# YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE

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1 YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999

2 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1 - P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face value P 0 = Pice N = numbe of days until matuity P P 1 0 * N 360 * d = P P P P 1 0 d * N 360 = P P 1 0 = holding peiod etun

3 The invoice on a bond (what you pay) is quoted pice plus accued inteest. 3

4 Govenment Bonds and Notes Calculating Accued Inteest Accued inteest is actual days/actual days Last settlement next coupon date coupon date date x Accued inteest = x/y times inteest payment Example: y (1) 10% coupon () \$100 inteest annually o \$50 semi-annual pe \$1000 face y = 183 x = [50] = \$7.3 4

5 Yield To Matuity 1. One coupon left [like T bill]. At coupon paying date C P o C C C C Pin P 0 = C [1 ] C [1 ] C [1 ] 3 C Pin 4 [1 ] 3. Between Coupon Dates P 0 A = C [1 ] w [1 C ] 1w [1 C ] w C pin [1 ] 3w 5

6 w = faction of yea to fist coupon actual days ove actual days w Note: Ignoe that intevals may be of uneven size because of Satudays o end of month. 6

7 Euobonds - will examine only bonds issued in dollas - inteest paid annually usually Calculating accued inteest uses 30 day months 360 day yeas Example 1: 1. Issue date Januay 8. Settlement date Mach 5 days Januay Febuay Mach [9, 30] 30 day [1,,3,4,5] x inteest = accued inteest 7

8 Example : 1. May 14 issue date. Sept 17 settlement date May 16 J, Ju Aug 90 Sept The 31st is the same as the 30th x inteest = accued inteest 8

9 Computing Yield To Matuity 1. With one payment emaining like T-bill but usually with 30-day 360 calenda.. Multiple payments Pice accued = C [1 ] v C [1 ] v1 C Pin... [1 ] n-1v is an annual ate v is faction of yea until payment and is done on 360 days in yea 30 day month calenda Most have options. We will discuss this late. 9

10 G N M A Definition: bonds issued with motgages as thei backing. Guaanties 1. Issue. If boowe fails to make a scheduled amotization payment in any month, issue makes good. If boowe defaults, issue must pomptly emit emaining motgage.. Govenment. Makes good payments if issue fails. Rate is motgage ate - 50 basis points e.g. 13% motgages 1.5% pool ate 44 basis points to issue 6 basis points to gov. 10

11 Timing of Payments Januay [Home ownes] Febuay [Home ownes] Pass though Mach Pass though If. No Pepayments constant amount paid each month detemining constant amount 100 = M [1 1 ] M [1 1 ]... [1 M 1 ] 360 M is scheduled amotized payment 8 1 m

12 Quoted Pices on GNMA Quoted as % of emaining pincipal balance Assume quoted pice is 95 1 million oiginal Pincipal value x.8 % still outstanding x.95 Quoted Pice \$760,000 Pice Accued inteest on GNMA Settlement day. Two business days afte tade date but fist settlement is usually thid Wednesday fo GNMA less than 9.5% and following Monday fo GNMA 9.5%. Reason: Pool factos not available until 10th of month. Accued inteest coupon 1 x numbe of days fom 1st 30 until settlement 1

13 Example: 13% GNMA 5 million face Feb. l5 settlement date.8 Pool facto Accued Inteest = x [.8x5x 13 1 ] = \$0, Note: Always use 30 days iespective of days in the month. Yield to Matuity Pice Accued = M (1 1 ) M (1 1 ) (1 M 1 )

14 LIBOR Inteest ates ae annual using simple inteest Inteest payment = pincipal x LIBOR x actual days to payment 360 Example: One million dolla deposit made June at Until Decembe we get actual days to payment = 183 inteest payment = 1,000,000 x x = 30,18 Late need semiannual ate. Adjustment is actual days of life 1 = 1 Libo x actual days Pound LIBOR is quoted on 365-day yea 14

