1、Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. 16CHAPTER 2PRICING OF BONDSCHAPTER SUMMARYThis 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 determine the appropr
2、iate yield at which to discount the expected cash flows. Among other aspects of a bond, we will look at the reasons why the price of a bond changesREVIEW 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 o
3、f money invested today is:Pn = P0(1+r)nwhere n = number of periods, Pn = future value n periods from now (in dollars), P0 = original principal (in dollars), r = interest rate per period (in decimal form), and the expression (1+r)n represents the future value of $1 invested today for n periods at a c
4、ompounding rate of r.When interest is paid more than one time per year, both the interest rate and the number of periods used to compute the future value must be adjusted as follows:r = annual interest rate / number of times interest paid per year, andn = number of times interest paid per year times
5、 number of years.The higher future value when interest is paid semiannually, as opposed to annually, reflects the greater opportunity for reinvesting the interest paid.Future Value of an Ordinary AnnuityWhen the same amount of money is invested periodically, it is referred to as an annuity. When the
6、 first investment occurs one period from now, it is referred to as an ordinary annuity.The equation for the future value of an ordinary annuity is:Pn = 1nrAwhere A is the amount of the annuity (in dollars).Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. 17Example of Future Value
7、of an Ordinary Annuity Using Annual Interest:If A = $2,000,000, r = 0.08, and n = 15, then Pn = P15 = 1nr A= $2,000,00027.152125 = $54,304.250.08.1723 ,02$Because 15($2,000,000) = $30,000,000 of this future value represents the total dollar amount of annual interest payments made by the issuer and i
8、nvested by the portfolio manager, the balance of $54,304,250 $30,000,000 = $24,304,250 is the interest earned by reinvesting these annual interest payments.Example of Future Value of an Ordinary Annuity Using Semiannual Interest:Consider the same example, but now we assume semiannual interest paymen
9、ts.If A = $2,000,000 / 2 = $1,000,000, r = 0.08 / 2 = 0.04, n = 2(15) = 30, thenPn = P30 = = =1nr04.1)( ,01$3 04.123 ,0$1,000,00056.085 = $56,085,000.The opportunity for more frequent reinvestment of interest payments received makes the interest earned of $26,085,000 from reinvesting the interest pa
10、yments greater than the $24,304,250 interest earned when interest is paid only one time per year.Present ValueThe present value is the future value process in reverse. We have:.1nPVrFor a given future value at a specified time in the future, the higher the interest rate (or discount rate), the lower
11、 the present value. For a given interest rate (discount rate), the further into the future that the future value will be received, then the lower its present value.Present Value of a Series of Future ValuesTo determine the present value of a series of future values, the present value of each future
12、value must first be computed. Then these present values are added together to obtain the present value of the entire series of future values.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. 18Present Value of an Ordinary AnnuityWhen the same dollar amount of money is received each
13、 period or paid each year, the series is referred to as an annuity. When the first payment is received one period from now, the annuity is called an ordinary annuity. When the first payment is immediate, the annuity is called an annuity due.The present value of an ordinary annuity is: 1n rPV =Awhere
14、 A is the amount of the annuity (in dollars).The term in brackets is the present value of an ordinary annuity of $1 for n periods.Example of Present Value of an Ordinary Annuity Using Annual Interest:If A = $100, r = 0.09, and n = 8, then: = =1n rPVA81 .09$= = $1005.534811 = $553.48.1 .9256$010.5867
15、$9Present Value When Payments Occur More Than Once Per YearIf the future value to be received occurs more than once per year, then the present value formula is modified so that (i) the annual interest rate is divided by the frequency per year, and (ii) the number of periods when the future value wil
16、l be received is adjusted by multiplying the number of years by the frequency per year.PRICING A BONDDetermining the price of any financial instrument requires an estimate of (i) the expected cash flows, and (ii) the appropriate required yield. The required yield reflects the yield for financial ins
17、truments with comparable risk, or alternative investments.The cash flows for a bond that the issuer cannot retire prior to its stated maturity date consist of periodic coupon interest payments to the maturity date, and the par (or maturity) value at maturity.Copyright 2010 Pearson Education, Inc. Pu
18、blishing as Prentice Hall. 19In general, the price of a bond can be computed using the following formula:.1ntnt=MCP + rwhere P = price (in dollars), n = number of periods (number of years times 2), C = semiannual coupon payment (in dollars), r = periodic interest rate (required annual yield divided
19、by 2), M = maturity value, and t = time period when the payment is to be received.Computing the Value of a Bond: An Example:Consider a 20-year 10% coupon 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 present
20、 value of the coupon payments is:= = = =1n rP 401 .5$01 8.53$010.7463$5= $802.31.0463. 5$The present value of the par or maturity value of $1,000 is: = = = 1nMr40$,.513.8,$117.46.The price of the bond (P) = present value coupon payments + present value maturity value = $802.31 + $117.46 = $919.77.Pr
21、icing Zero-Coupon BondsFor zero-coupon bonds, the investor realizes interest as the difference between the maturity value and the purchase price. The equation is: 1nMP rwhere M is the maturity value. Thus, the price of a zero-coupon bond is simply the present value of the maturity value.Copyright 20
22、10 Pearson Education, Inc. Publishing as Prentice Hall. 20Zero-Coupon Bond ExampleConsider the price of a zero-coupon bond that matures 15 years from now, if the maturity 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: = = = $252.
