Chapter Six Chapter 4, Part Bods, Bod Prices, Iterest Rates ad Holdig Period Retur Bod Prices 1. Zero-coupo or discout bod Promise a sigle paymet o a future date Example: Treasury bill. Coupo bod periodic iterest paymets + pricipal repaymet at maturity Example: U.S. Treasury Bods ad most corporate bods 3. Cosol periodic iterest paymets forever, pricipal ever repaid Example: U.K. govermet has some outstadig Zero-Coupo Bods Straightforward type of bod. U.S. Treasury bills (T-bills) are a good example - moey market istrumets that mature is less tha a year. Each T-bill represets a promise by the U.S. govermet to pay $00 at maturity o a fixed future date. No coupo paymets - zero-coupo bods Also called discout bods sice the bod price is less tha face value - they sell at a discout. $ FV P (1 i) 1
Zero-Coupo Bods How to price a $00 face value zero-coupo bod - $00 P (1 i) Assume i = 5%: Price of a Oe-Year Treasury Bill 00 $95.38 (1 0.05) Price of a Six-Moth Treasury Bill 00 (1 0.05) 1/ $975.90 Remember uits must match: aual iterest rate, therefore must be aual i the case ½ of a year. Zero-Coupo Bods Give the Price ad the Face Value, we ca compute the iterest rate usig the preset value formula. Suppose a 1-year T-Bill has a face value of $00 ad the price is $950. i = FV P -1 i = ($00/$950) - 1 = 0.056 = 5.6%. Price of a Coupo Bod C C C C FV PCB... 1 3 (1 i) Preset Value of Coupo Paymets P CB = Preset Value of yearly coupo paymets (C) + Preset Value of the Face Value (FV), where: i = iterest rate = time to maturity C is cotractually fixed. Preset Value of Pricipal Paymet
Example: Price of a %, -year Coupo Bod Coupo Paymet (C) =$0, Face value (FV) = $1,000, ad = time to maturity $0 PB 1 (1 i) $0 (1 i) $0... (1 i) $00 (1 i) Give values for i ad, we ca determie the bod price P CB Defiitio: Coupo Rate = Coupo Paymet / Face Value Example, Price of a -year Coupo Bod If =, i = 0., C = $0 ad FV = $00. P CB $0 $0 $0 $00 $00... 1 (1.1) (1.1) (1.1) (1.1) What s the Coupo Rate i this example? Excel Example Price of a -year Coupo Bod If =, i = 0.1, C = $0 ad FV = $00. P CB $0 $0 $0 $00 $887... 1 (1.1) (1.1) (1.1) (1.1) i goes up ad P CB goes dow What s the coupo rate for this bod? 3
Yield to Maturity -YTM yield to maturity: The yield bodholders receive if they hold the bod to its maturity whe the fial pricipal paymet is made. Simple example: Suppose a $0 face value, 1-yr, 5% coupo bod sells for $0: $5 $0 $5 P $0 (1 i) (1 i) (1 i) The value of i that solves the equatio is the yield to maturity (YTM). Yield to Maturity -YTM Example: 1 year, $0 FV, 5% coupo bod sellig for $99 Yield to maturity for this bod is 6.06 percet foud as the solutio to: $5 $99 (1 i) $5 ( 1 i) 1.0606 i 1.0606 1.0606 $99 Yield to Maturity - YTM If you pay $99 for a $0 face value bod, you will receive both the iterest paymet ad the icrease i value from $99 to $0. This rise i value is referred to as a capital gai ad is part of the retur o your ivestmet. Whe the price of a bod is higher tha face value, the bodholder icurs a capital loss. 4
Curret yield is C/P: Curret Yield 1 year, $0 FV, 5% coupo bod sellig for $99: 5 Curret Yield = 0.0505, or 5.05% 99 Recall: $5 YTM 1.0606 i 1.0606 1.0606, or 6.06% $99 6-13 YTM: -year Coupo Bod Suppose =, P B = $950, C = $0 ad FV = $00. P CB $0 $0 $0 $00 $950... 1 (1 i) What is the Coupo Rate? What is the Curret Yield? What s the YTM? - more complicated YTM =.85 or.85% Approx. YTM FV P C FV P C = the coupo paymet FV = Face Value P = Price = years to maturity 5
Usig the Approximatio Formula Previous example: =, P B = $950, C = $0 ad FV = $00. Curret Yield Yearly Coupo Paymet Price Paid For our -year, $00 FV bod with a $0 coupo paymet sellig at $950: Coupo rate = % Curret Yield =.5% YTM =.85% Approx. YTM =.77% Whe the coupo bod is priced at its face value, the yield to maturity equals the coupo rate The price of a coupo bod ad the yield to maturity are egatively related The yield to maturity is greater tha the coupo rate whe the bod price is below its face value 6