15 WHY PROBLEM WITH YIELD TO MATURITY Septembe

16 Reinvestment Assumption (Implicit in Yield to Matuity) Example: Pice = [1.05] 10 [1.05] [1.05] 4 = \$ Assume 1% 10 [1.06] [1.06] [1.06] /(1.05) 4 =

17 Assume 10% Reinvestment Ending Value 10(1.05) 3 10(1.05) 10(1.05) /(1.05) 4 = \$ Analytics: Pice = CF(1) (1 ) CF() (1 ) CF(3) (1 ) 3 CF(4) (1 ) 4 Pice = CF(1)(1 y ) 3 y CF()(1 (1 ) ) 4 CF(3)(1 y ) CF(4) Unless y =, these don't match. 17

18 Diffeent Bonds Assume Diffeent Reinvestment Rates Relative desiability depends on assumption about einvestment Coupon Pincipal Pice Matuity Fequency of payment Yield to matuity Bond A 10% 100 \$ yeas Annual 6% Bond B 3% 100 \$ yeas Annual 6.1% 18

19 Ending Value A B 5.8% Reinvestment Rate Usually one cossing point. 19

20 Yield to matuity on potfolio not weighted aveage on bonds that ae in potfolio. Illustating the Nonadditivity of Yields Bond Outlay (Pice) Peiods 1 3 Yield to Matuity Weighted Aveage Yield A B C A B B C A C % 6.00% 1.35% 11.9% 9.65% 13.71% 10.50% 9.04% 13.73% 0

21 Review I. Tems a. Accued inteest b. Quoted pice c. Invoice pice d. Yield to matuity II. Concepts a. That accued inteest calculations diffe acoss types of bonds b. That yield to matuity is a poo measue of elative value III. Calculations a. Accued inteest b. Yield to matuity 1

22 Poblems Assume that the settlement date is Decembe 1, 1997 and that the bond pays inteest on Septembe 15 and Mach 15. Futhe assume that the bond matues the following Septembe. The coupon on the bond is ten pecent and the face value is \$1, If the bond is a U.S. Teasuy bond, what is the accued inteest? Answe: Fo Teasuies, accued inteest is actual ove actual Since last inteest payment Septembe 15 Octobe 31 Novembe 30 Decembe 1 77 In Peiod Septembe 15 Octobe 31 Novembe 30 Decembe 31 Januay 31 Febuay 8 Mach

23 Accued inteest x \$50 = \$1.7. Assume the bond is a copoate. What is the accued inteest? Answe: Fo copoate bonds, months ae 30 days and 360- day yeas. Since last payment Septembe 15 Octobe 30 Novembe 30 Decembe 1 76 In peiod Septembe 15 Octobe 30 Novembe 30 Decembe 30 Januay 30 Febuay 30 Mach x \$50 = \$1.7 3

24 3. Assume bond is Italian govenment bond. What is the accued inteest? Answe: Same as. 4. What is the yield to matuity fo the bond in one if its pice is 99? Answe: Invoice Pice = = = % = 50 (1 ) (1 )

25 Accued Inteest Calculations Conventions Diffeent makets have diffeent conventions fo calculating accued inteest. The method used in each maket is denoted by specifying the method of calculating two values: d A y is the numbe of days fom the pevious coupon payment date to settlement date (o fom issue date to settlement date, if the next coupon payment if the fist); that is, the numbe of days ove which inteest has accued. is the assumed numbe of days in one yea. The paticula convention used can be Actual/Actual, Actual/365, Actual/360, o 30E/360. In the name of the convention, the fist pat of the name denotes the method of computing d; and the second pat of the name denotes the method of calculating A y. Computing the Accued Inteest Once d and A y have been calculated, accued inteest is computed by the fomula: I A is the accued inteest C is the annual coupon payment I A d = C A y Calculating d The thee methods of calculating d ae: 1. "Actual" - Calculate the actual numbe of days fom the pevious coupon payment date to the settlement date. 5

