[Code of Federal Regulations]
[Title 31, Volume 2]
[Revised as of July 1, 2008]
From the U.S. Government Printing Office via GPO Access
[CITE: 31CFR356 App B]
[Page 390-403]
TITLE 31--MONEY AND FINANCE: TREASURY
CHAPTER II--FISCAL SERVICE, DEPARTMENT OF THE TREASURY
PART 356_SALE AND ISSUE OF MARKETABLE BOOK-ENTRY TREASURY BILLS, NOTES, AND
Subpart D_Miscellaneous Provisions
Sec. Appendix B to Part 356--Formulas and Tables
I. Computation of Interest on Treasury Bonds and Notes.
II. Formulas for Conversion of Fixed-Principal Security Yields to
Equivalent Prices.
III. Formulas for Conversion of Inflation-Protected Security Yields to
Equivalent Prices.
IV. Computation of Adjusted Values and Payment Amounts for Stripped
Inflation-Protected Interest Components.
V. Computation of Purchase Price, Discount Rate, and Investment Rate
(Coupon-Equivalent Yield) for Treasury Bills.
The examples in this appendix are given for illustrative purposes
only and are in no way a prediction of interest rates on any bills,
notes, or bonds issued under this part.
[[Page 391]]
In some of the following examples, we use intermediate rounding for ease
in following the calculations. In actual practice, we generally do not
round prior to determining the final result.
If you use a multi-decimal calculator, we recommend setting your
calculator to at least 13 decimals and then applying normal rounding
procedures. This should be sufficient to obtain the same final results.
However, in the case of any discrepancies, our determinations will be
final.
I. Computation of Interest on Treasury Bonds and Notes
A. Treasury Fixed-Principal Securities
1. Regular Half-Year Payment Period. We pay interest on marketable
Treasury fixed-principal securities on a semiannual basis. The regular
interest payment period is a full half-year of six calendar months.
Examples of half-year periods are: (1) February 15 to August 15, (2) May
31 to November 30, and (3) February 29 to August 31 (in a leap year).
Calculation of an interest payment for a fixed-principal note with a par
amount of $1,000 and an interest rate of 8% is made in this manner:
($1,000 x .08)/2 = $40. Specifically, a semiannual interest payment
represents one half of one year's interest, and is computed on this
basis regardless of the actual number of days in the half-year.
2. Daily Interest Decimal. We compute a daily interest decimal in
cases where an interest payment period for a fixed-principal security is
shorter or longer than six months or where accrued interest is payable
by an investor. We base the daily interest decimal on the actual number
of calendar days in the half-year or half-years involved. The number of
days in any half-year period is shown in Table 1.
Table 1
----------------------------------------------------------------------------------------------------------------
Beginning and ending days are Beginning and ending days are
1st or 15th of the months the last days of the months
listed under interest period listed under interest period
Interest period (number of days) (number of days)
---------------------------------------------------------------
Regular year Leap year Regular year Leap year
----------------------------------------------------------------------------------------------------------------
January to July................................. 181 182 181 182
February to August.............................. 181 182 184 184
March to September.............................. 184 184 183 183
April to October................................ 183 183 184 184
May to November................................. 184 184 183 183
June to December................................ 183 183 184 184
July to January................................. 184 184 184 184
August to February.............................. 184 184 181 182
September to March.............................. 181 182 182 183
October to April................................ 182 183 181 182
November to May................................. 181 182 182 183
December to June................................ 182 183 181 182
----------------------------------------------------------------------------------------------------------------
Table 2 below shows the daily interest decimals covering interest
from \1/8\% to 20% on $1,000 for one day in increments of \1/8\ of one
percent. These decimals represent \1/181\, \1/182\, \1/183\, or \1/184\
of a full semiannual interest payment, depending on which half-year is
applicable.
Table 2
[Decimal for one day's interest on $1,000 at various rates of interest, payable semiannually or on a semiannual
basis, in regular years of 365 days and in years of 366 days (to determine applicable number of days, see table
1.)]
