[Federal Register Volume 69, Number 170 (Thursday, September 2, 2004)]
[Rules and Regulations]
[Pages 53619-53626]
From the Federal Register Online via the Government Publishing Office [www.gpo.gov]
[FR Doc No: 04-19999]


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DEPARTMENT OF THE TREASURY

Fiscal Service

31 CFR Part 356

[Department of the Treasury Circular, Public Debt Series No. 1-93]


Sale and Issue of Marketable Treasury Bills, Notes, and Bonds: 
Six-Decimal Pricing, Negative-Yield Bidding, Zero-Filling, and 
Noncompetitive Bidding and Award Limit Increase

AGENCY: Bureau of the Public Debt, Fiscal Service, Department of the 
Treasury.

ACTION: Final rule.

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SUMMARY: The Department of the Treasury (``Treasury,'' ``We,'' or 
``Us'') is issuing in final form an amendment to its regulations 
(Uniform Offering Circular for the Sale and Issue of

[[Page 53620]]

Marketable Book-Entry Treasury Bills, Notes, and Bonds). This amendment 
implements four policy changes and makes conforming changes to the 
formulas. First, this amendment changes the pricing convention for all 
marketable Treasury securities auctions from three decimal places to 
six decimal places. Second, this amendment allows for negative-yield 
bidding in Treasury inflation-protected securities (TIPS) auctions to 
accommodate circumstances in which the desired real yield is a negative 
number. Third, this amendment provides for ``zero-filling'' of 
competitive auction bids that are not expressed out to the required 
three decimals by modifying the bids to a three-decimal rate or yield 
that is mathematically equivalent to the rate or yield submitted. 
Finally, this amendment raises the noncompetitive bidding and award 
limit for all Treasury bill auctions from $1 million to $5 million, 
which is the current noncompetitive limit for all Treasury note and 
bond auctions.

EFFECTIVE DATE: September 20, 2004.

ADDRESSES: You may download this final rule from the Bureau of the 
Public Debt's Web site at http://www.publicdebt.treas.gov or from the 
Electronic Code of Federal Regulations (e-CFR) Web site at http://www.gpoaccess.gov/ecfr. It is also available for public inspection and 
copying at the Treasury Department Library, Room 1428, Main Treasury 
Building, 1500 Pennsylvania Avenue, NW., Washington, DC 20220. To visit 
the library, call (202) 622-0990 for an appointment.

FOR FURTHER INFORMATION CONTACT: Lori Santamorena (Executive Director), 
Chuck Andreatta or Lee Grandy (Associate Directors), Bureau of the 
Public Debt, Government Securities Regulations Staff, (202) 504-3632, 
or e-mail us at [email protected].

SUPPLEMENTARY INFORMATION: The Uniform Offering Circular, in 
conjunction with the offering announcement for each auction, provides 
the terms and conditions for the sale and issuance in an auction to the 
public of marketable Treasury bills, notes and bonds.\1\ In this 
notice, we describe the current rules and why we are changing them. 
Then we describe the final amendment to the Uniform Offering Circular.
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    \1\ The Uniform Offering Circular was published as a final rule 
on January 5, 1993 (58 FR 412). The circular, as amended, is 
codified at 31 CFR part 356.
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Background and Analysis

A. Six-Decimal Pricing

    It is a longstanding convention in marketable Treasury securities 
auctions that the prices at which we award securities to successful 
bidders are expressed in terms of price per 100 of par value to three 
decimal places, for example, 99.170. One result is that auctions of 
Treasury bills of less than 72 days currently do not result in price 
uniqueness for each discount rate bid.\2\ In other words, for these 
short-term Treasury bills, there may be multiple discount rates bid 
that result in the same three-decimal price. Furthermore, for extremely 
short-term Treasury bills, rounding the price to three decimals can 
result in the investment rate (the equivalent coupon-issue yield) being 
inaccurate. Treasury provides both the discount rate and the investment 
rate on its Treasury bill auction results announcements. Because the 
discount rate is based on a par value of $100, and the investment rate 
is based on the actual price paid per $100 of par, the discount rate 
should always be less than the investment rate. (The formula for 
calculating a purchase price from a discount rate is P = 100(1 - dr/
360), where d = the discount rate, in decimals, r = the number of days 
to maturity, and P = price per hundred (dollars). The formula for 
calculating an investment rate from a purchase price is
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    \2\ Price uniqueness occurs when each separate discount rate 
produces a different (unique) price, i.e., no two discount rates 
result in the same price. Price uniqueness is a function of the 
minimum bid increment allowed in auctions, price rounding 
conventions, and the number of days to maturity.
[GRAPHIC] [TIFF OMITTED] TR02SE04.004

