CHAPTER 2PRICING OF BONDSCHAPTER SUMMAR YThis chapter will focus on the time value of money and how to calculate the price of a bond. When pricing a bond it is necessary to estimate the expected cash flows and determ ine the appropriate yield at which to disco unt the expected cash flows. Among other aspects of a bond, we will look at the reas ons why the price of a bond cha ngesREVIEW OF TIME VALUE OF MONEYMoney has time value because of the opportunity to invest it at some interest rate.Future ValueThe future value of any sum of money in vested today is:P n = P o(1 + r)nwhere n = number of periods, P n = future value n periods from now (in dollars), P0 = orig inal prin cipal (in dollars), r = in terest rate per period (in decimal form), and the expressi on (1+ r)n represe nts the future value of $1 in vested today for n periods at a compo unding rate of r.When interest is paid more than one time per year, both the interest rate and thenu mber of periods used to compute the future value must be adjusted as follows:r = annual interest rate / number of times interest paid peryear, andn = number of times interest paid per year times number of years.The higher future value whe n in terest is paid semia nnu ally, as opposed to annu ally, reflectsthe greater opport unity for reinv esti ng the in terest paid. Future Value of an Ordi nary Ann uityWhen the same amount of money is invested periodically, it is referred to as an annuity.When the first investment occurs one period from now, it is referred to as an ordinary annu ity.The equation for the future value of an ordinary annuity is:where A is the amount of the annuity (in dollars).Example of Future Value of an Ordinary Annuity Using Annual Interest:If A = $2,000,000, r = 0.08, and n = 15, thenBecause 15($2,000,000) = $30,000,000 of this future value represents the total dollar amountof annual in terest payme nts made by the issuer and inv ested by the portfolio manager, thebalanee of $54,304,250 - $30,000,000 = $24,304,250 is the interest earned by reinv esti ngthese annual in terest payme nts.P n = A I (1 +r $ -1 I■. r 」$2,000,000 3.17217 -1IL 0.08 =$2,000,000[27.152125] = $54,304.250.P 15 =Example of Future Value of an Ordinary Annuity Using Semiannual Interest:Con sider the same example, but now we assume semia nnual in terest payme nts.If A = $2,000,000 / 2 = $1,000,000, r = 0.08 / 2 = 0.04, n = 2(15) = 30, the n$1,000,000[56.085] = $56,085,000. The opport unity for more freque nt rein vestme nt of in terest payme nts received makes thein terest earned of $26,085,000 from rein vesti ng the in terest payme nts greater tha n the$24,304,250 interest earned when interest is paid only one time per year. Prese nt ValueThe prese nt value is the future value process in reverse. We have:PVFor a give n future value at a specified time in the future, the higher the in terest rate (ordiscount rate), the lower the present value. For a given interest rate (discount rate), the furtherinto the future that the future value will be received, then the lower its present value. Prese nt Value of a Series of Future ValuesTo determ ine the prese nt value of a series of future values, the prese nt value of eachP n = A I (1 +r $ -1 P30=$1,000'000 ]忖] $1,000,000 「3.2434-1 ]= IL 0.04future value must first be computed. Then these prese nt values are added together to obta inthe prese nt value of the en tire series of future values. Prese nt Value of an Ordi nary Annu ityWhe n the same dollar amount of money is received each period or paid each year, the seriesis referred to as an annuity. When the first payment is received one period from now, theannuity is called an ordinary annuity. When the first payment is immediate, the ann uity iscalled an ann uity due.The prese nt value of an ordinary ann uity is:where A is the amount of the annuity (in dollars).The term in brackets is the prese nt value of an ordi nary annuity of $1 for n periods.Example of Present Value of an Ordinary Annuity Using Annual Interest:value formula is modified