31 CFR Appendix B to Part 356, Formulas and Tables
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.
1. Regular HalfYear Payment Period. We pay interest on marketable Treasury nonindexed 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 nonindexed 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 nonindexed 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 


0.003396739  0.003415301  0.003434066  0.003453039 

0.006793478  0.006830601  0.006868132  0.006906077 

0.010190217  0.010245902  0.010302198  0.010359116 

0.013586957  0.013661202  0.013736264  0.013812155 

0.016983696  0.017076503  0.017170330  0.017265193 

0.020380435  0.020491803  0.020604396  0.020718232 

0.023777174  0.023907104  0.024038462  0.024171271 
1  0.027173913  0.027322404  0.027472527  0.027624309 
1

0.030570652  0.030737705  0.030906593  0.031077348 
1

0.033967391  0.034153005  0.034340659  0.034530387 
1

0.037364130  0.037568306  0.037774725  0.037983425 
1

0.040760870  0.040983607  0.041208791  0.041436464 
1

0.044157609  0.044398907  0.044642857  0.044889503 
1

0.047554348  0.047814208  0.048076923  0.048342541 
1

0.050951087  0.051229508  0.051510989  0.051795580 
2  0.054347826  0.054644809  0.054945055  0.055248619 
2

0.057744565  0.058060109  0.058379121  0.058701657 
2

0.061141304  0.061475410  0.061813187  0.062154696 
2

0.064538043  0.064890710  0.065247253  0.065607735 
2

0.067934783  0.068306011  0.068681319  0.069060773 
2

0.071331522  0.071721311  0.072115385  0.072513812 
2

0.074728261  0.075136612  0.075549451  0.075966851 
2

0.078125000  0.078551913  0.078983516  0.079419890 
3  0.081521739  0.081967213  0.082417582  0.082872928 
3

0.084918478  0.085382514  0.085851648  0.086325967 
3

0.088315217  0.088797814  0.089285714  0.089779006 
3

0.091711957  0.092213115  0.092719780  0.093232044 
3

0.095108696  0.095628415  0.096153846  0.096685083 
3

0.098505435  0.099043716  0.099587912  0.100138122 
3

0.101902174  0.102459016  0.103021978  0.103591160 
3

0.105298913  0.105874317  0.106456044  0.107044199 
4  0.108695652  0.109289617  0.109890110  0.110497238 
4

0.112092391  0.112704918  0.113324176  0.113950276 
4

0.115489130  0.116120219  0.116758242  0.117403315 
4

0.118885870  0.119535519  0.120192308  0.120856354 
4

0.122282609  0.122950820  0.123626374  0.124309392 
4

0.125679348  0.126366120  0.127060440  0.127762431 
4

0.129076087  0.129781421  0.130494505  0.131215470 
4

0.132472826  0.133196721  0.133928571  0.134668508 
5  0.135869565  0.136612022  0.137362637  0.138121547 
5

0.139266304  0.140027322  0.140796703  0.141574586 
5

0.142663043  0.143442623  0.144230769  0.145027624 
5

0.146059783  0.146857923  0.147664835  0.148480663 
5

0.149456522  0.150273224  0.151098901  0.151933702 
5

0.152853261  0.153688525  0.154532967  0.155386740 
5

0.156250000  0.157103825  0.157967033  0.158839779 
5

0.159646739  0.160519126  0.161401099  0.162292818 
6  0.163043478  0.163934426  0.164835165  0.165745856 
6

0.166440217  0.167349727  0.168269231  0.169198895 
6

0.169836957  0.170765027  0.171703297  0.172651934 
6

0.173233696  0.174180328  0.175137363  0.176104972 
6

0.176630435  0.177595628  0.178571429  0.179558011 
6

0.180027174  0.181010929  0.182005495  0.183011050 
6

0.183423913  0.184426230  0.185439560  0.186464088 
6

0.186820652  0.187841530  0.188873626  0.189917127 
7  0.190217391  0.191256831  0.192307692  0.193370166 
7