where i = the investment rate, in decimals; P = price per hundred 
(dollars); r = number of days to maturity; and y = number of days in 
the year following the issue date (normally 365). See Section V of 
Appendix B.) However, this relationship does not always hold under our 
current three-decimal conventions.
    An example of the anomalies that can occur in very short-term 
Treasury bills occurred in Treasury's auction of four-day cash 
management bills on September 10, 2003. This bill was awarded at a 
discount rate of 0.940 percent and a three-decimal price of 99.990. 
Under the current bidding convention, 18 different discount rates could 
have been bid in the auction (from 0.860 percent to 0.945 percent), all 
having a corresponding rounded price of 99.990. In addition, the 
investment rate for the auction was 0.915 percent, which is less than 
the awarded discount rate of 0.940 percent.
    In the February 2004 Quarterly Refunding Statement, Treasury 
announced its intention to compute the price of awards in auctions to 
six decimal places per hundred.3-4 In an effort to make the 
transition as smooth as possible, the six-decimal pricing calculation 
formulas were made available at the Bureau of the Public Debt Website 
on March 4, 2004.\5\ In the May 2004 Quarterly Refunding Statement, 
Treasury reiterated its intention to change to the six-decimal pricing 
convention in the second half of the year.\6\
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    \3-4\ Treasury February Quarterly Refunding Statement, February 
4, 2004. Treasury stated its intention to implement six-decimal 
pricing later in the year.
    \5\ See Public Debt News Release on March 4, 2004. The formulas 
are available at http://www.publicdebt.treas.gov/of/ofcalc6decimal.htm.
    \6\ Treasury May Quarterly Refunding Statement, May 5, 2004.
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    Accordingly, to ensure price uniqueness for all discount rates or 
yields bid in all marketable Treasury securities auctions, we are 
amending the Uniform Offering Circular to calculate prices for awarded 
securities to six decimals per $100 of par value. Specifically, Sec.  
356.20(c) is being changed to state that price calculations for awarded 
securities will be rounded to six decimal places per hundred (rather 
than the current three decimals), for example, 99.954321. Calculating 
prices to six decimals will also make Treasury's pricing practice 
consistent with secondary market practices. As of the effective date of 
this amendment, this change will apply to all Treasury bill, note, and 
bond auctions.

B. Negative-Yield Bidding

    Treasury's current auction regulations do not expressly permit 
bidders in TIPS auctions to submit negative-yield bids. Since it is 
possible that under certain market conditions the yield desired by a 
competitive bidder in a TIPS auction would be a negative number, this 
amendment modifies the regulations to allow Treasury to accept 
negative-yield bids in TIPS auctions.
    The introduction of 5-year TIPS \7\ has increased the possibility 
that a Treasury TIPS auction could result in a negative-yield TIPS. 
However, a negative TIPS interest (coupon) rate is neither practical 
nor desirable. Therefore, if a TIPS auction produces a negative or zero 
yield, this amendment clarifies that

[[Page 53621]]

we will set the interest rate at zero and calculate the award price 
accordingly. Investors will receive the inflation-adjusted par amount 
at maturity. Therefore, Sec.  356.12(c)(1)(iii) is being modified to 
state that the real-yield bid submitted for a TIPS auction may be a 
positive number, a negative number, or zero. Also, Sec.  356.20(b) is 
being modified to state that if a TIPS auction produces a negative or 
zero yield, the interest rate will be set at zero, with successful 
bidders' award prices calculated accordingly.
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    \7\ Treasury May 2004 Quarterly Refunding Statement, May 5, 
2004. Treasury stated it would begin offering 5-year TIPS, with the 
first such offering to be conducted in October 2004.
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C. Zero-Filling

    When evaluating bids submitted in Treasury auctions, we currently 
reject any bid that does not adhere to the established three-decimal 
bidding format. Rejecting such bids reduces the number of competitive 
bids in Treasury auctions, which is counter to our objective of 
ensuring broad participation in Treasury auctions. Therefore, we have 
decided to accept competitive bids that are not expressed out to three 
decimals at a three-decimal rate or yield that is mathematically 
equivalent to the rate or yield that was submitted. For example, a bid 
of 5.32 will be treated as a bid of 5.320, a bid of 4.1 will be treated 
as a bid of 4.100, and a bid of 3 will be treated as a bid of 3.000. 
Accordingly, Sec. Sec.  356.12(c)(1)(i),(ii), and (iii) are being 
modified to state that any missing decimals in a competitive bid will 
be treated as zero.

D. Noncompetitive Bidding and Award Limit Increase for Treasury Bill 
Auctions

    In an October 25, 1991 Treasury News press release, Treasury 
announced it was increasing the maximum noncompetitive award in note 
and bond auctions from $1 million to $5 million, effective November 5, 
1991.\8\ The change was made to broaden participation in Treasury 
auctions, particularly to encourage bidding by smaller investors. The 
noncompetitive bid and award limit for Treasury bills remained at $1 
million. In an effort to make the maximum noncompetitive bid and award 
limit consistent for all marketable Treasury securities auctions, and 
to increase participation in Treasury auctions, Treasury is raising the 
noncompetitive bidding and award limit for Treasury bill auctions from 
$1 million to $5 million.
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    \8\ Treasury News press release dated October 21, 1991.
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    Accordingly, Sec.  356.12(b)(1) is being modified to provide 
generally that the maximum amount that can be bid noncompetitively in 
any Treasury securities auction is $5 million.\9\ Also, Sec.  356.22(a) 
is being modified to state that the maximum noncompetitive award to any 
bidder will be $5 million, which will apply to all Treasury auctions.
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    \9\ Paragraph 356.12(b)(1) also states that the maximum bid 
limitation does not apply if a bidder is bidding solely through a 
TreasuryDirect reinvestment request.
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E. Formulas and Effective Date

    Technical changes are being made to the formulas in Appendix B, 
Sections II, III, and V to conform with the changes we are making in 
the pricing conventions. To provide market participants and Treasury 
sufficient time to modify their settlement systems and to make any 
other operational changes that may be needed, we are providing a 
delayed effective date of September 20, 2004.

