Econ 102 Alan DeardorffWinter Term 1999 Bond Price HandoutPage 1 of 4Bond Prices and Interest RatesA bond is an IOU. That is, a bond is a promise to pay, in the future, fixed amounts thatare stated on the bond. The interest rate that a bond actually pays therefore depends onhow these payments compare to the price that is paid for the bond.1 That price isdetermined in a market so as to equate the implicit rate of interest paid on the bond to therate of interest that buyers could get on other bonds of comparable risk and time tomaturity. Figuring out what the interest rate on a bond is can be a quite tricky, since mostbonds make payments for several years and of different sizes. Less tricky is to go theother direction, from the interest rate to the price of the bond.This handout will work through two examples of how bond prices and interest rates wouldvary for two particularly simple kinds of bonds. Then it will provide the general formulafor the price of a bond.Example 1: A One-Year BondConsider a bond – I’ll call it B1 – with principal equal to $1000 and interest payment of$70, so that the bond is a promise to pay the principal plus interest, or $1000+$70=$1070,in one year. If the price of the bond is $1000, then clearly it is paying an interest rate forthat year of 7%. By buying the bond you would be, in effect, lending $1000 (the price ofthe bond), and getting repaid one year later both the amount that you lent (the principal)and interest of $70, which as a percent of what you lent is 70/1000 or 7%. That is, lettingPB1 be the price of the bond and i be the implicit interest rate, thenIf , then P iB11070 10001000701000007 7%= =−= = =$1000 .Now suppose instead that the price of the bond were higher – that you had to pay, say,$1010 for it. Then you would be lending more, but getting back the same, so that thepercentage return would be lower. How much lower? We can figure that out as follows:If , then P iB11070 10101010601010594%= =−= =$1010 .Notice that the price of the bond goes into the formula for i in two places. It is in the topof the fraction because, the higher is the price, the less will be the extra that you get back 1 Bonds routinely also have an interest rate stated on the bond itself, but this is hardly everthe actual interest rate. It would be the actual interest rate only if the price of the bondwere its face value – i.e., its principal – which it almost never is. The interest rate writtenon the bond itself is therefore pretty much meaningless, indicative perhaps of what theissuers of the bond hoped or expected the market rate would be.Econ 102 Alan DeardorffWinter Term 1999 Bond Price HandoutPage 2 of 4over and above what you paid in the first place. It is also in the bottom of the fractionbecause we want to calculate what you’ve earned as a percentage of what you paid.What if the price of the bond were less – say $990? ThenIf , then P iB11070 99099080990808%= =−= =$990 .One of the things this tells us is that, the higher is the price of the bond, the lower is theinterest rate that it pays. The reason is simply that the payment is fixed while the pricechanges. Since this is true also of more complicated bonds, it is a general property ofbond prices and interest rates. The higher are bond prices, the lower are interestrates, and vice versa.Suppose now that we do not know the price of the bond, but that we do know that othercomparable bonds are paying an interest rate of 5%. Then what must the price of thisbond be in order for it also to pay 5%? We can set up the same formula that we usedabove, but this time we know i and we don’t know PB1: iPPBB= = =−5% 005107011.We can solve the last equation here for PB1 by first multiplying both sides of it by PB1, thenadding it to both sides and solving: 005 1070005 1070105 107010701051 11 111....$1019P PP PPPB BB BBB=−+ === =Therefore, if other comparable bonds (similar risk and time to maturity) are paying 5%interest, then this bond will have to sell on the market for $1019 in order to pay the sameinterest rate of 5%.Using this reasoning more generally, any one-year bond that promises to make only asingle payment of $X in one year (called principal plus interest, but that does not matterfor the calculation) will have a price, call it PBX1, that depends on the market interest rate,i, as follows:PXiBX11=+Notice again that the bond price and the interest rate are inversely related: when one rises,the other falls.Econ 102 Alan DeardorffWinter Term 1999 Bond Price HandoutPage 3 of 4Example 2: A PerpetuityA perpetuity is a bond that pays the same amount every year forever, never paying backthe principal. Consider a perpetuity, call it B2, that pays $70 every year forever. Thispayment is the same as the interest payment of the one-year bond above, but here you getit every year. On the other hand, it never pays back the principal. So is it worth more orless than the one-year bond? That, it turns out, depends on the market interest rate.Again we will look at the implicit interest rate that this bond pays for several prices, thenturn this around to see what price is implied by any market interest rate.Suppose the price of the bond were $1000. Then by buying it you would again be lendingout $1000, and then you would get back interest payments every year of $70. Since $70is 7% of $1000, it seems clear that you are earning an interest rate of 7% per year, andthat’s right. The interest rate you get on a perpetuity is just the payment it makes, dividedby the price. In this caseIf , then P iB2701000007 7%= = = =$1000 .Now suppose that the price of the bond were higher, say $1100.2 To buy it you will nowhave to give up more, $1100, but you will get back only the same interest payments of $70a year. The interest rate is therefore lower:If , then P iB27011000064 64%= = = =$1100 . .Similarly, if the bond price were lower, the interest rate would be higher:If , then P iB270500014 14%= = = =$500 .In general, if a perpetuity pays $X per year, then its implicit interest rate is just the ratio ofX to its price, PXPerp.iXPXPerp=From this, solving the equation for PXPerp, you can see immediately that the price of aperpetuity is the ratio of the interest payment to the interest rate:PXiXPerp= 2 What would be the implicit interest rate on the one-year bond above if its price were $1100? Does thismake sense?Econ 102 Alan DeardorffWinter Term 1999 Bond Price HandoutPage 4 of 4Present