable to up-front and annual fees
paid to the government. Because these
fees are inflows to the Government, this subsidy component makes the total
subsidy either less positive or more negative.
It is calculated as the sum of the discounted fee-related cash flow
observations, before adjustment for defaults.
Other subsidy costs, defined
as the residual subsidy cost not attributed to financing, defaults net of
recoveries, or fees. It is calculated
as a residual.
Technical
considerations
The subsidy percentage and its components are derived from
several streams of cash flow observations, which are discounted to the “time of
disbursement” and aggregated. A
technical description of how this is done is contained in the appendices to
this paper:
Appendix A: Definitions from the Federal Credit Reform
Act and Related Materials provides some important definitions and
requirements.
Appendix B:
Discounting to the “Time of Disbursement” describes the
implementation of this requirement.
Appendix C:
Specifications and examples of the pro-rata method describes this
method for allocating aggregated data to disbursement periods.
Appendix D:
Specifications and examples of the reverse-spendout method describes
this method for allocating aggregated data to disbursement periods.
Changes from previous
methods
The methods in the revised Credit Subsidy Calculator
(CSC) are a significant departure from those used previously, though the effect
of these changes on the subsidy estimates will generally be minor. The important differences are:
The basket-of-zeros discounting
method is used in the place of a constant discount rate for cash flows
estimated to occur at different times.
A single effective rate is computed
to preserve the equality between the rates at which financing accounts earn or
pay interest and the rate used for discounting.
Cash flow observations may be
prepared in monthly, quarterly, and semi-annual frequencies. Previously, all
cash flow observations were in annual frequency only.
Cash flows are directly discounted to the exact time of disbursement. Previously they were indirectly discounted to the time of disbursement by discounting
cash flows and disbursements to the beginning of the fiscal year in which they
occurred. The previous method was
accurate when used with reverse-spendout discounting (which was used in the
previous model) but is not accurate when the basket-of-zeros discounting method
is used.
Appendix A:
Definitions from the Federal Credit Reform Act and Related Materials
The Federal Credit Reform Act
The Federal Credit Reform Act of 1990 (FCRA) made
fundamental changes in the budgetary treatment of direct loans and loan
guarantees. FCRA shifted the budget
basis from the amount of cash expected to flow into or out of the Treasury to
the estimated subsidy costs of the loans or guarantees.
As defined by the act, the subsidy cost of a direct loan or
a loan guarantee is the estimated long-term cost to the Federal Government
calculated on a present value basis, excluding administrative costs. The subsidy cost of a direct loan or a loan
guarantee is calculated by projecting the related cash flows to and from the
government over the life of the loans and then discounting those cash flows
back to the time of disbursement.
In 1997, section 502(5)(E) of the FCRA was amended to
require the use of the basket-of-zeros discounting method. This method defines the present value of a
series of payments as the price of a collection of zero-coupon bonds that, at
maturity, exactly match the stream of payments in amount and timing. (See the article entitled “Description of the “basket-of-zeros”
discounting method and the derivation of present value factors from the yield
curve” for details on this method.)
Compared to the former method, this method is more precise and will
yield identical cost estimates for credit programs that have identical cash
flows, even though the underlying loans may be of different maturity.
Cohorts and risk categories as the unit for subsidy calculation
OMB Circular A-11 defines a cohort as:
…all direct loans or loan guarantees
of a program for which a subsidy appropriation is provided for a given fiscal
year…. For direct loans and loan
guarantees for which a subsidy appropriation is provided for one fiscal year,
the cohort will be defined by that fiscal year. For direct loans and loan guarantees for which multi-year or
no-year appropriations are provided, the cohort is defined by the year of
obligation. Direct loans and loan
guarantees that are made from supplemental appropriations will be recorded in
the same cohort as those that are funded in annual appropriation acts. These rules apply even if the direct loans
or guaranteed loans are disbursed in subsequent years. (OMB Circular A-11, Preparation and Submission of Budget
Estimates, Transmittal Memorandum No. 72, July 12, 1999, page 283.)
