Problems and Solutions CHAPTER Problems. Problems on onds Exercise. On /04/0, consider a fixed-coupon bond whose features are the following: face value: $,000 coupon rate: 8% coupon frequency: semiannual maturity: 0/06/04 What are the future cash flows delivered by this bond? Solution.. The coupon cash flow is equal to $40 8% $,000 Coupon = = $40 It is delivered on the following future dates: 0/06/0, /06/0, 0/06/03, /06/03 and 0/06/04. The redemption value is equal to the face value $,000 and is delivered on maturity date 0/06/04. Exercise. Consider the same bond as in the previous exercise. We are still on /04/0.. Compute the accrued interest taking into account the Actual/Actual day-count basis.. Same question if we are now on 09/06/0. Solution.. The last coupon has been delivered on /06/0. There are 8 days between /06/0 and /04/0, and 8 days between the last coupon date /06/0 and the next coupon date 0/06/0. Hence, the accrued interest is equal to $6.88 Accrued Interest = 8 $40 = $6.88 8. The last coupon has been delivered on 0/06/0. There are 3 days between 0/06/0 and 09/06/0, and 84 days between the last coupon date 0/06/0 and the next coupon date /06/0. Hence, the accrued interest is equal to $6.739 Accrued Interest = 3 $40 = $6.739 84
Problems and Solutions Exercise.3 Solution.3 Exercise.4 Solution.4 An investor has a cash of $0,000,000 at disposal. He wants to invest in a bond with $,000 nominal value and whose dirty price is equal to 07.47%.. What is the number of bonds he will buy?. Same question if the nominal value and the dirty price of the bond are respectively $00 and 98.43%.. The number of bonds he will buy is given by the following formula Cash Number of bonds bought = Nominal Value of the bond dirty price Here, the number of bonds is equal to 9,306. n is equal to 0,6 n = 0,000,000,000 07.47% = 9,306.048 n = 0,000,000 00 98.43% = 0,7.3 On 0//99, consider a fixed-coupon bond whose features are the following: face value: Eur 00 coupon rate: 0% coupon frequency: annual maturity: 04//08 Compute the accrued interest taking into account the four different day-count bases: Actual/Actual, Actual/36, Actual/360 and 30/360. The last coupon has been delivered on 04//99. There are 93 days between 04//99 and 0//99, and 366 days between the last coupon date 04//99 and the next coupon date 04//00. The accrued interest with the Actual/Actual day-count basis is equal to Eur.73 93 0% Eur 00 = Eur.73 366 The accrued interest with the Actual/36 day-count basis is equal to Eur.88 93 0% Eur 00 = Eur.88 36 The accrued interest with the Actual/360 day-count basis is equal to Eur.36 93 0% Eur 00 = Eur.36 360 There are days between 04//99 and 04/30/99, months between May and September, and days between 09/30/99 and 0//99, so that there
Problems and Solutions 3 are 90 days between 04//99 and 0//99 on the 30/360 day-count basis + 30 + = 90 Exercise. inally, the accrued interest with the 30/360 day-count basis is equal to Eur.78 90 0% Eur 00 = Eur.78 360 Some bonds have irregular first coupons. A long first coupon is paid on the second anniversary date of the bond and starts accruing on the issue date. So, the first coupon value is greater than the normal coupon rate. A long first coupon with regular value is paid on the second anniversary date of the bond and starts accruing on the first anniversary date. So, the first coupon value is equal to the normal coupon rate. A short first coupon is paid on the first anniversary date of the bond and starts accruing on the issue date. The first coupon value is smaller than the normal coupon rate. A short first coupon with regular value is paid on the first anniversary date of the bond and has a value equal to the normal coupon rate. Consider the four following bonds with nominal value equal to million euros and annual coupon frequency: ond : issue date 0//96, coupon %, maturity date 0//0, long first coupon, redemption value 00%; ond : issue date 0//96, coupon %, maturity date 0//0, long first coupon with regular value, redemption value 99%; ond 3: issue date //9, coupon 3%, maturity date 3 years and months, short first coupon, redemption value 00%; ond 4: issue date 08//9, coupon 4.