so that (i) the annual in terest rate is divided by the freque ncy peryear, and (ii) the number of periods when the future value will be received isadjusted by multiplying the number of years by the frequency per year.PRICING A BONDPV = A卜(1+r )nIf A = $100, r = 0.09, and n = 8, then: - 1-1 1 - 1- 1 1 PV = A ■ (1 + r y =$100 . (1.09 丫 r 一 0.09 一 $100 $100[5.534811] = $553.48.1.99256 -0.09Determining the price of any financial instrument requires an estimate of (i) the expected cashflows, and (ii) the appropriate required yield. The required yield reflects the yield for finan cial instrume nts with comparable risk, or alter native inv estme nts.The cash flows for a bond that the issuer cannot retire prior to its stated maturity date con sistof periodic coup on in terest payme nts to the maturity date, and the par (or maturity) value atmaturity.In gen eral, the price of a bond can be computed using the follow ing formula:where P = price (in dollars), n = number of periods (number of years times 2), C =semia nnual coup on payme nt (in dollars), r = periodic in terest rate (required annual yielddivided by 2), M = maturity value, and t = time period whe n the payme nt is to be received.Computing the Value of a Bond: An Example:Con sider a 20-year 10% coup on bond with a par value of $1,000 and a required yield of 11%.Given C = 0.1($1,000) / 2 = $50, n = 2(20) = 40 and r = 0.11 / 2 = 0.055, the prese nt value ofthe coup on payme nts is:t=1C t + M1 r 七 1 + r n$50 16.046131 = $802.31.The prese nt value of the par or maturity value of $1,000 is: =迪=$117.46. 8.51331 The price of the bond (P) = prese nt value coup on payme nts + prese nt value maturity value =$802.31 + $117.46 = $919.77. Prici ng Zero-Coup on BondsFor zero-coup on bon ds, the inv estor realizes in terest as the differe nee betwee n thematurity value and the purchase price. The equatio n is: M工 n(1 +r ) where M is the maturity value. Thus, the price of a zero-coup on bond is simply the prese ntvalue of the maturity value. Zero-Coup on Bond ExampleCon sider the price of a zero-coup on bond that matures 15 years from now, if thematurity value is $1,000 and the required yield is 9.4%. Given M = $1,000, r = 0.094 /2 = 0.047, and n = 2(15) = 30, we have: P J =“1,00： = $1,000 = $252.12. (1+r ：n (1.047 F 3.99644 Price-Yield Relati on shipA fun dame ntal property of a bond is that its price cha nges in the opposite directi on from thecha nge in the required yield. The reas on is that the price of the bond is the prese nt value of$50匕空竺= IL 0.055$1,00040 (1.055)the cash flows.Relati on ship Betwee n Coup on Rate, Required Yield, and PriceWhe n yields in the marketplace rise above the coup on rate at a give n point in time , the price of the bond falls so that an investor buying the bond can realizes capital appreciation. The appreciation represents a form of interest to a new investor to compe nsate for a coup on rate that is lower tha n the required yield. Whe n a bond sells below its par value, it is said to be selling at a discount. A bond whose price is above its par value is said to be selling at a premium.Relation ship Betwee n Bond Price and Time if In terest Rates Are UnchangedFor a bond selli ng at par value, the coup on rate is equal to the required yield. As the bond moves closer to maturity, the bond will continue to sell at par value. Its price will rema in con sta nt as the bond moves toward the maturity date.The price of a bond will not rema in con sta nt for a bond selli ng at a premium or a disco unt. The disco unt bond in creases in price as it approaches maturity, assu ming that the required yield does not cha nge. For a premium bond, the opposite occurs. For both bon ds, the price will equal par value at the maturity date.Reas ons for the Change in the Price of a BondThe price of a bond can cha nge for three reas ons: (i) there is a cha nge in the required yield owing to cha nges in the credit quality of the issuer; (ii) there is a cha nge in the price of the bond selling at a premium or a