OMB Circular A-11 defines risk categories as:
… subdivisions of a cohort of direct
loans or loan guarantees into groups that are relatively homogeneous in cost,
given the facts known at the time of obligation or commitment. They are developed by agencies in
consultation with the OMB representative with primary budget responsibility for
the credit account. The number will depend on the size of the difference in
subsidy cost between categories and the ability to predict it statistically
based on facts known at origination.
Risk categories will group all
direct loans or loan guarantees within a cohort that share characteristics
predictive of defaults and other costs.
They may be defined by characteristics or combinations of
characteristics of the loan, the project financed, and/or the borrower. Examples of characteristics or indicators
that may predict cost include:
The loan-to-value
ratio;
The relationship between
the loan interest rate and relevant market rates;
Type of school attended
for education loans;
Country risk categories
for international loans; and
Various asset or income
ratios.
Statistical evidence must be
presented, based on historical analysis of program data or comparable credit
data, concerning the likely costs of defaults, other deviations from contract,
or other costs that are expected to be associated with the loans in that
category. (OMB Circular A-11, Preparation
and Submission of Budget Estimates, Transmittal Memorandum No. 72, July 12,
1999, pages 289-90.)
Appendix B:
Discounting to the “Time of Disbursement”
The Federal Credit Reform Act (Section 502(5)(B) and (C))
requires discounting all cash flow observations to the point of disbursement of
the loan to the borrower:
(B) The cost of a direct loan shall
be the net present value, at the time
when the direct loan is disbursed, of the following estimated cash flows:
(i)
loan disbursements;
(ii)
repayments of principal; and
(iii) payments of interest and other
payments by or to the Government over the life of the loan after adjusting for
estimated defaults, prepayments, fees, penalties, and other recoveries;
including the effects of changes in
loan terms resulting from the exercise by the borrower of an option included in
the loan contract.
(C) The cost of a loan guarantee
shall be the net present value at the
time when the guaranteed loan is disbursed, of the following estimated cash
flows:
(i) payments by the
Government to cover defaults and delinquencies, interest subsidies, or other
payments;
(ii) payments to the
Government including origination and other fees, penalties, and recoveries;
including the effects of changes in
loan terms resulting from the exercise by the guaranteed lender of an option
included in the loan guarantee contract, or by the borrower of an option
included in the guaranteed loan contract. (Federal
Credit Reform Act of 1990, Section 502(5), “Definitions” Boldface added for
emphasis.)
In combination with the requirement to use the
“basket-of-zeros” method for discounting, the results of discounting to the
“time of disbursement” as opposed to, say, the beginning of the first fiscal
year of the cohort, can be substantial.
Consider the following example:
A series of three loans are made at
the beginning of three successive years.
Each loan has a balloon payment after three years with 5 percent
interest, compounded annually. The cash
flows and present value factors (which are calculated from the interest rates
in the economic assumptions for the FY 1999 Budget) are:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
Year 6
|
|
Loan 1:
|
|
|
|
|
|
|
|
Disbursement…
|
100.00
|
|
|
|
|
|
|
Repayment…...
|
|
|
|
115.76
|
|
|
|
Loan 2:
|
|
|
|
|
|
|
|
Disbursement…
|
|
100.00
|
|
|
|
|
|
Repayment…...
|
|
|
|
|
115.76
|
|
|
Loan 3:
|
|
|
|
|
|
|
|
Disbursement…
|
|
|
100.00
|
|
|
|
|
Repayment…...
|
|
|
|
|
|
115.76
|
|
Totals:
|
|
|
|
|
|
|
|
Disbursement…
|
100.00
|
100.00
|
100.00
|
|
|
|
|
Repayment…...
|
|
|
|
115.76
|
115.76
|
115.76
|
|
PV factors
|
1.00000
|
.950495
|
.900567
|
.852296
|
.805735
|
.761002
|
When discounted to the point of disbursement, the present
values of the cash flows and subsidies for the loans are equal, as follows:
PV
disbursement = 100.00 • 1.000000 = 100.00
PV
repayment = 115.76 • 0.852296 = 98.66
Subsidy = 100.00 - 98.66 = 1.34
(also 1.34 percent)
The subsidy for the three loans combined would be:
3 • 1.34 = 4.02 (in dollars)
and
4.02 /
300.0 = 1.34 percent (let’s call this
result “A”)
If, instead, the aggregated cash flows were discounted to
the beginning of year 1, the result would be:
PV
disbursements = 100.00 • 1.000000 +
100.00 • 0.950495
+ 100.00 • .900567
or 285.11
PV
repayments = 115.76 • 0.852296 + 115.76 • 0.805735 + 115.76 • 0.761002
or 280.02
The subsidy would be:
285.11 - 280.02 = 5.09 (in dollars)
and
5.09 /
285.11 = 1.78 percent (let’s call this
result “B”)
Why do the subsidies in result “A” and “B” differ? In “A,” all repayments are discounted to the
time of disbursement using a present value factor derived from the 3-year spot
rate. In “B,” each repayment is
discounted to the time of disbursement using a present value factor derived
from the 3-year forward rate.