----------------------------------------------------------------------------------------------------------------
Half-year of Half-year of Half-year of Half-year of
Rate per annum (percent) 184 days 183 days 182 days 181 days
----------------------------------------------------------------------------------------------------------------
\1/8\................................................... 0.003396739 0.003415301 0.003434066 0.003453039
\1/4\................................................... 0.006793478 0.006830601 0.006868132 0.006906077
\3/8\................................................... 0.010190217 0.010245902 0.010302198 0.010359116
\1/2\................................................... 0.013586957 0.013661202 0.013736264 0.013812155
\5/8\................................................... 0.016983696 0.017076503 0.017170330 0.017265193
\3/4\................................................... 0.020380435 0.020491803 0.020604396 0.020718232
\7/8\................................................... 0.023777174 0.023907104 0.024038462 0.024171271
1....................................................... 0.027173913 0.027322404 0.027472527 0.027624309
1\1/8\.................................................. 0.030570652 0.030737705 0.030906593 0.031077348
1\1/4\.................................................. 0.033967391 0.034153005 0.034340659 0.034530387
1\3/8\.................................................. 0.037364130 0.037568306 0.037774725 0.037983425
1\1/2\.................................................. 0.040760870 0.040983607 0.041208791 0.041436464
1\5/8\.................................................. 0.044157609 0.044398907 0.044642857 0.044889503
1\3/4\.................................................. 0.047554348 0.047814208 0.048076923 0.048342541
[[Page 392]]
1\7/8\.................................................. 0.050951087 0.051229508 0.051510989 0.051795580
2....................................................... 0.054347826 0.054644809 0.054945055 0.055248619
2\1/8\.................................................. 0.057744565 0.058060109 0.058379121 0.058701657
2\1/4\.................................................. 0.061141304 0.061475410 0.061813187 0.062154696
2\3/8\.................................................. 0.064538043 0.064890710 0.065247253 0.065607735
2\1/2\.................................................. 0.067934783 0.068306011 0.068681319 0.069060773
2\5/8\.................................................. 0.071331522 0.071721311 0.072115385 0.072513812
2\3/4\.................................................. 0.074728261 0.075136612 0.075549451 0.075966851
2\7/8\.................................................. 0.078125000 0.078551913 0.078983516 0.079419890
3....................................................... 0.081521739 0.081967213 0.082417582 0.082872928
3\1/8\.................................................. 0.084918478 0.085382514 0.085851648 0.086325967
3\1/4\.................................................. 0.088315217 0.088797814 0.089285714 0.089779006
3\3/8\.................................................. 0.091711957 0.092213115 0.092719780 0.093232044
3\1/2\.................................................. 0.095108696 0.095628415 0.096153846 0.096685083
3\5/8\.................................................. 0.098505435 0.099043716 0.099587912 0.100138122
3\3/4\.................................................. 0.101902174 0.102459016 0.103021978 0.103591160
3\7/8\.................................................. 0.105298913 0.105874317 0.106456044 0.107044199
4....................................................... 0.108695652 0.109289617 0.109890110 0.110497238
4\1/8\.................................................. 0.112092391 0.112704918 0.113324176 0.113950276
4\1/4\.................................................. 0.115489130 0.116120219 0.116758242 0.117403315
4\3/8\.................................................. 0.118885870 0.119535519 0.120192308 0.120856354
4\1/2\.................................................. 0.122282609 0.122950820 0.123626374 0.124309392
4\5/8\.................................................. 0.125679348 0.126366120 0.127060440 0.127762431
4\3/4\.................................................. 0.129076087 0.129781421 0.130494505 0.131215470
4\7/8\.................................................. 0.132472826 0.133196721 0.133928571 0.134668508
5....................................................... 0.135869565 0.136612022 0.137362637 0.138121547
5\1/8\.................................................. 0.139266304 0.140027322 0.140796703 0.141574586
5\1/4\.................................................. 0.142663043 0.143442623 0.144230769 0.145027624
5\3/8\.................................................. 0.146059783 0.146857923 0.147664835 0.148480663
5\1/2\.................................................. 0.149456522 0.150273224 0.151098901 0.151933702
5\5/8\.................................................. 0.152853261 0.153688525 0.154532967 0.155386740
5\3/4\.................................................. 0.156250000 0.157103825 0.157967033 0.158839779