0.193614130  0.194672131  0.195741758  0.196823204 
7

0.197010870  0.198087432  0.199175824  0.200276243 
7

0.200407609  0.201502732  0.202609890  0.203729282 
7

0.203804348  0.204918033  0.206043956  0.207182320 
7

0.207201087  0.208333333  0.209478022  0.210635359 
7

0.210597826  0.211748634  0.212912088  0.214088398 
7

0.213994565  0.215163934  0.216346154  0.217541436 
8  0.217391304  0.218579235  0.219780220  0.220994475 
8

0.220788043  0.221994536  0.223214286  0.224447514 
8

0.224184783  0.225409836  0.226648352  0.227900552 
8

0.227581522  0.228825137  0.230082418  0.231353591 
8

0.230978261  0.232240437  0.233516484  0.234806630 
8

0.234375000  0.235655738  0.236950549  0.238259669 
8

0.237771739  0.239071038  0.240384615  0.241712707 
8

0.241168478  0.242486339  0.243818681  0.245165746 
9  0.244565217  0.245901639  0.247252747  0.248618785 
9

0.247961957  0.249316940  0.250686813  0.252071823 
9

0.251358696  0.252732240  0.254120879  0.255524862 
9

0.254755435  0.256147541  0.257554945  0.258977901 
9

0.258152174  0.259562842  0.260989011  0.262430939 
9

0.261548913  0.262978142  0.264423077  0.265883978 
9

0.264945652  0.266393443  0.267857143  0.269337017 
9

0.268342391  0.269808743  0.271291209  0.272790055 
10  0.271739130  0.273224044  0.274725275  0.276243094 
10

0.275135870  0.276639344  0.278159341  0.279696133 
10

0.278532609  0.280054645  0.281593407  0.283149171 
10

0.281929348  0.283469945  0.285027473  0.286602210 
10

0.285326087  0.286885246  0.288461538  0.290055249 
10

0.288722826  0.290300546  0.291895604  0.293508287 
10

0.292119565  0.293715847  0.295329670  0.296961326 
10

0.295516304  0.297131148  0.298763736  0.300414365 
11  0.298913043  0.300546448  0.302197802  0.303867403 
11

0.302309783  0.303961749  0.305631868  0.307320442 
11

0.305706522  0.307377049  0.309065934  0.310773481 
11

0.309103261  0.310792350  0.312500000  0.314226519 
11

0.312500000  0.314207650  0.315934066  0.317679558 
11

0.315896739  0.317622951  0.319368132  0.321132597 
11

0.319293478  0.321038251  0.322802198  0.324585635 
11

0.322690217  0.324453552  0.326236264  0.328038674 
12  0.326086957  0.327868852  0.329670330  0.331491713 
12

0.329483696  0.331284153  0.333104396  0.334944751 
12

0.332880435  0.334699454  0.336538462  0.338397790 
12

0.336277174  0.338114754  0.339972527  0.341850829 
12

0.339673913  0.341530055  0.343406593  0.345303867 
12

0.343070652  0.344945355  0.346840659  0.348756906 
12

0.346467391  0.348360656  0.350274725  0.352209945 
12

0.349864130  0.351775956  0.353708791  0.355662983 
13  0.353260870  0.355191257  0.357142857  0.359116022 
13

0.356657609  0.358606557  0.360576923  0.362569061 
13

0.360054348  0.362021858  0.364010989  0.366022099 
13

0.363451087  0.365437158  0.367445055  0.369475138 
13

0.366847826  0.368852459  0.370879121  0.372928177 
13

0.370244565  0.372267760  0.374313187  0.376381215 
13

0.373641304  0.375683060  0.377747253  0.379834254 
13

0.377038043  0.379098361  0.381181319  0.383287293 
14  0.380434783  0.382513661  0.384615385  0.386740331 
14

