Pt. 356, App. B
Appendix B to Part
356—Formulas and Tables
I. Computation of Interest on Treasury Bonds and Notes.
II. Formulas for Conversion of FixedPrincipal Security Yields to Equivalent Prices.
III. Formulas for Conversion of InflationProtected Security Yields to Equivalent Prices.
IV. Computation of Adjusted Values and Payment Amounts for Stripped InflationProtected Interest Components.
V. Computation of Purchase Price, Discount Rate, and Investment Rate (CouponEquivalent 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. 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 multidecimal 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 FixedPrincipal Securities
1. Regular HalfYear Payment Period. We pay interest on marketable Treasury fixedprincipal securities on a semiannual basis. The regular interest payment period is a full halfyear of six calendar months. Examples of halfyear 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 fixedprincipal note with a par amount of $1,000 and an interest rate of 8% is made in this manner: ($1,000 × .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 halfyear.
2. Daily Interest Decimal. We compute a daily interest decimal in cases where an interest payment period for a fixedprincipal 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 halfyear or halfyears involved. The number of days in any halfyear period is shown in Table 1.
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 
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 halfyear 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.)]
Rate per annum (percent) 
Halfyear of 184 days 
Halfyear of 183 days 
Halfyear of 182 days 
Halfyear of 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 
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 
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 
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 fixedprincipal security covers less than a full halfyear 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 2year note paying 83/8% interest was issued on July 2, 1990, with the first interest payment on December 31, 1990. The number of days in the full halfyear 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 83/8%, under the column for halfyear 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 halfyear 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 halfyear before the interest payment date. We add that amount to the regular interest amount for the full halfyear 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 5year 2month note paying 77/8% interest was issued on December 3, 1990, with the first interest payment due on August 15, 1991. Interest for the regular halfyear 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 184day halfyear (August 15, 1990, to February 15, 1991). Accordingly, the daily interest decimal for 77/8% 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 InflationProtected Securities
1. Indexing Process. We pay interest on marketable Treasury inflationprotected securities on a semiannual basis. We issue inflationprotected securities with a stated rate of interest that remains constant until maturity. Interest payments are based on the security's inflationadjusted 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:
Where Date = valuation date
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:
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:
where D = 30, t = 15
Ref CPIApril 1, 1996 = 154.40, the nonseasonally adjusted CPIU for January 1996.
Ref CPIMay 1, 1996 = 154.90, the nonseasonally adjusted CPIU for February 1996.
(iii) Putting these values in the equation in paragraph (ii) above:
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 inflationprotected 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 inflationprotected security is outstanding, the applicable CPI is:
• Discontinued,
• In the judgment of the Secretary, fundamentally altered in a manner materially 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 twelvemonth 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:
(b) Generalizing for the last reported CPI issued N months prior to month M:
(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 HalfYear Payment Period. Interest on marketable Treasury inflationprotected securities is payable on a semiannual basis. The regular interest payment period is a full halfyear or six calendar months. Examples of halfyear 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 inflationadjusted principal, in current dollars, for the date on which it is paid. We will calculate interest payments by multiplying onehalf of the specified annual interest rate for the inflationprotected securities by the inflationadjusted principal for the interest payment date.
Specifically, we compute a semiannual interest payment on the basis of onehalf of one year's interest regardless of the actual number of days in the halfyear.
Example
A 10year inflationprotected note paying 37/8% 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 inflationadjusted 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.
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 fixedprincipal security, if accrued interest covers a fractional portion of a full halfyear period, the number of days in the full halfyear 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 halfyear periods. A separate daily interest decimal must be multiplied by the respective number of days in each halfyear 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 inflationprotected security, we calculate accrued interest as shown in section III, paragraphs A and B of this appendix.