5\7/8\.................................................. 0.159646739 0.160519126 0.161401099 0.162292818
6....................................................... 0.163043478 0.163934426 0.164835165 0.165745856
6\1/8\.................................................. 0.166440217 0.167349727 0.168269231 0.169198895
6\1/4\.................................................. 0.169836957 0.170765027 0.171703297 0.172651934
6\3/8\.................................................. 0.173233696 0.174180328 0.175137363 0.176104972
6\1/2\.................................................. 0.176630435 0.177595628 0.178571429 0.179558011
6\5/8\.................................................. 0.180027174 0.181010929 0.182005495 0.183011050
6\3/4\.................................................. 0.183423913 0.184426230 0.185439560 0.186464088
6\7/8\.................................................. 0.186820652 0.187841530 0.188873626 0.189917127
7....................................................... 0.190217391 0.191256831 0.192307692 0.193370166
7\1/8\.................................................. 0.193614130 0.194672131 0.195741758 0.196823204
7\1/4\.................................................. 0.197010870 0.198087432 0.199175824 0.200276243
7\3/8\.................................................. 0.200407609 0.201502732 0.202609890 0.203729282
7\1/2\.................................................. 0.203804348 0.204918033 0.206043956 0.207182320
7\5/8\.................................................. 0.207201087 0.208333333 0.209478022 0.210635359
7\3/4\.................................................. 0.210597826 0.211748634 0.212912088 0.214088398
7\7/8\.................................................. 0.213994565 0.215163934 0.216346154 0.217541436
8....................................................... 0.217391304 0.218579235 0.219780220 0.220994475
8\1/8\.................................................. 0.220788043 0.221994536 0.223214286 0.224447514
8\1/4\.................................................. 0.224184783 0.225409836 0.226648352 0.227900552
8\3/8\.................................................. 0.227581522 0.228825137 0.230082418 0.231353591
8\1/2\.................................................. 0.230978261 0.232240437 0.233516484 0.234806630
8\5/8\.................................................. 0.234375000 0.235655738 0.236950549 0.238259669
8\3/4\.................................................. 0.237771739 0.239071038 0.240384615 0.241712707
8\7/8\.................................................. 0.241168478 0.242486339 0.243818681 0.245165746
9....................................................... 0.244565217 0.245901639 0.247252747 0.248618785
9\1/8\.................................................. 0.247961957 0.249316940 0.250686813 0.252071823
9\1/4\.................................................. 0.251358696 0.252732240 0.254120879 0.255524862
9\3/8\.................................................. 0.254755435 0.256147541 0.257554945 0.258977901
9\1/2\.................................................. 0.258152174 0.259562842 0.260989011 0.262430939
9\5/8\.................................................. 0.261548913 0.262978142 0.264423077 0.265883978
9\3/4\.................................................. 0.264945652 0.266393443 0.267857143 0.269337017
9\7/8\.................................................. 0.268342391 0.269808743 0.271291209 0.272790055
10...................................................... 0.271739130 0.273224044 0.274725275 0.276243094
10\1/8\................................................. 0.275135870 0.276639344 0.278159341 0.279696133
10\1/4\................................................. 0.278532609 0.280054645 0.281593407 0.283149171
10\3/8\................................................. 0.281929348 0.283469945 0.285027473 0.286602210
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10\1/2\................................................. 0.285326087 0.286885246 0.288461538 0.290055249
10\5/8\................................................. 0.288722826 0.290300546 0.291895604 0.293508287
10\3/4\................................................. 0.292119565 0.293715847 0.295329670 0.296961326
10\7/8\................................................. 0.295516304 0.297131148 0.298763736 0.300414365
11...................................................... 0.298913043 0.300546448 0.302197802 0.303867403
11\1/8\................................................. 0.302309783 0.303961749 0.305631868 0.307320442
11\1/4\................................................. 0.305706522 0.307377049 0.309065934 0.310773481
11\3/8\................................................. 0.309103261 0.310792350 0.312500000 0.314226519
11\1/2\................................................. 0.312500000 0.314207650 0.315934066 0.317679558
11\5/8\................................................. 0.315896739 0.317622951 0.319368132 0.321132597
11\3/4\................................................. 0.319293478 0.321038251 0.322802198 0.324585635
