December 1, 1999
The BasketofZeros Approach to Discounting
The revised Credit Subsidy Calculator implements an improved
method of discounting, called the "basket of zeros." Previously, credit subsidies are calculated
using the “similar maturity” method that was adopted when credit reform was
first enacted. Under the similar
maturity method, all cash flows are discounted using the interest rate (more
technically called the “yieldtomaturity” rate) on a Treasury security of
similar maturity to the term of the loan.
For example, the cash flows for a 10year loan are discounted using the
rate on a 10year Treasury security, and the cash flows for a 30year loan are
discounted using the rate on a 30year Treasury security.
The distinguishing feature of the basketofzeros method is
that each cash flow is discounted using the interest rate on a zero coupon
Treasury (explained below) with the same maturity as that cash flow, regardless
of the term of the loan. Cash flows
that would occur exactly at the end of one year are discounted using the
interest rate on a Treasury zero that would mature in exactly one year. Cash flows that would occur exactly at the
end of the fifth year are discounted using the interest rate on a Treasury zero
that would mature in exactly five years.
Cash flows that would occur exactly at the end of five years and one
month would be discounted using the interest rate on a Treasury zero that would
mature in exactly five years one month.
And so on. The basketofzeros method,
therefore, defines the present value of any collection of future cash flows as
the market price of a collection (or “basket”) of Treasury zeros that, at
maturity, exactly matches the cash flows.
The basketofzeros method provides a more precise measure
of present value because it permits matching discount rates with the timing of
cash flows. A zero coupon bond pays all
interest and principal at maturity. The
term “zero” distinguishes these securities from other Treasury notes and bonds
that make semiannual coupon payments of interest and a payment of principal at
maturity. The interest rate on a zero
is a rate for a single payment at a particular point in time. In contrast, the interest rate on a 10year
Treasury note is a rate applicable to 20 semiannual coupon payments of
interest. The yieldtomaturity rate,
therefore, is a blending of rates for 20 points in time. Unless the cash flows for a direct loan or
loan guarantee match the cash flows on a Treasury security that makes coupon
payments, using the yieldtomaturity rate as the discount rate provides an
imperfect measure of present value.
Example 1 compares the two methods for a hypothetical loan
guarantee program. The loans are for 10 years.
The Government is assumed to pay claims of $10,000 per year (line
1). The interest rates for Treasury
zeros are shown on line 2a. Line 2b
shows the yieldtomaturity rate for a 10year Treasury note. The rates are based on the Treasury yield
curve for 1997.
Example 1























Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

Year 7

Year 8

Year 9

Year 10


Assumptions













1

Guarantee
claim payments

10,000

10,000

10,000

10,000

10,000

10,000

10,000

10,000

10,000

10,000

2

Interest
rates:













a.
Treasury zeros

5.71%

6.12%

6.25%

6.34%

6.42%

6.48%

6.54%

6.58%

6.60%

6.61%



b.
10‑year "yield‑to‑maturity"

6.56%




























Subsidy Calculation



Using similar maturity (10‑year rate):




Using basket of zeros:


3


PV of Year 1 claims (@10 year rate)

9,384



PV of Year 1 claims (@ 1 year rate)

9,460

4


PV of Year 2 claims (@10 year rate)

8,806



PV of Year 2 claims (@ 2 year rate)

8,880

5


PV of Year 3 claims (@10 year rate)

8,264



PV of Year 3 claims (@ 3 year rate)

8,337

6


PV of Year 4 claims (@10 year rate)

7,755



PV of Year 4 claims (@ 4 year rate)

7,820

7


PV of Year 5 claims (@10 year rate)

7,277



PV of Year 5 claims (@ 5 year rate)

7,326

8


PV of Year 6 claims (@10 year rate)

7,326



PV of Year 6 claims (@ 6 year rate)

7,326

9


PV of Year 7 claims (@10 year rate)

6,408



PV of Year 7 claims (@ 7 year rate)

6,418

10


PV of Year 8 claims (@10 year rate)

6,013



PV of Year 8 claims (@ 8 year rate)

6,006

11


PV of Year 9 claims (@10 year rate)

5,643



PV of Year 9 claims (@ 9 year rate)

5,626

12


PV of Year 10 claims (@10 year rate)

5,295



PV of Year 10 claims (@10 year rate)

5,273

13


Total subsidy

71,673



Total subsidy

72,007
























Difference =333



















The present value of each of the cash flows is shown on
lines 312, with the total on line 13.
The left bank shows the present values using the yieldtomaturity rate,
which is the 10year rate (6.56%) in this example. The right bank shows the present values using the rates on
Treasury zeros. Using these rates, the present value of the $10,000 payment in
year 1 is discounted at 5.71%, the payment in year 5 is discounted using 6.42%,
and the payment in year 10 is discounted using 6.61%. Because the yield curve is not flat – usually it is upward
sloping – the Treasury zero rate differs from the yieldtomaturity rate in
every year. The present values are
therefore different in every year. For
example, the present value of the 4th year payment is $7,820, using the rate on
a 4year Treasury zero, and it is $7,755, using the yieldtomaturity
rate. As a result, the total subsidy
cost estimates differ by $333.
The basketofzeros is an improvement over similar maturity
because it is more accurate. Each 7cash
flow is discounted by the discount rate that is defined for the term of that
cash flow, not the term of the final contractual cash flow of the loan. For example, the basket of zeros produces
the same subsidy cost estimate for loans and loan guarantees that have
identical cash flows, regardless of the contractual term of the loan, whereas
the similar maturity approach produces different cost estimates. Example 2 illustrates this for the following
two Governmentguaranteed loans: one loan has a one year term and the other has
a 10 year term, both default at the end of the first year, and the Government
pays a $10,000 guarantee claim for each.
Since the cash flows are identical, the subsidy cost should be the
same. The similar maturity approach
(left bank) would yield different subsidy estimates for the two guarantees,
because the 1year rate would be used for the loan with a term of one year, and
the 10year rate would be used for the loan with a term of ten years. The basket of zeros approach (right bank)
would yield the same estimate of subsidy cost for both guarantees, because both
default payments would be discounted using the 1year rate.
Example 2























Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

Year 7

Year 8

Year 9

Year 10


















Assumptions












1

Guarantee
claim payments:













1 year loan

10,000












10 year loan

10,000










2

Interest
rates:













a.
Treasury zeros

5.71%

6.12%

6.25%

6.34%

6.42%

6.48%

6.54%

6.58%

6.60%

6.61%



b.
10‑year "yield‑to‑maturity"

6.56%




























Subsidy Calculation



Using similar maturity (10‑year rate):





Using basket of zeros:



3


1 year loan (discount rate = 10 year
rate)

9,460



1 year loan (discount rate = 1 year
rate)

9,460

4


10 year loan (discount rate = 10 year
rate)

9,384



10 year loan (discount rate = 1 year
rate)

9,460

5



Difference

76




Difference

0











To simplify the calculations,
similar maturity is defined by broad categories: loans maturing in 1 year or
less, more the 1 year and less than 5 years, 5 years or more and less than 10
years, 10 years or more and less than 20 years, and 20 years or more. The discount rate for each category is the
average interest rate on Treasury securities with remaining maturities within
each of these categories.
The revised Credit Subsidy
Calculator will be able to use discount rates that are stated in twicemonthly
intervals.