Which calculation is correct? If all discounting were done on the basis of the yield curve
prevailing at the time when the cohort began, result “B” would be correct.
Under this assumption, forward rates are the implicit forecast for future rates
and, with upward-sloping yield curves, are assumed, generally, to be rising
over time. (Forward rates, implicitly,
assume that the upward slope in the yield curve arises entirely from
expectations that rates will rise over time.
In fact, yield curves may have an upward slope for several reasons other
than such expectations, including the increased risk inherent in longer-term
lending, market segmentation, time preferences, and so forth.)
But Federal credit programs do not use the forward rates as
a forecast of future rates. Instead,
future interest rates are assumed explicitly for budget purposes. In particular, the yield curve assumed to
prevail at the beginning of the cohort is assumed to prevail in all subsequent
periods for the preparation of budget estimates. Later, when actual rates are available, they are substituted for
the assumed rates.
Given this explicit assumption about future rates, the
calculation that gave result “A” is appropriate for estimates of credit
subsidies in prospective lending and is used by the CSC.
Considerations in
implementing the “time of disbursement” provision
A useful implementation of the “time of disbursement”
provision must deal with two items:
Timing. The amount by which
a payment should be discounted depends on the span of time between the time
when the loan was disbursed and the time when the payment was made. If a payment occurs exactly one year after
disbursement, it should be discounted by using a present value factor for one
year, exactly. To make this calculation, cash flow
estimates must be associated with a particular disbursement and the time of
that disbursement must be known.
Multiple
disbursements. Few, if any, Federal
credit programs make individual loans in isolation. For virtually all programs, the authority to make loans is
sufficient to make many loans and often over a period of more than one
year. Thus, the relationship between
the stream of loan disbursements and the stream of inflows and outflows related
to those loans is a “many-to-many” relationship. In these circumstances a method is needed to associate specific
inflows and outflows with disbursements made at a specific time. Without a method for doing so, it would be
impossible to determine the relationship between the time of disbursement and
the time a payment is made.
The methods used in the CSC for these considerations are
described in the following sections.
Timing
considerations
An individual set of cash flow estimates may be
provided in any of the following frequencies:
Annual
Semi-annual
Quarterly
Monthly.
Within any of these frequencies, cash flow estimates may be
specified as taking place at the:
Beginning
of the period
Middle of
the period
End of the
period.
In addition, each set of cash flow estimates has a specific
starting date, which, by default, is the date when the authority is first
available to make or guarantee loans.
These start dates are translated into an “elapsed time” value in
twice-monthly units. For example,
October 1 of the year when authority first becomes available would have an
“elapsed time” of zero. Twelve months later,
on September 30 of the same fiscal year, the “elapse time” would be 24.
Finally, an “offset” is computed based on the relationship
between the “elapsed time” of the beginning of the cash flow estimates and the
“elapsed time” of the disbursement to which these cash flow estimates are
attributed (more on how this is determined below). For example, if the
disbursement took place in the middle of the first year (“elapse time” value of
12) and the first cash flow observation took place at year-end (“elapse time”
of 24), the “offset” for that stream of cash flow observations would be 12.
Later, when the present value of the cash flows needs to be
calculated, the offset and frequency are used to select present value factors
to use in determining the present value.
The present value factors are available on a twice-monthly
frequency. Thus the offset is the
index, on a twice-monthly basis, of the first present value factor to use (an
index value of zero pertains to October 1 of the first fiscal year of the risk
category).
