Chapter 3 answers

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 Now Cash 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 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 satisfies the equation:where CF t= 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 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 rate of 10% gives the following present value:Years from Now Promised Annual Payments(Cash Flow to Investor)Present Valueof 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 rate 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 Now Promised Annual Payments(Cash Flow to Investor)Present Valueof Cash Flow at 13%1$2,000$1,769.912$2,000$1,566.293 $2,500 $1,732.634 $4,000 $2,453.27Present 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. Trying an annual interest rate of 12% gives the following present value:Years from Now Promised Annual Payments(Cash Flow to Investor)Present Valueof Cash Flow at 12%1$2,000$1,785.712$2,000$1,594.393 $2,500 $1,779.454 $4,000 $2,542.07Present value = $7,701.62Using 12%, the present value of the cash flow is $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 periodic payments in a year. The generalized formula for determining the yield is2. What is the effective annual yield if the semiannual periodic interest rate is 4.3%?To obtain an effective annual yield associated with 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 numbers, 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, the yield to maturity is found by first computing the periodic interest rate, y , which satisfies the relationship:where P = price of the bond, C = semiannua l coupon interest (in dollars), M = maturity value (in dollars), and n = number of periods (number of years times 2).It is much easier to compute the yield to maturity for a zero-coupon bond because we can use:1 = /1-⎥⎦⎤⎢⎣⎡P M y n.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 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 basis 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.Bond Coupon Rate (%) Number of Years to Maturity PriceW 7 5 $884.20X 8 7 $948.90Y 9 4 $967.70Z 0 10 $456.39Bond 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 periods 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 $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.Period Cash Flowfor Bond WCash Flowfor Bond 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 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 relationshipwhere 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 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 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 periods 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 seen below.Years fromNow Promised 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 $635.40Present value = $884.17Using 5%, the present value of the cash flow is $884.17, which 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 9.99928%).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 calculator 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 $967.70. Therefore, the yield is close to 5%. The precise periodic interest rate using Excel or a financial calculator is5.11078%. 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 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 maturity 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 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 bonds 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 expected 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 with 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 loss—negative dollar return) when the bond matures, is called, or is sold; and (iii) interest income 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 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-interestcomponent.If we are going to compute a potential yield to make a decision, we should be aware of the fact that any measure of a bond’s 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 any 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 theinterest-on-interest component. Additionally, implicit in the yield-to-maturity computation is the assumption that the coupon payments can 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 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 which bond might give the highest actual realized rate. Thus, we cannot compare them based upon this criterion. However, if the portfolio manager is risk inv erse in the sense that she or he doesn’t want to buy a longer term bond, which will likely have more variability in its return, then the manager might 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 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 reinvestment 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. Using 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 different interest-rate scenarios for reinvestment rates and future market yields. Only by investigating multiple scenarios can the portfolio manager see how sensitive the bond’s performanc e will be to each scenario. This can help the manager choose between the two bond choices.