Procedural Requirements

    It has been determined that this final rule is not a significant 
regulatory action for purposes of Executive Order 12866. The notice and 
public procedures requirements of the Administrative Procedure Act do 
not apply.
    Since no notice of proposed rulemaking is required, the provisions 
of the Regulatory Flexibility Act (5 U.S.C. 601 et seq.) do not apply.
    The Office of Management and Budget has approved the collections of 
information in this final rule amendment in accordance with the 
Paperwork Reduction Act of 1995. This final rule is technical in nature 
and imposes no additional burdens on auction bidders.

List of Subjects in 31 CFR Part 356

    Bonds, Federal Reserve System, Government Securities, Securities.

0
For the reasons stated in the preamble, 31 CFR part 356 is amended as 
follows:

PART 356--SALE AND ISSUE OF MARKETABLE BOOK-ENTRY TREASURY BILLS, 
NOTES AND BONDS (DEPARTMENT OF THE TREASURY CIRCULAR, PUBLIC DEBT 
SERIES NO. 1-93)

0
1. The authority citation for part 356 continues to read as follows:

    Authority: 5 U.S.C. 301; 31 U.S.C. 3102 et seq.; 12 U.S.C. 391.


0
2. Section 356.12 is amended by revising paragraphs (b)(1) and 
(c)(1)(i),(ii), and (iii) to read as follows:


Sec.  356.12  What are the different types of bids and do they have 
specific requirements or restrictions?

* * * * *
    (b) Noncompetitive bids. (1) Maximum bid. You may not bid 
noncompetitively for more than $5 million. The maximum bid limitation 
does not apply if you are bidding solely through a TreasuryDirect 
reinvestment request. A request for reinvestment of securities maturing 
in TreasuryDirect is a noncompetitive bid.
* * * * *
    (c) Competitive bids.--(1) Bid format (i) Treasury bills. A 
competitive bid must show the discount rate bid, expressed with three 
decimals in .005 increments. The third decimal must be either a zero or 
a five, for example, 5.320 or 5.325. We will treat any missing decimals 
as zero, for example, a bid of 5.32 will be treated as 5.320.
    (ii) Treasury fixed-principal securities. A competitive bid must 
show the yield bid, expressed with three decimals, for example, 4.170. 
We will treat any missing decimals as zero, for example, a bid of 4.1 
will be treated as 4.100.
    (iii) Treasury inflation-protected securities. A competitive bid 
must show the real yield bid, expressed with three decimals, for 
example, 3.070. We will treat any missing decimals as zero, for 
example, a bid of 3 will be treated as 3.000. The real yield may be a 
positive number, a negative number, or zero.
* * * * *
0
3. Section 356.20 is amended by revising paragraphs (b) and (c) to read 
as follows:


Sec.  356.20  How does the Treasury determine auction awards?

* * * * *
    (b) Determining the interest rate for new note and bond issues. We 
set the interest rate at a \1/8\ of one percent increment. If a 
Treasury inflation-protected securities auction results in a negative 
or zero yield, the interest rate will be set at zero, and successful 
bidders' award prices will be calculated accordingly (See Appendix B to 
this part for formulas).
    (1) Single-price auctions. The interest rate we establish produces 
the price closest to, but not above, par when evaluated at the yield of 
awards to successful competitive bidders.
    (2) Multiple-price auctions. The interest rate we establish 
produces the price closest to, but not above, par when evaluated at the 
weighted-average yield of awards to successful competitive bidders.
    (c) Determining purchase prices for awarded securities. We round 
price calculations to six decimal places on the basis of price per 
hundred, for example, 99.954321 (See Appendix B to this part).

[[Page 53622]]

    (1) Single-price auctions. We award securities to both 
noncompetitive and competitive bidders at the price equivalent to the 
highest accepted discount rate or yield at which bids were accepted. 
For inflation-protected securities, the price for awarded securities is 
the price equivalent to the highest accepted real yield.
    (2) Multiple-price auctions--(i) Competitive bids. We award 
securities to competitive bidders at the price equivalent to each yield 
or discount rate at which their bids were accepted.
    (ii) Noncompetitive bids. We award securities to noncompetitive 
bidders at the price equivalent to the weighted average yield or 
discount rate of accepted competitive bids.

0
4. Section 356.22 is amended by revising paragraph (a) to read as 
follows:


Sec.  356.22  Does the Treasury have any limitations on auction awards?