%, maturity date 08//00, short first coupon with regular value, redemption value 00%. Compute the future cash flows of each of these bonds. Solution. ond pays 00,000 euros on 0//98, 0,000 euros on 0//99, 0//00, 0//0 and,00,000 euros on 0//0. ond pays 0,000 euros on 0//98, 0//99, 0//00, 0//0 and,040,000 euros on 0//0. ond 3 pays,000 euros on 0//96, 30,000 euros on 0//97, 0//98 and,030,000 euros on 0//99. ond 4 pays 4,000 euros on 08//96, 08//97, 08//98, 08//99 and,04,000 euros on 08//00. Exercise.8 Solution.8 An investor wants to buy a bullet bond of the automotive sector. He has two choices: either invest in a US corporate bond denominated in euros or in a rench corporate bond with same maturity and coupon. Are the two bonds comparable? The answer is no. irst, the coupon and yield frequency of the US corporate bond is semiannual, while it is annual for the rench corporate bond. To compare the yields
4 Problems and Solutions on the two instruments, you have to convert either the semiannual yield of the US bond into an equivalently annual yield or the annual yield of the rench bond into an equivalently semiannual yield. Second, the two bonds do not necessarily have the same rating, that is, the same credit risk. Third, they do not necessarily have the same liquidity. Exercise.3 Treasury bills are quoted using the yield on a discount basis or on a moneymarket basis.. The yield on a discount basis denoted by y d is computed as y d = P n where is the face value, P the price, the year-basis 36 or 360 and n is the number of calendar days remaining to maturity. Prove in this case that the price of the T-bill is obtained using the equation P = n y d. The yield on a money-market basis denoted by y m is computed as y m = y d n y d Prove in this case that the price of the T-bill is obtained using the equation 3. Show that P = + n y m y d = y m + n y m Solution.3. rom the equation we find and finally, we obtain. rom the equation y d = P n y d P = n = P n y d y m = y d n y d
Problems and Solutions we find Then, we have inally, we obtain 3. rom the equation y m = n y m P n P = P P n n = = P P P = + n y m n P P = P we find Then, we have y m = y d n y d y m n y d y d = 0 y d n y m = y m inally, we obtain y d = y m + n y m Exercise. What is the price P of the certificate of deposit issued by bank X on 06/06/00, with maturity 08//00, face value $0,000,000, an interest rate at issuance of % falling at maturity and a yield of 4.% as of 07/3/00? Solution. Recall that the price P of such a product is given by + c n c P = + ym n m where is the face value, c the interest rate at issuance, n c is the number of days between issue and maturity, is the year-basis 360 or 36, y m is the yield on a money-market basis and n m is the number of days between settlement and maturity. Then, the price P of the certificate of deposit issued by bank X is equal to + % 360 80 P = $0,000,000 = $0,079,6.3 + 4.% 360 Indeed, there are 80 calendar days between 06/06/00 and 08//00, and calendar days between 07/3/00 and 08//00
6 Problems and Solutions Exercise.6 Solution.6 On 0/03/00, an investor buys $ million US T-ill with maturity date 06/7/00 and discount yield.76% on the settlement date.. What is the price of the T-ill?. What is the equivalent money-market yield?. The settlement date of the transaction is 0/04/00 trading date plus working day. There are 74 calendar days between the settlement date and the maturity date. The price P of the T-ill is equal to 00.76% 74 = 99.493 360. The equivalent money-market yield is equal to.77%.76% 360 360 74.76% =.77% CHAPTER Problems Exercise. Suppose the -year continuously compounded interest rate is %. What is the effective annual interest rate? Solution. The effective annual interest rate is R = e 0. = 0.7 =.7%. Exercise. Solution. If you deposit $,00 in a bank account that earns 8% annually on a continuously compounded basis, what will be the account balance in 7.4 years? The account balance in 7.4 years will be $,00.e 8% 7.4 = $4,4.98 Exercise.3 Solution.3 Exercise.7 If an investment has a cumulative 63.4% rate of return over 3.78 years, what is the annual continuously compounded rate of return? The annual continuously compounded rate of return R is such that We find R c = ln.634/3.78 = 3%..634 = e 3.78Rc. What is the price of a -year bond with a nominal value of $00, a yield to maturity of 7% with annual compounding frequency, a 0% coupon rate and an annual coupon frequency?. Same question for a yield to maturity of 8%, 9% and 0%. Conclude.