discount, without any change in the required yield, simply because the bond is movi ng toward maturity; or, (iii) there is a cha nge in the required yield owing to a cha nge in the yield on comparable bonds (i.e., a cha nge in the yield required by the market).COMPLICATIONSThe framework for prici ng a bond assumes the follow ing: (i) the n ext coup on payme nt isexactly six mon ths away; (ii) the cash flows are known; (iii) the appropriate required yield canbe determ in ed; and, (iv) one rate is used to disco unt all cash flows. Next Coup on Payme nt Due in Less tha n Six Mon thsWhe n an inv estor purchases a bond whose n ext coup on payme nt is due in less tha n sixmon ths, the accepted method for computi ng the price of the bond is as follows:where v = (days betwee n settleme nt and n ext coup on) / (days in six-m onth period). Cash Flows May Not Be KnownFor most bon ds, the cash flows are not known with certa in ty. This is because anissuer may call a bond before the stated maturity date. Determi ning the Appropriate Required YieldAll required yields are benchmarked off yields offered by Treasury securities. there, we must still decompose the required yield for a bond into its component parts. One Disco unt Rate Applicable to All Cash FlowsA bond can be viewed as a package of zero-coup on bon ds, in which case a uniquedisco unt rate should be used to determ ine the prese nt value of each cash flow.n C2(1 + 泌1 + 十 + Mv n -J (1 + r) (1 + r) FromPRICING FLOATING-RATE AND INVERSE-FLOATING-RATE SECURITIESThe cash flow is not known for either a float in g-rate or an inv erse-floati ng-rate security; it will depend on the reference rate in the future.Price of a FloaterThe coup on rate of a float in g-rate security (or floater) is equal to a refere nce rate plus some spread or margin. The price of a floater depe nds on (i) the spread over the refere nce rate and (ii) any restricti ons that may be imposed on the resetti ng of the coup on rate.Price of an Inv erse FloaterIn general, an inverse floater is created from a fixed-rate security. The security fromwhich the inv erse floater is created is called the collateral. From the collateral twobonds are created: a floater and an inv erse floater.The price of a floater depe nds on (i) the spread over the refere nee rate and (ii) any restrict ions that may be imposed on the resett ing of the coup on rate. For example, a floater may have a maximum coup on rate called a cap or a minimum coup on ratecalled a floor. The price of a floater will trade close to its par value as long as the spread above the reference rate that the market requires is unchanged, and neither the cap nor the floor is reached.The price of an inverse floater equals the collateral ' s price minus the floater PRICE QUOTES AND ACCRUED INTERESTPrice QuotesA bond selling at par is quoted as 100, meaning 100% of its par value. A bond selling at a discount will be selling for less than 100; a bond selling at a premium will be selling for more tha n 100.Accrued In terestWhe n an inv estor purchases a bond betwee n coup on payme nts, the inv estor must compe nsate the seller of the bond for the coup on in terest earned from the time of the last coup on payme nt to the settleme nt date of the bond. This amount is called accrued in terest. For corporate and muni cipal bon ds, accrued in terest is based on a 360-day year, with each month having 30 days.The amount that the buyer pays the seller is the agreed-upon price plus accrued interest. This is often referred to as the full price or dirty price. The price of a bond without accrued interest is called the clean price.The exceptions are bonds that are in default. Such bonds are said to be quoted flat, that is, without accrued interest.ANSWERS TO QUESTIONS FOR CHAPTER 2(Questions are in bold print followed by answers.)1. A pension fund man ager inv ests $10 milli on in a debt obligati on that promises to pay 7.3% per year for four years. What is the future value of the $10 milli on?To determ ine the future value