0.383831522  0.385928962  0.388049451  0.390193370 
14

0.387228261  0.389344262  0.391483516  0.393646409 
14

0.390625000  0.392759563  0.394917582  0.397099448 
14

0.394021739  0.396174863  0.398351648  0.400552486 
14

0.397418478  0.399590164  0.401785714  0.404005525 
14

0.400815217  0.403005464  0.405219780  0.407458564 
14

0.404211957  0.406420765  0.408653846  0.410911602 
15  0.407608696  0.409836066  0.412087912  0.414364641 
15

0.411005435  0.413251366  0.415521978  0.417817680 
15

0.414402174  0.416666667  0.418956044  0.421270718 
15

0.417798913  0.420081967  0.422390110  0.424723757 
15

0.421195652  0.423497268  0.425824176  0.428176796 
15

0.424592391  0.426912568  0.429258242  0.431629834 
15

0.427989130  0.430327869  0.432692308  0.435082873 
15

0.431385870  0.433743169  0.436126374  0.438535912 
16  0.434782609  0.437158470  0.439560440  0.441988950 
16

0.438179348  0.440573770  0.442994505  0.445441989 
16

0.441576087  0.443989071  0.446428571  0.448895028 
16

0.444972826  0.447404372  0.449862637  0.452348066 
16

0.448369565  0.450819672  0.453296703  0.455801105 
16

0.451766304  0.454234973  0.456730769  0.459254144 
16

0.455163043  0.457650273  0.460164835  0.462707182 
16

0.458559783  0.461065574  0.463598901  0.466160221 
17  0.461956522  0.464480874  0.467032967  0.469613260 
17

0.465353261  0.467896175  0.470467033  0.473066298 
17

0.468750000  0.471311475  0.473901099  0.476519337 
17

0.472146739  0.474726776  0.477335165  0.479972376 
17

0.475543478  0.478142077  0.480769231  0.483425414 
17

0.478940217  0.481557377  0.484203297  0.486878453 
17

0.482336957  0.484972678  0.487637363  0.490331492 
17

0.485733696  0.488387978  0.491071429  0.493784530 
18  0.489130435  0.491803279  0.494505495  0.497237569 
18

0.492527174  0.495218579  0.497939560  0.500690608 
18

0.495923913  0.498633880  0.501373626  0.504143646 
18

0.499320652  0.502049180  0.504807692  0.507596685 
18

0.502717391  0.505464481  0.508241758  0.511049724 
18

0.506114130  0.508879781  0.511675824  0.514502762 
18

0.509510870  0.512295082  0.515109890  0.517955801 
18

0.512907609  0.515710383  0.518543956  0.521408840 
19  0.516304348  0.519125683  0.521978022  0.524861878 
19