Examples—(1) Treasury FixedPrincipal Securities—(i) Involving One HalfYear: A note paying interest at a rate of 63/4%, originally issued on May 15, 2000, as a 5year note with a first interest payment date of November 15, 2000, was reopened as a 4year 9month 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 HalfYears:
A 103/4% bond, originally issued on July 2, 1985, as a 20year 1month bond, with a first interest payment date of February 15, 1986, was reopened as a 19year 10month bond on November 4, 1985. Interest had accrued for 44 days, from July 2 to August 15, 1985, during a 181day halfyear (February 15 to August 15); and for 81 days, from August 15 to November 4, during a 184day halfyear (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 FixedPrincipal 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.
v^{n} = 1 / [1 (i/2)] ^{n} = present value of 1 due at the end of n periods.
an = (1 − v^{n}) / (i/2) = v v^{2} v^{3} ... v^{n} = present value of 1 per period for n periods
Special Case: If i = 0, then an⌉ = n. Furthermore, when i = 0, an⌉ cannot be calculated using the formula: (1 − v^{n})/(i/2). In the special case where i = 0, an⌉ must be calculated as the summation of the individual present values (i.e., v v^{2} v^{3} ... v^{n}). Using the summation method will always confirm that an⌉ = n when i = 0.
A = accrued interest.
A. For fixedprincipal securities with a regular first interest payment period:
Formula:
P[1 (r/s)(i/2)] = (C/2)(r/s) (C/2)an⌉ 100v^{n}.
Example:
For an 83/4% 30year 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⌉ = (1 − .0779403508) / .0442, or 20.8610780353.
Resolution:
P[1 (r/s)(i/2)] = (C/2)(r/s) (C/2)an⌉ 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 fixedprincipal securities with a short first interest payment period:
Formula:
P[1 (r/s)(i/2)] = (C/2)(r/s) (C/2)an⌉ 100v^{n}.
Example:
For an 81/2% 2year 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⌉ = (1 − .8814740565) / .04295, or 2.7596261590.
Resolution:
P[1 (r/s)(i/2)] = (C/2)(r/s) (C/2)an⌉ 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 fixedprincipal securities with a long first interest payment period:
Formula:
P[1 (r/s)(i/2)] = [(C/2)(r/s)]v (C/2)an⌉ 100v^{n}.
Example:
For an 81/2% 5year 2month 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⌉ = (1−.658589)/.04265, or 8.0049454082.
Resolution:
P[1 (r/s)(i/2)] = [(C/2)(r/s)]v (C/2)an⌉ 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 fixedprincipal securities reopened during a regular interest period where the purchase price includes predetermined accrued interest.
(2) For new fixedprincipal 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⌉ 100v^{n}.
Where:
A = [(s−r)/s](C/2).
Example:
For a 91/2% 10year 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⌉ = (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⌉ 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 fixedprincipal securities reopened during the regular portion of a long first payment period:
Formula:
(P A)[1 (r/s)(i/2)] = (r′s″)(C/2) C/2 (C/2)an⌉ 100v^{n}.
Where:
A = AI′ AI,
AI′ = (r′/s″)(C/2),
AI = [(s−r) / s](C/2), and
r = number of days from the reopening date to the first interest payment date,
s = number of days in the semiannual period for the regular portion of the first interest payment period,
r′ = number of days in the fractional portion (or “initial short period”) of the first interest payment period,
s″ = number of days in the semiannual period ending with the commencement date of the regular portion of the first interest payment period.
Example:
A 103/4% 19year 9month bond due August 15, 2005, is issued on July 2, 1985, and reopened on November 4, 1985, with interest payments on February 15 and August 15 (first payment on February 15, 1986), solve for the price per 100 (P) at a yield of 10.47%. Accrued interest is calculated from July 2 to November 4.
Definitions:
C = 10.75.
i = .1047.
n = 39.
r = 103 (November 4, 1985, to February 15, 1986).
s = 184 (August 15, 1985, to February 15, 1986).
r′ = 44 (July 2 to August 15, 1985).
s″ = 181 (February 15 to August 15, 1985).
v^{n} = 1 / [(1 .1047 / 2)]^{39}, or .1366947986.
an⌉ = (1 − .1366947986) / .05235, or 16.4910258142.
AI′ = (44 / 181)(10.75 / 2), or 1.306630.
AI = [(184 − 103) / 184](10.75 / 2), or 2.366168.
A = AI′ AI, or 3.672798.
Resolution:
(P A)[1 (r/s)(i/2)] = (r′/s″)(C/2) C/2 (C/2)an⌉ 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 fixedprincipal securities reopened during a short first payment period:
Formula:
(P A)[1 (r/s)(i/2)] = (r′/s)(C/2) (C/2)an⌉ 100v ^{n}.