11\7/8\................................................. 0.322690217 0.324453552 0.326236264 0.328038674
12...................................................... 0.326086957 0.327868852 0.329670330 0.331491713
12\1/8\................................................. 0.329483696 0.331284153 0.333104396 0.334944751
12\1/4\................................................. 0.332880435 0.334699454 0.336538462 0.338397790
12\3/8\................................................. 0.336277174 0.338114754 0.339972527 0.341850829
12\1/2\................................................. 0.339673913 0.341530055 0.343406593 0.345303867
12\5/8\................................................. 0.343070652 0.344945355 0.346840659 0.348756906
12\3/4\................................................. 0.346467391 0.348360656 0.350274725 0.352209945
12\7/8\................................................. 0.349864130 0.351775956 0.353708791 0.355662983
13...................................................... 0.353260870 0.355191257 0.357142857 0.359116022
13\1/8\................................................. 0.356657609 0.358606557 0.360576923 0.362569061
13\1/4\................................................. 0.360054348 0.362021858 0.364010989 0.366022099
13\3/8\................................................. 0.363451087 0.365437158 0.367445055 0.369475138
13\1/2\................................................. 0.366847826 0.368852459 0.370879121 0.372928177
13\5/8\................................................. 0.370244565 0.372267760 0.374313187 0.376381215
13\3/4\................................................. 0.373641304 0.375683060 0.377747253 0.379834254
13\7/8\................................................. 0.377038043 0.379098361 0.381181319 0.383287293
14...................................................... 0.380434783 0.382513661 0.384615385 0.386740331
14\1/8\................................................. 0.383831522 0.385928962 0.388049451 0.390193370
14\1/4\................................................. 0.387228261 0.389344262 0.391483516 0.393646409
14\3/8\................................................. 0.390625000 0.392759563 0.394917582 0.397099448
14\1/2\................................................. 0.394021739 0.396174863 0.398351648 0.400552486
14\5/8\................................................. 0.397418478 0.399590164 0.401785714 0.404005525
14\3/4\................................................. 0.400815217 0.403005464 0.405219780 0.407458564
14\7/8\................................................. 0.404211957 0.406420765 0.408653846 0.410911602
15...................................................... 0.407608696 0.409836066 0.412087912 0.414364641
15\1/8\................................................. 0.411005435 0.413251366 0.415521978 0.417817680
15\1/4\................................................. 0.414402174 0.416666667 0.418956044 0.421270718
15\3/8\................................................. 0.417798913 0.420081967 0.422390110 0.424723757
15\1/2\................................................. 0.421195652 0.423497268 0.425824176 0.428176796
15\5/8\................................................. 0.424592391 0.426912568 0.429258242 0.431629834
15\3/4\................................................. 0.427989130 0.430327869 0.432692308 0.435082873
15\7/8\................................................. 0.431385870 0.433743169 0.436126374 0.438535912
16...................................................... 0.434782609 0.437158470 0.439560440 0.441988950
16\1/8\................................................. 0.438179348 0.440573770 0.442994505 0.445441989
16\1/4\................................................. 0.441576087 0.443989071 0.446428571 0.448895028
16\3/8\................................................. 0.444972826 0.447404372 0.449862637 0.452348066
16\1/2\................................................. 0.448369565 0.450819672 0.453296703 0.455801105
16\5/8\................................................. 0.451766304 0.454234973 0.456730769 0.459254144
16\3/4\................................................. 0.455163043 0.457650273 0.460164835 0.462707182
16\7/8\................................................. 0.458559783 0.461065574 0.463598901 0.466160221
17...................................................... 0.461956522 0.464480874 0.467032967 0.469613260
17\1/8\................................................. 0.465353261 0.467896175 0.470467033 0.473066298
17\1/4\................................................. 0.468750000 0.471311475 0.473901099 0.476519337
17\3/8\................................................. 0.472146739 0.474726776 0.477335165 0.479972376
17\1/2\................................................. 0.475543478 0.478142077 0.480769231 0.483425414
17\5/8\................................................. 0.478940217 0.481557377 0.484203297 0.486878453
17\3/4\................................................. 0.482336957 0.484972678 0.487637363 0.490331492
17\7/8\................................................. 0.485733696 0.488387978 0.491071429 0.493784530