0.519701087  0.522540984  0.525412088  0.528314917 
19

0.523097826  0.525956284  0.528846154  0.531767956 
19

0.526494565  0.529371585  0.532280220  0.535220994 
19

0.529891304  0.532786885  0.535714286  0.538674033 
19

0.533288043  0.536202186  0.539148352  0.542127072 
19

0.536684783  0.539617486  0.542582418  0.545580110 
19

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 nonindexed 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.
A 2year 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 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 8 3/8%, 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.
A 5year 2month 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 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 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).
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:
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:
(ii) For example, the Ref CPI for April 15, 1996 is calculated as follows:
(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:
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.
A 10year inflationprotected 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 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.
1. Indexing and Interest Payment Process. We issue floating rate notes with a daily interest accrual feature. This means that the interest rate “floats” based on changes in the representative index rate. We pay interest on a quarterly basis. The index rate is the High Rate of the 13week Treasury bill auction announced on the auction results press release that has been converted into a simpleinterest money market yield computed on an actual/360 basis and rounded to nine decimal places. Interest payments are based on the floating rate note's variable interest rate from, and including, the dated date or last interest payment date to, but excluding, the next interest payment or maturity date. We make quarterly interest payments by accruing the daily interest amounts and adding those amounts together for the interest payment period.
2. Interest Rate. The interest rate on floating rate notes will be the spread plus the index rate (as it may be adjusted on the calendar day following each auction of 13week bills).
3. Interest Accrual. In general, accrued interest for a particular calendar day in an accrual period is calculated by using the index rate from the most recent auction of 13week bills that took place before the accrual day, plus the spread determined at the time of a new floating rate note auction, divided by 360, subject to a zeropercent minimum daily interest accrual rate. However, a 13week bill auction that takes place in the twobusinessday period prior to a settlement date or interest payment date will be excluded from the calculation of accrued interest for purposes of the settlement amount or interest payment. Any changes in the index rate that would otherwise have occurred during this twobusinessday period will occur on the first calendar day following the end of the period.
4. Index Contingencies.
(i) If Treasury were to discontinue auctions of 13week bills, the Secretary has authority to determine and announce a new index for outstanding floating rate notes.
(ii) If Treasury were to not conduct a 13week bill auction in a particular week, then the interest rate in effect for the notes at the time of the last 13week bill auction results announcement will remain in effect until such time, if any, as the results of a 13week Treasury auction are again announced by Treasury. Treasury reserves the right to change the index rate for any newly issued floating rate note.
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 nonindexed 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 Nonindexed Securities  (i) Involving One HalfYear: A note paying interest at a rate of 6 3/4%, 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 10 3/4% 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).
6. For a floating rate note, if accrued interest covers a portion of a full quarterly interest payment period, we calculate accrued interest as shown in section IV, paragraphs C and D of this appendix.
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. For nonindexed securities with a regular first interest payment period:
For an 8 3/4% 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%.
B. For nonindexed securities with a short first interest payment period:
For an 8 1/2% 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%.
C. For nonindexed securities with a long first interest payment period:
For an 8 1/2% 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%.
D. (1) For nonindexed securities reopened during a regular interest period where the purchase price includes predetermined accrued interest.
(2)For new nonindexed 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.
For a 9 1/2% 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).
E. For nonindexed securities reopened during the regular portion of a long first payment period:
A 10 3/4% 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.
F. For nonindexed securities reopened during a short first payment period:
For a 10 1/2% 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.
G. For nonindexed securities reopened during the fractional portion (initial short period) of a long first payment period:
For a 9 3/4% 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.
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.
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:
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 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.)
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.
We issued a 3 5/8% 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.
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.
A. For newly issued floating rate notes issued at par:
The purpose of this example is to demonstrate how a floating rate note price is derived at the time of original issuance. Additionally, this example depicts the association of the July 31, 2012 issue date and the twobusinessday lockout period. For a new twoyear floating rate note auctioned on July 25, 2012, and issued on July 31, 2012, with a maturity date of July 31, 2014, and an interest accrual rate on the issue date of 0.215022819% (index rate of 0.095022819% plus a spread of 0.120%), solve for the price per 100 (P). This interest accrual rate is used for each daily interest accrual over the life of the security for the purposes of this example. In a new issuance (not a reopening) of a floating rate note, the discount margin determined at auction will be equal to the spread.
As of the issue date the latest 13week bill, auctioned at least two days prior, has the following information:
Table 1  13Week Bill Auction Data
Auction date  Issue date  Maturity date  Auction
clearing price 
Auction high rate  Index rate 

7/23/2012  7/26/2012  10/25/2012  99.975986  0.095%  0.095022819% 
The rationale for using a 13week bill auction that has occurred at least two days prior to the issue date is due to the twobusinessday lockout period. This lockout period applies only to the issue date and interest payment dates, thus any 13week bill auction that occurs during the twoday lockout period is not used for calculations related to the issue date and interest payment dates. The following sample calendar depicts this relationship for the floating rate note issue date.
The following table presents the future interest payment dates and the number of days between them.
Table 2  Payment Dates
Dates  Days between dates 

Issue Date: 

1st Interest Date: 

2nd Interest Date: 

3rd Interest Date: 

4th Interest Date: 

5th Interest Date: 

6th Interest Date: 

7th Interest Date: 

8th Interest & Maturity Dates: 