Where:
A = [(r′ − r)/s](C/2) and
r′ = number of days from the original issue date to the first interest payment date.
Example:
For a 101/2% 8year note due May 15, 1991, originally issued on May 16, 1983, and reopened on August 15, 1983, with interest payments on November 15 and May 15 (first payment on November 15, 1983), solve for the price per 100 (P) at a yield of 10.53%. Accrued interest is calculated from May 16 to August 15.
Definitions:
C = 10.50.
i = .1053.
n = 15.
r = 92 (August 15, 1983, to November 15, 1983).
s = 184 (May 15, 1983, to November 15, 1983).
r′ = 183 (May 16, 1983, to November 15, 1983).
v ^{n} = 1/[(1 .1053/2)]^{15}, or .4631696332.
an⌉ = (1 − .4631696332) / .05265, or 10.1962082956.
A = [(183 − 92) / 184](10.50 / 2), or 2.596467.
Resolution:
(P A)[1 (r/s)(i/2)] = (r′/s)(C/2) (C/2)an⌉ 100v ^{n} or
(P 2.596467)[1 (92/184)(.1053/2)] = (183/184)(10.50/2) (10.50/2)(10.1962082956) 100(.4631696332).
(1) (P 2.596467)[1 .026325] = 5.2214673913 53.5300935520 46.31696332.
(2) (P 2.596467)[1.026325] = 105.0685242633.
(3) (P 2.596467) = 105.0685242633 / 1.026325.
(4) (P 2.596467) = 102.373541.
(5) P = 102.373541 − 2.596467.
(6) P = 99.777074.
G. For fixedprincipal securities reopened during the fractional portion (initial short period) of a long first payment period:
Formula:
(P A)[1 (r/s)(i/2)] = [(r′/s)(C/2)]v (C/2)an⌉ 100v ^{n}.
Where:
A = [(r′ − r)/s](C/2), and
r = number of days from the reopening date to the end of the short period.
r′ = number of days in the short period.
s = number of days in the semiannual period ending with the end of the short period.
Example:
For a 93/4% 6year 2month note due December 15, 1994, originally issued on October 15, 1988, and reopened on November 15, 1988, with interest payments on June 15 and December 15 (first payment on June 15, 1989), solve for the price per 100 (P) at a yield of 9.79%. Accrued interest is calculated from October 15 to November 15.
Definitions:
C = 9.75.
i = .0979.
n = 12.
r = 30 (November 15, 1988, to December 15, 1988).
s = 183 (June 15, 1988, to December 15, 1988).
r′ = 61 (October 15, 1988, to December 15, 1988).
v = 1 / (1 .0979/2), or .9533342867.
v ^{n} = [1 / (1 .0979/2)]^{12}, or .5635631040.
an⌉ = (1 − .5635631040)/.04895, or 8.9159733613.
A = [(61 − 30)/183](9.75/2), or .825820.
Resolution:
(P A)[1 (r/s)(i/2)] = [(r′/s)(C/2)]v (C/2)an⌉ 100v ^{n} or
(P .825820)[1 (30/183)(.0979/2)] = [(61/183)(9.75/2)](.9533342867) (9.75/2)(8.9159733613) 100(.5635631040).
(1) (P .825820)[1 .00802459] = 1.549168216 43.4653701362 56.35631040.
(2) (P .825820)[1.00802459] = 101.3708487520.
(3) (P .825820) = 101.3708487520 / 1.00802459.
(4) (P .825820) = 100.563865.
(5) P = 100.563865 −. 825820.
(6) P = 99.738045.
III. Formulas for Conversion of InflationIndexed Security Yields to Equivalent Prices
Definitions
P = unadjusted or real price per 100 (dollars).
Padj = inflation adjusted price; P × Index RatioDate.
A = unadjusted accrued interest per $100 original principal.
Aadj = inflation adjusted accrued interest; A× Index RatioDate.
SA = settlement amount including accrued interest in current dollars per $100 original principal; Padj Aadj.
r = days from settlement date to next coupon date.
s = days in current semiannual period.
i = real yield, expressed in decimals (e.g., 0.0325).