Discounting, to the time of disbursement, then is calculated
as follows:
n-1
Present
value = å ( Xi • P(i•f)+offset
)
i=0
where:
Xi Cash flow observation in
period i, where the first period is 0, the second period is 1, and so forth,
regardless of the frequency of the observations.
Pi Present value factor for
period i, where i is stated in twice-monthly intervals.
offset is the “offset” as described
above
f Frequency, expressed in twice-monthly periods
(e.g., annual frequency would be 24, monthly frequency would be 2)
n Number of observations
Special handling is needed when the index of a present value
factor is negative. When this happens, the index is negated and the reciprocal
of the present value factor (for the negated index) is used instead.
This could occur, for example, if the disbursement by the
private lender took place on the October 1 when the cohort began and the fees
were collected on, say, the previous June 1.
In this case, the present value of those fees would be higher than their
cash value to reflect the interest that would accrue on them from June 1 to
October 1.
Multiple
disbursements
When all disbursements take place at the same time, there is
no ambiguity in computing the “offset” as described above and in calculating a
present value. When disbursements are
made in two or more periods, an ambiguity arises regarding the point in time to
which a cash flow observation should be discounted. For example, if a direct loan program makes disbursements at the
beginning of two successive years, should the borrower’s payments received in
the second year be discounted to the beginning of the first year or to the
beginning of the second?
The CSC provides two methods for resolving the ambiguity:
Explicit association. Cash
flow observations can be prepared in a way that specifically associates cash
flow observations with the periods in which the underlying disbursement was
made. This method is the most flexible
and accurate approach for attributing cash flow observations to specific
disbursement periods. It also requires
more work on the part of analysts in preparing cash flow estimates.
Estimated association. When
the agency does not associate cash flow observations with disbursements in a particular
period, the CSC divides aggregate cash flow observations into separate groups
that can be attributed to disbursements in individual periods. Two methods are available for doing this:
the “pro-rata” method (described in Appendix C) and the “reverse-spendout”
method (described in Appendix D). The
“reverse spendout” method is more accurate, potentially, but does not work well
in all instances. The “pro-rata” method
is generally less accurate, but produces generally useful results in all
circumstances. These methods are used
as follows:
If the spreadsheet contains a
specification to “force” the pro-rata method, then the “pro-rata” method is
used in all instances where aggregate data must be distributed by disbursement
year.
If the “force pro-rata”
specification is not used, the CSC tries the “reverse-spendout” method
first. If it succeeds (the test for
success is given at the end of Appendix D), it is used. If not, the “pro-rata” method is used.
The use of alternative methods for
associating cash flows with disbursement years raises some concerns about the
consistency of results when small changes in cash flow observations cause the
“reverse-spendout” method to fail and the “pro-rata” method to be used in its
place. Such a switch in methods can
result in a change in the results that are out of proportion to the change in
input data. When comparability of methods is important, the specification can
be used to force the use of the “pro-rata” method in all circumstances, even
when the “reverse-spendout” method would be more accurate.
Appendix C
Specifications and examples of the pro-rata method
This appendix describes the pro-rata method that is used to
associate cash flow observations with specific disbursement periods. This method is less refined than the
“reverse-spendout” method, described in Appendix D. However, where the “reverse-spendout” method has limited
applicability, this method will produce generally useful, if somewhat rough,
approximations in all circumstances.
An example
Consider a loan guarantee program, with disbursements by
private lenders, fees paid to the government, and payments from the government
to the private lender when the borrower defaults, as follows:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Disbursements
|
15,000
|
15,000
|
|
|
|
|
Upfront fees
|
150
|
150
|
|
|
|
|
Annual fees
|
15
|
30
|
30
|
30
|
15
|
|
Default pmts
|
|
|
|
250
|
250
|
If our knowledge were limited to the values in the table,
plus the fact that transactions take place at the beginning of each year, how
would we attribute a portion of the annual fees, upfront fees, and default
payments to each disbursement year? The
basis for this allocation is the following:
Year 1: The fees are attributed to year 1
disbursements. They could not be
attributed to disbursements not yet made.
Year 2: The upfront fees in year 2 are most
likely to be related to year 2 disbursements.
The annual fees are attributable to both years equally.
Year 3: The annual fees are attributable to
both years equally.