Table 3  Projected Cash Flows and Compound Factors





1  0.000597286  0.054950312  1.000549503 
2  0.000597286  0.054950312  1.000549503 
3  0.000597286  0.053158454  1.000531584 
4  0.000597286  0.054950312  1.000549503 
5  0.000597286  0.054950312  1.000549503 
6  0.000597286  0.054950312  1.000549503 
7  0.000597286  0.053158454  1.000531584 
8  0.000597286  100.054950312  1.000549503 
The price is computed as follows:
B. For newly issued floating rate notes issued at a premium:
The purpose of this example is to demonstrate how a floating rate note auction can result in a price at a premium given a negative discount margin and spread at auction. For a new twoyear floating rate note auctioned on July 25, 2012, and issued on July 31, 2012, with a maturity date of July 31, 2014, solve for the price per 100 (P). In a new issue (not a reopening) of a floating rate note, the discount margin established at auction will be equal to the spread. In this example, the discount margin determined at auction is −0.150%, but the floating rate note is subject to a daily interest rate accrual minimum of 0.000%.
As of the issue date the latest 13week bill, auctioned at least two days prior, has the following information:
Table 1  13Week Bill Auction Data
Auction date  Issue date  Maturity date  Auction
clearing price 
Auction high rate  Index rate 

7/23/2012  7/26/2012  10/25/2012  99.975986  0.095%  0.095022819% 
The following table presents the future interest payment dates and the number of days between them.
Table 2  Payment Dates
Dates  Days between dates 

Issue Date: 

1st Interest Date: 

2nd Interest Date: 

3rd Interest Date: 

4th Interest Date: 

5th Interest Date: 

6th Interest Date: 

7th Interest Date: 

8th Interest & Maturity Dates: 

Ai represents the projected cash flow the floating rate note holder will receive, for a $100 par value, at the future interest payment date Ti, where i = 1,2, . . ., 8. Ti − Ti−1 is the number of days between the future interest payment dates Ti−1 and Ti. To account for the payback of the par value, the variable 1{i=8} takes the value 1 if the payment date is the maturity date, or 0 otherwise. For example:
Bi represents the projected compound factor between the future dates Ti−1 and Ti, where i = 1,2, . . ., 8. All Bi's are computed using the discount margin m = −0.150% (equals the spread obtained at the auction), and the fixed index rate of r = 0.095022819% applicable to the issue date (7/31/2012). For example:
The following table shows the projected daily accrued interests for $100 par value (ai's), cash flows at interest payment dates (Ai's), and the compound factors between payment dates (Bi's).
Table 3  Projected Cash Flows and Compound Factors





1  0.000000000  0.000000000  0.999859503 
2  0.000000000  0.000000000  0.999859503 
3  0.000000000  0.000000000  0.999864084 
4  0.000000000  0.000000000  0.999859503 
5  0.000000000  0.000000000  0.999859503 
6  0.000000000  0.000000000  0.999859503 
7  0.000000000  0.000000000  0.999864084 
8  0.000000000  100.000000000  0.999859503 
The price is computed as follows:
C. Pricing and accrued interest for reopened floating rate notes
The purpose of this example is to determine the floating rate note prices with and without accrued interest at the time of the reopening auction. For a twoyear floating rate note that was originally auctioned on July 25, 2012, with an issue date of July 31, 2012, reopened in an auction on August 30, 2012 and issued on August 31, 2012, with a maturity date of July 31, 2014, solve for accrued interest per 100 (AI), the price with accrued interest per 100 (PD) and the price without accrued interest per 100 (PC). Since this is a reopening of an original issue from the prior month, Table 2 as shown in the example is used for accrued interest calculations. In the case of floating rate note reopenings, the spread on the security remains equal to the spread that was established at the original auction of the floating rate notes.
The following table shows the past results for the 13week bill auction.
Table 1  13Week Bill Auction Data
Auction date  Issue date  Maturity date  Auction
clearing price 
Auction
high rate (percent) 
Index rate
(percent) 