C = real annual coupon, payable semiannually, in terms of real dollars paid on $100 initial, or real, principal of the security.
n = number of full semiannual periods from issue date to maturity date, except that, if the issue date is a coupon frequency date, n will be one less than the number of full semiannual periods remaining until maturity. Coupon frequency dates are the two semiannual dates based on the maturity date of each note or bond issue. For example, a security maturing on July 15, 2026 would have coupon frequency dates of January 15 and July 15.
v ^{n} = 1/(1 i/2)^{n} = present value of 1 due at the end of n periods.
an⌉ = (1 − v ^{n}) /(i/2) = v v ^{2} v ^{3} ^{...} v ^{n} = present value of 1 per period for n periods.
Special Case: If i = 0, then an⌉ = n. Furthermore, when i = 0, an⌉ cannot be calculated using the formula: (1 − v ^{n})/(i/2). In the special case where i = 0, an⌉ must be calculated as the summation of the individual present values (i.e., v v ^{2} v ^{3} ^{...} v ^{n}). Using the summation method will always confirm that an⌉ = n when i = 0.
Date = valuation date.
D = the number of days in the month in which Date falls.
t = calendar day corresponding to Date.
CPI = Consumer Price Index number.
CPIM = CPI reported for the calendar month M by the Bureau of Labor Statistics.
Ref CPIM = reference CPI for the first day of the calendar month in which Date falls (also equal to the CPI for the third preceding calendar month), e.g., Ref CPIApril 1 is the CPIJanuary.
Ref CPIM 1 = reference CPI for the first day of the calendar month immediately following Date.
Ref CPIDate = Ref CPIM − [(t − 1)/D][Ref CPIM 1Ref 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 inflationindexed securities with a regular first interest payment period:
Formulas:
Padj = P × Index RatioDate.
A = [(s−r)/s] × (C/2).
Aadj = A × Index RatioDate.
SA = Padj Aadj
Index RatioDate = Ref CPIDate/Ref CPIIssueDate.
Example:
We issued a 10year inflationindexed 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 CPIDate = 164.
Ref CPIIssueDate = 164.
Resolution:
Index RatioDate = Ref CPIDate / Ref CPIIssueDate = 164/164 = 1.
A = [(181 − 181)/181] × 3.875/2 = 0.
Aadj = 0 × 1 = 0.
v^{n} = 1/(1 i/2)^{n} = 1/(1 .03898/2)^{19} = 0.692984572.
an⌉ = (1 − v^{n})/(i/2) = (10.692984572) / (.03898/2) = 15.752459107.
Formula:
P = 99.811030.
Padj = P × Index RatioDate.
Padj = 99.811030 × 1 = 99.811030.
SA = Padj × 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 inflationindexed securities reopened during a regular interest period where the purchase price includes predetermined accrued interest.
(2) For new inflationindexed 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:
Padj = P × Index RatioDate.
A = [(s−r)/s] × (C/2).
Aadj = A × Index RatioDate.
SA = Padj Aadj.
Index RatioDate = Ref CPIDate/Ref CPIIssueDate.
Example:
We issued a 35/8% 10year inflationindexed 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⌉ = (1−v^{n})/(i/2) = (1 − 0.722138438)/(.0365/2) = 15.225291068.
Formula:
P = 100.703267 − 0.906250.
P = 99.797017.
Padj = P × Index RatioDate.
Padj = 99.797017 × 1.01074 = 100.86883696.
Padj = 100.868837.
A = [(184−92)/184] × 3.625/2 = 0.906250.
Aadj = A × Index RatioDate.
Aadj = 0.906250 × 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 inflationadjusted 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 InflationProtected Interest Components
Note:
Valuing an interest component stripped from an inflationprotected 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 inflationprotected 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 inflationprotected security.
Definitions:
c = C/100 = the regular annual interest rate, payable semiannually, e.g., .03625 (the decimal equivalent of a 35/8% 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 10year inflationprotected note paying 37/8% 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.
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 (CouponEquivalent 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 13week 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 26week 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 pricebasis 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 (couponequivalent yield) for Treasury bills:
1. For bills of not more than one halfyear 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 halfyear 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 52week 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 = .082373244or
(6)i = 8.237%.