1、Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-1Chapter 3 Measuring Yield Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-2Learning ObjectivesAfter reading this chapter, you will understandhow to calculate the yield on any investmenthow to calculate the curre
2、nt yield, yield to maturity, yield to call, yield to put, and cash flow yieldhow to calculate the yield for a portfoliohow to calculate the discount margin for a floating-rate securitythe three potential sources of a bonds returnCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-3Le
3、arning Objectives (continued)After reading this chapter, you will understandhow to calculate the yield on any investmentwhat reinvestment risk isthe limitations of conventional yield measureshow to calculate the total return for a bondwhy the total return is superior to conventional yield measuresho
4、w to use horizon analysis to assess the potential returnCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-4Computing the Yield or Internal Rate of Return on any InvestmentqThe yield on any investment is the interest rate that will make the present value of the cash flows from the i
5、nvestment equal to the price (or cost) of the investment.qMathematically, the yield on any investment, y, is the interest rate that satisfies the equation. where P = price of the investment, CF = cash flow in year t =1,2,3, N, y = yield calculated from this relationship (and also called the internal
6、 rate of return), N = number of years.312231111NNCFCFCFCFP.yrrrCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-5Computing the Yield or Internal Rate of Return on any Investment (continued)qAbsent a financial calculator or computer software, solving for the yield (y) requires a tr
7、ial-and-error (iterative) procedure. qThe objective is to find the yield that will make the present value of the cash flows equal to the price.qKeep in mind that the yield computed is the yield for the period.qThat is, if the cash flows are semiannual, the yield is a semiannual yield.qIf the cash fl
8、ows are monthly, the yield is a monthly yield.qTo compute the simple annual interest rate, the yield for the period is multiplied by the number of periods in the year.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-6Computing the Yield or Internal Rate of Return on any Investment
9、 (continued)q Special Case: Investment with Only One Future Cash FlowWhen the case where there is only one future cash flow, it is not necessary to go through the time-consuming trial-and-error procedure to determine the yield.We can use the below equation:11nnCFyPCopyright 2010 Pearson Education, I
10、nc. Publishing as Prentice Hall3-7Computing the Yield or Internal Rate of Return on any Investment (continued)q Annualizing YieldsTo obtain an effective annual yield associated with a periodic interest rate, the following formula is used:effective annual yield = (1 + periodic interest rate)m 1 where
11、 m is the frequency of payments per year. To illustrate, if interest is paid quarterly and the periodic interest rate is 0.08 / 4 = 0.02, then we have: effective annual yield = (1.02)4 1 = 1.0824 1 = 0.0824 or 8.24%Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-8Computing the Yi
12、eld or Internal Rate of Return on any Investment (Continued)q Annualizing YieldsWe can also determine the periodic interest rate that will produce a given annual interest rate by solving the effective annual yield equation for the periodic interest rate .Solving, we find thatperiodic interest rate =
13、 (1 + effective annual yield )1/m 1To illustrate, if the periodic quarterly interest rate that would produce an effective annual yield of 12%, then we have: periodic interest rate = (1.12)1/4 1 = 1.0287 1 = 0.0287 or 2.87%Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-9Computing
14、 the Yield or Internal Rate of Return on any Investment (continued)q Annualizing YieldsWe can also determine the periodic interest rate that will produce a given annual interest rate by solving the effective annual yield equation for the periodic interest rate .Solving, we find thatperiodic interest
15、 rate = (1 + effective annual yield )1/m 1To illustrate, if the periodic quarterly interest rate that would produce an effective annual yield of 12%, then we have : periodic interest rate = (1.12)1/4 1 = 1.0287 1 = 0.0287 or 2.87%Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-10
16、Conventional Yield Measuresq Bond yield measures commonly quoted by dealers and used by portfolio managers are:1) Current Yield2) Yield To Maturity3) Yield To Call4) Yield To Put5) Yield To Worst6) Cash Flow Yield7) Yield (Internal Rate of Return) for a Portfolio8) Yield Spread Measures for Floating
17、-Rate SecuritiesCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-11Conventional Yield Measures (continued)1) Current YieldCurrent yield relates the annual coupon interest to the market price.The formula for the current yield is:current yield = annual dollar coupon interest / price