7/23/2012  7/26/2012  10/25/2012  99.975986  0.095  0.095022819 
7/30/2012  8/2/2012  11/1/2012  99.972194  0.110  0.110030595 
8/6/2012  8/9/2012  11/8/2012  99.974722  0.100  0.100025284 
8/13/2012  8/16/2012  11/15/2012  99.972194  0.110  0.110030595 
8/20/2012  8/23/2012  11/23/2012  99.973167  0.105  0.105028183 
8/27/2012  8/30/2012  11/29/2012  99.973458  0.105  0.105027876 
The following table shows the index rates applicable for the accrued interest.
Table 2  Applicable Index Rate
Accrual starts  Accrual ends  Number of days in accrual period  Applicable floating rate  

Auction date  Index rate
(percent) 

7/31/2012  7/31/2012  1  7/23/2012  0.095022819 
8/1/2012  8/6/2012  6  7/30/2012  0.110030595 
8/7/2012  8/13/2012  7  8/6/2012  0.100025284 
8/14/2012  8/20/2012  7  8/13/2012  0.110030595 
8/21/2012  8/27/2012  7  8/20/2012  0.105028183 
8/28/2012  8/30/2012  3  8/27/2012  0.105027876 
The accrued interest as of the new issue date (8/31/2012) for a $100 par value is:
The following table presents the future interest payment dates and the number of days between them.
Table 3  Payment Dates
Dates  Days between dates 

Original Issue Date: 

New Issue Date: 

1st Interest Date: 

2nd Interest Date: 

3rd Interest Date: 

4th Interest Date: 

5th Interest Date: 

6th Interest Date: 

7th Interest Date: 

8th Interest & Maturity Dates: 

ai represents the daily projected interest, for a $100 par value, that will accrue between the future interest payment dates Ti−1 and Ti, where i = 1,2,...,8. ai's are computed using the spread s = 0.120% obtained at the original auction, and the fixed index rate of r = 0.105027876% applicable to the new issue date (8/31/2012). For example:
Ai represents the projected cash flow the floating rate note holder will receive, less accrued interest, for a $100 par value, at the future interest payment date Ti, where i = 1,2,...,8. Ti − Ti1 is the number of days between the future interest payment dates Ti−1 and Ti. To account for the payback of the par value, the variable 1{i = 8} takes the value 1 if the payment date is the maturity date, or 0 otherwise. For example:
and
Let
Bi represents the projected compound factor between the future dates Ti−1 and Ti, where i = 1,2,...,8. All Bi's are computed using the discount margin m = 0.100% obtained at the reopening auction, and the fixed index rate of r = 0.105027876% applicable to the new issue date (8/31/2012). For example:
The following table shows the projected daily accrued interests for $100 par value (ai's), cash flows at interest payment dates (Ai's), and the compound factors between payment dates (Bi's).
Table 4  Projected Cash Flows and Compound Factors





1  0.000625077  0.038129697  1.000347408 
2  0.000625077  0.057507084  1.000523960 
3  0.000625077  0.055631853  1.000506874 
4  0.000625077  0.057507084  1.000523960 
5  0.000625077  0.057507084  1.000523960 
6  0.000625077  0.057507084  1.000523960 
7  0.000625077  0.055631853  1.000506874 
8  0.000625077  100.057507084  1.000523960 
The price with accrued interest is computed as follows:
D. For calculating interest payments:
For a new issue of a twoyear floating rate note auctioned on July 25, 2012, and issued on July 31, 2012, with a maturity date of July 31, 2014, and a first interest payment date of October 31, 2012, calculate the quarterly interest payments (IPi) per 100. In a new issuance (not a reopening) of a new floating rate note, the discount margin determined at auction will be equal to the spread. The interest accrual rate used for this floating rate note on the issue date is 0.215022819% (index rate of 0.095022819% plus a spread of 0.120%) and this rate is used for each daily interest accrual over the life of the security for the purposes of this example.
Example 1: Projected interest payment as of the original issue date.
As of the issue date the latest 13week bill, auctioned at least two days prior, has the following information:
Table 1  13Week Bill Auction Data
Auction date  Issue date  Maturity date  Auction
clearing price 
Auction high rate  Index rate 

