Chapter3answers

1、chapter 3measuring yieldanswers to questions for chapter 3(questions are in bold print followed by answers.)1. a debt obligation offers the following payments:years from nowcash flow to investor1$2,0002$2,0003$2,5004$4,000suppose that the price of this debt obligation is $7,704. what is the yield or

2、 internal rate of return offered by this debt obligation?the yield on any investment is the interest rate that will make the present value of the cash flows from the investment equal to the price (or cost) of the investment.mathematically, the yield on any investment, y, is the interest rate that sa

3、tisfies the equation:p = where cft = cash flow in year t, p = price of the investment, and n = number of years. the yield calculated from this relationship is also called the internal rate of return. to solve for the yield (y), we can use a trial-and-error (iterative) procedure. the objective is to

4、find the interest rate that will make the present value of the cash flows equal to the price. to compute the yield for our problem, different interest rates must be tried until the present value of the cash flows is equal to $7,704 (the price of the financial instrument). trying an annual interest r

5、ate of 10% gives the following present value:years from nowpromised annual payments(cash flow to investor)present value of cash flow at 10%1$2,000$1,818.182$2,000$1,652.893$2,500$1,878.294$4,000$2,732.05 present value = $8,081.41because the present value of $8,081.41 computed using a 10% interest ra

6、te exceeds the price of $7,704, a higher interest rate must be used, to reduce the present value. trying an annual interest rate of 13% gives the following present value:years from nowpromised annual payments(cash flow to investor)present value of cash flow at 13%1$2,000$1,769.912$2,000$1,566.293$2,

7、500$1,732.634$4,000$2,453.27 present value = $7,522.10because the present value of $7,522.10 computed using a 13% interest rate is below the price of $7,704, a lower interest rate must be used, to reduce the present value. thus, to increase the present value, a lower interest rate must be tried. try

8、ing an annual interest rate of 12% gives the following present value:years from nowpromised annual payments(cash flow to investor)present value of cash flow at 12%1$2,000$1,785.712$2,000$1,594.393$2,500$1,779.454$4,000$2,542.07 present value = $7,701.62using 12%, the present value of the cash flow i

9、s $7,701.62, which is almost equal to the price of the financial instrument of $7,704. therefore, the yield is close to 12%. the precise yield using excel or a financial calculator is 11.987%.although the formula for the yield is based on annual cash flows, it can be generalized to any number of per

10、iodic payments in a year. the generalized formula for determining the yield iswhere cft = cash flow in period t, and n = number of periods.keep in mind that the yield computed is the yield for the period. that is, if the cash flows are semiannual, the yield is a semiannual yield. if the cash flows a

11、re monthly, the yield is a monthly yield. to compute the simple annual interest rate, the yield for the period is multiplied by the number of periods in the year.2. what is the effective annual yield if the semiannual periodic interest rate is 4.3%?to obtain an effective annual yield associated with

12、 a periodic interest rate, the following formula is used:effective annual yield = (1 + periodic interest rate)m 1where m is the frequency of payments per year. in our problem, the periodic interest rate is a semiannual rate of 4.3% and the frequency of payments is twice per year. inserting these num

13、bers, we have:effective annual yield = (1.043)2 1 = 1.087849 1 = 0.087849 or about 8.785%.3. what is the yield to maturity of a bond?the 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, th

14、e yield to maturity is found by first computing the periodic interest rate, y, which satisfies the relationship:p = where p = price of the bond, c = semiannual coupon interest (in dollars), m = maturity value (in dollars), and n = number of periods (number of years times 2).it is much easier to comp

15、ute the yield to maturity for a zero-coupon bond because we can use:.the yield-to-maturity calculation takes into account not only the current coupon income but also any capital gain or loss that the investor will realize by holding the bond to maturity. in addition, the yield to maturity considers

16、the timing of the cash flows.4. what is the yield to maturity calculated on a bond-equivalent basis?for a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield to maturity, which understates the effective annual yield. the yield to maturity computed on the bas

17、is of this market convention is called the bond-equivalent yield.5. answer the following questions.(a) show the cash flows for the following four bonds, each of which has a par value of $1,000 and pays interest semiannually.bondcoupon rate(%)number of yearsto maturitypricew75$884.20x87$948.90y94$967

18、.70z010$456.39 bond w has cash flows of 0.07($1,000) / 2 = $35 for semiannual periods from periods 1 to 10. at the end of period 10, bond w also pays back the par of $1,000 for a total payment of $1,000 + $35 = $1,035.bond x has cash flows of 0.08($1,000) / 2 = $40 for semiannual periods from period

19、s 1 to 14. at the end of period 14, bond x also pays back the par of $1,000 for a total payment of $1,000 + $40 = $1,040.bond y has cash flows of 0.08($1,000) / 2 = $45 for semiannual periods from periods 1 to 8. at the end of period 8, bond y also pays back the par of $1,000 for a total payment of