23、12.1nMPr30$,.47964.,1$Price-Yield RelationshipA fundamental property of a bond is that its price changes in the opposite direction from the change in the required yield. The reason is that the price of the bond is the present value of the cash flows.Relationship Between Coupon Rate, Required Yield,
24、and PriceWhen yields in the marketplace rise above the coupon rate at a given 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 compensate for a coupon rate that is low
25、er than the required yield. When 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.Relationship Between Bond Price and Time if Interest Rates Are UnchangedFor a bond selling at par value, the coupon
26、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 remain constant as the bond moves toward the maturity date.The price of a bond will not remain constant for a bond selling at a premium or a discount. The discount b
27、ond increases in price as it approaches maturity, assuming that the required yield does not change. For a premium bond, the opposite occurs. For both bonds, the price will equal par value at the maturity date.Reasons for the Change in the Price of a BondThe price of a bond can change for three reaso
28、ns: (i) there is a change in the required yield owing to changes in the credit quality of the issuer; (ii) there is a change 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 moving toward maturity; or, (iii) there is a
29、change in the required yield owing to a change in the yield on comparable bonds (i.e., a change in the yield required by the market).Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. 21COMPLICATIONSThe framework for pricing a bond assumes the following: (i) the next coupon payment
30、is exactly six months away; (ii) the cash flows are known; (iii) the appropriate required yield can be determined; and, (iv) one rate is used to discount all cash flows.Next Coupon Payment Due in Less than Six MonthsWhen an investor purchases a bond whose next coupon payment is due in less than six
31、months, the accepted method for computing the price of the bond is as follows:n11t=1 ( +r)vt vnMCP ( r+ )where v = (days between settlement and next coupon) / (days in six-month period).Cash Flows May Not Be KnownFor most bonds, the cash flows are not known with certainty. This is because an issuer
32、may call a bond before the stated maturity date.Determining the Appropriate Required YieldAll required yields are benchmarked off yields offered by Treasury securities. From there, we must still decompose the required yield for a bond into its component parts.One Discount Rate Applicable to All Cash
33、 FlowsA bond can be viewed as a package of zero-coupon bonds, in which case a unique discount rate should be used to determine the present value of each cash flow.PRICING FLOATING-RATE AND INVERSE-FLOATING-RATE SECURITIESThe cash flow is not known for either a floating-rate or an inverse-floating-ra
34、te security; it will depend on the reference rate in the future.Price of a FloaterThe coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some spread or margin. The price of a floater depends on (i) the spread over the reference rate and (ii) any restrictions that
35、may be imposed on the resetting of the coupon rate.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. 22Price of an Inverse FloaterIn general, an inverse floater is created from a fixed-rate security. The security from which the inverse floater is created is called the collateral. F
36、rom the collateral two bonds are created: a floater and an inverse floater.The price of a floater depends on (i) the spread over the reference rate and (ii) any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater may have a maximum coupon rate called a cap or
37、 a minimum coupon rate called 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 collaterals price minus the fl
38、oaters price.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 than 100.Accrued InterestWhen an investor purchases a
39、bond between coupon payments, the investor must compensate the seller of the bond for the coupon interest earned from the time of the last coupon payment to the settlement date of the bond. This amount is called accrued interest. For corporate and municipal bonds, accrued interest is based on a 360-
40、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
41、default. Such bonds are said to be quoted flat, that is, without accrued interest.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. 23ANSWERS TO QUESTIONS FOR CHAPTER 2(Questions are in bold print followed by answers.)1. A pension fund manager invests $10 million in a debt obligati
42、on that promises to pay 7.3% per year for four years. What is the future value of the $10 million?To determine the future value of any sum of money invested today, we can use the future value equation, which is: Pn = P0 (1 + r)n where n = number of periods, Pn = future value n periods from now (in d
43、ollars), P0 = 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 insurance company has guaranteed a payment of $14 million to a pension fund
44、4.5 years from now. If the life insurance company receives a premium 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 sufficient funds from this investment to meet the $14 million obligation?To determine the futu
45、re value of any sum of money invested today, we can use the future value equation, which is: Pn = P0 (1 + r)n where n = number of periods, Pn = 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 value
46、s, we have: P4.5 = $10,400,000(1.0625)4.5 = $10,400,000(1.313651676) = $13,661,977.43. Thus, it will be short $13,661,977.43 $14,000,000 = $338,022.57.3. Answer the below questions.(a) The portfolio manager of a tax-exempt fund is considering investing $500,000 in a debt instrument that pays an annu
47、al interest rate of 5.7% for four years. At the end of four years, the portfolio manager plans to reinvest the proceeds for three more 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,
48、 the portfolio managers amount is given by: Pn = P0 (1 + r)n. Inserting in our values, we have P4 = $500,000(1.057)4 = $500,000(1.248245382) = $624,122.66. In three more years at the end of year seven, the manager amount is given by: P7 = P4(1 + r)3. Inserting in our values, we have: P7 = $624,122.6
49、6(1.072)3 = $624,122.66(1.231925248) = $768,872.47.(b) Suppose that the portfolio manager in Question 3, part a, has the opportunity to invest the $500,000 for seven years in a debt obligation that promises to pay an annual interest rate of 6.1% compounded semiannually. Is this investment alternative more attractive than the one in Question 3, part a?At the end of year seven, the port