Problems and Solutions 7 Solution.7. The price P of a bond is given by the formula n N c P = + y i + N + y n which simplifies into P = N c y i= [ ] + y n + N + y n where N, c, y and n are respectively the nominal value, the coupon rate, the yield to maturity and the number of years to maturity of the bond. Here, we obtain for P P = 0 7% [ ] + 7% + 00 + 7% P is then equal to.30% of the nominal value or $.30. Note that we can also use the Excel function Price to obtain P.. Prices of the bond for different yields to maturity YTM are given in the following table YTM % Price $ 8 07.98 9 03.890 0 00 ond prices decrease as rates increase. Exercise.0 Solution.0. What is the yield to maturity of a -year bond with a nominal value of $00, a 0% coupon rate, an annual coupon frequency and a price of 97.86?. Same question for a price of 00 and 0.4.. The yield to maturity y of this bond is the solution to the following equation [ ] + P = N c y + y n N + y n where N, c, P and n are respectively the nominal value, the coupon rate, the price and the number of years to maturity of the bond. Here, y is solution to 97.86 = 0 y [ ] + y + 00 + y Using, for example, Newton s three points method or the Solver function in Excel, we obtain 0.74%. Note that we can also use the Excel function Yield to obtain y.. Yields to maturity YTM of the bond for different prices are given in the following table
8 Problems and Solutions Price YTM % 00 0 0.4 8.63 Exercise.3 Solution.3 Consider the following bond: annual coupon %, maturity years, annual compounding frequency.. What is its relative price change if its required yield increases from 0% to %?. What is its relative price change if its required yield increases from % to 6%? 3. What conclusion can you draw from these examples? Explain why.. The initial price P is equal to P = + 0% + + 0% + + 0% 3 + + 0% 4 + 0 + 0% = 8.046 After the yield change, the price becomes P = + % + + % + + % 3 + + % 4 + 0 + % = 77.8 Hence, the bond price has decreased by P P = 3.97% P. The initial price P is equal to P = + % + + % + + % 3 + + % 4 + 0 + % = 00 After the yield change, the price becomes P = + 6% + + 6% + + 6% 3 + + 6% 4 + 0 + 6% = 9.788 Hence, the bond price has decreased by P P = 4.% P 3. In low interest-rate environments, the relative price volatility of a bond is higher than in high interest-rate environments for the same yield change here, in our example +%. This is due to the convexity relationship between the price of a bond and its yield.
