Source: https://www.law.cornell.edu/cfr/text/31/appendix-B_to_part_356
Timestamp: 2017-02-22 06:28:19
Document Index: 199140556

Matched Legal Cases: ['art 356', 'art 356', 'art_356', 'art 356', 'art 356', 'art 356']

31 CFR Appendix B to Part 356, Formulas and Tables | US Law | LII / Legal Information Institute
CFR › Title 31 › Subtitle B › Chapter II › Subchapter A › Part 356 › Subpart D › Appendix B_to_part_356 31 CFR Appendix B to Part 356, Formulas and Tables
Appendix B to Part 356 - Formulas and Tables I. Computation of Interest on Treasury Bonds and Notes.
Table 1 Interest period Beginning and ending days are 1st or 15th of the months listed under interest period
(number of days) Beginning and ending days are the last days of the months listed under interest period
(number of days) Regular year Leap year Regular year Leap year January to July
182 February to August
184 March to September
183 April to October
184 May to November
183 June to December
184 July to January
184 August to February
182 September to March
183 October to April
182 November to May
183 December to June
A 2-year note paying 8
3/8% interest was issued on July 2, 1990, with the first interest payment on December 31, 1990. The number of days in the full half-year period of June 30 to December 31, 1990, was 184 (See Table 1.). The number of days for which interest actually accrued was 182 (not including July 2, but including December 31). The daily interest decimal, $0.227581522 (See Table 2, line for 8
3/8%, under the column for half-year of 184 days.), was multiplied by 182, resulting in a payment of $41.419837004 per $1,000. For $20,000 of these notes, $41.419837004 would be multiplied by 20, resulting in a payment of $828.39674008 ($828.40).
A 5-year 2-month note paying 7
7/8% interest was issued on December 3, 1990, with the first interest payment due on August 15, 1991. Interest for the regular half-year portion of the payment was computed to be $39.375 per $1,000 par amount. The fractional portion of the payment, from December 3 to February 15, fell in a 184-day half-year (August 15, 1990, to February 15, 1991). Accordingly, the daily interest decimal for 7
7/8% was $0.213994565. This decimal, multiplied by 74 (the number of days from but not including December 3, 1990, to and including February 15), resulted in interest for the fractional portion of $15.835597810. When added to $39.375 (the normal interest payment portion ending on August 15, 1991), this produced a first interest payment of $55.210597810, or $55.21 per $1,000 par amount. For $7,000 par amount of these notes, $55.210597810 would be multiplied by 7, resulting in an interest payment of $386.474184670 ($386.47).
D = the number of days in the month in which Date falls t = the calendar day corresponding to Date CPIM = CPI reported for the calendar month M by the Bureau of Labor Statistics Ref CPIM = Ref CPI for the first day of the calendar month in which Date falls, e.g., Ref CPIApril1 is the CPIJanuary
Ref CPIM 1 = Ref CPI for the first day of the calendar month immediately following Date (ii) For example, the Ref CPI for April 15, 1996 is calculated as follows:
Index RatioApril 16, 1996 = 154.65000/154.63333 = 1.000107803. This value truncated to six decimals is 1.000107; rounded to five decimals it is 1.00011.
A 10-year inflation-protected note paying 3
7/8% interest was issued on January 15, 1999, with the first interest payment on July 15, 1999. The Ref CPI on January 15, 1999 (Ref CPIIssueDate) was 164, and the Ref CPI on July 15, 1999 (Ref CPIDate) was 166.2. For a par amount of $100,000, the inflation-adjusted principal on July 15, 1999, was (166.2/164) × $100,000, or $101,341. This amount was multiplied by .03875/2, or .019375, resulting in a payment of $1,963.48.
4. We round all accrued interest computations to five decimal places for a $1,000 par amount, using normal rounding procedures. We calculate accrued interest for a par amount of securities greater than $1,000 by applying the appropriate multiple to accrued interest payable for a $1,000 par amount, rounded to five decimal places. We calculate accrued interest for a par amount of securities less than $1,000 by applying the appropriate fraction to accrued interest payable for a $1,000 par amount, rounded to five decimal places. 5. For an inflation-protected security, we calculate accrued interest as shown in section III, paragraphs A and B of this appendix.
Examples - (1)Treasury Non-indexed Securities - (i) Involving One Half-Year: A note paying interest at a rate of 6
3/4%, originally issued on May 15, 2000, as a 5-year note with a first interest payment date of November 15, 2000, was reopened as a 4-year 9-month note on August 15, 2000. Interest had accrued for 92 days, from May 15 to August 15. The regular interest period from May 15 to November 15, 2000, covered 184 days. Accordingly, the daily interest decimal, $0.183423913, multiplied by 92, resulted in accrued interest payable of $16.874999996, or $16.87500, for each $1,000 note purchased. If the notes have a par amount of $150,000, then 150 is multiplied by $16.87500, resulting in an amount payable of $2,531.25.