Year 4: The annual fees are attributable to
both years equally. The default payment is attributed to disbursements in year
1.
Year 5: For symmetry with the treatment of
the first year, the annual fee is attributed to the year 2 disbursements
only. The default payment is attributed
to the last disbursement.
Cash flows, by disbursement year, would be estimated as
follows:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Year 1:
|
|
|
|
|
|
|
Disbursements
|
15,000
|
|
|
|
|
|
Upfront fees
|
150
|
|
|
|
|
|
Annual fees
|
15
|
15
|
15
|
15
|
|
|
Default pmts
|
|
|
|
250
|
|
|
Year 2:
|
|
|
|
|
|
|
Disbursements
|
|
15,000
|
|
|
|
|
Upfront fees
|
|
150
|
|
|
|
|
Annual fees
|
|
15
|
15
|
15
|
15
|
|
Default pmts
|
|
|
|
|
250
|
The cash flows associated with year 1 disbursements would be
discounted to the beginning of year 1; those associated with year 2
disbursements, to the beginning of year 2.
Based on the specification that transactions occur at the beginning of
the year and a constant discount rate of 6 percent, the present values would be
determined as follows:
Year 1,
upfront fees: 150 (they occur at the
beginning of the year)
Annual
fees: ( 15 / 1.060 ) + (15
/ 1.061 ) + (15 / 1.062 ) + (15 / 1.063)
Or: 15.00 + 14.15 + 13.35 + 12.59
Or: 55.09
The present value of the year 2 cash flows would be
identical. The upfront fee occurs at
the time of disbursement and is therefore not discounted. The four receipts of
annual fees occur at the time of disbursement, after one year, after two years,
and after three years.
In this example, the pro-rata approach works reasonably
well. It would not work well for all
circumstances, however. See
“Limitations of the pro-rata method” at the end of this appendix.
Description of the
method
The pro-rata method attributes cash flow observations to
disbursement years in proportion to the amounts disbursed in each year. The
following example describes how pro-rata factors are developed to allocate cash
flow observations from seven periods to disbursements in three periods.
The first step is to build a matrix from which distribution
factors can be calculated. The matrix,
with seven columns for the cash flow periods and three rows for the
disbursement periods, is populated with the amounts disbursed. Clearly, cash
flow observations in the first period could only be related to disbursements in
the first period; cash flow observations in the second period could be
attributed to disbursements in the first and second periods; and so forth. To prevent the disbursements in the first
period from having excessive importance, a symmetrical pattern is used for the
last cash flow observation: the last cash flow observation is attributed
entirely to the last disbursement; the second to the last is attributed to the
last two disbursements; and so forth.
Thus, each disbursement will have a share of, at most, NCF -
ND + 1 periods (NCF is the number of cash flow
observations and ND is the number of disbursement periods). In our example, each disbursement would have
a share of the five cash flow observations.
The matrix would look like this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
CF 7
|
|
D 1
|
100.0
|
100.0
|
100.0
|
100.0
|
100.0
|
|
|
|
D 2
|
|
100.0
|
100.0
|
100.0
|
100.0
|
100.0
|
|
|
D 3
|
|
|
100.0
|
100.0
|
100.0
|
100.0
|
100.0
|
|
Total
|
100.0
|
200.0
|
300.0
|
300.0
|
300.0
|
200.0
|
100.0
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
(It may be useful to highlight the fact that disbursement
amounts, rather than the amounts of cash flow observations, are used to compute
distribution factors. A simple example
will make the reason for this obvious.
What would happen if a direct loan program made disbursements in three
successive years in the amounts of 100, 0, 100?)
The above matrix is converted to distribution factors by
dividing each cell by the column totals.
The result looks like this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
CF 7
|
|
D 1
|
1.000
|
0.500
|
0.333
|
0.333
|
0.333
|
|
|
|
D 2
|
|
0.500
|
0.333
|
0.333
|
0.333
|
0.500
|
|
|
D 3
|
|
|
0.333
|
0.333
|
0.333
|
0.500
|
1.000
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
The cash flows attributable to disbursements in the first
year would be estimated by multiplying the aggregate cash flows for all
disbursements by the factors in the first row.
Cash flow observations attributable to disbursements in the second and
third years would be similarly obtained.
