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.

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 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 govsecreg@bpd.treas.gov.

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. 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.

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. 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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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. 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. 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.

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, § 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 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 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, § 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, § 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.

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, §§ 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. 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.

Accordingly, § 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. Also, § 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.

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

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)

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.

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

3. Section 356.20 is amended by revising paragraphs (b) and (c) to read as follows:

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

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 61/8 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.

vⁿ = 1 / [1 + (i/2)]ⁿ = present value of 1 due at the end of n periods.

a_n⌉ = (1−vⁿ) / (i/2) = v + v² + v³ + ... vⁿ = present value of 1 per period for n periods.

Special Case: If i = 0, then a_n⌉ = n. Furthermore, when i = 0, a_n⌉ cannot be calculated using the formula: (1 − v_n)/(i/2). In the special case where i = 0, a_n⌉ must be calculated as the summation of the individual present values (i.e., v + v² + v³ + ... + vⁿ). Using the summation method will always confirm that a_n⌉ = 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)a_n⌉ + 100vⁿ.

Example:

For an 83/4% 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.)

vⁿ = 1 / [(1 + .0884 / 2)]⁵⁹, or .0779403508.

a_n⌉ = (1 − .0779403508) / .0442, or 20.8610780353.

Resolution:

P[1 + (r/s)(i/2)] = (C/2)(r/s) + (C/2)a_n⌉ + 100vⁿ  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)a_n⌉ + 100vⁿ.

Example:

For an 81/2% 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).

vⁿ = 1 / [(1 + .0859 / 2)]³, or .8814740565.

a_n⌉ = (1 − .8814740565) / .04295, or 2.7596261590.

Resolution:

P[1 + (r/s)(i/2)] = (C/2)(r/s) + (C/2)a_n⌉ + 100vⁿ 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)a_n⌉ + 100vⁿ.

Example:

For an 81/2% 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.

vⁿ = 1 / (1 + .0853/2)¹⁰, or .6585890783.

a_n⌉ = (1−.658589)/.04265, or 8.0049454082.

Resolution:

P[1 + (r/s)(i/2)] = [(C/2)(r/s)]v + (C/2)a_n⌉ + 100vⁿ 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)a_n⌉ + 100vⁿ.

Where:

A = [(s−r)/s](C/2).

Example:

For a 91/2% 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.

r = 167 (November 29, 1985, to May 15, 1986).

s = 181 (November 15, 1985, to May 15, 1986).

vⁿ = 1 / [(1 + .0954/2)]¹⁹, or .4125703996.

a_n⌉ = (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)a_n⌉ + 100vⁿ 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)a_n⌉ + 100vⁿ.

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 103/4% 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).

vⁿ = 1 / [(1 + .1047 / 2)]³⁹, or .1366947986.

a_n⌉ = (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)a_n⌉ + 100vⁿ 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)a_n⌉ + 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 101/2% 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.

a_n⌉ = (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)a_n⌉ + 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)a_n⌉ + 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 93/4% 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.

a_n⌉ = (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)a_n⌉ + 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).

P_adj = inflation adjusted price; P × Index Ratio_Date.

A = unadjusted accrued interest per $100 original principal.

A_adj = inflation adjusted accrued interest; A× Index Ratio_Date.

SA = settlement amount including accrued interest in current dollars per $100 original principal; P_adj + A_adj.

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 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.

a_n⌉ = (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 a_n⌉ = n. Furthermore, when i = 0, a_n⌉ cannot be calculated using the formula: (1 − v ⁿ)/(i/2). In the special case where i = 0, a_n⌉ 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 a_n⌉ = 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.

CPI_M = CPI reported for the calendar month M by the Bureau of Labor Statistics.

Ref CPI_M = 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 CPI_{April 1} is the CPI_January.

Ref CPI_M+1 = reference CPI for the first day of the calendar month immediately following Date.

Ref CPI_Date = Ref CPI_M − [(t − 1)/D][Ref CPI_M+1-Ref CPI_M].

Index Ratio_Date = Ref CPI_Date / Ref CPI_IssueDate.

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

A. For inflation-indexed securities with a regular first interest payment period:

Formulas:

P_adj = P × Index Ratio_Date.

A = [(s−r)/s] × (C/2).

A_adj = A × Index Ratio_Date.

SA = P_adj + A_adj.

Index Ratio_Date = Ref CPI_Date/Ref CPI_IssueDate.

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 37/8% 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 CPI_Date = 164.

Ref CPI_IssueDate = 164.

Resolution:

Index Ratio_Date = Ref CPI_Date / Ref CPI_IssueDate = 164/164 = 1.

A = [(181 − 181)/181] × 3.875/2 = 0.

A_adj = 0 × 1 = 0.

vⁿ = 1/(1 + i/2)ⁿ = 1/(1 + .03898/2)¹⁹ = 0.692984572.

a _n ⌉ = (1 − vⁿ)/(i/2) = (1-0.692984572) / (.03898/2) = 15.752459107.

Formula:

P = 99.811030.

P_adj = P × Index Ratio_Date.

P_adj = 99.811030 × 1 = 99.811030.

SA = P_adj × A_adj.

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 × (132/120), or $11,000.

Formulas:

P_adj = P × Index Ratio_Date.

A = [(s-r)/s] × (C/2).

A_adj = A × Index Ratio_Date.

SA = P_adj + A_adj.

Index Ratio_Date = Ref CPI_Date/Ref CPI_IssueDate.

Example:

We issued a 35/8% 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 CPI_Date = 163.29032.

Ref CPI_IssueDate = 161.55484.

Resolution:

Index Ratio_Date = Ref CPI_Date/Ref CPI_IssueDate = 163.29032/161.55484 = 1.01074.

vⁿ = 1/(1 + i/2)ⁿ = 1/(1 + .0365/2)¹⁸ = 0.722138438.

a_n = ⌉ 1−vⁿ)/(i/2) = (1 − 0.722138438)/(.0365/2) = 15.225291068.

Formula:

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P = 100.703267 − 0.906250.

P = 99.797017.

P_adj = P × Index Ratio_Date.

P_adj = 99.797017 × 1.01074 = 100.86883696.

P_adj = 100.868837.

A = [(184−92)/184] × 3.625/2 = 0.906250.

A_adj = A × Index Ratio_Date.

A_adj = 0.906250 × 1.01074 = 0.91598313.

A_adj = 0.915983.

SA = P_adj + A_adj = 100.868837 + 0.915983.

SA = 101.784820.

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

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:

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:

(2) d = [.04065433 × 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:

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:

(2) i = [.004425 × 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:

and solving for “i” produces:

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.

b = 364 / 365, or .997260274.

a = (364 / 730) − .25, or .248630137.

c = (92.265 − 100) / 92.265, or −.083834607.

Resolution:

(3) i = (−.997260274 + 1.038221216) / .497260274.

(4) i = .040960942 / .497260274.

(5) i = .082373244 or

(6) i = 8.237%.