18、The current yield calculation takes into account only the coupon interest and no other source of return that will affect an investors yield.The time value of money is also ignored.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-12Conventional Yield Measures (continued)2) Yield To
19、 MaturityThe yield to maturity is the interest rate that will make the present value of the cash flows equal to the price (or initial investment). For a semiannual pay bond, the yield to maturity is found by first computing the periodic interest rate, y, which satisfies the relationship:where P = pr
20、ice of the bond, C = semiannual coupon interest (in dollars), M = maturity value (in dollars), and n = number of periods (number of years multiplied by 2).2311111nnCCCCMP + + + . . .+ + yyyyyCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-13Conventional Yield Measures (Continued)
21、2) Yield To Maturity (continued) For a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield to maturity. The yield to maturity computed on the basis of this market convention is called the bond-equivalent yield The yield-to-maturity calculation takes into acc
22、ount (i) the current coupon income, (ii) any capital gain or loss realized by holding the bond to maturity, and (iii) the timing of the cash flows.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-14Conventional Yield Measures (Continued)3) Yield To CallThe call price is the price
23、at which the bond may be called .There is a call schedule that specifies a call price for each call date.The yield to call assumes that the issuer will call the bond at some assumed call date and the call price is specified in the call schedule.The yield to call can be expressed as follows:where M *
24、 = call price (in dollars) and n* = number of periods until the assumed call date (number of years times 2)For a semiannual pay bond, doubling the periodic interest rate (y) gives the yield to call on a bond-equivalent basis.12311111* = nnCCCCMP + + + . . .+ + yyyyyCopyright 2010 Pearson Education,
25、Inc. Publishing as Prentice Hall3-15Conventional Yield Measures (Continued)4) Yield To PutIf an issue is putable, it means that the bondholder can force the issuer to buy the issue at a specified price. The put schedule specifies when the issue can be put and the put price. When an issue is putable,
26、 a yield to put is calculated.The yield to put is the interest rate that makes the present value of the cash flows to the assumed put date plus the put price on that date as set forth in the put schedule equal to the bonds price.The formula for the yield to put is the same as for the yield to call,
27、but M * is now defined as the put price and n* is the number of periods until the assumed put date.The procedure is the same as calculating the yield to maturity and the yield to call.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-16Conventional Yield Measures (Continued)5) Yiel
28、d To WorstA practice in the industry is for an investor to calculate the yield to maturity, the yield to every possible call date, and the yield to every possible put date. The minimum of all of these yields is called the yield to worst.6) Cash Flow Yield Amortizing securities involve cash flows tha
29、t include interest plus principal repayment and the cash flow each period consists of three components: (i) coupon interest, (ii) scheduled principal repayment, and (iii) prepayments.For amortizing securities, market participants calculate a cash flow yield, which is the interest rate that will make
30、 the present value of the projected cash flows equal to the market price.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-17Conventional Yield Measures (Continued)7) Yield (Internal Rate of Return) for a PortfolioThe yield for a portfolio of bonds is not simply the average or weig
31、hted average of the yield to maturity of the individual bond issues in the portfolio. It is computed by determining the cash flows for the portfolio and determining the interest rate that will make the present value of the cash flows equal to the market value of the portfolio.Copyright 2010 Pearson
32、Education, Inc. Publishing as Prentice Hall3-18Conventional Yield Measures (Continued)8) Yield Spread Measures for Floating-Rate SecuritiesThe coupon rate for a floating-rate security changes periodically based on the coupon reset formula.This formula consists of the reference rate and the quoted ma
33、rgin.Since the future value for the reference rate is unknown, it is not possible to determine the cash flows.This means that a yield to maturity cannot be computed.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-19Conventional Yield Measures (Continued)8) Yield Spread Measures f
34、or Floating-Rate SecuritiesInstead, there are several conventional measures used as margin or spread measures cited by market participants for floaters.These include spread for life (or simple margin), adjusted simple margin, adjusted total margin, and discount margin.The most popular of these measu
