October 22, 1999

** **

**Description of
the “basket-of-zeros” discounting method**

**and the
derivation of present value factors**

**from the yield
curve**

# A few definitions

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
methods used to derive present value factors from the yield curve and use them
to determine present value are described below.

*Coupon or “bond-equivalent” rate**.*
Treasury notes and bonds pay interest every six months. This payment,
called the “coupon” for historical reasons, is computed using the “coupon rate”
divided by 2. Thus, a Treasury bond in
the amount of $1,000 with a coupon rate of 6 percent would make a coupon
payment of $30 every six-months. The
coupon rate does not take into account that the interest earnings might be
reinvested.

** Effective annual yield**. This is the yield the bond holder would
receive if interest earnings were reinvested at the coupon rate. For example,
using the example above and assuming that the $30 coupon payment can be
reinvested at 6 percent, the annual yield would be 6.09 percent. The effective annual yield is computed as [
1 + ( coupon rate / 2) ]

^{2 }– 1.

** Zero-coupon bond**. If a bond has a
coupon rate of zero percent (pays no explicit interest) and the face value is
paid at maturity, it is called a zero-coupon bond. Ordinarily, Treasury limits
the issuance of such securities to periods of less than one year. However, they are available indirectly
available through the “STRIPS” (Separate Trading of Registered Interest and
Principal Securities) program, by which holders of eligible securities may
trade coupon or principal payments separately in the secondary market or
combine them.

** Spot rates**. The spot rate for a
particular maturity is the bond-equivalent rate on a zero coupon bond of the
same remaining maturity. Spot rates can
be obtained by direct observation of zero-coupon bonds in the secondary market. Also, they can be derived from the observed
yields on unstripped notes and bonds, as described below. Because the market for “STRIPS” is not as
extensive as the market for Treasury securities in unstripped form, the second
method is generally used to obtain spot rates. Spot rates are stated as
bond-equivalents to facilitate comparison with other Treasury rates.

** Bond price**. The theoretical price of a bond in the
secondary market is determined by calculating the present value of each coupon
payment and, at maturity, the payment of principal, using a different spot rate
for each payment. A payment in 6 months
would be discounted using the 6-month spot rate, a payment in 12 months would
be discounted using the 12-month spot rate, and so forth. If the principal payment were to be received
at the end of 10 years, it would be discounted using the 10-year spot
rate. The sum of the present values of
the individual payments, so calculated, is the theoretical price of the bond.

** Yield-to-maturity**. This is the bond-equivalent rate that an
investor would need to receive on all coupon payments to make the price of a
bond equal to par.

* *

** Present Value factors**. To determine the present value of a single
payment, the payment is multiplied by the appropriate present value
factor. This factor, in turn, is
derived from the spot rate for the maturity of that payment. For example, the present value of a single
payment of $1,000 at the end of three years would be computed by first deriving
a present value factor by:

* *

Present value factor = 1 / [ 1 + (
spot rate / 2) ] ^{6}

and, multiplying the payment by that present value factor. If the spot rate at the three-year point is 8 percent, the present value would be calculated as follows:

Present value factor = 1 / [ 1 + ( 8.00 / 200 ) ] ^{6}

= 0.79031

Present value = Cash value • 0.79031

= 1000 • 0.79031

resulting in a present value of $790.31.

** Present value**. For a
series of payments, which may vary in amount and timing, the present value is
the price of a collection of zero-coupon bonds that, at maturity, exactly match
the series of payments. Present values can also be calculated using a single
discount rate; however, without a clear basis for selecting a discount rate
(for example, “similar maturity” is open to varying interpretation), the
relationship between nominal and present values is less clear.

* *

** Forward rate**. The forward rate is the reinvestment rate
(generally on six-month bills) that would equalize earnings on a long-term
security (usually at higher rates) with series of short-term (generally
six-month) securities. For example, if
the six-month rate is 6.00 percent and the one-year rate is 6.50 percent, then
the forward rate in six months would be 7.00 percent (calculated from the ratio
of exponentiated spot rates, or ( ( 1.0325

^{2}/ 1.03 ) - 1 ) • 200).

* *

# Conventions

The following conventions are used in this document:

Interest rates are consistently shown in decimal form (“0.06” rather than “6.0 percent”).

