Bond Prices and Interest Rates
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1 Winer erm 1999 Bond rice Handou age 1 of 4 Bond rices and Ineres Raes A bond is an IOU. ha is, a bond is a promise o pay, in he fuure, fixed amouns ha are saed on he bond. he ineres rae ha a bond acually pays herefore depends on how hese paymens compare o he price ha is paid for he bond. 1 ha price is deermined in a marke so as o equae he implici rae of ineres paid on he bond o he rae of ineres ha buyers could ge on oher bonds of comparable risk and ime o mauriy. Figuring ou wha he ineres rae on a bond is can be a quie ricky, since mos bonds make paymens for several years and of differen sizes. Less ricky is o go he oher direcion, from he ineres rae o he price of he bond. his handou will work hrough wo examples of how bond prices and ineres raes would vary for wo paricularly simple kinds of bonds. hen i will provide he general formula for he price of a bond. Example 1: A One-Year Bond Consider a bond I ll call i wih principal equal o $1000 and ineres paymen of $, so ha he bond is a promise o pay he principal plus ineres, or $1000+$=$10, in one year. he price of he bond is $1000, hen clearly i is paying an ineres rae for ha year of 7%. By buying he bond you would be, in effec, lending $1000 (he price of he bond), and geing repaid one year laer boh he amoun ha you len (he principal) and ineres of $, which as a percen of wha you len is /1000 or 7%. ha is, leing be he price of he bond and i be he implici ineres rae, hen = $1000, hen i = = = = 7% Now suppose insead ha he price of he bond were higher ha you had o pay, say, $1010 for i. hen you would be lending more, bu geing back he same, so ha he percenage reurn would be lower. How much lower? We can figure ha ou as follows: = $1010, hen i = = = 594% Noice ha he price of he bond goes ino he formula for i in wo places. I is in he op of he fracion because, he higher is he price, he less will be he exra ha you ge back 1 Bonds rouinely also have an ineres rae saed on he bond iself, bu his is hardly ever he acual ineres rae. I would be he acual ineres rae only if he price of he bond were is face value i.e., is principal which i almos never is. he ineres rae wrien on he bond iself is herefore prey much meaningless, indicaive perhaps of wha he issuers of he bond hoped or expeced he marke rae would be.
2 Winer erm 1999 Bond rice Handou age of 4 over and above wha you paid in he firs place. I is also in he boom of he fracion because we wan o calculae wha you ve earned as a percenage of wha you paid. Wha if he price of he bond were less say $990? hen = $990, hen i = = = 8. 08% One of he hings his ells us is ha, he higher is he price of he bond, he lower is he ineres rae ha i pays. he reason is simply ha he paymen is fixed while he price changes. Since his is rue also of more complicaed bonds, i is a general propery of bond prices and ineres raes. he higher are bond prices, he lower are ineres raes, and vice versa. Suppose now ha we do no know he price of he bond, bu ha we do know ha oher comparable bonds are paying an ineres rae of 5%. hen wha mus he price of his bond be in order for i also o pay 5%? We can se up he same formula ha we used above, bu his ime we know i and we don know : B i = 5% = 0 05 = We can solve he las equaion here for by firs muliplying boh sides of i by, hen adding i o boh sides and solving: = = = = = $ herefore, if oher comparable bonds (similar risk and ime o mauriy) are paying 5% ineres, hen his bond will have o sell on he marke for $1019 in order o pay he same ineres rae of 5%. Using his reasoning more generally, any one-year bond ha promises o make only a single paymen of $ in one year (called principal plus ineres, bu ha does no maer for he calculaion) will have a price, call i B1, ha depends on he marke ineres rae, i, as follows: B1 = 1 + i Noice again ha he bond price and he ineres rae are inversely relaed: when one rises, he oher falls.
3 Winer erm 1999 Example : A erpeuiy Bond rice Handou age 3 of 4 A perpeuiy is a bond ha pays he same amoun every year forever, never paying back he principal. Consider a perpeuiy, call i B, ha pays $ every year forever. his paymen is he same as he ineres paymen of he one-year bond above, bu here you ge i every year. On he oher hand, i never pays back he principal. So is i worh more or less han he one-year bond? ha, i urns ou, depends on he marke ineres rae. Again we will look a he implici ineres rae ha his bond pays for several prices, hen urn his around o see wha price is implied by any marke ineres rae. Suppose he price of he bond were $1000. hen by buying i you would again be lending ou $1000, and hen you would ge back ineres paymens every year of $. Since $ is 7% of $1000, i seems clear ha you are earning an ineres rae of 7% per year, and ha s righ. he ineres rae you ge on a perpeuiy is jus he paymen i makes, divided by he price. In his case B = $1000, hen i = = = 7% 1000 Now suppose ha he price of he bond were higher, say $1100. o buy i you will now have o give up more, $1100, bu you will ge back only he same ineres paymens of $ a year. he ineres rae is herefore lower: B = $1100, hen i = = = 6. 4% 1100 Similarly, if he bond price were lower, he ineres rae would be higher: B = $500, hen i = = 014. = 14% 500 In general, if a perpeuiy pays $ per year, hen is implici ineres rae is jus he raio of o is price, erp. i = erp From his, solving he equaion for erp, you can see immediaely ha he price of a perpeuiy is he raio of he ineres paymen o he ineres rae: erp = i Wha would be he implici ineres rae on he one-year bond above if is price were $1100? Does his make sense?
4 Winer erm 1999 resen Value Bond rice Handou age 4 of 4 All of his is really jus special cases of he more general idea of presen value. he presen value of an amoun of money ha will be paid in he fuure any fuure paymen or receip is defined as he amoun ha could be len oday o ge back ha amoun in he fuure (or equivalenly he amoun ha could be borrowed oday and paid back wih ha money in he fuure) a he prevailing ineres rae. We ve already seen ha he formula for he price of a one-year bond is really, hen, jus he presen value of wha he bond will pay back one year from now. ha is, if you lend an amoun Y oday for one year a ineres rae i, you will ge back 1 =Y+iY=(1+Y a year from now. hus 1 Y = is he presen value of paymen, 1, one year in he fuure. By similar reasoning, if you lend ou an amoun Y oday for wo years, i is as hough you len i ou wice, bu geing ineres on he ineres he second ime. ha is, you ge back 1 afer he firs year, hen = 1 +i 1 afer he second. Now =(1+ 1 =(1+(1+Y=(1+ Y. herefore Y = is he presen value of a paymen wo years from now. In general, for any sream of paymens = 1,,,, for any number of years, he presen value of is given by he formula 1 V ( ) = More simply, for hose who are familiar wih he symbol Σ for a summaion, V ( ) = We now see wha bond prices really are: he price of a bond is he presen value of he paymens promised by he bond. a bond promises a sream of paymens as above, hen he price of he bond is V(). Acual bonds ypically promise a fixed ineres paymen, called he coupon paymen, C, each year unil mauriy, hen pay back he enire principal, 0, in he year he bond maures. he erm o mauriy of he bond is denoed, hen he price (presen value) of he bond is B C = 0 + ( 1+ i ) i ) = 1 C C C 0 =
5 Winer erm 1999 Bond rice Handou age 5 of 4 Noice again ha because he ineres rae, i, is always in he boom of hese fracions, if he ineres rae goes up, he price of he bond will go down, and vice versa.
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