Problems and Solutions 9 Exercise.4 Solution.4 We consider the following zero-coupon curve: Maturity years Zero-Coupon Rate % 4.00 4.0 3 4.7 4 4.90.00. What is the price of a -year bond with a $00 face value, which delivers a % annual coupon rate?. What is the yield to maturity of this bond? 3. We suppose that the zero-coupon curve increases instantaneously and uniformly by 0.%. What is the new price and the new yield to maturity of the bond? What is the impact of this rate increase for the bondholder? 4. We suppose now that the zero-coupon curve remains stable over time. You hold the bond until maturity. What is the annual return rate of your investment? Why is this rate different from the yield to maturity?. The price P of the bond is equal to the sum of its discounted cash flows and given by the following formula P = + 4% + + 4.% + + 4.7% 3 + + 4.9% 4 + 0 + % = $00.36. The yield to maturity R of this bond verifies the following equation 4 00.36 = + R i + 0 + R Using the Excel function Yield, we obtain 4.9686% for R. 3. The new price P of the bond is given by the following formula: P = i= + 4.% + + % + +.% 3 + +.4% 4 + 0 +.% = $97.999 The new yield to maturity R of this bond verifies the following equation 4 97.999 = + R i + 0 + R i= Using the Excel function yield, we obtain.468% for R. The impact of this rate increase is an absolute capital loss of $.37 for the bondholder. Absolute Loss = 97.999 00.36 = $.37
0 Problems and Solutions and a relative capital loss of.34% Relative Loss =.37 00.36 =.34% 4. efore maturity, the bondholder receives intermediate coupons that he reinvests in the market: after one year, he receives $ that he reinvests for 4 years at the 4-year zerocoupon rate to obtain on the maturity date of the bond + 4.9% 4 = $6.044 after two years, he receives $ that he reinvests for 3 years at the 3-year zerocoupon rate to obtain on the maturity date of the bond + 4.7% 3 = $.7469 after three years, he receives $ that he reinvests for years at the -year zero-coupon rate to obtain on the maturity date of the bond + 4.% = $.460 after four years, he receives $ that he reinvests for year at the -year zerocoupon rate to obtain on the maturity date of the bond + 4% = $. after five years, he receives the final cash flow equal to $0. The bondholder finally obtains $7.464 five years later 6.044 +.7469 +.460 +. + 0 = $7.464 which corresponds to a 4.944% annual return rate 7.464 / = 4.944% 00.36 This return rate is different from the yield to maturity of this bond 4.9686% because the curve is not flat at a 4.9686% level. With a flat curve at a 4.9686% level, we obtain $7.608 five years later 6.0703 +.789 +.09 +.484 + 0 = $7.608 which corresponds exactly to a 4.9686% annual return rate. 7.608 / = 4.9686% 00.36 Exercise. Let us consider the two following rench Treasury bonds whose characteristics are the following:
Problems and Solutions Name Maturity Coupon Price years Rate % ond 6 00 ond 0 0 3.8 Your investment horizon is 6 years. Which of the two bonds will you select? Solution. Exercise.8 It depends on the level of the reinvestment rate, at which you can reinvest the coupons of ond, as well as on the yield to maturity of ond at horizon. If you suppose, for example, that the reinvestment rate is equal to the yield to maturity of ond at horizon, then the total return of ond will decrease as the reinvestment rate increases, as opposed to ond. Indeed, while the unique source of return for ond is its reinvested coupons, it lies for ond in its price appreciation. ond and ond will yield nearly the same annualized return.% for a reinvestment rate of 6.36%. We consider three bonds with the following features ond Maturity years Annual Coupon Price ond 0 06.6 ond 8 06.0 ond 3 3 8 06.4. ind the -year, -year and 3-year zero-coupon rates from the table above.. We consider another bond with the following features ond Maturity Annual Coupon Price ond 4 3 years 9 09.0 Use the zero-coupon curve to price this bond. 3. ind an arbitrage strategy. Solution.8. The -year zero-coupon rate denoted by R0,, verifies 0 + R0, = 06.6 We find the expression R0, = 0 06.6 = 3.8% The -year zero-coupon rate denoted by R0,, verifies 8 + 3.8% + 08 + R0, = 06.0 We find the expression / 08 R0, = = 4.738% 06. 8 +3.8%
Problems and Solutions The 3-year zero-coupon rate denoted by R0, 3, verifies 8 + 3.8% + 8 + 4.738% + 08 + R0, 3 3 = 06.4 We find the expression /3 08 R0, 3 = 8 06.4 +3.8% =.78% 8 +4.738%. The price P of ond 4 using the zero-coupon curve is given by the following formula: 9 P = + 3.8% + 9 + 4.738% + 09 +.78% 3 = 09.77 3. This bond is underpriced by the market compared to its theoretical value. There is an arbitrage if the market price of this bond reverts to the theoretical value. We have to simply buy the bond at a $09.0 price and hope that it is mispriced by the market and will soon revert to around $09.77. Exercise.0 We consider two bonds with the following features ond Maturity years Coupon Rate % Price YTM % ond 0 0,3..39 ond 0 964.3.473 YTM stands for yield to maturity. These two bonds have a $,000 face value, and an annual coupon frequency.. An investor buys these two bonds and holds them until maturity. Compute the annual return rate over the period, supposing that the yield curve becomes instantaneously flat at a.4% level and remains stable at this level during 0 years.. What is the rate level such that these two bonds provide the same annual return rate? In this case, what is the annual return rate of the two bonds? Solution.0. We consider that the investor reinvests its intermediate cash flows at a unique.4% rate. or ond, the investor obtains the following sum at the maturity of the bond 9 00 +.4% i +,00 =,8. i= which corresponds exactly to a.3703% annual return rate.,8. /0 =.3703%,3.