The results might look like this:
|
Original values
|
50.0
|
100.0
|
150.0
|
150.0
|
150.0
|
100.0
|
50.0
|
Original values allocated to disbursement years:
|
D 1
|
50.0
|
50.0
|
50.0
|
50.0
|
50.0
|
|
|
|
D 2
|
|
50.0
|
50.0
|
50.0
|
50.0
|
50.0
|
|
|
D 3
|
|
|
50.0
|
50.0
|
50.0
|
50.0
|
50.0
|
(Note: Dn are the
disbursement periods)
A couple of observations:
The cash flow observations for each
year of disbursement are identical, except for the difference in the timing of
disbursements;
The period of time from the first to
the last cash flow observations is the same for each disbursement year, though
all occur over a period of time shorter than the original values
An example with
varying amounts disbursed
What happens when the disbursements are not in equal
amounts? Consider an example where the
disbursements in the three years are 100.0, 200.0, and 300.0. The matrix of disbursements-based weights
would look like this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
CF 7
|
|
D 1
|
100.0
|
100.0
|
100.0
|
100.0
|
100.0
|
|
|
|
D 2
|
|
200.0
|
200.0
|
200.0
|
200.0
|
200.0
|
|
|
D 3
|
|
|
300.0
|
300.0
|
300.0
|
300.0
|
300.0
|
|
Total
|
100.0
|
300.0
|
600.0
|
600.0
|
600.0
|
500.0
|
300.0
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
And the matrix of factors (after dividing by the
totals) would look like this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
CF 7
|
|
D 1
|
1.000
|
0.333
|
0.167
|
0.167
|
0.167
|
|
|
|
D 2
|
|
0.667
|
0.333
|
0.333
|
0.333
|
0.400
|
|
|
D 3
|
|
|
0.500
|
0.500
|
0.500
|
0.600
|
1.000
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
Results when some
years are missing
This procedure can result in some patterns that might look
odd at first. For example, consider
five periods of cash flow observations and three periods of disbursements, in
which disbursements are 100.0, 0.0, and 100.0.
The matrix of disbursements-based weights would look like
this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
|
D 1
|
100.0
|
100.0
|
100.0
|
100.0
|
|
|
|
D 2
|
|
0.0
|
0.0
|
0.0
|
0.0
|
|
|
D 3
|
|
|
100.0
|
100.0
|
100.0
|
100.0
|
|
Totals
|
100.0
|
100.0
|
200.0
|
200.0
|
100.0
|
100.0
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
And the matrix of factors (after dividing by the
totals) would look like this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
|
D 1
|
1.000
|
1.000
|
0.500
|
0.500
|
|
|
|
D 2
|
|
0.0
|
0.0
|
0.0
|
0.0
|
|
|
D 3
|
|
|
0.500
|
0.500
|
1.000
|
1.000
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
A diagonal matrix
Another example of irregular inputs occurs when there are as
many periods of disbursements as periods of cash flow observations:
The matrix of disbursements-based weights would look like
this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
|
D 1
|
100.0
|
|
|
|
|
|
|
D 2
|
|
100.0
|
|
|
|
|
|
D 3
|
|
|
100.0
|
|
|
|
|
D 4
|
|
|
|
100.0
|
|
|
|
D 5
|
|
|
|
|
100.0
|
|
|
D 6
|
|
|
|
|
|
100.0
|
|
Totals
|
100.0
|
100.0
|
100.0
|
100.0
|
100.0
|
100.0
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
And the matrix of factors (after dividing by the
totals) would look like this:
|
|
CF 1
|
CF 2
|
CF 3
|
CF 4
|
CF 5
|
CF 6
|
|
D 1
|
1.000
|
|
|
|
|
|
|
D 2
|
|
1.000
|
|
|
|
|
|
D 3
|
|
|
1.000
|
|
|
|
|
D 4
|
|
|
|
1.000
|
|
|
|
D 5
|
|
|
|
|
1.000
|
|
|
D 6
|
|
|
|
|
|
1.000
|
(Note: Dn are the
disbursement periods; CFn are the periods with cash flow observations)
Treatment of
truncated cash flow observations
If a set of cash flow observations has annual observations
for loan disbursements and upfront fees and there are disbursements in three
successive years, but upfront fees in just two years, there is a problem. What portion of the fees should be
attributed to disbursements in the third year?