    (a) Awards to noncompetitive bidders. The maximum award to any 
bidder is $5 million. This limit does not apply to bidders bidding 
solely through TreasuryDirect reinvestment requests.
* * * * *

0
5. Appendix B to part 356, sections II and III are revised to read as 
follows:

Appendix B to Part 356--Formulas and Tables

* * * * *

II. Formulas for Conversion of Fixed-Principal Security Yields to 
Equivalent Prices

Definitions

P = price per 100 (dollars), rounded to six places, using normal 
rounding procedures.
C = the regular annual interest per $100, payable semiannually, 
e.g., 6.125 (the decimal equivalent of a 6\1/8\ interest rate).
i = nominal annual rate of return or yield to maturity, based on 
semiannual interest payments and expressed in decimals, e.g., .0719.
n = number of full semiannual periods from the issue date to 
maturity, except that, if the issue date is a coupon frequency date, 
n will be one less than the number of full semiannual periods 
remaining to maturity. Coupon frequency dates are the two semiannual 
dates based on the maturity date of each note or bond issue. For 
example, a security maturing on November 15, 2015, would have coupon 
frequency dates of May 15 and November 15.
r = (1) number of days from the issue date to the first interest 
payment (regular or short first payment period), or (2) number of 
days in fractional portion (or ``initial short period'') of long 
first payment period.
s = (1) number of days in the full semiannual period ending on the 
first interest payment date (regular or short first payment period), 
or (2) number of days in the full semiannual period in which the 
fractional portion of a long first payment period falls, ending at 
the onset of the regular portion of the first interest payment.

v\n\ = 1 / [1 + (i/2)] \n\ = present value of 1 due at the end of n 
periods.
an[rceil] = (1-vn) / (i/2) = v + v\2\ + v\3\ + 
... v\n\ = present value of 1 per period for n periods.

    Special Case: If i = 0, then an[rceil] = n. 
Furthermore, when i = 0, an[rceil] cannot be calculated 
using the formula: (1 - vn)/(i/2). In the special case 
where i = 0, an[rceil] must be calculated as the 
summation of the individual present values (i.e., v + v\2\ + v\3\ + 
... + v\n\). Using the summation method will always confirm that 
an[rceil] = n when i = 0.

A = accrued interest.

    A. For fixed-principal securities with a regular first interest 
payment period:

Formula:

P[1 + (r/s)(i/2)] = (C/2)(r/s) + (C/2)an[rceil] + 
100v\n\.

Example:

    For an 8\3/4\% 30-year bond issued May 15, 1990, due May 15, 
2020, with interest payments on November 15 and May 15, solve for 
the price per 100 (P) at a yield of 8.84%.

Definitions:

C = 8.75.
i = .0884.
r = 184 (May 15 to November 15, 1990).
s = 184 (May 15 to November 15, 1990).
n = 59 (There are 60 full semiannual periods, but n is reduced by 1 
because the issue date is a coupon frequency date.)
v\n\ = 1 / [(1 + .0884 / 2)]\59\, or .0779403508.
an[rceil] = (1 - .0779403508) / .0442, or 20.8610780353.

Resolution:

P[1 + (r/s)(i/2)] = (C/2)(r/s) + (C/2)an[rceil] + 100v\n\ 
or
P[1 + (184/184)(.0884/2)] = (8.75/2)(184/184) + (8.75/
2)(20.8610780353) + 100(.0779403508).
(1) P[1 + .0442] = 4.375 + 91.2672164044 + 7.7940350840.
(2) P[1.0442] = 103.4362514884.
(3) P = 103.4362514884 / 1.0442.
(4) P = 99.057893.

    B. For fixed-principal securities with a short first interest 
payment period:

Formula:

P[1 + (r/s)(i/2)] = (C/2)(r/s) + (C/2)an[rceil] + 
100v\n\.

Example:

    For an 8\1/2\% 2-year note issued April 2, 1990, due March 31, 
1992, with interest payments on September 30 and March 31, solve for 
the price per 100 (P) at a yield of 8.59%.

Definitions:

C = 8.50.
i = .0859.
n = 3.
r = 181 (April 2 to September 30, 1990).
s = 183 (March 31 to September 30, 1990).
v\n\ = 1 / [(1 + .0859 / 2)]\3\, or .8814740565.
an[rceil] = (1 - .8814740565) / .04295, or 2.7596261590.

Resolution:

P[1 + (r/s)(i/2)] = (C/2)(r/s) + (C/2)an[rceil] + 100v\n\ 
or
P[1 + (181/183)(.0859/2)] = (8.50/2)(181/183) + (8.50/
2)(2.7596261590) + 100(.8814740565).
(1) P[1 + .042480601] = 4.2035519126 + 11.7284111757 + 88.14740565.
(2) P[1.042480601] = 104.0793687354.
(3) P = 104.0793687354 / 1.042480601.
(4) P = 99.838183.

    C. For fixed-principal securities with a long first interest 
payment period:

Formula:

P[1 + (r/s)(i/2)] = [(C/2)(r/s)]v + (C/
2)an[
] + 100v\n\.

Example:

    For an 8\1/2\% 5-year 2-month note issued March 1, 1990, due May 
15, 1995, with interest payments on November 15 and May 15 (first 
payment on November 15, 1990), solve for the price per 100 (P) at a 
yield of 8.53%.

Definitions:

C = 8.50.
i = .0853.
n = 10.
r = 75 (March 1 to May 15, 1990, which is the fractional portion of 
the first interest payment).
s = 181 (November 15, 1989, to May 15, 1990).
v = 1 / (1 + .0853/2), or .9590946147.
v\n\ = 1 / (1 + .0853/2)\10\, or .6585890783.
an[rceil] = (1-.658589)/.04265, or 8.0049454082.

Resolution:

P[1 + (r/s)(i/2)] = [(C/2)(r/s)]v + (C/2)an[rceil] + 
100v\n\ or
P[1 + (75/181)(.0853/2)] = [(8.50/2)(75/181)].9590946147 + (8.50/
2)(8.0049454082) + 100(.6585890783).
(1) P[1 + .017672652] = 1.6890133062 + 34.0210179850 + 
65.8589078339.
(2) P[1.017672652] = 101.5689391251.
(3) P = 101.5689391251 / 1.017672652.
(4) P = 99.805118.