35、res is the discount margin, which estimates the average margin over the reference rate that the investor can expect to earn over the life of the security.Exhibit 3-1 shows the calculation of the discount margin for a six-year floating-rate security. (See Overhead 3-20.)Copyright 2010 Pearson Educati
36、on, Inc. Publishing as Prentice Hall3-20Exhibit 3-1 Calculation of the Discount Margin for a Floating-Rate SecurityFloating-rate securityMaturity: six yearsCoupon rate: reference rate + 80 basis pointsReset every six monthsPresent Value of Cash Flow at Assumed Annual Margin (basis points) PeriodRefe
37、renceRateCash Flowa 808488961001 10%5.45.12335.12245.12145.11955.11852105.44.86094.85904.85724.85354.85163105.44.61184.60924.60664.60134.59874105.44.37554.37224.36894.36234.35905105.44.15144.14744.14354.13564.13176105.43.93873.93423.92973.92083.91637105.43.73693.73193.72703.71713.71228105.43.54543.5
38、4013.53473.52403.51869105.43.36383.35803.35233.34093.335210105.43.19143.18543.17943.16733.161311105.43.02793.02163.01533.00282.99651210105.456.072955.945455.818255.564755.4385Present Value =100.000099.826999.654199.309899.1381aFor periods 111: cash flow = 100 (reference rate + assumed margin)(0.5);
39、for period 12: cash flow = 100 (reference rate + assumed margin)(0.5) + 100.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-21Potential Sources of a Bonds Dollar Returnq An investor who purchases a bond can expect to receive a dollar return from one or more of these sources:i.the
40、 periodic coupon interest payments made by the issuerii.any capital gain (or capital lossnegative dollar return) when the bond matures, is called, or is soldiii. interest income generated from reinvestment of the periodic cash flowsq The current yield considers only the coupon interest payments.q Th
41、e yield to maturity, yield to call, and cash flow yield all take into account the three components.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-22Potential Sources of a Bonds Dollar Return(continued)q Determining the Interest-On-Interest Dollar ReturnThe interest-on-interest c
42、omponent can represent a substantial portion of a bonds potential return. The coupon interest plus interest on interest can be found by using the following equation:whereC is the coupon interestr is the semiannual reinvestment raten is the number of periods11nrCr coupon interest + interest on intere
43、st Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-23Potential Sources of a Bonds Dollar Return(continued)qDetermining the Interest-On-Interest Dollar ReturnThe total dollar amount of coupon interest is found by multiplying the semiannual coupon interest by the number of periods:
44、total coupon interest = nCwhere C is the coupon interest, r is the semiannual reinvestment rate, and n is the number of periods.The interest-on-interest component is then the difference between the coupon interest plus interest on interest and the total dollar coupon interest, as expressed by the fo
45、rmula11nrCnCr interest on interest Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall3-24Potential Sources of a Bonds Dollar Return(continued)qDetermining the Interest-On-Interest Dollar ReturnExample. Assume that the coupon interest (C) is $50, the semiannual reinvestment rate (r) i
46、s 4.5%, and the number of periods (n) is 40. What is the interest-on-interest?Using our equation for interest on interest and inserting in our given values we get: 4010.045 150 40 5050 107.03032 1,4000.045 $7,351.5211nrCnCr interest on interest Copyright 2010 Pearson Education, Inc. Publishing as Pr
47、entice Hall3-25Potential Sources of a Bonds Dollar Return(continued)q Yield To Maturity and Reinvestment RiskThe investor realizes the yield to maturity only if the bond is held to maturity and the coupon payments can be reinvested at the computed yield to maturity.Reinvestment risk is the risk that
48、 future reinvestment rates will be less than the yield to maturity at the time the bond is purchased.There are two characteristics of a bond that determine the importance of the interest-on- interest component and therefore the degree of reinvestment risk: maturity and coupon.Copyright 2010 Pearson
49、Education, Inc. Publishing as Prentice Hall3-26Potential Sources of a Bonds Dollar Return(continued)q Yield To Maturity and Reinvestment RiskFor a given yield to maturity and a given coupon rate, the longer the maturity, the more dependent the bonds total dollar return is on the interest-on-interest
50、 component in order to realize the yield to maturity at the time of purchase.For a given maturity and a given yield to maturity, higher coupon rates will make the bonds total dollar return more dependent on the reinvestment of the coupon payments in order to produce the yield to maturity anticipated