(b) Suppose that the portfolio 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 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 three years) and render coupon payments for a six-year periodthat 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 manager wou ld choose the bond that fits the firm’s 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 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 years. Bond A will only lock in a 10% cash flow per dollar invested for three years. However, bondB 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 required to meet its five-year guaranteed investment contract.7. Consider the following bond:Suppose 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 ) gives the yield to maturity on a bond-equivalent basis. Taking 4.5385% times two gives us a yield to maturity equal to 9.077%.We can also verify that the yield to maturity is 9.077% by using this rate to compute the value of the bond to determine if it is $1,168.99. In doing this, we first compute the present value of the coupon payments where C is the annuity coupon payment and N is the number of periods. We have:()111N y C y ⎡⎤-⎢⎥+⎢⎥⎢⎥⎣⎦= ()3611 1 .045385$55 0.045385⎡⎤-⎢⎥⎢⎥⎣⎦ = $55[17.57569] = $966.663.We next compute the present value of the maturity value where M is the par value of $1,000. We get:()11N M y ⎡⎤⎢⎥+⎣⎦= ()361$1,000 1.045385 ⎡⎤⎢⎥⎣⎦= $1,000[0.2023273] = $202.327.When a 9.077% / 2 = 4.5285% semiannual interest rate is used, the present value of the cash flows is $966.663 + $202.327 = $1,168.99 or about $1,169. Thus, the yield to maturity for this bond is 9.077%.(b) Show that the yield to first par call is 8.793%.First of all, we could compute the internal return based upon the cash flows if the bond is held for 13 years. We would get 4.39651%. For a semiannual pay bond, doubling the periodic interest rate (y ) gives the yield to call on a bond-equivalent basis. Taking 4.39651% times two gives us a yield to first par call of 8.793%.We can also verify the yield to call is 8.793% by using this rate to compute the value of the bond to determine if it is $1,168.99. Doing this, we first compute the present value of the coupon payments where N is now the number of periods until the bond is assumed to be called. We get:()111N y C y ⎡⎤-⎢⎥+⎢⎥⎢⎥⎣⎦ = 2611 (1 .043965)$55 0.043965⎡⎤-⎢⎥⎢⎥⎣⎦= $55[15.314173] = $842.280.We next compute the present value of the maturity value under the assumption it will be called in 13 years where M is now M* (which is the call price in dollars), and n is now n * (which is the number of periods until the assumed call date, e.g., number of years times 2). We get:()*1*1N M y ⎡⎤⎢⎥+⎢⎥⎣⎦= ()261$1,000 1.043965 ⎡⎤⎢⎥⎣⎦= $1,000[0.3267123] = $326.712.When a semiannual interest rate of 8.793% / 2 = 4.3965% is used, the present value of the cash flows is $842.280 + $326.712 = $1,168.99 or about $1,169. Thus, the yield to call for this bond is8.793%.(c) Show that the yield to put is 6.942%.First of all, we could compute the internal return based upon the cash flows if the bond is held for 5 years. We would get 3.4710%. For a semiannual pay bond, doubling the periodic interest rate (y ) gives the yield to put on a bond-equivalent basis. Taking 3.4710% times two gives us a yield to first par put of 6.9420%.We can also verify the yield to put is 6.942% by using this rate to compute the value of the bond to determine if it is $1,168.99. Doing this, we first compute the present value of the coupon payments where N is now the number of periods until the bond is assumed to be sold. We get:()111N y C y ⎡⎤-⎢⎥+⎢⎥⎢⎥⎣⎦= 1011 (1 .03471)$55 0. 03471⎡⎤-⎢⎥⎢⎥⎣⎦= $55[8.328775] = $458.083.We next compute the present value of the maturity value under the assumption the put will be exercised five years where M is now M * (which is the put price in dollars), and n is now n* (which is the number of periods until the put date, e.g., number of years times 2). We get:()*1*1N M y ⎡⎤⎢⎥+⎢⎥⎣⎦= ()101$1,000 1.03471 ⎡⎤⎢⎥⎣⎦ = $1,000[0.7109769] = $710.977.When a 6.942% / 2 = 3.471 % semiannual interest rate is used, the present value of the cash flows is $458.083 + $710.977 = $1,169.06 or about $1,169. Thus, the yield to put for this bond is6.942%.(d) Suppose that the call schedule for this bond is as follows:Can be called in eight years at $1,055.Can be called in 13 years at $1,000.And suppose this bond can only be put in five years and assume that the yield to first par call is 8.535%. What is the yield to worst for this bond?A 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. If the bond is called in eight years at $1,055, then we can compute the yield to maturity and get 9.986% or about 10%. If the bond is called in thirteen years at $1,000, then we can compute the yield to maturity and get 11.00%. If the bond is put in five years, the yield to maturity is 8.535%. Since the latter is the lowest the yield to worse is for this bond is 8.535%.8. (a) What is meant by an amortizing security?Amortized securities are fixed income securities whose cash flows include scheduled principal repayments prior to maturity. That is, the cash flow in each period includes interest plus principal repayment.For amortizing securities, reinvestment risk is even greater than for nonamortizing securities. The reason is that the investor must now reinvest the periodic principal repayments in addition to the periodic coupon interest payments. Moreover, the cash flows are monthly, not semiannually as with nonamortizing securities. In brief, the investor must not only reinvest periodic coupon interest payments and principal, but must do it more often. This increases reinvestment risk.(b) What are the three components of the cash flow for an amortizing security?As stated in part (a), an amortizing security includes both interest plus principal repayment.However, we must also note that the amount the borrower can repay in principal may exceed thescheduled amount. This excess amount of principal repayment over the amount scheduled is called a prepayment. Thus, for amortizing securities, the cash flow each period consists of three components: (1) coupon interest, (2) scheduled principal repayment, and (3) prepayments.