20、$1,000 + $45 = $1,045.bond z has cash flows of 0 ($1,000) / 2 = $0 for semiannual periods from periods 1 to 20. at the end of period 20, bond z also pays back the $1,000 for a total payment of $1,000 + $0 = $1,000.below we show these cash flows in table format.periodcash flowfor bond wcash flowfor b

21、ond xcash flowfor bond ycash flowfor bond z1$35$40$45$02$35$40$45$03$35$40$45$04$35$40$45$05$35$40$45$06$35$40$45$07$35$40$45$08$35$40$1,045$09$35$40$010$1,035$40$011$40$012$40$013$40$014$1,040$015$016$017$018$019$020$1,000(b) calculate the yield to maturity for the four bonds.the yield to maturity

22、is computed in the same way as the internal rate of return; the cash flows are those that the investor would realize by holding the bond to maturity. for a semiannual pay bond, the yield to maturity is found by first computing the periodic interest rate, y, which satisfies the relationshipp = where

23、p = price of the bond, c = semiannual coupon interest (in dollars), m = maturity value (in dollars), and n = number of periods (number of years times 2).for a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield to maturity. however, annualizing the yield by

24、doubling the periodic interest rate understates the effective annual yield. despite this, the market convention is to compute the yield to maturity by doubling the periodic interest rate, y, that satisfies our equation. the yield to maturity computed on the basis of this market convention is called

25、the bond-equivalent yield.the computation of the yield to maturity requires a trial-and-error procedure. to illustrate the computation, we first look at bond w. the cash flows for this bond are ten coupon payments of $35 every six months and the principal of $1,000 to be paid in ten six-month period

26、s from now.to get y using our equation given above, different interest rates must be tried until the present value of the cash flows is equal to the price. in doing this, we get the following yield to maturities for the four bonds.for bond w, we get a periodic interest rate real close to 5%. this is

27、 seen below.years fromnowpromised annual payments(cash flow to investor)present valueof cash flow at 5%1$35 $33.332$35 $31.753$35 $30.234$35 $28.795$35 $27.426$35 $26.127$35 $24.878$35 $23.689$35 $22.5610$1,035$635.40 present value = $884.17using 5%, the present value of the cash flow is $884.17, wh

28、ich is almost equal to the price of the financial instrument of $884.20. therefore, the periodic interest rate is close to 5%. the precise yield using excel or a financial calculator is 4.99964%. doubling the periodic interest rate of 5% gives a yield to maturity of 10% (doubling 4.99964% gives 9.99

29、928%).for bond x, we get an interest rate real close to 4.50%. using this rate, the value of the cash flow is $951.59, which is almost equal to the price of the financial instrument of $948.90. therefore, the yield is close to 4.5%. the precise periodic interest rate using excel or a financial calcu

30、lator is 4.5271%. doubling the periodic interest rate of 4.5% gives a yield to maturity of 9% (doubling 4.5271% gives 9.0542%).for bond y, we get an interest rate close to 5%. using this rate, the value of the cash flow is $967.68, which is almost equal to the price of the financial instrument of $9

31、67.70. therefore, the yield is close to 5%. the precise periodic interest rate using excel or a financial calculator is 5.11078%. doubling the periodic interest rate of 5% gives a yield to maturity of 10% (doubling 5.11083% gives 10.2215%).for bond z, we get an interest rate close to 4%. using this

32、rate, the value of the cash flow is $456.39, which is equal to the price of the financial instrument of $456.39. therefore, the yield is virtually 4%. the precise periodic interest rate using excel or a financial calculator is 3.99965%. doubling the periodic interest rate of 4% gives a yield to matu

33、rity of 8% (doubling 3.99965% gives 7.9993%).6. a portfolio manager is considering buying two bonds. bond a matures in three years and has a coupon rate of 10% payable semiannually. bond b, of the same credit quality, matures in 10 years and has a coupon rate of 12% payable semiannually. both bonds

34、are priced at par.(a) suppose that the portfolio manager plans to hold the bond that is purchased for three years. which would be the best bond for the portfolio manager to purchase?the shorter term bond will pay a lower coupon rate but it will likely cost less for a given market rate. since the bon

35、ds are of equal risk in terms of credit quality (the maturity premium for the longer term bond should be greater), the question when comparing the two bond investments is: what investment will be expected to give the highest cash flow per dollar invested? in other words, which investment will be exp

36、ected to give the highest effective annual rate of return. in general, holding the longer term bond should compensate the investor in the form of a maturity premium and a higher expected return. however, as seen in the discussion below, the actual realized return for either investment is not known w