7/23/2012  7/26/2012  10/25/2012  99.975986  0.095%  0.095022819% 
The following table presents the future interest payment dates and the number of days between them.
Table 2  Payment Dates
Dates  Days between dates 

Issue Date: 

1st Interest Date: 

2nd Interest Date: 

3rd Interest Date: 

4th Interest Date: 

5th Interest Date: 

6th Interest Date: 

7th Interest Date: 

8th Interest & Maturity Dates: 

Using the spread s = 0.120%, and the fixed index rate of r = 0.095022819% applicable to the issue date (7/31/2012), the first and seventh projected interest payments are computed as follows:
The following table shows all projected interest payments as of the issue date.
Table 3  Projected Interest Payments

Dates 


1  10/31/2012  0.054950312 
2  1/31/2013  0.054950312 
3  4/30/2013  0.053158454 
4  7/31/2013  0.054950312 
5  10/31/2013  0.054950312 
6  1/31/2014  0.054950312 
7  4/30/2014  0.053158454 
8  7/31/2014  0.054950312 
Example 2: Projected interest payment as of the reopening issue date (intermediate values, including rates in percentage terms, are rounded to nine decimal places).
This example demonstrates the calculations required to determine the interest payment due when the reopened floating rate note is issued. This example also demonstrates the need to calculate accrued interest at the time of a floating rate reopening auction. Since this is a reopening of an original issue from 31 days prior, Table 5 as shown in the example is used for accrued interest calculations. For a twoyear floating rate note originally auctioned on July 25, 2012 with an original issue date of July 31, 2012, reopened by an auction on August 30, 2012 and issued on August 31, 2012, with a maturity date of July 31, 2014, calculate the quarterly interest payments (IPI) per 100. T−1 is the dated date if the reopening occurs before the first interest payment date, or otherwise the latest interest payment date prior to the new issue date.
The following table shows the past results for the 13week bill auction.
Table 4  13Week Bill Auction Data
Auction date  Issue date  Maturity date  Auction
clearing price 
Auction
high rate (percent) 
Index rate
(percent) 

7/23/2012  7/26/2012  10/25/2012  99.975986  0.095  0.095022819 
7/30/2012  8/2/2012  11/1/2012  99.972194  0.110  0.110030595 
8/6/2012  8/9/2012  11/8/2012  99.974722  0.100  0.100025284 
8/13/2012  8/16/2012  11/15/2012  99.972194  0.110  0.110030595 
8/20/2012  8/23/2012  11/23/2012  99.973167  0.105  0.105028183 
8/27/2012  8/30/2012  11/29/2012  99.973458  0.105  0.105027876 
The following table shows the index rates applicable for the accrued interest.
Table 5  Applicable Index Rate
Accrual starts  Accrual ends  Number of days in
accrual period 
Applicable floating rate  

Auction date  Index rate
(percent) 

7/31/2012  7/31/2012  1  7/23/2012  0.095022819 
8/1/2012  8/6/2012  6  7/30/2012  0.110030595 
8/7/2012  8/13/2012  7  8/6/2012  0.100025284 
8/14/2012  8/20/2012  7  8/13/2012  0.110030595 
8/21/2012  8/27/2012  7  8/20/2012  0.105028183 
8/28/2012  8/30/2012  3  8/27/2012  0.105027876 
The accrued interest as of 8/31/2012 for a $100 par value is:
The following table presents the future interest payment dates and the number of days between them.
Table 6  Payment Dates
Dates  Days between dates 

Original Issue Date: 

New Issue Date: 

1st Interest Date: 

2nd Interest Date: 

3rd Interest Date: 

4th Interest Date: 

5th Interest Date: 

6th Interest Date: 

7th Interest Date: 

8th Interest & Maturity Dates: 

Using the original spread s = 0.120% (obtained on 7/25/2012), and the fixed index rate of r = 0.105027876% applicable to the new issue date (8/31/2012), the first and eighth projected interest payments are computed as follows:
The following table shows all projected interest payments as of the new issue date.
Table 7  Projected Interest Payments