    D. (1) For fixed-principal securities reopened during a regular 
interest period where the purchase price includes predetermined 
accrued interest.
    (2) For new fixed-principal securities accruing interest from 
the coupon frequency date immediately preceding the issue date, with 
the interest rate established in the auction being used to determine 
the accrued interest payable on the issue date.

Formula:

(P + A)[1 + (r/s)(i/2)] = C/2 + (C/2)an[rceil] + 100v\n\.
Where:

    A = [(s-r)/s](C/2).

Example:

    For a 9\1/2\% 10-year note with interest accruing from November 
15, 1985, issued November 29, 1985, due November 15, 1995, and 
interest payments on May 15 and November 15, solve for the price per 
100 (P) at a yield of 9.54%. Accrued interest is from November 15 to 
November 29 (14 days).

Definitions:

C = 9.50.
i = .0954.
n = 19.

[[Page 53623]]

r = 167 (November 29, 1985, to May 15, 1986).
s = 181 (November 15, 1985, to May 15, 1986).
v\n\ = 1 / [(1 + .0954/2)]\19\, or .4125703996.
an[rceil] = (1 - .4125703996) / .0477, or 12.3150859630.
A = [(181 - 167) / 181](9.50/2), or .367403.

Resolution:

(P+A)[1 + (r/s)(i/2)] = C/2 + (C/2)an[rceil] + 100v\n\ or
(P + .367403)[1 + (167/181)(.0954/2)] = (9.50/2) + (9.50/
2)(12.3150859630) + 100(.4125703996).
(1) (P + .367403)[1 + .044010497] = 4.75 + 58.4966583243 + 
41.25703996.
(2) (P + .367403)[1.044010497] = 104.5036982843.
(3) (P + .367403) = 104.5036982843 / 1.044010497.
(4) (P + .367403) = 100.098321.
(5) P = 100.098321 -.367403.
(6) P = 99.730918.

    E. For fixed-principal securities reopened during the regular 
portion of a long first payment period:

Formula:

(P + A)[1 + (r/s)(i/2)] = (r's'')(C/2) + C/2 + (C/
2)an[rceil] + 100v\n\.

Where:

A = AI' + AI,
AI' = (r'/s'')(C/2),
AI = [(s-r) / s](C/2), and

r = number of days from the reopening date to the first interest 
payment date,
s = number of days in the semiannual period for the regular portion 
of the first interest payment period,
r' = number of days in the fractional portion (or ``initial short 
period'') of the first interest payment period,
s'' = number of days in the semiannual period ending with the 
commencement date of the regular portion of the first interest 
payment period.

Example:

    A 10\3/4\% 19-year 9-month bond due August 15, 2005, is issued 
on July 2, 1985, and reopened on November 4, 1985, with interest 
payments on February 15 and August 15 (first payment on February 15, 
1986), solve for the price per 100 (P) at a yield of 10.47%. Accrued 
interest is calculated from July 2 to November 4.

Definitions:

C = 10.75.
i = .1047.
n = 39.
r = 103 (November 4, 1985, to February 15, 1986).
s = 184 (August 15, 1985, to February 15, 1986).
r' = 44 (July 2 to August 15, 1985).
s'' = 181 (February 15 to August 15, 1985).
v\n\ = 1 / [(1 + .1047 / 2)]\39\, or .1366947986.
an[rceil] = (1 - .1366947986) / .05235, or 16.4910258142.
AI' = (44 / 181)(10.75 / 2), or 1.306630.
AI = [(184 - 103) / 184](10.75 / 2), or 2.366168.
A = AI' + AI, or 3.672798.

Resolution:

(P + A)[1 + (r/s)(i/2)] = (r'/s'')(C/2) + C/2 + (C/
2)an[rceil] + 100v\n\ or
(P + 3.672798)[1 + (103/184)(.1047/2)] = (44/181)(10.75/2) +10.75/2 
+ (10.75/2)(16.4910258142) + 100(.1366947986).
(1) (P + 3.672798)[1 + .02930462] = 1.3066298343 + 5.375 + 
88.6392637512 + 13.6694798628.
(2) (P + 3.672798)[1.02930462] = 108.9903734482.
(3) (P + 3.672798) = 108.9903734482 / 1.02930462.
(4) (P + 3.672798) = 105.887384.
(5) P = 105.887384 -3.672798.
(6) P = 102.214586.

    F. For fixed-principal securities reopened during a short first 
payment period:

Formula:

(P + A)[1 + (r/s)(i/2)] = (r'/s)(C/2) + (C/
2)an[
] + 100v \n\.

Where:

A = [(r' - r)/s](C/2) and

r' = number of days from the original issue date to the first 
interest payment date.

Example:

    For a 10\1/2\% 8-year note due May 15, 1991, originally issued 
on May 16, 1983, and reopened on August 15, 1983, with interest 
payments on November 15 and May 15 (first payment on November 15, 
1983), solve for the price per 100 (P) at a yield of 10.53%. Accrued 
interest is calculated from May 16 to August 15.