of any sum of money inv ested today, we can use thefuture value equation, which is: P n = P0 (1 + r)n where n = number of periods, P n =future value n periods from now (in dollars), P o = original principal (in dollars) and r = interest rate per period (in decimal form). Inserting in our values, we have: P4 = $10,000,000(1.073) 4 = $10,000,000(1.325558466) = $13,255,584.66 .2. Suppose that a life in sura nce compa ny has guara nteed a payme nt of $14 millio nto a pension fund 4.5 years from no w. If the life in sura nce compa ny receives apremium of $10.4 million from the pension fund and can invest the entire premium for 4.5 years at an annual interest rate of 6.25%, will it have sufficientfunds from this in vestme nt to meet the $14 millio n obligati on?To determ ine the future value of any sum of money inv ested today, we can use thefuture value equation, which is: P n = P0 (1 + r)n where n = number of periods, P n =future value n periods from now (in dollars), P0 = original principal (in dollars)and r = interest rate per period (in decimal form). Inserting in our values, we have: P4.5 = $10,400,000(1.0625)处=$10,400,000(1.313651676)= $13,661,977.43. Thus, it will beshort $13,661,977.43 -$14,000,000 = -$338,022.57 .3. An swer the below questi ons.(a) The portfolio man ager of a tax-exempt fund is con sideri ng in vest ing $500,000in a debt instrument that pays an annual interest rate of 5.7% for four years. At theend of four years, the portfolio man ager pla ns to reinv est the proceeds for threemore years and expects that for the three-year period, an annual interest rate of 7.2% can be earned. What is the future value of this investment? At the end of year four, the portfolio manager' s aPmo= nPis(givenr)5y ：Inserting in our values, we have P 4 = $500,000(1.057) 4 = $500,000(1.248245382)= $624,122.66. In threemore years at the end of year seven, the manager amount is given by: P 7 = P 4(1 + r)3. Inserting in our values, we have: P 7 = $624,122.66(1.072)3 = $624,122.66(1.231925248) = $768,872.47 .(b) Suppose that the portfolio man ager in Questi on 3, part a, has the opport unity to inv est the $500,000 for seve n years in a debt obligati on that promises to pay an annual interest rate of 6.1% compoundedsemiannually.Is this investmentalter native more attractive tha n the one in Questio n 3, part a?At the end of year seve n, the portfolio man agerequati on, which adjusts for semia nnual compo unding. We have:Inserting in our values, we haveP 7 = $500,000(1 + 0.061/2)$500,000(1.522901960) = $761,450.98 . Thus, this investment alternative is not more attractive. It is less by the amount of $761,450.98-$768,872.47 = -$7,421.49 .4. Suppose that a portfolio manager purchases $10 million of par value of aneight-year bond that has a coup on rate of 7% and pays in terest once per year. The first annual coup on payme nt will be made one year from now. How much will the portfolio manager have if she (1) holds the bond until it matures eight years from now, and (2) can reinv est all the annual in terest payme nts at an annual in terest rate of 6.2%?At the end of year eight, the portfolio man ager equati on, which adjusts for annual compo unding. We have:' s amount is given by the following P n = P 0(1 + r/2)2(n ).2(7)= $500,000(1.0305) 14 =s amount is give n by the follow ingwhere A = coup on rate times par value. In sert ing in our values, we have:P = 0.07($10,000,000) |(1 + 0.062 -1I +[0.062$10,000,000 = $6,978,160.38 + $10,000,000 =$16,978,160.385. An swer the below questi ons.(a) If the discount rate that is used to calculate the present value of a debt obligati on cash flow is in creased, what happe ns to the price of that debt obligati on?A fun dame ntal property of a bond is that its price cha nges in the opposite directi on from the cha nge in the required yield. The reas on is that the price of the bond is the prese nt value of the cash flows. As the required yield in creases, the prese nt value of the cash flow decreases; hence the price decreases. The opposite is true whe n the required yield decreases: The prese nt value of the cash flows in creases, and therefore the price of the bond in creases.