26 Examples: Thee ae 10 days fom 1/3/93 to 1/13/93 Thee ae 41 days fom 1/3/93 to /13/93 Thee ae 31 days fom 1/1/93 to /1/93 Thee ae 8 days fom /1/93 to 3/1/93 Thee ae 9 days fom /1/9 to 3/1/9. "30" - Calculate the numbe of days fom the pevious coupon payment date to the settlement date by assuming 30-day months, as follows: Let the two dates by M 1 /D 1 /Y 1 and M /D /Y. If D 1 is 31, change it to 30. If D is 31, change it to fist of next month. Then d = 360(Y -Y 1 ) 30(M -M 1 ) (D -D 1 ). Examples: (Note: Accoding to this fomula, Febuay will always appea to be a 30-day month). Thee ae 10 days fom 1/3/93 to 1/13/93 Thee ae 30 days fom 1/1/93 to /1/93 Thee ae 30 days fom /1/93 to 3/1/93 Thee ae 30 days fom /1/9 to 3/1/9 Thee ae 4 days fom /7/93 to 3/1/93 Thee ae 1 days fom 5/30/93 to 5/31/93 Thee ae days fom 5/9/93 to 5/31/93 This "30" method is used in the U.S. agencies, copoates, and municipals makets. 6

27 3. "30E" - Calculate the numbe of days fom the pevious coupon payment date to the settlement date by assuming 30-day months, as follows: Let the two dates by M 1 /D 1 /Y 1 and M /D /Y. If D 1 is 31, change it to 30. If D is 31, change it to 30. Then d = 360(Y -Y 1 ) 30(M -M 1 ) (D -D 1 ). Examples: (Note: Accoding to this fomula, Febuay will always appea to be a 30-day month.) Thee ae 30 days fom /1/93 to 3/1/93 Thee ae 30 days fom /1/9 to 3/1/9 Thee ae 4 days fom /7/93 to 3/1/93 Thee ae 0 days fom 5/30/93 to 5/31/93 Thee is 1 day fom 5/9/93 to 5/31/93 This method, slightly diffeent fom the "30" method above, is used in the Euobond maket and many Euopean govenment and copoate makets. Calculating A y The thee method of calculating A y ae: (1) "365" - A y is equal to 365. () "360" - A y is equal to 360. (3) "Actual - A y is equal to the numbe of days in the cuent coupon peiod times the numbe of coupon payments pe yea. Fo a semi-annual coupon, the numbe of days in the coupon peiod can ange fom 181 to 184, so A y can ange fom 36 to 368. Examples: (1) A 6.75% bond paying semi-annually is taded to settle on July, The pevious coupon paid on Mach 15, 1993, and the next coupon pays on Septembe 15, If the bond accues accoding to the Actual/Actual convention, what is the accued inteest at settlement? Thee ae 184 actual days in the cuent coupon peiod, and thee ae 109 actual days fom the last coupon payment to settlement. Theefoe, the accued inteest is: 109 x 6.75= pe 100 of x184 face value () Assume that the bond in example (1) accues accoding to the 30/360 convention, instead of Actual/Actual. What would be the accued inteest? Thee ae 107 "30" days fom the last coupon payment to settlement. Theefoe, the accued inteest is: 7

28 Accued Inteest - Maket Conventions x 675= pe 100 of face value Instument Type Accual Convention Coupons pe Yea Notes Domestic: US Govt. Teasuy Bonds US Govt. Agency Bonds Copoate Bonds Municipal Bonds GNMA/FNMA/FLHMC Actual/Actual 30/360 30/360 30/360 Actual/360 1 Intenational: Euobonds Geman Govt. and Cop. Bonds Italian Govt. and Cop. Bonds Japanese Govt. and Cop. Bonds Swiss Govt. and Cop. Bonds UK Govt. and Cop. Bonds 30E/360 30E/360 30E/360 Actual/365 30/360 Actual/ Notes: 1. In the Japanese makets, Febuay 9 is neve counted. 8

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