Definitions:

C = 10.50.
i = .1053.
n = 15.
r = 92 (August 15, 1983, to November 15, 1983).
s = 184 (May 15, 1983, to November 15, 1983).
r' = 183 (May 16, 1983, to November 15, 1983).
v \n\ = 1/[(1 + .1053/2)]\15\, or .4631696332.
an[] 
= (1 - .4631696332) / .05265, or 10.1962082956.
A = [(183 - 92) / 184](10.50 / 2), or 2.596467.

Resolution:

(P + A)[1 + (r/s)(i/2)] = (r'/s)(C/2) + (C/
2)an[
] + 100v \n\ or
(P + 2.596467)[1+(92/184)(.1053/2)] = (183/184)(10.50/2) + (10.50/
2)(10.1962082956) + 100(.4631696332).
(1) (P + 2.596467)[1 + .026325] = 5.2214673913 + 53.5300935520 + 
46.31696332.
(2) (P + 2.596467)[1.026325] = 105.0685242633.
(3) (P + 2.596467) = 105.0685242633 / 1.026325.
(4) (P + 2.596467) = 102.373541.
(5) P = 102.373541 - 2.596467.
(6) P = 99.777074.

    G. For fixed-principal securities reopened during the fractional 
portion (initial short period) of a long first payment period:

Formula:

(P + A)[1 + (r/s)(i/2)] = [(r'/s)(C/2)]v + (C/
2)an[
] + 100v \n\.

Where:

A = [(r' - r)/s](C/2), and
r = number of days from the reopening date to the end of the short 
period.
r' = number of days in the short period.
s = number of days in the semiannual period ending with the end of 
the short period.

Example:

    For a 9\3/4\% 6-year 2-month note due December 15, 1994, 
originally issued on October 15, 1988, and reopened on November 15, 
1988, with interest payments on June 15 and December 15 (first 
payment on June 15, 1989), solve for the price per 100 (P) at a 
yield of 9.79%. Accrued interest is calculated from October 15 to 
November 15.

Definitions:

C = 9.75.
i = .0979.
n = 12.
r = 30 (November 15, 1988, to December 15, 1988).
s = 183 (June 15, 1988, to December 15, 1988).
r' = 61 (October 15, 1988, to December 15, 1988).
v = 1 / (1 + .0979/2), or .9533342867.
v \n\ = [1 / (1 + .0979/2)]\12\, or .5635631040.
an[] 
= (1 - .5635631040)/.04895, or 8.9159733613.
A = [(61 - 30)/183](9.75/2), or .825820.

Resolution:

(P + A)[1 + (r/s)(i/2)] = [(r'/s)(C/2)]v + (C/
2)an[
] + 100v \n\ or
(P + .825820)[1 + (30/183)(.0979/2)] = [(61/183)(9.75/
2)](.9533342867) + (9.75/2)(8.9159733613) + 100(.5635631040).
(1) (P + .825820)[1+ .00802459] = 1.549168216 + 43.4653701362 + 
56.35631040.
(2) (P + .825820)[1.00802459] = 101.3708487520.
(3) (P + .825820) = 101.3708487520 / 1.00802459.
(4) (P + .825820) = 100.563865.
(5) P = 100.563865 -. 825820.
(6) P = 99.738045.

III. Formulas for Conversion of Inflation-Indexed Security Yields to 
Equivalent Prices

Definitions

P = unadjusted or real price per 100 (dollars).
Padj = inflation adjusted price; P x Index 
RatioDate.
A = unadjusted accrued interest per $100 original principal.
Aadj = inflation adjusted accrued interest; Ax Index 
RatioDate.
SA = settlement amount including accrued interest in current dollars 
per $100 original principal; Padj + Aadj.
r = days from settlement date to next coupon date.
s = days in current semiannual period.
i = real yield, expressed in decimals (e.g., 0.0325).
C = real annual coupon, payable semiannually, in terms of real 
dollars paid on $100 initial, or real, principal of the security.
n = number of full semiannual periods from issue date to maturity 
date, except that, if the issue date is a coupon frequency date, n 
will be one less than the number of full semiannual periods 
remaining until maturity. Coupon frequency dates are the two 
semiannual dates based on the maturity date of each note or bond

[[Page 53624]]

issue. For example, a security maturing on July 15, 2026 would have 
coupon frequency dates of January 15 and July 15.
v \n\ = 1/(1 + i/2)\n\ = present value of 1 due at the end of n 
periods.
an[] 
= (1 - v \n\) /(i/2) = v + v \2\ + v \3\ + \...\ + v \n\ = present 
value of 1 per period for n periods.

    Special Case: If i = 0, then 
an[] 
= n. Furthermore, when i = 0, 
an[] 
cannot be calculated using the formula: (1 - v \n\)/(i/2). In the 
special case where i = 0, 
an[] 
must be calculated as the summation of the individual present values 
(i.e., v + v \2\ + v \3\ + \...\ + v \n\). Using the summation 
method will always confirm that 
an[] 
= n when i = 0.

Date = valuation date.
D = the number of days in the month in which Date falls.
t = calendar day corresponding to Date.
CPI = Consumer Price Index number.
CPIM = CPI reported for the calendar month M by the 
Bureau of Labor Statistics.
Ref CPIM = reference CPI for the first day of the 
calendar month in which Date falls (also equal to the CPI for the 
third preceding calendar month), e.g., Ref CPIApril 1 is 
the CPIJanuary.
Ref CPIM+1 = reference CPI for the first day of the 
calendar month immediately following Date.
Ref CPIDate = Ref CPIM - [(t - 1)/D][Ref 
CPIM+1-Ref CPIM].
Index RatioDate = Ref CPIDate / Ref 
CPIIssueDate.