(b) Suppose that the disco unt rate used to calculate the prese nt value of a debt obligati on ' s cash flowxi%. Suppose also that the only cash flows for this debt obligati on are $200,000 four years from now and $200,000 five years from now.For which of these cash flows will the prese nt value be greater?Cash flows that come earlier will have a greater value. As long as x% is positive and the amount is the same, the prese nt value will be greater for the $200,000 four years from now compared to five years from now. This can also be seen by noting that if x > 0 thenPn=A「(1 + r )n—1 l r+ Par ValueJ$10,000,000 =$700,000[9.9688005]implies $2,000 ―. $2,000 ―l(1 + x )」|(1 + x )」will hold.6. The pension fund obligation of a corporation is calculated as the present value of the actuarially projected ben efits that will have to be paid to ben eficiaries. Why is the in terest rate used to disco unt the projected ben efits importa nt?The prese nt value in creases as the disco unt rate decreases and decreases as the disco unt rate in creases. Thus, in order to project the ben efits accurately, we n eed an accurate estimate of the disco unt rate. If we un derestimate the disco unt rate the n we will be projecting more available pension funds than we will actually have.7. A pension fund manager knows that the following liabilities must be satisfied:Years from Now Liability (in millio ns)1 2.02 3.03 5.4 45.8Suppose that the pension fund man ager wants to inv est a sum of money that will satisfy this liability stream. Assuming that any amount that can be invested today can earn an annual in terest rate of 7.6%, how much must be inv ested today to satisfy this liability stream?To satisfy year one ' snliab1)tith(e pension fund manager must invest an amounttoday that is equal to the future value of $2.0 milli on at 7.6%. We have:——1_5 . The latter_1 xin equality P5 —L n _1 + r=$2,000,000,1(.076) 一=$2,000,0000.929368030 = $1,858,736.06 .To satisfy year two' n =a )ilitly& pension fund manager must invest an amounttoday that is equal to the future value of $3.0 milli on at 7.6%. We have:PVJ+J =$3,000,000| 1J = $3,000,0000.863724935】= $2,591,174.80 ."(1.076)To satisfy year threen =s3gbheyp (e nsion fund man ager must in vest an amount today that is equal to the future value of $5.4 milli on at 7.6%. We have:To satisfy year four ' n = kabtlhte pension fund man ager must in vest an amounttoday that is equal to the future value of $5.8 million at 7.6%. We have:If we add the four present values, we get $1,858,736.06 + $2,591,174.80 +$4,334,679.04 + $4,326,920.42= $13,111,510.32 , which is the amou nt the pen sionfund man ager n eeds to inv est today to cover the liability stream for the n ext four years.8. Calculate for each of the following bonds the price per $1,000 of par value assu ming semia nnual coup on payme nts.Bo nd Coupon Rate (%)Years to MaturityRequired Yield (%)A 8 9 7B 9 20 9 C6 15 10 D148P ^P4d + rf _ $5,400,000$5,400,0000.802718341】=$4,334,679.04=$5,800,000“6)4$5,800,0000.74602076^1 = $4,326,920.42Con sider a 9-year 8% coup on bond with a par value of $1,000 and a required yield of 7%. Give n C = 0.08($1,000) / 2 = $40, n = 2(40) = 80 and r = 0.07 / 2 = 0.035, the prese nt value of the coup on payme nts is:=$63.793. Thus, the price of the bond (P) = prese nt value of coup on payme nts + present value of par value = $1,069.951 + $63.793 =$1,133.74 .Con sider a 20-year 9% coup on bond with a par value of $1,000 and a required yield of 9%. Give n C = 0.09($1,000) / 2 = $45, n = 2(20) = 40 and r = 0.09 / 2 = 0.045, the prese nt value of the coup on payme nts is:$45[18.401584] = $828.071.$40$401 . 80(1.035 ) -0.035 一$40 - 115.67573750.035r1- b.063792852[ = $40 26.7487757】=$1,069.951.0.035The present value of the par or maturity $1,000 15.6757375value of $1,000 is:M1+ r n$1,000- 80(1.035 )丨 (1 +r $1 1=$45 —匸-0.045一=$45 1 一。
