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A B C D E F G 1 3/19/2009 2 3 Chapter 4. Tool Kit for Bond Valuation 4 5 The value of any financial asset is the present value of the asset's expected future cash flows. The key inputs are (1) the 6 expected cash flows and (2) the appropriate discount rate, given the bond's risk, maturity, and other characteristics. The 7 model developed here analyzes bonds in various ways. 8 9 BOND VALUATION (Section 4.3) 10 11 A bond has a 15-year maturity, a 10% annual coupon, and a $1,000 par value. The required rate of return (or the yield to 12 maturity) on the bond is 10%, given its risk, maturity, liquidity, and other rates in the economy. What is a fair value for the 13 bond, i.e., its market price? 14 15 First, we list the key features of the bond as "model inputs": 16 Years to Mat: 15 17 Coupon rate: 10% 18 Annual Pmt: $100 19 Par value = FV: $1,000 20 Required return, rd: 10% 21 22 The easiest way to solve this problem is to use Excel's PV function. Click fx, then financial, then PV. Then fill in 23 the menu items as shown in our snapshot in the screen shown just below. 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 Value of bond = $1,000.00 Thus, this bond sells at its par value. That situation always exists if the going 42 rate is equal to the coupon rate. 43 44 45 The PV function can only be used if the payments are constant, but that is normally the case for bonds. 46 A B C D E F G 47 Bond Prices on Actual Dates 48 49 50 Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for 51 new issues, but it is generally not correct for outstanding bonds. However, Excel has several date and time functions, and a 52 bond valuation function that uses the calendar, so we can get exact valuations on any given date. 53 54 Here is the data for MicroDrive's bond as of the day it was issued. 55 56 Settlement date (day on which you find bond price) = 1/5/2010 57 Maturity date = 1/5/2025 58 Coupon rate = 10.00% 59 Required return, rd = 10.00% Redemption (100 means the bond pays 100% of its 60 face value at maturity) = 100 61 Frequency (# payments per year) = 1 62 Basis (1 is for actual number of days in month and year) 1 63 64 Click on fx on the formula bar (or click Insert and then Function). This gives you the "Insert Function" dialog box. To find 65 a bond's price, use the PRICE function (found in the "Financial" category of the "Insert Function dialog box). The PRICE 66 function returns the price per $100 dollars of face value. 67 68 Using PRICE function with inputs that are cell references: 69 Value of bond based on $100 face value = $100.00 70 Value of bond in dollars based on $1,000 face value = $1,000.00 71 72 Using the PRICE function with inputs that are not cell references: 73 Value of bond based on $100 face value = =PRICE(DATE(2009,1,5),DATE(2024,1,5),10%,10%,100,1,1) 74 Value of bond based on $100 face value = 100.0000 75 Value of bond in dollars based on $1,000 face value = $1,000.00 76 77 78 Interest Rate Changes and Bond Prices 79 80 Suppose the going interest rate changed from 10%, falling to 5% or rising to 15%. How would those changes affect the 81 value of the bond? 82 83 84 We could simply go to the input data section shown above, change the value for r from 10% to 5% and then 15%, and 85 observe the changed values. An alternative is to set up a data table to show the bond's value at a range of rates, i.e., to show 86 the bond's sensitivity to changes in interest rates. This is done below, and the values at 5% and 15% are boldfaced. 87 Bond Value 88 Going rate, r: $1,000 To make the data table, first type the headings, then type the rates in 89 0% $2,500.00 cells A87:A91, and then put the formula =B41 in cell B86, then select 90 5% $1,518.98 the range A86:B912. Then click Data and then Table to get the 91 10% $1,000.00 menu. The input data are in a column, so put the cursor on column 92 15% $707.63 and enter C20 the place where the going rate is inputted. Click OK 93 20% $532.45 to complete the operation and get the table. 94 95 We can use the data table to construct a graph that shows the bond's 96 sensitivity to changing rates. 97 A B C D E F G 98 99 Interest Rate Sensitivity 100 101 $3,000 102 $2,500 $2,000 103 $1,500 104 $1,000 105 $500 106 $0 107 0% 5% 10% 15% 20% 108 109 110 111 BOND YIELDS (Section 4.4) 112 113 Yield to Maturity 114 115 The YTM is defined as the rate of return that will be earned if a bond makes all scheduled payments and is held to 116 maturity. The YTM is the same as the total rate of return discussed in the chapter, and it can also be interpreted as the 117 "promised rate of return," or the return to investors if all promised payments are made. The YTM for a bond that sells at 118 par consists entirely of an interest yield. However, if the bond sells at any price other than its par value, the YTM consists 119 of the interest yield together with a positive or negative capital gains yield. The YTM can be determined by solving the 120 bond value formula for I. However, an easier method for finding it is to use Excel's Rate function. Since the price of a bond 121 is simply the sum of the present values of its cash flows, so we can use the time value of money techniques to solve these 122 problems. 123 124 Problem: Suppose that you are offered a 14-year, 10% annual coupon, $1,000 par value bond at a price of $1,494.93. What 125 is the Yield to Maturity of the bond? 126 127 Use the Rate function to solve the problem. 128 129 Years to Mat: 14 130 Coupon rate: 10% 131 Annual Pmt: $100.00 Going rate, r =YTM: 5.00% 132 Current price: $1,494.93 133 Par value = FV: $1,000.00 134 135 The yield-to-maturity is the same as the expected rate of return only if (1) the probability of default is zero, and (2) the bond 136 can not be called. If there is any chance of default, then there is a chance some payments may not be made. In this case, the 137 expected rate of return will be less than the promised yield-to-maturity. 