37、ith certainty.to begin with, an investor who purchases a bond can expect to receive a dollar return from (i) the periodic coupon interest payments made by the issuer; (ii) any capital gain (or capital lossnegative dollar return) when the bond matures, is called, or is sold; and (iii) interest income

38、 generated from reinvestment of the periodic cash flows. the last component of the potential dollar return is referred to as reinvestment income. for a standard bond (our situation) that makes only coupon payments and no periodic principal payments prior to the maturity date, the interim cash flows

39、are simply the coupon payments. consequently, for such bonds the reinvestment income is simply interest earned from reinvesting the coupon interest payments. for these bonds, the third component of the potential source of dollar return is referred to as the interest-on-interest component.if we are g

40、oing to compute a potential yield to make a decision, we should be aware of the fact that any measure of a bonds potential yield should take into consideration each of the three components described above. the current yield considers only the coupon interest payments. no consideration is given to an

41、y capital gain (or loss) or interest on interest. the yield to maturity takes into account coupon interest and any capital gain (or loss). it also considers the interest-on-interest component. additionally, implicit in the yield-to-maturity computation is the assumption that the coupon payments can

42、be reinvested at the computed yield to maturity. the yield to maturity is a promised yield and will be realized only if the bond is held to maturity and the coupon interest payments are reinvested at the yield to maturity. if the bond is not held to maturity and the coupon payments are reinvested at

43、 the yield to maturity, then the actual yield realized by an investor can be greater than or less than the yield to maturity.given the facts that (i) one bond, if bought, will not be held to maturity, and (ii) the coupon interest payments will be reinvested at an unknown rate, we cannot determine wh

44、ich bond might give the highest actual realized rate. thus, we cannot compare them based upon this criterion. however, if the portfolio manager is risk inverse in the sense that she or he doesnt want to buy a longer term bond, which will likely have more variability in its return, then the manager m

45、ight prefer the shorter term bond (bond a) of three years. this bond also matures when the manager wants to cash in the bond. thus, the manager would not have to worry about any potential capital loss in selling the longer term bond (bond b). the manager would know with certainty what the cash flows

46、 are. if these cash flows are spent when received, the manager would know exactly how much money could be spent at certain points in time.finally, a manager can try to project the total return performance of a bond on the basis of the planned investment horizon and expectations concerning reinvestme

47、nt rates and future market yields. this permits the portfolio manager to evaluate which of several potential bonds considered for acquisition will perform best over the planned investment horizon. as we just argued, this cannot be done using the yield to maturity as a measure of relative value. usin

48、g total return to assess performance over some investment horizon is called horizon analysis. when a total return is calculated over an investment horizon, it is referred to as a horizon return. the horizon analysis framework enables the portfolio manager to analyze the performance of a bond under d

49、ifferent interest-rate scenarios for reinvestment rates and future market yields. only by investigating multiple scenarios can the portfolio manager see how sensitive the bonds performance will be to each scenario. this can help the manager choose between the two bond choices.(b) suppose that the po

50、rtfolio manager plans to hold the bond that is purchased for six years instead of three years. in this case, which would be the best bond for the portfolio manager to purchase?similar to our discussion in part (a), we do not know which investment would give the highest actual realized return in six

51、years when we consider reinvesting all cash flows. if the manager buys a three-year bond, then there would be the additional uncertainty of now knowing what three-year bond rates would be in three years. the purchase of the ten-year bond would be held longer than previously (six years compared to th

52、ree years) and render coupon payments for a six-year period that are known. if these cash flows are spent when received, the manager will know exactly how much money could be spent at certain points in time. not knowing which bond investment would give the highest realized return, the portfolio mana

53、ger would choose the bond that fits the firms goals in terms of maturity.(c) suppose that the portfolio manager is managing the assets of a life insurance company that has issued a five-year guaranteed investment contract (gic). the interest rate that the life insurance company has agreed to pay is

54、9% on a semiannual basis. which of the two bonds should the portfolio manager purchase to ensure that the gic payments will be satisfied and that a profit will be generated by the life insurance company?the portfolio manager needs to generate a semiannual cash flow of 9% semiannual basis for five ye

55、ars. bond a will only lock in a 10% cash flow per dollar invested for three years. however, bond b will lock in a 12% cash flow per dollar invested for ten years. thus, the portfolio manager would choose bond b and hopefully we able buy as many of these bonds s are needed to generate the cash flows

56、required to meet its five-year guaranteed investment contract.7. consider the following bond:coupon rate = 11%maturity = 18 yearspar value = $1,000first par call in 13 yearsonly put date in five years and putable at par valuesuppose that the market price for this bond $1,169.(a) show that the yield to maturity for this bond is 9.077%.first of all, we could compute the internal return based upon the cash flows if the bond is held to maturity. we would get 4.5385%. for a semiannual pay bond, doubling the periodic interest rate (y)