(2)Involving Two Half-Years:
3/4% bond, originally issued on July 2, 1985, as a 20-year 1-month bond, with a first interest payment date of February 15, 1986, was reopened as a 19-year 10-month bond on November 4, 1985. Interest had accrued for 44 days, from July 2 to August 15, 1985, during a 181-day half-year (February 15 to August 15); and for 81 days, from August 15 to November 4, during a 184-day half-year (August 15, 1985, to February 15, 1986). Accordingly, $0.296961326 was multiplied by 44, and $0.292119565 was multiplied by 81, resulting in products of $13.066298344 and $23.661684765 which, added together, resulted in accrued interest payable of $36.727983109, or $36.72798, for each $1,000 bond purchased. If the bonds have a par amount of $11,000, then 11 is multiplied by $36.72798, resulting in an amount payable of $404.00778 ($404.01).
C = the regular annual interest per $100, payable semiannually, e.g., 6.125 (the decimal equivalent of a 6
1/8% interest rate).
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
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
n). Using the summation method will always confirm that an⌉ = n when i = 0.
P[1 (r/s)(i/2)] = (C/2)(r/s) (C/2)an⌉ 100v
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%.
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.
(1) P[1 .0442] = 4.375 91.2672164044 7.7940350840.
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%.
n = 1 / [(1 .0859 / 2)]
3, or .8814740565.
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.
P[1 (r/s)(i/2)] = [(C/2)(r/s)]v (C/2)an⌉ 100v
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:
n = 1 / (1 .0853/2)
10, or .658589
Resolution: P[1 (r/s)(i/2)] = [(C/2)(r/s)]v (C/2)an⌉ 100v
(1) P[1 .017672652] = 1.6890133062 34.0210179850 65.8589078339.
(2)For new non-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.
(P A)[1 (r/s)(i/2)] = C/2 (C/2)an⌉ 100v
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).
n = 1 / [(1 .0954/2)]
19, or .4125703996.
Resolution: (P A)[1 (r/s)(i/2)] = C/2 (C/2)an⌉ 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.
(P A)[1 (r/s)(i/2)] = (r′s″)(C/2) C 2 (C/2)an⌉ 100v
A = AI′ AI, AI′ = (r′/s″)(C/2), AI = [(s−r) / s](C/2), and
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.
n = 1 / [(1 .1047 / 2)]
39, or .1366947986.
Resolution: (P A)[1 (r/s)(i/2)] = (r′/s″)(C/2) C/2 (C/2)an⌉ 100v
(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.
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.
v n = 1/[(1 .1053/2)]
15, or .4631696332.
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.
v n = [1 / (1 .0979/2)]
12, or .5635631040.
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.
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.
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:
n = 1/(1 i/2)
n = 1/(1 .03898/2)
19 = 0.692984572.
an⌉ = (1 − v
n)/(i/2) = (1-0.692984572) / (.03898/2) = 15.752459107.
(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.
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.
n = 1/(1 .0365/2)
18 = 0.722138438.
an⌉ = (1−v
n)/(i/2) = (1 − 0.722138438)/(.0365/2) = 15.225291068.
Table 1 - 13-Week Bill Auction Data
99.975986
0.095022819%
Table 2 - Payment Dates
Issue Date: T
0 = 7/31/2012
1st Interest Date: T
1 = 10/31/2012
0 = 92
2nd Interest Date: T
2 = 1/31/2013
3rd Interest Date: T
3 = 4/30/2013
2 = 89
4th Interest Date: T
4 = 7/31/2013
4 − T
3 = 92
5th Interest Date: T
5 = 10/31/2013
5 − T
4 = 92
6th Interest Date: T
6 = 1/31/2014
6 − T
7th Interest Date: T
7 = 4/30/2014
7 − T
8th Interest & Maturity Dates: T
8 = 7/31/2014
8 − T
7 = 92
ai = 100 × max(r s,0)/360
Ai = ai × (Ti − Ti−1) 100 × 1{i = 8}
a1 = 100 × max(0.00095022819 0.00120,0)/360 = 0.000597286
A8 = 92 × 0.000597286 100 = 100.054950312
Bi = 1 (r m) × (Ti − Ti − 1)/360
B3 = 1 (0.00095022819 0.00120) × 89/360 = 1.000531584.
Table 3 - Projected Cash Flows and Compound Factors
0.000597286
0.054950312
1.000549503
0.053158454
1.000531584
100.054950312
Ai = ai × (Ti−Ti− 1) 100 × 1{i = 8}
ai Represents the daily projected interest, for a $100 par value, that will accrue between the future interest payment dates Ti− 1 and Ti where i = 1,2, . . . ,8. ai's are computed using the spread s = − 0.150%, and the fixed index rate of r = 0.095022819% applicable to the issue date (7/31/2012). For example:
A8 = 92 × 0.000000000 100 = 100.000000000
Bi = 1 (r m) × (Ti-Ti−1)/360
B3 = 1 (0.00095022819−0.00150) × 89/360 = 0.999864084.