Multiplication is indicated by the “•” symbol. For example, 2 • 3 = 6.

Symbols (time periods denoted by the subscript “n” are semiannual periods unless otherwise stated):

C_{i} Coupon payment, per $1,000
of principal, in period i (if the coupon rate is 5.4 percent, C_{2} is
$27.00)

F_{i} Forward rate for the time
period i

L_{i} Logarithmic factor in
period i where L_{1} is log(1) and has a value of zero, L_{2}
is log(2) and has a value of 0.301, and L_{n} would have a value of
log(n).

P_{i} Present value factor for
semiannual period i, computed as:

_{i}

P_{i} = P [ 1/(1 + ( F_{k} / 2 )) ]

^{k=1 }

If F_{1} is 0.052 ( 5.2
percent ), P_{1} would be .9747.

Present value factors can also be
calculated directly from spot rates where the present value factor in
semiannual period n would be calculated from the spot rate, stated in
bond-equivalent terms, for time period n (S_{n}) by:

P_{n}
= 1 / [ 1 + ( S_{n} / 200 ) ]^{n}

P_{fti} Present value factor for
period i, with frequency f, and timing t.
Frequency can be monthly, quarterly, semiannually, or annually. Timing can be beginning, middle, or end of
the period, or throughout the period (equivalent to occurring mid-period).

X_{fti } Cash flow observation in
period i, with frequency f, and timing t.
Frequency and timing as defined above.

Y_{i} Yield-to-maturity rate for
period i

# Published yield curve data

The starting point for the derivation of spot rates and discount factors is the published points on the Treasury yield curve. This yield curve shows yield-to-maturity rates, derived from the prices at which securities are traded in the secondary market, arrayed against the remaining maturity of the securities. The Department of the Treasury and the Federal Reserve routinely publish quotations for selected points on the yield curve. The publications are cited below.

As of this writing, the published yield curve points are:

3 months

6 months

12 months

2 years

3 years

5 years

7 years

10 years

30 years

The sections below describe how these published yield-to-maturity observations are translated into a collection of present value factors that can be used in the credit subsidy calculator.

# Discounting requirements

The credit subsidy calculator needs to discount cash flow estimates that:

May be stated in annual, semiannual, quarterly, or monthly intervals;

May, within such periods, have activity concentrated at the beginning, middle, or end of those periods; and,

May extend for up to 100 years (a
little over twice the term of the Federal credit program with the longest term
(48 years)). See the *Federal Credit
Supplement* to the *Budget of the
United States Government, Fiscal Year 2000,* tables 3 and 4, for details.

** **

**Overview of the
derivation of present value factors**

The conversion of published yield curve points to twice-monthly forward rates and present value factors takes place in several steps.

For the interval from 6 months to 30 years, forward rates and their associated present value factors are derived at semiannual points from the published yield curve;

Semiannual spot rates are derived from semiannual present value factors;

In the same interval, twice-monthly spot rates are interpolated from the semiannual spot rates and twice-monthly present value factors are derived;

For the interval from 30 years to 100 years, forward rates are held constant (at the value for the forward rate at 30 years) and present value factors and spot rates are derived; and

For the interval from 0 months to 6 months, spot rates are interpolated (3 months to 6 months) and estimated (0 months to 3 months).

The methods used in each of these steps are discussed below.

**How forward
rates and present value factors are derived from the yield-to-maturity curve in
the interval from 6 months to 30 years**

The six-month forward rate and six-month spot rates are equal to the observed six-month yield rate.

The one-year present value factor and forward rate are
derived from the values of P_{1} (present value factor, first
semiannual period) and C_{2} (coupon rate, second semiannual period),
which are known from the published data.
The value of P_{2} can be determined from the following
condition.

1000 = ( 1000 • P_{2} ) + [
C_{2} • ( P_{1} + P_{2} ) ]

which states that the present value of the stream of payments produced by the security is par. Because the observed points are based on trading in the secondary market, the condition is reasonable.

Rearranging,

P_{2} = [ 1000 – ( C_{2}
• P_{1} ) ] / ( 1000 + C_{2} )

Given the rates above:

P_{2} = [ 1000 – ( 27 •
.9747 ) ] / ( 1000 + 27 )

or 0.9481

The present value factor, P_{2}, can be converted to
the forward rate, F_{2}, as follows:

F_{2}
= [ ( P_{1} / P_{2} ) - 1 ] • 2

or 0.05611 or 5.611 percent.