    Note: When the Issue Date is different from the Dated Date, the 
denominator is the Ref CPIDatedDate.

    A. For inflation-indexed securities with a regular first 
interest payment period:
Formulas:
[GRAPHIC] [TIFF OMITTED] TR02SE04.005

Padj = P x Index RatioDate.
A = [(s-r)/s] x (C/2).
Aadj = A x Index RatioDate.
SA = Padj + Aadj.
Index RatioDate = Ref CPIDate/Ref 
CPIIssueDate.

Example:

    We issued a 10-year inflation-indexed note on January 15, 1999. 
The note was issued at a discount to yield of 3.898% (real). The 
note bears a 3\7/8\% real coupon, payable on July 15 and January 15 
of each year. The base CPI index applicable to this note is 164. (We 
normally derive this number using the interpolative process 
described in Appendix B, section I, paragraph B.)

Definitions:

C = 3.875.
i = 0.03898.
n = 19 (There are 20 full semiannual periods but n is reduced by 1 
because the issue date is a coupon frequency date.).
r = 181 (January 15, 1999 to July 15, 1999).
s = 181 (January 15, 1999 to July 15, 1999).
Ref CPIDate = 164.
Ref CPIIssueDate = 164.

Resolution:

Index RatioDate = Ref CPIDate / Ref 
CPIIssueDate = 164/164 = 1.
A = [(181 - 181)/181] x 3.875/2 = 0.
Aadj = 0 x 1 = 0.
vn = 1/(1 + i/2)n = 1/(1 + .03898/2)\19\ = 
0.692984572.
an[rceil] = (1 - vn)/(i/2) = (1-0.692984572) / (.03898/2) 
= 15.752459107.

Formula:
[GRAPHIC] [TIFF OMITTED] TR02SE04.006

P = 99.811030.
Padj = P x Index RatioDate.
Padj = 99.811030 x 1 = 99.811030.
SA = Padj x Aadj.
SA = 99.811030 + 0 = 99.811030.
    Note: For the real price (P), we have rounded to six places. 
These amounts are based on 100 par value.
    B. (1) For inflation-indexed securities reopened during a 
regular interest period where the purchase price includes 
predetermined accrued interest.
    (2) For new inflation-indexed securities accruing interest from 
the coupon frequency date immediately preceding the issue date, with 
the interest rate established in the auction being used to determine 
the accrued interest payable on the issue date.
    Bidding: The dollar amount of each bid is in terms of the par 
amount. For example, if the Ref CPI applicable to the issue date of 
the note is 120, and the reference CPI applicable to the reopening 
issue date is 132, a bid of $10,000 will in effect be a bid of 
$10,000 x (132/120), or $11,000.

Formulas:
[GRAPHIC] [TIFF OMITTED] TR02SE04.007

Padj = P x Index RatioDate.
A = [(s-r)/s] x (C/2).
Aadj = A x Index RatioDate.
SA = Padj + Aadj.
Index RatioDate = Ref CPIDate/Ref 
CPIIssueDate.

Example:

    We issued a 3\5/8\% 10-year inflation-indexed note on January 
15, 1998, with interest payments on July 15 and January 15. For a 
reopening on October 15, 1998, with inflation compensation accruing 
from January 15, 1998 to October 15, 1998, and accrued interest 
accruing from July 15, 1998 to October 15, 1998 (92 days), solve for 
the price per 100 (P) at a real yield, as determined in the 
reopening auction, of 3.65%. The base index applicable to the issue 
date of this note is 161.55484 and the reference CPI applicable to 
October 15, 1998, is 163.29032.

Definitions:

C = 3.625.
i = 0.0365.
n = 18.
r = 92 (October 15, 1998 to January 15, 1999).
s = 184 (July 15, 1998 to January 15, 1999).
Ref CPIDate = 163.29032.
Ref CPIIssueDate = 161.55484.

Resolution:

Index RatioDate = Ref CPIDate/Ref 
CPIIssueDate = 163.29032/161.55484 = 1.01074.
vn = 1/(1 + i/2)n = 1/(1 + .0365/
2)18 = 0.722138438.
an = 
[] 1-
vn)/(i/2) = (1 - 0.722138438)/(.0365/2) = 15.225291068.

Formula:


[[Page 53625]]


[GRAPHIC] [TIFF OMITTED] TR02SE04.008

P = 100.703267 - 0.906250.
P = 99.797017.
Padj = P x Index RatioDate.
Padj = 99.797017 x 1.01074 = 100.86883696.
Padj = 100.868837.
A = [(184-92)/184] x 3.625/2 = 0.906250.
Aadj = A x Index RatioDate.
Aadj = 0.906250 x 1.01074 = 0.91598313.
Aadj = 0.915983.
SA = Padj + Aadj = 100.868837 + 0.915983.
SA = 101.784820.