18...................................................... 0.489130435 0.491803279 0.494505495 0.497237569
18\1/8\................................................. 0.492527174 0.495218579 0.497939560 0.500690608
18\1/4\................................................. 0.495923913 0.498633880 0.501373626 0.504143646
18\3/8\................................................. 0.499320652 0.502049180 0.504807692 0.507596685
18\1/2\................................................. 0.502717391 0.505464481 0.508241758 0.511049724
18\5/8\................................................. 0.506114130 0.508879781 0.511675824 0.514502762
18\3/4\................................................. 0.509510870 0.512295082 0.515109890 0.517955801
18\7/8\................................................. 0.512907609 0.515710383 0.518543956 0.521408840
19...................................................... 0.516304348 0.519125683 0.521978022 0.524861878
[[Page 394]]
19\1/8\................................................. 0.519701087 0.522540984 0.525412088 0.528314917
19\1/4\................................................. 0.523097826 0.525956284 0.528846154 0.531767956
19\3/8\................................................. 0.526494565 0.529371585 0.532280220 0.535220994
19\1/2\................................................. 0.529891304 0.532786885 0.535714286 0.538674033
19\5/8\................................................. 0.533288043 0.536202186 0.539148352 0.542127072
19\3/4\................................................. 0.536684783 0.539617486 0.542582418 0.545580110
19\7/8\................................................. 0.540081522 0.543032787 0.546016484 0.549033149
20...................................................... 0.543478261 0.546448087 0.549450549 0.552486188
----------------------------------------------------------------------------------------------------------------
3. Short First Payment Period. In cases where the first interest
payment period for a Treasury fixed-principal security covers less than
a full half-year period (a ``short coupon''), we multiply the daily
interest decimal by the number of days from, but not including, the
issue date to, and including, the first interest payment date. This
calculation results in the amount of the interest payable per $1,000 par
amount. In cases where the par amount of securities is a multiple of
$1,000, we multiply the appropriate multiple by the unrounded interest
payment amount per $1,000 par amount.
Example
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).
4. Long First Payment Period. In cases where the first interest
payment period for a bond or note covers more than a full half-year
period (a ``long coupon''), we multiply the daily interest decimal by
the number of days from, but not including, the issue date to, and
including, the last day of the fractional period that ends one full
half-year before the interest payment date. We add that amount to the
regular interest amount for the full half-year ending on the first
interest payment date, resulting in the amount of interest payable for
$1,000 par amount. In cases where the par amount of securities is a
multiple of $1,000, the appropriate multiple should be applied to the
unrounded interest payment amount per $1,000 par amount.
Example
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).
B. Treasury Inflation-Protected Securities
1. Indexing Process. We pay interest on marketable Treasury
inflation-protected securities on a semiannual basis. We issue
inflation-protected securities with a stated rate of interest that
remains constant until maturity. Interest payments are based on the
security's inflation-adjusted principal at the time we pay interest. We
make this adjustment by multiplying the par amount of the security by
the applicable Index Ratio.
2. Index Ratio. The numerator of the Index Ratio, the Ref
CPIDate, is the index number applicable for a specific day.
The denominator of the Index Ratio is the Ref CPI applicable for the
original issue date. However, when the dated date is different from the
original issue date, the denominator is the Ref CPI applicable for the
dated date. The formula for calculating the Index Ratio is:
[GRAPHIC] [TIFF OMITTED] TR28JY04.000
Where Date = valuation date
[[Page 395]]
3. Reference CPI. The Ref CPI for the first day of any calendar
month is the CPI for the third preceding calendar month. For example,
the Ref CPI applicable to April 1 in any year is the CPI for January,
which is reported in February. We determine the Ref CPI for any other
day of a month by a linear interpolation between the Ref CPI applicable
to the first day of the month in which the day falls (in the example,
January) and the Ref CPI applicable to the first day of the next month
(in the example, February). For interpolation purposes, we truncate
calculations with regard to the Ref CPI and the Index Ratio for a
specific date to six decimal places, and round to five decimal places.
Therefore the Ref CPI and the Index Ratio for a particular date will
be expressed to five decimal places.