Calculation of the present value factor for 1.5 years (P_{3})
is more complicated because there is no direct observation for the third period
and P_{3 }must be found by interpolation.

Interpolation is simplified by the observation that successive forward rates generally follow a logarithmic pattern (a proportional increase in term is associated with an equal proportional increase in the rate)[1].

The first point to be interpolated is the 18-month (F_{3})
point, which is derived from the to-be-determined value of F_{4 }as
follows:

F_{3} = F_{2} + [ ( F_{4} – F_{2}
) • ( L_{3} – L_{2} ) / ( L_{4} – L_{2} ) ]

Though the value of F_{4} is unknown, it can be
determined from the price of a two-year security with a stated
yield-to-maturity rate that should result in a bond price of par. Specifically:

1000 = ( C_{4}
• P_{1 }) + (C_{4} • P_{2 }) + (C_{4} • P_{3
}) + [ ( 1000 + C_{4}) • P_{4 }]

Where:

P_{3}
= [ 1 / ( 1 + F_{3} / 2 )] • P_{2}

P_{4}
= [ 1 / ( 1 + F_{4} / 2 )] • P_{3}

There is no direct way to solve these equations. The solution must be determined by a
trial-and-error method in which successive values are tried for the two-year
forward rate, F_{4}. For each
test value, a value of F_{3 }and present value factors, P_{3 }and
P_{4},_{ }are computed, and the price of the two-year security
is found. If the price is below par,
the value of F_{4 }is lowered; if above par, it is raised until a value
for F_{4 }is found that makes the price exactly equal to par. This process is repeated until a solution is
found[2].

This method is repeated for the interval between the next pair of published yield curve points (2 years and 3 years), then the next (3 years to 5 years), and so forth, until the semiannual forward rates at all semiannual points, from 6 months to 30 years, have been found.

# How twice-monthly present value factors are calculated

Once a full set of forward rates has been calculated (semiannual from 6 months to 30 years), the spot rates are derived from the present value factors, as follows:

S_{n}
= [ 2 • (1/P_{n}) ^{(1/n)} ] - 2

Twice-monthly spot rates are then interpolated from the semiannual spot rates using the logarithmic method described above (z is a fraction of a period between sequential periods x and y):

S_{z} = S_{x} + [ (
S_{y} – S_{x} ) • ( L_{z} – L_{x} ) / ( L_{y}
– L_{x} ) ]

The twice-monthly spot rates are then converted back to present values:

P_{z}
= [ 1 / ( 1 + S_{z} / 2 )^{z} ]

How present value factors are projected from 30 years to 100 years

Forward rates after 30 years are held constant at the calculated 30-year forward rate. The corresponding twice-monthly present value factors are calculated as follows:

P_{n}
= P_{n-1 }• ( 1 / (( 1 + F_{60} / 2 )^{( 1 /} ^{12 )}))

**How spot rates
and present value factors prior to 6 months are derived**

** **

The spot rates from 3 months to 6 months are interpolated from the observed 13- and 26-week rates. Present value factors are derived from these spot rates.

Prior to 3 months, the twice-monthly spot rates are calculated through a series of linear equations. The equations were calculated through regression analysis of historical Treasury Bill quotes available in major newspapers. Twelve equations were calculated representing weeks one through 12. Six of these equations (or equations directly related to them) are used to calculate spot rates at the desired twice-monthly points.

The data for the regression estimates were drawn from a random sample of 100 daily observations (13 rates per observation). The sample frame was the quotations in the Wednesday newspapers from January 1968 to June 1998[3]. Because Treasury Bills mature on Thursday, the quotes in the Wednesday papers gave data points at exactly 8 days, 15 days, … and 85 days to maturity. When consecutive quotes (i.e., rates on securities maturing one week apart) differed by more than eight percent, the observation for that day was dropped. This relatively modest constraint eliminated 28 percent of the observations in the sample. Apart from the desire to remove observations with erratic period-to-period changes, there was no specific basis for choosing the 8 percent change rule. Obviously, a different rule would have a different effect on the data used for analysis, though, within reasonable limits, little effect on the estimated parameters.