When this occurs, the CSC attributes nothing to the third year.
In the general case, a diagonal matrix, equal in size to the
number of cash flow observations and with values of 1.00 in the diagonal cells,
will be used whenever the number of periods of cash flow observations is equal
to or less than the number of periods of disbursements.
Limitations of the
pro-rata method
The pro-rata method will produce generally reasonable
results when cash flow observations have a relatively constant relationship to
disbursements. The pro-rata method has
increasingly severe limitations when the relationship between cash flow
observations and disbursements varies over time.
In addition, interest payments could pose a special
problem. The magnitude of interest
payments is related to the loan balance outstanding, the interest rate charged,
and the time since the last payment.
With these considerations alone, we might choose a way to allocate
interest payments that would differ from other kinds of payments. An example might help illustrate why this is
so.
Consider a loan program in which disbursements are made in
the middle of the first two periods.
When the loan is disbursed, interest to the end of the first period is
collected. Afterwards, interest
payments are due at the end of each period.
A balloon payment of principal is due with the final interest payment.
Assuming an interest rate of five percent, the cash flow observations would
look like this.
|
|
Period 1
|
Period 2
|
Period 3
|
Period 4
|
Period 5
|
Period 6
|
|
Loan 1
|
|
|
|
|
|
|
|
Disbursement…...
|
100.0
|
|
|
|
|
|
|
Interest pmt……..
|
2.5
|
5.0
|
5.0
|
5.0
|
5.0
|
|
|
Principal pmt…...
|
|
|
|
|
100.0
|
|
|
|
|
|
|
|
|
|
|
Loan 2
|
|
|
|
|
|
|
|
Disbursement…...
|
|
100.0
|
|
|
|
|
|
Interest pmt……..
|
|
2.5
|
5.0
|
5.0
|
5.0
|
5.0
|
|
Principal pmt…...
|
|
|
|
|
|
100.0
|
|
|
|
|
|
|
|
|
|
Total
|
|
|
|
|
|
|
|
Disbursement…...
|
100.0
|
100.0
|
|
|
|
|
|
Interest pmt……..
|
2.5
|
7.5
|
10.0
|
10.0
|
10.0
|
5.0
|
|
Principal pmt…...
|
|
|
|
|
100.0
|
100.0
|
If the allocation approach described above were used, the
allocation matrix would be:
|
|
Period 1
|
Period 2
|
Period 3
|
Period 4
|
Period 5
|
Period 6
|
|
Disb. period 1
|
1.000
|
0.500
|
0.500
|
0.500
|
0.500
|
|
|
Disb. period 2
|
|
0.500
|
0.500
|
0.500
|
0.500
|
1.000
|
and the interest payments would be distributed as follows:
|
|
Period 1
|
Period 2
|
Period 3
|
Period 4
|
Period 5
|
Period 6
|
|
Disb. period 1
|
2.50
|
3.75
|
5.00
|
5.00
|
5.00
|
|
|
Disb. period 2
|
|
3.75
|
5.00
|
5.00
|
5.00
|
5.00
|
The distribution in the second period is simply wrong.
The problem is that the allocation of interest payments
(assuming identical interest rates) should be related to the outstanding
balance of the loan and the time since the last interest payment. On that basis, the first loan would get a
larger, rather than equal, share of the second year total.
The problem in making such an accommodation is the variety
of loan contracts that might be considered.
Loans may have prepayments of interest, grace periods, capitalization of
unpaid interest, and sliding scales of interest rates charged. Each of these would call for a different
kind of adjustment and some would require extensive specification of the terms
of the loans.
A similar problem exists with principal payments. If the loan repayments are in constant
amounts, with interest and principal, the principal payment of each successive
payment is slightly larger. The
allocation methods described above do not take that varying proportion into
account, nor is there any plausible way to do so in the absence of the full
details of the terms of the loan.
In both this case and the case of interest payments,
agencies need to prepare disbursement period details, based on the
characteristics of the program, rather than relying on the approximations used
by the CSC when disbursement period details are not provided.