(i) The formula for the Ref CPI for a specific date is:
[GRAPHIC] [TIFF OMITTED] TR28JY04.001
Where Date = valuation date
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:
[GRAPHIC] [TIFF OMITTED] TR28JY04.002
where D = 30, t = 15
Ref CPIApril 1, 1996 = 154.40, the non-seasonally adjusted
CPI-U for January 1996.
Ref CPIMay 1, 1996 = 154.90, the non-seasonally adjusted CPI-
U for February 1996.
(iii) Putting these values in the equation in paragraph (ii) above:
[GRAPHIC] [TIFF OMITTED] TR28JY04.003
This value truncated to six decimals is 154.633333; rounded to five
decimals it is 154.63333.
(iv) To calculate the Index Ratio for April 16, 1996, for an
inflation-protected security issued on April 15, 1996, the Ref
CPIApril 16, 1996 must first be calculated. Using the same
values in the equation above except that t=16, the Ref
CPIApril 16, 1996 is 154.65000.
The Index Ratio for April 16, 1996 is:
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.
4. Index Contingencies.
(i) If a previously reported CPI is revised, we will continue to use
the previously reported (unrevised) CPI in calculating the principal
value and interest payments.
If the CPI is rebased to a different year, we will continue to use
the CPI based on the base reference period in effect when the security
was first issued, as long as that CPI continues to be published.
(ii) We will replace the CPI with an appropriate alternative index
if, while an inflation-protected security is outstanding, the applicable
CPI is:
Discontinued,
In the judgment of the Secretary, fundamentally
altered in a manner materially
[[Page 396]]
adverse to the interests of an investor in the security, or
In the judgment of the Secretary, altered by
legislation or Executive Order in a manner materially adverse to the
interests of an investor in the security.
(iii) If we decide to substitute an alternative index we will
consult with the Bureau of Labor Statistics or any successor agency. We
will then notify the public of the substitute index and how we will
apply it. Determinations of the Secretary in this regard will be final.
(iv) If the CPI for a particular month is not reported by the last
day of the following month, we will announce an index number based on
the last available twelve-month change in the CPI. We will base our
calculations of our payment obligations that rely on that month's CPI on
the index number we announce.
(a) For example, if the CPI for month M is not reported timely, the
formula for calculating the index number to be used is:
[GRAPHIC] [TIFF OMITTED] TR28JY04.004
(b) Generalizing for the last reported CPI issued N months prior to
month M:
[GRAPHIC] [TIFF OMITTED] TR28JY04.005
(c) If it is necessary to use these formulas to calculate an index
number, we will use that number for all subsequent calculations that
rely on the month's index number. We will not replace it with the actual
CPI when it is reported, except for use in the above formulas. If it
becomes necessary to use the above formulas to derive an index number,
we will use the last CPI that has been reported to calculate CPI numbers
for months for which the CPI has not been reported timely.
5. Computation of Interest for a Regular Half-Year Payment Period.
Interest on marketable Treasury inflation-protected securities is
payable on a semiannual basis. The regular interest payment period is a
full half-year or six calendar months. Examples of half-year periods are
January 15 to July 15, and April 15 to October 15. An interest payment
will be a fixed percentage of the value of the inflation-adjusted
principal, in current dollars, for the date on which it is paid. We will
calculate interest payments by multiplying one-half of the specified
annual interest rate for the inflation-protected securities by the
inflation-adjusted principal for the interest payment date.
Specifically, we compute a semiannual interest payment on the basis
of one-half of one year's interest regardless of the actual number of
days in the half-year.
Example
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) x $100,000, or $101,341. This amount was
multiplied by .03875/2, or .019375, resulting in a payment of $1,963.48.
C. Accrued Interest
1. You will have to pay accrued interest on a Treasury bond or note
when interest accrues prior to the issue date of the security. Because
you receive a full interest payment despite having held the security for
only a portion of the interest payment period, you must compensate us
through the payment of accrued interest at settlement.
2. For a Treasury fixed-principal security, if accrued interest
covers a fractional portion of a full half-year period, the number of
days in the full half-year period and the stated interest rate will
determine the daily interest decimal to use in computing the accrued
interest. We multiply the decimal by the number of days for which
interest has accrued.