Twelve equations were then calculated by regressing each of the weekly rates on the 13-week rate using the form y = ax + b. The parameters, based on the 2-, 4-, 9-, and 11-week rates, are used to calculate directly the twice-monthly spot rates. (The 1.5-month rate is calculated from an equation representing the average of the 6- and 7-month equations. An over-night rate (time zero) equation was calculated from the slope of the 1- and 2-week equations.). The twice-monthly regression equations are:

Rate[4] Equation

Time
zero = 1.0078 • S_{.25} -
0.57439

0.5
months = .96868 • S_{.25} -
0.26107

1.0
month = .94362 • S_{.25} -
0.07258

1.5
months = .97198 • S_{.25} -
0.10305

2.0
months = .98595 • S_{.25} -
0.06110

2.5
months = .99551 • S_{.25} -
0.04864

where S_{.25} is the 13-week spot rate.

At this point, present value factors are available for the beginning, middle, and end of each month, for 100 years.

# How present value factors are used to compute present values

Cash flow observations may have monthly, quarterly, semiannual, or annual frequencies. Within those frequencies, activity may be concentrated at the beginning of the period, end of the period, or throughout the period (equivalent to middle of period). Thus, from this collection of twice-monthly discount factors, 12 subsets are used for computing present values:

**Frequency Within-period
timing**

Monthly Beginning of the month

Throughout the month (or mid-month)

End of the month

Quarterly Beginning of the quarter

Throughout the quarter (or mid-quarter)

End of the quarter

Semiannually Beginning of the semiannual period

Throughout the semiannual period (or mid-semiannual

period)

End of the semiannual period

Annually Beginning of the year

Throughout the year (or mid-year)

End of the year

The present value of the cash flow observations is computed by summing the products of the observation for a particular frequency (f), timing (t), and period (n) by the present value factor for the same frequency, timing, and period:

Present
value = ( X_{ft1 }•_{ }P_{ft1
}) + ( X_{ft2 }•_{ }P_{ft2
}) + ... + ( X_{ftn }•_{
}P_{ftn })

The present value, in this computation, is the market price of a collection of zero coupon bonds that, at maturity, exactly match the amounts and maturities of the cash flow observations; hence, the term “basket-of-zeros.”

This method is superior to using a constant rate for discounting because the present value of each payment is based on an instrument of unambiguously “similar maturity” for that payment. This calculation avoids an obvious anomaly produced by the previous method in which two loan guarantee programs with identical cash flows would result in different subsidies whenever the loans they guaranteed had differing maturity. With the basket-of-zeros method, two programs with identical cash flows would have identical subsidies.

**Sources of Information on Treasury Interest Rates and References**

Market yields on Treasury marketable securities are published by the Board of Governors of the Federal Reserve System and by the Department of the Treasury:

Department of the Treasury,
Financial Management Service, *Treasury
Bulletin*, Government Printing Office.
This quarterly publication is available from the Government Printing
Office bookstores and from the web site of the Financial Management Service (www.fms.treas.gov).

Board of Governors of the Federal Reserve System, Federal Reserve Statistical Release H.15, “Selected Interest Rates.” This weekly release may be obtained from the FRB website (www.bog.frb.fed.us).

There are a number of references that can be consulted for more information on discounting practices. The following is one of the many available:

Frank J. Fabozzi, ed., *The Handbook of Fixed Income Securities*,
fifth edition, Irwin Professional Publishing, 1997.

In addition, numerous articles on this and closely related subjects have been published in academic and professional journals. The following list is by no means complete:

Peter A. Abken, “Innovations in
Modeling the Term Structure of Interest Rates,” *Economic Review* of the Federal Reserve Bank of Atlanta,
July/August, 1990, pp. 2-27.

Michael G. Bradley and Stephen A.
Lumpkin, “The Treasury Yield Curve as a Cointegrated System,” *Journal of Financial and Quantitative
Analysis*, September 1992, pp. 449-463.

Michael J. Brennan and Eduardo S.
Schwartz, “An Equilibrium Model of Bond Pricing and a Test for Market
Efficiency,” *Journal of Financial and
Quantitative Analysis*, March 1982.