Dates 


1  10/31/2012  0.057562689 
2  1/31/2013  0.057507084 
3  4/30/2013  0.055631853 
4  7/31/2013  0.057507084 
5  10/31/2013  0.057507084 
6  1/31/2014  0.057507084 
7  4/30/2014  0.055631853 
8  7/31/2014  0.057507084 
E. Pricing and accrued interest for new issue floating rate notes with an issue date that occurs after the dated date
The purpose of this example is to demonstrate how a floating rate note can have a price without accrued interest of less than $100 par value when the issue date occurs after the dated date. An original issue twoyear floating rate note is auctioned on December 29, 2011, with a dated date of December 31, 2011, an issue date of January 3, 2012, and a maturity date of December 31, 2013.
As of the issue date the latest 13week bill, auctioned at least two days prior, has the following information:
Table 1  13WEEK BILL AUCTION DATA
Auction date  Issue date  Maturity date  Auction
clearing price 
Auction high rate  Index rate 

12/27/2011  12/29/2011  3/29/2012  99.993681  0.025%  0.025001580% 
The following table shows the index rates applicable for the accrued interest.
Table 2  Applicable Index Rate
Accrual starts  Accrual ends  Number of days in
accrual period 
Applicable floating rate  

Auction date  Index rate  
12/31/2011  1/2/2012  3  12/27/2011  0.025001580% 
The accrued interest as of the new issue date (1/3/2012) for a $100 par value is:
The following table presents the future interest payment dates and the number of days between them.
Table 3  Payment Dates
Dates  Days between dates 

Dated Date: = 

Issue Date: 

1st Interest Date: 

2nd Interest Date: 

3rd Interest Date: 

4th Interest Date: 

5th Interest Date: 

6th Interest Date: 

7th Interest Date: 

8th Interest & Maturity Dates: 

ai represents the daily projected interest, for a $100 par value, that will accrue between the future interest payment dates Ti−1 and Ti, where i = 1,2,...,8. ai's are computed using the spread s = 1.000% obtained at the auction, and the fixed index rate of r = 0.025001580% applicable to the issue date (1/3/2012). For example:
Ai represents the projected cash flow the floating rate note holder will receive, less accrued interest, for a $100 par value, at the future interest payment date Ti, where i = 1,2,...,8. Ti − Ti−1 is the number of days between the future interest payment dates Ti1 and Ti. To account for the payback of the par value, the variable 1{i = 8} takes the value 1 if the payment date is the maturity date, or 0 otherwise. For example:
Bi represents the projected compound factor between the future dates Ti−1 and Ti, where i = 1,2,...,8. All Bi's are computed using the discount margin m = 1.000% (equals the spread obtained at the auction), and the fixed index rate of r = 0.025001580% applicable to the issue date (1/3/2012). For example:
The following table shows the projected daily accrued interests for $100 par value (ai 's), cash flows at interest payment dates (Ai 's), and the compound factors between payment dates (Bi's).
Table 4  Projected Cash Flows and Compound Factors





1  0.002847227  0.250555976  1.002505559 
2  0.002847227  0.259097657  1.002590976 
3  0.002847227  0.261944884  1.002619448 
4  0.002847227  0.261944884  1.002619448 
5  0.002847227  0.256250430  1.002562504 
6  0.002847227  0.259097657  1.002590976 
7  0.002847227  0.261944884  1.002619448 
8  0.002847227  100.261944884  1.002619448 
The price with accrued interest is computed as follows:
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.
A. Conversion of the discount rate to a purchase price for Treasury bills of all maturities:
For a bill issued November 24, 1989, due February 22, 1990, at a discount rate of 7.610%, solve for price per 100 (P).
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.
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.
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:
For a 26week bill issued December 30, 1982, due June 30, 1983, with a price of $95.934567, solve for the discount rate (d).
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:
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).
2. For bills of more than one halfyear to maturity:
This formula must be solved by using the quadratic equation, which is:
Therefore, rewriting the bill formula in the quadratic equation form gives:
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).