0.999859503
0.999864084
0.095022819
99.972194
0.110030595
99.974722
0.100025284
99.973167
0.105028183
99.973458
0.105027876
Table 2 - Applicable Index Rate
AI = 1 × 100 × max (0.00095022819 0.00120,0)/360
6 × 100 × max (0.00110030595 0.00120,0)/360
7 × 100 × max (0.00100025284 0.00120,0)/360
7 × 100 × max (0.00110030595 0.00120,0)/360
7 × 100 × max (0.00105028183 0.00120,0)/360
3 × 100 × max (0.00105027876 0.00120,0)/360
6 × 0.000638974
7 × 0.000611181
7 × 0.000638974
7 × 0.000625078
3 × 0.000625077
AI = 0.000597286 0.003833844 0.004278267 0.004472818 0.004375546 0.001875231
Table 3 - Payment Dates
Original Issue Date: T
−1 = 7/31/2012
New Issue Date: T
0 = 8/31/2012
0 = 61
ai = 100 × max(r s, 0)/360
ai = 100 × max(0.00105027876 0.00120,0)/360 = 0.000625077
A8 = 92 × 0.000625077 100 = 100.057507084
Bi = 1 (r m) × (Ti − Ti-1)/360
B3 = 1 (0.00105027876 0.00100) × 89/360 = 1.000506874
Table 4 - Projected Cash Flows and Compound Factors
0.000625077
0.038129697
1.000347408
0.057507084
1.000523960
0.055631853
1.000506874
100.057507084
IP1 = 92 × [100 × max(0.00095022819 0.00120,0)/360]
IP7 = 89 × [100 × max(0.00095022819 0.00120,0)/360]
Table 3 - Projected Interest Payments
Table 4 - 13-Week Bill Auction Data
Table 5 - Applicable Index Rate
Table 6 - Payment Dates
1 = 7/31/2012
IP1 = 0.019432992 61 × [100 × max (0.00105027876 0.00120,0)/360]
IP1 = 0.019432992 61 × 0.000625077
IP1 = 0.019432992 0.038129697 = 0.057562689
IP8 = 92 × [100 × max (0.00105027876 0.00120,0)/360]
Table 7 - Projected Interest Payments
0.057562689
99.993681
0.025001580%
AI = 3 × 100 × max (0.00025001580 0.01000,0)/360
Dated Date: = T
−1 = 12/31/2011
0 = 1/3/2012
1 = 3/31/2012
2 = 6/30/2012
3 = 9/30/2012
4 = 12/31/2012
5 = 3/31/2013
6 = 6/30/2013
7 = 9/30/2013
6 = 92
8 = 12/31/2013
a1 = 100 × max(0.00025001580 0.01000,0)/360 = 0.002847227
A8 = 92 × 0.002847227 100 = 100.261944884
Bi = 1 (r m) × (Ti − Ti−1)/360
B3 = 1 (0.00025001580 0.01000) × 92/360 = 1.002619448
0.002847227
0.250555976
1.002505559
0.259097657
1.002590976
0.261944884
1.002619448
0.256250430
1.002562504
100.261944884
V. Computation of Adjusted Values and Payment Amounts for Stripped Inflation-Protected Interest Components Note:
c = C/100 = the regular annual interest rate, payable semiannually, e.g., .03625 (the decimal equivalent of a 3
5/8% interest rate) Par = par amount of the security to be stripped Ref CPIIssueDate = reference CPI for the original issue date (or dated date, when the dated date is different from the original issue date) of the underlying (unstripped) security Ref CPIDate = reference CPI for the maturity date of the interest component AV = adjusted value of the interest component PA = payment amount at maturity by Treasury
7/8% interest was issued on January 15, 1999, with the second interest payment on January 15, 2000. The Ref CPI of January 15, 1999 (Ref CPIIssueDate) was 164.00000, and the Ref CPI on January 15, 2000 (Ref CPIDate) was 168.24516. Calculate the adjusted value and the payment amount at maturity of the interest component.
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:
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:
2. For bills of more than one half-year to maturity: Formula:
ax 2 bx c = 0. Therefore, rewriting the bill formula in the quadratic equation form gives:
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:
(3) i = (−.997260274 1.038221216) / .497260274.
[ 69 FR 45202, July 28, 2004, as amended at 69 FR 52967, Aug. 30, 2004; 69 FR 53622, Sept. 2, 2004; 73 FR 14939, Mar. 20, 2008; 78 FR 46428, 46430, July 31, 2013; 78 FR 50335, Aug. 19, 2013; 78 FR 52857, Aug. 27, 2013; 78 FR 59228-59230, Sept. 26, 2013; 81 FR 43070, July 1, 2016]
At 78 FR 59228-59230, Sept. 26, 2013, appendix B to part 356 was amended; however, portions of the amendment could not be incorporated due to inaccurate amendatory instructions.