Michael J. Brennan and Eduardo S.
Schwartz, “Analyzing Convertible Bonds,” *Journal
of Financial and Quantitative Analysis* 15, (1980), pp. 907-929.

Michael J. Brennan and Eduardo S.
Schwartz, “Bond Pricing and Market Efficiency,” *Financial Analysis Journal*, September-October 1982.

John Y. Campbell, “A Defense of Traditional
Hypotheses about the Term Structure of Interest Rates,” *Journal of Finance*, March 1986, pp. 183-193.

John Y. Campbell and Robert J.
Shiller, “Yield Spreads and Interest Rate Movements: A Bird’s Eye View,” *Review of Economic Studies*, May 1991, pp.
495-514.

Donald R. Chambers, Willard T.
Carleton, and Donald W. Wakeman, “A New Approach to Estimation of the Term
Structure of Interest Rates,” *Journal of
Financial and Quantitative Analysis*, September 1984, pp. 233-252. Polynomial spot rate function method.

K. C. Chan, G. Andrew Karolyi,
Francis A. Longstaff, and Anthony B. Sanders, “An Empirical Comparison of
Alternative Models of the Short-term Interest Rate,” *Journal of Finance*, July 1992, pp. 1209-1227.

Thomas S. Coleman, Lawrence Fisher,
and Roger G. Ibbotson, “Estimating the Term Structure of Interest Rates from
Data that include the Prices of Coupon Bonds,” *The Journal of Fixed Income*, September 1995, pp. 85-116. Piecewise-constant forward rate method.

Georges Courtadon, “The Pricing of
Options on Default-Free Bonds,” *Journal
of Financial and Quantitative Analysis* 17 (1982), pp. 75-100.

John Cox, Jonathan Ingersoll, Jr.,
and Stephen Ross, “A Re-examination of Traditional Hypotheses about the Term
Structure of Interest Notes,” *Journal of
Finance*, September 1981, pp. 769-99.

John M. Culbertson, “The Term
Structure of Interest Rates,” *Quarterly
Journal of Economics*, November, 1957, pp. 485-517.

Phillip R. Daves and Michael C.
Ehrhardt, “Liquidity, Reconstitution, and the Value of US Treasury Strips,” *Journal of Finance*, March 1993, pp.
315-329.

Werner M. De Bondt and May M.
Bange, “Inflation Forecast Errors and Time Variation in Term Premia,” *Journal of Financial and Quantitative
Analyses*, December 1992, pp. 479-496.

Paul Diamant, “Semi-Empirical
Smooth Fit to Treasury Yield Curve,” *The
Journal of Fixed Income*, July 1996, pp. 55-70.

Michael E. Echols and Jan Walter
Elliott, “A Quantitative Yield Curve Model for Estimating the Term Structure of
Interest Rates,” *Journal of Financial and
Quantitative Analysis*, March 1976, pp. 87-114.

Tom Engsted, “The Term Structure of
Interest Rates in Denmark, 1982-89,” *Bulletin
of Economic Research*, January 1993, pp. 19-37.

Frank J. Fabozzi and Gifford Fong, *Advanced Fixed Income Portfolio Management*
(Chicago: Probus Publishing Company, 1994), Appendix A. Exponential spline
discount method.

Eugene Fama, “The Information in
the Term Structure,” *Journal of Financial
Economics*, December 1984, pp. 509-528.

Irving Fisher, “Appreciation and
Interest,” *Publications of the American
Economic Association*, August, 1986.

Mark Fisher, Douglas Nychka, and David Zervos, “Fitting the Term Structure of Interest Rates with Smothing Splines,” Finance and Economics Division Series, Division of Research and Statistics, Division of Monetary Affairs, Federal Reserve Board, Washington, DC, January 1995.

Kenneth Froot, “New Hope for the
Expectations Hypothesis of the Term Structure of Interest Rates,” *Journal of Finance*, June 1989.

John Hull, *Options, Futures, and Other Derivative Securities*, second edition,
(Englewood Cliffs, NJ: Prentice-Hall, 1993), Chapter 15, “Interest Rate
Derivative Securities.”

Terence C. Langetieg, “A
Multivariate Model of the Term Structure,” *Journal
of Finance*, March 1980.