3. If a reopened bond or note has a long first interest payment
period (a ``long coupon''), and the dated date for the reopened issue is
less than six full months before the first interest payment, the accrued
interest will fall into two separate half-year periods. A separate daily
interest decimal must be multiplied by the respective number of days in
each half-year period during which interest has accrued.
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 Fixed-Principal 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
[[Page 397]]
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:
A 10\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).
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 = (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[rceil] = n. Furthermore,
when i = 0, an[rceil] cannot be calculated using the formula:
(1 - v\n\)/(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.
[[Page 398]]
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[rceil] + 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 .658589
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.
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[min]s[sec])(C/2) + C/2 + (C/
2)an[rceil] + 100v\n\.
Where:
A = AI[min] + AI,
AI[min] = (r[min]/s[sec])(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[min] = number of days in the fractional portion (or ``initial short
period'') of the first interest payment period,
s[sec] = 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[min] = 44 (July 2 to August 15, 1985).
s[sec] = 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[min] = (44 / 181)(10.75 / 2), or 1.306630.
AI = [(184 - 103) / 184](10.75 / 2), or 2.366168.
[[Page 399]]
A = AI[min] + AI, or 3.672798.
Resolution:
(P + A)[1 + (r/s)(i/2)] = (r[min]/s[sec])(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[min]/s)(C/2) + (C/2)an[rceil] +
100v \n\.
Where:
A = [(r[min] - r)/s](C/2) and
r[min] = 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[min] = 183 (May 16, 1983, to November 15, 1983).
v \n\ = 1/[(1 + .1053/2)]\15\, or .4631696332.
an[rceil] = (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[min]/s)(C/2) + (C/2)an[rceil] +
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[min]/s)(C/2)]v + (C/2)an[rceil]
+ 100v \n\.
Where:
A = [(r[min] - r)/s](C/2), and
r = number of days from the reopening date to the end of the short
period.
r[min] = 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[min] = 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[rceil] = (1 - .5635631040)/.04895, or 8.9159733613.
A = [(61 - 30)/183](9.75/2), or .825820.
Resolution:
(P + A)[1 + (r/s)(i/2)] = [(r[min]/s)(C/2)]v + (C/2)an[rceil]
+ 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
[[Page 400]]
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.
an[rceil] = (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[rceil] = n. Furthermore,
when i = 0, an[rceil] cannot be calculated using the formula:
(1 - v \n\)/(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.
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.
v\n\ = 1/(1 + i/2)\n\ = 1/(1 + .03898/2)\19\ = 0.692984572.
an[rceil] = (1 - v\n\)/(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
[[Page 401]]
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.
v\n\ = 1/(1 + i/2)\n\ = 1/(1 + .0365/2)\18\ = 0.722138438.
an[rceil] = (1-v\n\)/(i/2) = (1 - 0.722138438)/(.0365/2) =
15.225291068.
Formula:
[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.
IV. Computation of Adjusted Values and Payment Amounts for Stripped
Inflation-Protected Interest Components
Note: Valuing an interest component stripped from an inflation-
protected security at its adjusted value enables this interest component
to be interchangeable (fungible) with other interest components that
have the same maturity date, regardless of the underlying inflation-
protected security from which the interest components were stripped. The
adjusted value provides for fungibility of these various interest
components when buying, selling, or transferring them or when
reconstituting an inflation-protected security.
Definitions:
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
Formulas:
AV = Par(C/2)(100/Ref CPIIssueDate) (rounded to 2 decimals
with no intermediate rounding)
PA = AV(Ref CPIDate/100) (rounded to 2 decimals with no
intermediate rounding)
Example:
A 10-year inflation-protected note paying 3\7/8\% interest was issued on
January 15, 1999, with the second interest payment on January 15, 2000.
The Ref CPI of January
[[Page 402]]
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.
Definitions:
c = .03875
Par = $1,000,000
Ref CPIIssueDate = 164.00000
Ref CPIDate = 168.24516
Resolution:
For a par amount of $1 million, the adjusted value of each stripped
interest component was $1,000,000(.03875/2)(100/164.00000), or
$11,814.02 (no intermediate rounding).
For an interest component that matured on January 15, 2000, the payment
amount was $11,814.02 (168.24516/100), or $19,876.52 (no intermediate
rounding).
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
[[Page 403]]
$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.
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%.
[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]