Problems and Solutions 3 or ond, the investor obtains the following sum at the maturity of the bond 9 0 +.4% i +,00 =,640.76 i= which corresponds exactly to a.489% annual return rate.,640.76 /0 =.489% 964.3. We have to find the value R, such that 00 9 i= + R i +,00,3. = 0 9 i= + R i +,00 964.3 Using the Excel solver, we finally obtain 6.4447% for R. The annual return rate of the two bonds is equal to.664% 00 9i= + 6.4447 i /0 +,00 =.664%,3. Exercise.4 Assume that the following bond yields, compounded semiannually: 6-month Treasury Strip:.00%; -year Treasury Strip:.%; 8-month Treasury Strip:.7%.. What is the 6-month forward rate in six months?. What is the -year forward rate in six months? 3. What is the price of a semiannual 0% coupon Treasury bond that matures in exactly 8 months? Solution.4.. + R 0, = + R 0, 0..06 =.0 + 0, 0., 0. 0, 0., 0. =.003% + R 0,. 3 = + R 0, 0..087 3 =.0 + 0, 0., 0. + 0,0., + 0, 0., 0, 0., = 6.60% 3. The cash flows are coupons of % in six months and a year, and coupon plus principal payment of 0% in 8 months. We can discount using the spot rates
4 Problems and Solutions that we are given: 0 P = + + 0.0 + 3 = 06.066 + 0.0 + 0.07 Exercise.6 Solution.6 Consider a coupon bond with n = 0 semesters i.e., 0 years to maturity, an annual coupon rate c = 6.% coupons are paid semiannually, and nominal value N = $,000. Suppose that the semiannually compounded yield to maturity YTM of this bond is y =.%.. Compute the current price of the bond using the annuity formula.. Compute the annually compounded YTM and the current yield of the bond. Compare them with y. 3. If the yield to maturity on the bond does not change over the next semester, what is the Holding Period Return HPR obtained from buying the bond now and selling it one semester from now, just after coupon payment? At what price will the bond sell one semester from now just after coupon payment?. or the current price of the bond, we use the formula P 0 = N c + y so that P 0 =,000 6.%.% + y / n + 0.07 0 + N + y / n,000 =,076.4 + 0.07 0. The annually compounded yield to maturity YTM denoted by y and the current yield denoted by y c are obtained using the following formulas: y = + y = + 0.0 = 0.076 y c = cn 0.06,000 = = 0.06040 P 0,076.4 Therefore, they are both larger than y. 3. irst, we compute P, the price of the bond one semester from now: P = N c N + y =,000 6.%.% =,073. + y / n + 0.07 9 + y / n +,000 + 0.07 9 The Holding Period Return from buying the bond now and selling it one semester from now is then: HPR = P P 0 + cn,073.0,076.4 + 3. = =.7% P 0,076.4