Appendix D
Specifications and examples of the reverse-spendout method
This appendix describes the reverse-spendout method that is
used to associate cash flow observations with specific disbursement
periods. This method is more refined
than the “pro-rata” method, described in Appendix C, but is somewhat fragile,
as explained below.
An example
Consider a loan guarantee program, with disbursements by
private lenders. Fees are paid to the
government on the basis of outstanding loan balances and decline over
time. The fees paid to the government
might look like this:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Disbursements
|
10,000
|
45,000
|
|
|
|
|
Disb 1 balance
|
10,000
|
7,500
|
5,000
|
2,500
|
|
|
Disb 1 fees
|
100
|
75
|
50
|
25
|
|
|
Disb 2 balance
|
|
45,000
|
33,750
|
22,500
|
11,250
|
|
Disb 2 fees
|
|
450
|
338
|
225
|
112
|
|
Total fees
|
100
|
525
|
388
|
250
|
112
|
If these total fees were distributed using the pro-rata
method (see Appendix C), the result would be:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Disb 1 fees
|
100
|
95
|
61
|
45
|
|
|
Disb 2 fees
|
|
430
|
327
|
205
|
112
|
As explained in Appendix C, the pro-rata method is
approximate and, in this case, the deviations from the actuals are
obvious. Is there any better method to
associating cash flow observations with disbursement years? Yes, though not necessarily in all
circumstances.
The reverse-spendout method assumes some proportionality in
the attribution of cash flow observations to disbursement years. Specifically, the proportionate relationship
of the first period disbursement to the first period cash flow observation
(which can be observed) is the same as the second period disbursement to its
share of the second period cash flow observation (which cannot be
observed). This “constant” relationship
can be used to derive a more proportionate distribution of cash flow
observations to disbursement years.
For the fees shown above, the “reverse-spendout” method
would be applied as follows:
The first fee payment would be
attributed to first period disbursements (same as the pro-rata method).
The portion of the second period
fees attributed to the second period disbursements would be based on the ratio
of the first period fees to first period disbursements:
[ ( 100 /
10,000 ) • 45,000 ] or 450
and the remainder, 75, would be
attributed to the first period.
The portion of third year fees
attributed to the second year of disbursements would be calculated in the same
way:
[ ( 75 /
10,000 ) • 45,000 ] or 338
and the remainder, 50, would be
attributed to the first period.
The process would be reversed for
the final observations. The last fee
observation would be attributed entirely to the last disbursement and a process
similar to the above for the second to last.
The result would be as follows:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Disb 1 fees
|
100
|
75
|
50
|
25
|
|
|
Disb 2 fees
|
|
450
|
338
|
225
|
112
|
This result exactly matches fees that were based on the
declining loan balance.
Some problems
As mentioned earlier, the reverse-spendout method can be
fragile. This is best shown by example. Let’s take the example above and change a
single number, in the second year, to one (obviously, this could be a data
entry error; however, the CSC does not have enough information to evaluate data
quality):
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Disbursements
|
10,000
|
45,000
|
|
|
|
|
Total fees
|
100
|
1
|
388
|
250
|
112
|
When the reverse-spendout method is applied, the portion of
second year fees attributed to second year disbursements would be the same as
above
[ ( 100 /
10,000 ) • 45,000 ] or 450
and the residual, -449, would be attributed to the first
year.
The portion of the third year fees attributed to the second
year disbursements would be
[ ( -449 /
10,000 ) • 45,000 ] or -2,020.50
and the residual, 2,408.50, would be attributed to the first
year.
When the allocation is finished, it would look like this:
|
|
Year 1
|
Year 2
|
Year 3
|
Year 4
|
Year 5
|
|
Disb 1 fees
|
100
|
-449
|
2408.50
|
25
|
|
|
Disb 2 fees
|
|
450
|
-2020.50
|
225
|
112
|
The reverse-spendout method fails here because the aggregate
fees are not in proportion to
disbursements. When a calculation is
made assuming that they are, it should not be surprising that the results are
unsatisfactory.
Test for
applicability
The CSC assumes that the reverse-spendout method is
applicable any time it results in distributions of cash flow observations to
disbursement years in which all amounts distributed have the same sign as the
aggregate amount. If not, the CSC uses
the more robust (though less precise) pro-rata method.