Miles Livingston, *Money and Capital Markets* (Englewood
Cliffs, NJ: Prentice-Hall, 1990), pp. 254-56, “Appendix: Local versus Unbiased
Expectations Hypotheses.”

F. A. Lutz, “The Structure of
Interest Rates,” *Quarterly Journal of
Economics*, November 1940, pp. 36-63.

Reuben Kessel, “The Cyclical
Behavior of the Term Structure of Interest Rates,” in *Essays in Applied Price Theory* (Chicago: University of Chicago
Press, 1965).

J. Houston McCulloch, “The
Tax-Adjusted Yield Curve,” *Journal of
Finance*, June 1975.

J. Houston McCulloch, “Measuring
the Term Structure of Interest Rates,” *Journal
of Business*, January 1971, pp. 19-31.
Cubic spline discount method.

David Meiselman, *The Term Structure of Interest Rates, *(Englewood
Cliffs, NJ: Prentice-Hall, 1962).

Robert C. Merton, “The Rational Theory
of Options Pricing,” *Bell Journal of
Economics and Management Science*, 1973, pp. 141-183.

Franco Modigliani and Richard
Sutch, “Innovations in Interest Rate Policy,” *American Economic Review*, May 1966, pp. 178-197.

Nelson and Siegel, “Parsimonious
Modeling of Yield Curves,” *Journal of
Business*, October 1987, pp. 473-489.

Joseph P. Ogden, “An Analysis of
Yield Curve Notes,” *Journal of Finance*,
March 1987, pp. 99-110.

Risk Metrics Technical Document, Morgan Guaranty Trust Company, Global Research, New York, May 26, 1995.

Stephen M. Schaefer, “Tax-Induced
Clientele Effects in the Market for British Government Securities,” *Economic Journal*, June 1981, pp.
415-438. Bernstein polynomial spline
discount function method.

Robert J. Shiller, “The Term Structure
of Interest Rates,” with an appendix by J. Houston McCulloch, in Benjamin
Friedman and Frank H. Hahn, editors, *Handbook
of Monetary Economics*, Vol. 1 (Amsterdam: North Holland, 1990), chapter 13,
pp. 627-722.

Mark P. Taylor, “Modeling the Yield
Curve,” *Economic Journal*, May 1992,
pp. 524-532.

James C. Van Horne, *Financial Market Rates and Flows*, 4^{th}
edition (Englewood Cliffs, NJ: Prentice-Hall, 1994). Covers the alternative
term-structure theories in detail.

Richard B. Worley and Stanley
Diller, “Interpreting the Yield Curve,” *Financial
Analysis Journal,* November-December 1976, pp. 37-45.

[1] Obviously, there have been times when the forward rate curves were inverted (short-term rates above long-term rates) or dome-shaped (medium-term rates were higher than either short- or long-term rates). However, these exceptions have been infrequent. In addition, the interpolation methods, described below, perform reasonably well for the intervals between published yield curve points even when the overall curve has an unusual shape.

[2] The
algorithm used for this trial-and-error method is interval bisection
method. The target rate is assumed to
occur between a lower limit of -199.0 percent and an upper limit equal to the
largest value that can be represented in an 8-byte floating point format or
approximately 10^{308}. The
midpoint of that interval is found and tested.
If the midpoint estimate is too high, the upper half is discarded and
the process repeated on the remaining interval. A solution is found when the mid-point value test results in a
par-valued security or when the difference between the upper and lower limits
of the remaining interval is zero or too small to be represented in 8-byte
floating point format (roughly 15 decimal digits of precision). This method is described in more detail in *How the single effective rate is calculated*,
which has several references on numerical methods.

[3] In order to verify that the regression equations were not biased by the use of Wednesday observations, a separate (smaller) sample of Tuesday observations were collected and analyzed. The regression equations produced by the Tuesday data were compared to the Wednesday equations via Chow tests (1 vs. 1, 2 vs. 2, etc.). No significant differences were found between the equation coefficients.

[4] These are approximations because the effort needed to obtain quotes that are exactly 0.5 months, 1.0 months, …, and 2.5 months to maturity would exceed the value that the added precision might offer. Thus, 15 days is used as a reasonable approximation of 0.5 months; 29 days for 1 month; the average of 43 and 50 days for 1.5 months; 64 days for 2 months; and 78 days for 2.5 months.