(c) What is meant by a cash flow yield?For amortizing securities, market participants calculate a cash flow yield. It is the interest rate that will make the present value of the projected cash flows equal to the market price. The difficulty in computing a cash flow yield is projecting what the prepayment will be in each period.9. How is the internal rate of return of a portfolio calculated?The yield for a portfolio of bonds is not simply the average or weighted 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. Mathematically, the yield, y, on a portfolio (like any investment) is the interest rate that satisfies the equation:Solving for the yield (y) requires a trial-and-error (iterative) procedure. The objective is to find the interest rate that will make the present value of the cash flows equal to the price. An example demonstrates how this is done. Suppose that all investments in the portfolio selling for $903.10 promises to make the following annual payments:Years fromNow Promised Annual Payments (Cash Flow to Investor)1 $1002 $1003 $1004 $1,000To compute the portfolio internal rate of return, different interest rates must be tried until the present value of the cash flows is equal to $903.10 (the price of the financial instrument). Tryingan annual interest rate of 10% gives the following present value:Years fromNow Promised Annual Payments(Cash Flow to Investor)Present Valueof Cash Flow at 10%1$ 100$ 90.912$ 100$ 82.643 $ 100 $ 75.134 $1,000 $ 683.01Present value = $ 931.69Because the present value computed using a 10% interest rate exceeds the price of $903.10, a higher interest rate must be used, to reduce the present value. If a 12% interest rate is used, the present value is $875.71, computed as follows:Years fromNow Promised Annual Payments(Cash Flow to Investor)Present Valueof Cash Flow at 12%1$ 100$ 89.292$ 100$ 79.723 $ 100 $ 71.184 $1,000 $ 635.52Present value = $ 875.71Using 12%, the present value of the cash flow is less than the price of the financial instrument. Therefore, a lower interest rate must be tried, to increase the present value. Using an 11% interest rate:Years from Now Promised Annual Payments(Cash Flow to Investor)Present Valueof Cash Flow at 11%1$ 100$ 90.092$ 100$ 81.163 $ 100 $ 73.124 $1,000 $ 658.73Present value = $ 903.10Using 11%, the present value of the cash flow is equal to the price of the portfolio. Therefore, the yield is 11%.Keep in mind that the yield computed is now the yield for the period. That is, if the cash flows are semiannual, the yield is a semiannual yield. If the cash flows are 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.10. What is the limitation of using the internal rate of return of a portfolio as a measure of the portfolio’s yield?Implicit in the internal rate of return computation is the assumption that the portfolio cash flowscan be reinvested at the computed internal rate of return. Also, when we compute an internal rate of return, we annualized interest rates by multiplying by the number of periods in a year (we call the resulting value the simple annual interest rate).For example, multiplying by 2 annualizes a semiannual yield. Alternatively, an annual interest rate is converted to a semiannual interest rate by dividing by 2. This simplified procedure for computing the annual interest rate given a periodic (weekly, monthly, quarterly, semiannually, and so on) interest rate is not accurate. To obtain an effective annual yield associated with 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. For example, suppose that the periodic interest rate is 4% and the frequency of payments is twice per year. Inserting in the values we get:effective annual yield = (1.04)2– 1 = 1.0816 – 1 = 0.0816 or 8.16%.This is different from 8.00%, which we get by multiplying 4.00% times two.11. Suppose that the coupon rate of a floating-rate security resets every six months at a spread of 70 basis points over the reference rate. If the bond is trading at below par value, explain whether the discount margin is greater than or less than 70 basis points.If the bond is trading below par value, then the discount margin or assumed annual spread (basis points) will be greater than 70 basis points. This is because the spread must increase to make the present value of the cash flows less than the par value. This is illustrated in Exhibit 3-1 where the bond is trading below par and the spread (basis points) had to increase in order for the present value of the cash flows to fall to a level to equal the current trading value.12. An investor is considering the purchase of a 20-year 7% coupon bond selling for $816 and a par value of $1,000. The yield to maturity for this bond is 9%.Answer the below questions.(a) What would be the total future dollars if this investor invested $816 for 20 years earning 9% compounded semiannually?To determine the future value of any sum of money invested today, we use the below equation: 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: P n= P0(1 + r)n= $816(1.045)40= $816(5.8163645) = $4,746.15.(b) What are the total coupon payments over the life of this bond?The total dollar amount of coupon interest is found by multiplying the semiannual coupon interest by the number of periods: total coupon interest = nC. Thus, the total coupon payments are: nC = 40($35) = $1,400.00.(c) What would be the total future dollars from the coupon payments and the repayment of。