multiplied by .03/2, or .015, resulting in a payment of $1,650.00. C. Accrued Interest Accrued interest will be payable by the purchaser of a Treasury bond or note when interest accrues prior to the issue date of the security. Because the purchaser receives a full interest payment despite having held the security for only a portion of the interest payment period, the Department is compensated through the payment of accrued interest at settlement. For a fixed-principal security, if accrued interest covers a fractional portion of a full half-year period, the number of days in the full half-year period and the stated interest rate will determine the daily interest decimal to be used in computing the accrued interest. The decimal is multiplied by the number of days for which interest has accrued. If a reopened fixed-principal security has a long first interest payment period (a "long coupon"), and the dated date for the reopened issue is less than six full months before the first interest payment, the accrued interest will fall into two separate half-year periods, and a separate daily interest decimal must be multiplied by the respective number of days in each half-year period during which interest has accrued. All accrued interest computations are rounded to five decimal places for a $1,000 inflationadjusted principal, using normal rounding procedures. Accrued interest for a par amount of securities greater than $1,000 is calculated by applying the appropriate multiple to accrued interest payable for $1,000 par amount, rounded to five decimal places. For an inflation-indexed security, accrued interest will be calculated as shown in section III, paragraphs A and B of this appendix. Examples. (1) Fixed-Principal Securities (i) Involving One Half-Year: A bond paying interest at a rate of 8%, originally issued on August 15, 1990, as a 30-year bond with a first interest payment date of February 15, 1991, was reopened as a 29-year 9-month bond on November 15, 1990. Interest had accrued for 92 days, from August 15 to November 15. The regular interest period from August 15 to February 15, 1991, covered 184 days. Accordingly, the daily interest decimal, $0.237771739, multiplied by 92, resulted in accrued interest payable of $21.874999988, or $21.87500, for each $1,000 bond purchased. If the bonds have a par amount of $150,000, then 150 is multiplied by $21.87500, resulting in an amount payable of $3,281.25. (ii) Involving Two Half-Years: A 104% bond, originally issued on July 2, 1985, as a 20-year 1-month bond, with a first interest payment date of February 15, 1986, was reopened as a 19-year 10-month bond on November 4, 1985. Interest had accrued for 44 days, from July 2 to August 15, 1985, during a 181-day half-year (February 15 to August 15); and for 81 days, from August 15 to November 4, during a 184day half-year (August 15, 1985, to February 15, 1986). Accordingly, $0.296961326 was multiplied by 44, and $0.292119565 was multiplied by 81, resulting in products of $13.066298344 and $23.661684765 which, added together, resulted in accrued interest payable of $36.727983109, or $36.72798, for each $1,000 bond purchased. If the bonds have a par amount of $11,000, then 11 is multiplied by $36.72798, resulting in an amount payable of $404.00778 ($404.01). II. FORMULAS FOR CONVERSION OF FIXEDPRINCIPAL SECURITY YIELDS TO EQUIVALENT PRICES Definitions P=price per 100 (dollars), rounded to three places, using normal rounding procedures C=the regular annual interest per $100, payable semiannually, e.g., 10.125 (the dollar equivalent of a 10%% 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, 1995, 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"=1/[1+(1/2)]"=present value of 1 due at the end of n periods a=(1-v")/(1/2)=v+v2+y3+ +V"-present value of 1 per period for n periods A-accrued interest A. For fixed-principal securities with a regular first interest payment period: Formula: P[1+(r/s)(i/2)]=(C/2)(r/s)+(C/2)a2+100 vo Example: For an 84% 30-year bond, issued May 15, 1990, due May 15, 2020, with interest payments on November 15 and May 15, solve for the price per 100 (P) at a yield of 8.84%. Definitions: C-8.75 i=.0884 r=184 (May 15 to November 15, 1990) s=184 (May 15 to November 15, 1990) n=59 (There are 60 full semiannual periods, but n is reduced by 1 because the issue date is a coupon frequency date.) y=1/[(1+.0884/2)], or .077940 a-(1-.077940)/.0442, or 20.861086 Resolution: P[1+(r/s)(i/2)]=(C/2)(r/s)+(C/2)an+100 vo or 2)(20.861086)+100(.077940) (1) P[1+.0442]=4.375+91.267251+7.7940. (2) P[1.0442]=103.436251 (3) P=103.436251+1.0442 (4) P-99.057892 (5) P=99.058 B. For fixed-principal securities with a short first interest payment period: Formula: P[1+(r/s)(i/2)]=(C/2)(r/s)+(C/2)a+100 vn Example: For an 82% 2-year note, issued April 2, 1990, due March 31, 1992, with interest payments on September 30 and March 31, solve for the price per 100 (P) at a yield of 8.59%. Definitions: C=8.50 i=.0859 n=3 r=181 (April 2 to September 30, 1990) s=183 (March 31 to September 30, 1990) V=1/[(1+.0859/2)]3, or .881474 a-(1-.881474)/.04295, or 2.759627 Resolution: P[1+(r/s)(i/2)]=(C/2)(r/s)+(C/2)a,+100 va or 2)(2.759627)+100(.881474) (1) P[1+.042481]=4.203552+ 11.728415+88.1474 (2) P[1.042481]=104.079367 (3) P=104.079367+1.042481 (4) P=99.838143 (5) P=99.838 C. For fixed-principal securities with a long first interest payment period: Formula: P[1+(r/s)(i/2)]=[(C/2)(r/s)]v+(C/2)a,+100 vn Example: For an 82% 5-year 2-month note, issued March 1, 1990, due May 15, 1995, with interest payments on November 15 and May 15 (first payment on November 15, 1990), solve for the price per 100 (P) at a yield of 8.53%. Definitions: C-8.50 i=.0853 n=10 r=75 (March 1 to May 15, 1990, which is the fractional portion of the first interest payment) 8-181 (November 15, 1989, to May 15, 1990) v=1/(1+.0853/2), or .959095 vn=1/(1+.0853/2)10, or .658589 a-(1-.658589)/.04265, or 8.004947 Resolution: P[1+(r/s)(i/2)]=[(C/2)(r/s)]v+(C/2)a+100 vn or P[1+(75/181)(.0853/2)]=[(8.50/2)(75/181)] .959095+(8.50/2)(8.004947)+100(.658589) (1) P[1+.017673]=1.689014+ 34.021025+65.8589 (2) P[1.017673]=101.568939 (3) P=101.568939+1.017673 (4) P=99.805084 (5) P=99.805 D. (1) For fixed-principal securities reopened during a regular interest period where the purchase price includes predetermined accrued interest. (2) For new fixed-principal securities accruing interest from the coupon frequency date immediately preceding the issue date, with the interest rate established in the auction being used to determine the accrued interest payable on the issue date. Formula: (P+A)[1+(r/s)(i/2)]=C/2+(C/2)a+100 va For a 92% 10-year note, interest accruing from November 15, 1985, issued November 29, 1985, due November 15, 1995, with 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). 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 10% 19-year 9-month bond due August 15, 2005, is issued on July 2, 1985, and reopened on November 4, 1985, with interest payments on February 15 and August 15 (first payment on February 15, 1986), solve for the price per 100 (P) at a yield of 10.47%. Accrued interest is calculated from July 2 to November 4. Definitions: C=10.75 i=.1047 n=39 r=103 (November 4, 1985, to February 15, 1986) 8-184 (August 15, 1985, to February 15, 1986) r=44 (July 2 to August 15, 1985) s"=181 (February 15 to August 15, 1985) v=1/[(1+.1047/2)], or .136695 a-(1-.136695).05235, or 16.491022 AI=(44/181)(10.75/2), or 1.306630 AT=[(184-103)/184](10.75/2), or 2.366168 A=AI'+AI, or 3.672798 Resolution: ment 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) a=(1-.463170) / .05265, or 10.196201 (P + A)[1 + (r/s)(i/2)] = (r'/s)(C/2) + (C/2)a, + 100 vn or (P + 2.596467)[1 + (92/184)(.1053/2)] = (183/ 184)(10.50/2) + (10.50/2)(10.196201) + 100(.463170) (1) (P + 2.596467)[1 + .026325] = 5.221467 + 53.530055 + 46.3170 (2) (P+2.596467)[1.026325]=105.068522 (3) (P+2.596467)+105.068522 +1.026325 (4) (P+2.596467)=102.373539 (5) P=102.373539-2.596467 (6) P=99.777072 (7) P=99.777 G. For fixed-principal securities reopened during the fractional portion (initial short period) of a long first payment period: Formula: (P+A)[1+(r/s)(i/2)]=[(r'/s)(C/2)]v+(C/2)a,+100 vn 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 94% 6-year 2-month note due December 15, 1994, originally issued on October 15, 1988, and reopened on November 15, 1988, with interest payments on June 15 and December 15 (first payment on June 15, 1989), solve for the price per 100 (P) at a yield of 9.79%. Accrued interest is calculated from October 15 to November 15. Definitions: C=9.75 i=.0979 n=12 r-30 (November 15, 1988, to December 15, 1988) V" [1/(1+.0979/2)]12, or .563563 Resolution: A adj SA = inflation adjusted accrued interest; A x Index RatioDate settlement amount including accrued interest in current dollars per $100 original principal; Padj + Aadj r = days from settlement date to next coupon date 8 = 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, Padj= P x Index RatioDate A = [(sr)/s] × (C/2) Aadj = A x Index RatioDate Index RatioDate = Ref CPIDate/Ref CPIssue Date Example. The Treasury issues a 10-year inflation-indexed note on July 15, 1996. The note is issued at a discount to yield 3.1% (real). The note bears a 3% real coupon, payable on January 15 and July 15 of each year. The base CPI index applicable to this note is 120.1 Calculate the settlement amount. C = 3.00 Definitions: 'This number is normally derived using the interpolative process described in appendix B, section I, paragraph B. A = [(sr)/s] × (C/2) Aadj = A x Index RatioDate SA = Padi + Aadj Index RatioDate = Ref CPIDate/Ref CPIIssue Date Example. A 3% 10-year inflation-indexed note was issued July 15, 1996, due July 15, 2006, with interest payments on January 15 and July 15. For a reopening on April 15, 1997, with inflation compensation accruing from July 15, 1996 to April 15, 1997, and accrued interest accruing from January 15, 1997 to April 15, 1997 (90 days), solve for the price per 100 (P) at a real yield, as determined in the reopening auction, of 3.40%. The base index applicable to the issue date of this note is 120 and the reference CPI applicable to April 15, 1997, is 132. Definitions: C = 3.00 i = 0.0340 n = 18 r = 91 (April 15, 1997 to July 15, 1997) s = 181 (January 15, 1997 to July 15, 1997) Ref CPIDate = 132 Ref CPIIssue Date = 120 Resolution: Index RatioDate = Ref CPIDate/Ref CPIIssue Date = 132/120 = 1.100 vn = · 1/(1 + i/2)" = 1/(1 + .0340/2) 18 = 0.73828296 a1=(1-v" (i/2)=(1−0.73828296)/(.0340/2) = 15.39512000 |