    Note: For the real price (P), and the inflation-adjusted price 
(Padj), we have rounded to six places. For accrued 
interest (A) and the adjusted accrued interest (Aadj), we 
have rounded to six places. These amounts are based on 100 par 
value.
* * * * *

0
6. Appendix B to Part 356, Section V, is revised to read as follows:

V. Computation of Purchase Price, Discount Rate, and Investment Rate 
(Coupon-Equivalent Yield) for Treasury Bills

    A. Conversion of the discount rate to a purchase price for 
Treasury bills of all maturities:

Formula:

P = 100 (1 - dr / 360).

Where:

d = discount rate, in decimals.
r = number of days remaining to maturity.
P = price per 100 (dollars).

Example:

    For a bill issued November 24, 1989, due February 22, 1990, at a 
discount rate of 7.610%, solve for price per 100 (P).

Definitions:

d = .07610.
r = 90 (November 24, 1989 to February 22, 1990).

Resolution:

P = 100 (1 - dr / 360).
(1) P = 100 [1 - (.07610)(90) / 360].
(2) P = 100 (1 - .019025).
(3) P = 100 (.980975).
(4) P = 98.097500.

    Note: Purchase prices per $100 are rounded to six decimal 
places, using normal rounding procedures.

    B. Computation of purchase prices and discount amounts based on 
price per $100, for Treasury bills of all maturities:
    1. To determine the purchase price of any bill, divide the par 
amount by 100 and multiply the resulting quotient by the price per 
$100.

Example:

    To compute the purchase price of a $10,000 13-week bill sold at 
a price of $98.098000 per $100, divide the par amount ($10,000) by 
100 to obtain the multiple (100). That multiple times 98.098000 
results in a purchase price of $9,809.80.
    2. To determine the discount amount for any bill, subtract the 
purchase price from the par amount of the bill.

Example:

    For a $10,000 bill with a purchase price of $9,809.80, the 
discount amount would be $190.20, or $10,000 - $9,809.80.

    C. Conversion of prices to discount rates for Treasury bills of 
all maturities:

Formula:
[GRAPHIC] [TIFF OMITTED] TR02SE04.009

Where:

P = price per 100 (dollars).
d = discount rate.
r = number of days remaining to maturity.

Example:

    For a 26-week bill issued December 30, 1982, due June 30, 1983, 
with a price of $95.934567, solve for the discount rate (d).

Definitions:

P = 95.934567.
r = 182 (December 30, 1982, to June 30, 1983).

Resolution:
[GRAPHIC] [TIFF OMITTED] TR02SE04.010

(2) d = [.04065433 x 1.978021978].
(3) d = .080415158.
(4) d = 8.042%.

    Note: Prior to April 18, 1983, we sold all bills in price-basis 
auctions, in which discount rates calculated from prices were 
rounded to three places, using normal rounding procedures. Since 
that time, we have sold bills only on a discount rate basis.
    D. Calculation of investment rate (coupon-equivalent yield) for 
Treasury bills:
    1. For bills of not more than one half-year to maturity:

Formula:
[GRAPHIC] [TIFF OMITTED] TR02SE04.011

Where:

i = investment rate, in decimals.
P = price per 100 (dollars).
r = number of days remaining to maturity.
y = number of days in year following the issue date; normally 365 
but, if the year following the issue date includes February 29, then 
y is 366.

Example:

    For a cash management bill issued June 1, 1990, due June 21, 
1990, with a price of $99.559444 (computed from a discount rate of 
7.930%), solve for the investment rate (i).

Definitions:

P = 99.559444.
r = 20 (June 1, 1990, to June 21, 1990).
y = 365.

Resolution:
[GRAPHIC] [TIFF OMITTED] TR02SE04.012

(2) i = [.004425 x 18.25].
(3) i = .080756.
(4) i = 8.076%.

    2. For bills of more than one half-year to maturity:

Formula:

P [1 + (r - y/2)(i/y)] (1 + i/2) = 100.

    This formula must be solved by using the quadratic equation, 
which is:

ax 2 + bx + c = 0.

    Therefore, rewriting the bill formula in the quadratic equation 
form gives:
[GRAPHIC] [TIFF OMITTED] TR02SE04.013

and solving for ``i'' produces:
[GRAPHIC] [TIFF OMITTED] TR02SE04.014

Where:

i = investment rate in decimals.
b = r/y.
a = (r/2y) - .25.
c = (P-100)/P.
P = price per 100 (dollars).
r = number of days remaining to maturity.
y = number of days in year following the issue date; normally 365, 
but if the year following the issue date includes February 29, then 
y is 366.

Example:

    For a 52-week bill issued June 7, 1990, due June 6, 1991, with a 
price of $92.265000 (computed from a discount rate of 7.65%), solve 
for the investment rate (i).

Definitions:

r = 364 (June 7, 1990, to June 6, 1991).
y = 365.
P = 92.265000.

[[Page 53626]]

b = 364 / 365, or .997260274.
a = (364 / 730) - .25, or .248630137.
c = (92.265 - 100) / 92.265, or -.083834607.

Resolution:
[GRAPHIC] [TIFF OMITTED] TR02SE04.015

(3) i = (-.997260274 + 1.038221216) / .497260274.
(4) i = .040960942 / .497260274.
(5) i = .082373244 or
(6) i = 8.237%.
* * * * *

Donald V. Hammond,
Fiscal Assistant Secretary.
[FR Doc. 04-19999 Filed 9-1-04; 8:45 am]
BILLING CODE 4810-39-P