More o Zero Coupo or Discout Bods Defiitio: A discout bod is sold at some price P, ad pays a larger amout (FV) after t years. There is o periodic iterest paymet. Let P B = price of the bod, i= iterest rate, = years to maturity, ad FV = Face Value (the value at maturity): P B FV ( 1i) Zero Coupo Bods - Price Price of a Oe-Year Treasury Bill at 4% ad FV = $1,000: P B 00 $961.53 (1 0.04) Price of a Six-Moth Treasury Bill at 4% ad FV = $1,000: P B 00 (1 0.04) 1/ $980.58 Price of a 0-Year zero coupo bod at 8% ad FV = $0,000: $0000 P $4,90.96 0 (1 0.08) B YTM - Zero Coupo Bods P FV ( 1i) FV (1 i) P (1/ ) (1/ ) FV FV 1 i i 1 P P 7
Zero Coupo Bods - YTM For a discout bod with FV = $15,000 ad P = $4,00, ad = 0, the iterest rate (or yield to maturity) would be: (1/ ) FV i 1 P i 15,000 4,00 (1/ 0) 1 i = 1.0657-1=> i =6.57% Note: This is the formula for compoud aual rate of growth Zero Coupo Bods - YTM For a discout bod with FV = $,000 ad P = $6,491, ad = 7, the iterest rate (or yield to maturity) would be:,000 i 6,491 (1/ 7) 1 i = 1.06368 1 =.06368 or 6.368% http://olie.wsj.com/mdc/public/p age/_300-tstrips.html From a Coupo Bod to Zero Coupo Bods Zero Coupo Bods are called Strips here s why. C C PB 1 (1 i) (1 i) C (1 i) 3 C FV... (1 i) (1 i) Create +1 discout bods http://www.treasurydirect.gov/istit /marketables/strips/strips.htm 8
Cosols Cosols or perpetuities, are like coupo bods whose coupo paymets last forever. The borrower pays oly iterest, ever repayig the pricipal. The U.S. govermet sold cosols oce i 1900, but the Treasury has bought them all back. The price of a cosol is the preset value of all future coupo paymets. P Cosol Yearly Coupo Paymet i Yearly Coupo Paymet i YTM P cosol Holdig Period Retur The holdig period retur is the retur to holdig a bod ad sellig it before maturity. The holdig period retur ca differ from the yield to maturity. Oe Year Holdig Period Retur Example: year bod 6% coupo rate Purchase at face value, $0 Hold for oe year ad the sell it What additioal iformatio do you eed to aswer this questio? 9
Holdig Period Retur Suppose market iterest rates at the time of the sale fall to 5%? 1-yr Holdig Period Retur = $6 $0 $7.11 $0 $13.11 0.1311 $0 $0 Curret Yield + Capital Gai The ivestor eared $13.11 o a $0 ivestmet. The 1-yr Holdig Period Retur = 13.11% Holdig Period Retur What if market iterest rates at the time of the sale rise to 7%? $6 $0 $93.48 $0 $0.5 0.005 $0 $0 1-yr Holdig Period Retur = - 0.5% Holdig Period Retur The oe-year holdig period retur is the sum of the yearly coupo paymet divided by the price paid for the bod ad the chage i the price divided by the price paid. Yearly Coupo Paymet Price Paid Chage i Price of the Bod Price of the Bod = Curret Yield + Capital Gai (as a %)
Holdig Period Retur You purchase coupo bod ad sell oe year later. RET ( t t 1) C ( Pt 1 Pt) C Pt 1 Pt P P P t t t Curret Yield Capital Gai Aother Example You purchase a -year, % coupo bod, face value = $1,000. If the iterest rate oe year later is the same at %: Oe year holdig period retur = $0 $00 $00 $00 $00 or.0% $0 $00. Holdig Period Retur If the iterest rate oe year later is lower, say at 8%: Oe year holdig period retur = $0 $115 $00 $5.5 $00 $00 $00 or.5% Where did the $115 come from? I the previous example, where did the $7.11 come from? 11
Holdig Period Returs If the iterest rate i oe year is higher at 1%: Oe year holdig period retur = $0 $00 or -.70% $893 $00 $00 $7 $00.007 Where did the $893 come from. You eed to kow how to calculate the $115 ad $893 Key Coclusios From Table The retur equals the yield to maturity (YTM) oly if the holdig period equals the time to maturity. A rise i iterest rates is associated with a fall i bod prices, resultig i a capital loss if the holdig period is less tha the time to maturity. The greater the percetage price chage associated with a iterest-rate chage, the more distat the maturity. 1
Iterest-Rate Risk Chage i bod price due to chage i iterest rate Prices ad returs for log-term bods are more volatile tha those for shorter-term bods There is o iterest-rate risk for a bod whose time to maturity matches the holdig period Reivestmet (iterest rate) Risk If ivestor s holdig period exceeds the term to maturity proceeds from sale of bod are reivested at ew iterest rate the ivestor is exposed to reivestmet risk The ivestor beefits from risig iterest rates, ad suffers from fallig iterest rates 13