138 139 Finding the Yield to Maturity on Actual Dates 140 141 Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for 142 new issues, but it is generally not correct for outstanding bonds. However, Excel has a function that uses the actual 143 calendar when finding yields. Consider the bond above, with 14 years until maturity. Suppose the actual current date is 144 1/5/2011, so the bond matures on 1/5/2025. 145 146 Here is the data for the bond. 147 148 Settlement date (day on which you find bond price) = 01/05/11 149 Maturity date = 01/05/25 A B C D E F G 150 Coupon rate = 10.00% 151 Price = bond price per $100 par value = $149.49 Redemption (100 means the bond pays 100% of its 152 face value at maturity) = 100 153 Frequency (# payments per year) = 1 154 Basis (1 is for actual number of days in month and year) 0 155 156 Using the YIELD function with inputs that are cell references: 157 Yield to maturity = 5.0% 158 159 160 Yield to Call 161 The yield to call is the rate of return investors will receive if their bonds are called. If the issuer has the right to call the 162 bonds, and if interest rates fall, then it would be logical for the issuer to call the bonds and replace them with new bonds 163 that carry a lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to 164 maturity is replaced with years to call, and the maturity value is replaced with the call price. 165 166 Problem: Suppose you purchase a 15-year, 10% annual coupon, $1,000 par value bond with a call provision after 10 years 167 at a call price of $1,100. One year later, interest rates have fallen from 10% to 5% causing the value of the bond to rise to 168 $1,494.93. What is the bond's YTC? Note that this is the same bond as in the previous question, but now we assume it can 169 be called. 170 171 Use the Rate function to solve the problem. 172 173 Years to call: 9 174 Coupon rate: 10% 175 Annual Pmt: $100.00 Rate = I = YTC = 4.21% 176 Current price: $1,494.93 177 Call price = FV $1,100.00 178 Par value $1,000.00 179 180 This bond's YTM is 5%, but its YTC is only 4.21%. Which would an investor be more likely to actually earn? 181 182 183 This company could call the old bonds, which pay $100 per year, and replace them with bonds that pay somewhere in the 184 vicinity of $50 (or maybe even only $42.10) per year. It would want to save that money, so it would in all likelihood call the 185 bonds. In that case, investors would earn the YTC, so the YTC is the expected return on the bonds. 186 187 Current Yield 188 The current yield is the annual interest payment divided by the bond's current price. The current yield provides 189 information regarding the amount of cash income that a bond will generate in a given year. However, it does not account 190 for any capital gains or losses that will be realized fi the bond is held to maturity or call. 191 192 Problem: What is the current yield on a $1,000 par value, 10% annual coupon bond that is currently selling for 193 $985? 194 195 Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we would 196 still use the annual interest. 197 198 Par value $1,000.00 199 Coupon rate: 10% Current Yield = 10.15% 200 Annual Pmt: $100.00 201 Current price: $985.00 A B C D E F G 202 203 The current yield provides information on a bond's cash return, but it gives no indication of the bond's total return. To see 204 this, consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no interest income. 205 However, the zero appreciates through time, and its total return clearly exceeds zero. 206 207 208 CHANGES IN BOND VALUES OVER TIME (Section 4.4) 209 210 What happens to a bond price over time? To set up this problem, we will enter the different interest rates, and use the array 211 of cash flows above. The following example operates under the precept that the bond is issued at par ($1,000) in year 0. 212 From this point, the example sets three conditions for interest rates to follow: interest rates stay constant at 10%, interest 213 rates fall to 5%, or interest rates rise to 15%. Then the price of the bond over the fifteen years of its life is determined for 214 each of the scenarios. 215 216 Suppose interest rates rose to 15% or fell to 5% immediately after the bond was issued, and they remained at the new level 217 for the next 15 years. What would happen to the price of the bond over time? 218 219 We could set up data tables to get the data for this problem, but instead we simply inserted the PV formula into the 220 following matrix to calculate the value of the bond over time. Note that the formula takes the interest rate from the column 221 heads, and the value of N from the left column. Note that the N = 0 values for the 5% and 15% rates are consistent with the 222 results in the data table above. We can also plot the data, as shown in the graph below. 223 A B C D E F G 224 Value of Bond in Given Year: 225 N 5% 10% 15% 226 0 $1,519 $1,000 $708 227 1 $1,495 $1,000 $714 228 2 $1,470 $1,000 $721 229 3 $1,443 $1,000 $729 230 4 $1,415 $1,000 $738 231 5 $1,386 $1,000 $749 232 6 $1,355 $1,000 $761 233 7 $1,323 $1,000 $776 234 8 $1,289 $1,000 $792 235 9 $1,254 $1,000 $811 236 10 $1,216 $1,000 $832 237 11 $1,177 $1,000 $857 238 12 $1,136 $1,000 $886 239 13 $1,093 $1,000 $919 240 14 $1,048 $1,000 $957 241 15 $1,000 $1,000 $1,000 242 243 244 Price of Bond Over Time 245 246 $1,600 247 248 $1,400 249 $1,200 250 $1,000 Rate Drops to 5% 251 $800 Rate Stays at 10% 252 $600 Rate Rises to 15% 253 $400 254 255 $200 256 $0 257 0 5 10 15 258 259 260 If rates fall, the bond goes to a premium, but it moves toward par as maturity approaches. The reverse hold if rates rise 261 and the bond sells at a discount. If the going rate remains equal to the coupon rate, the bond will continue to sell at par. 262 Note that the above graph assumes that interest rates stay constant after the initial change. That is most unlikely--interest 263 rates fluctuate, and so do the prices of outstanding bonds. 264 265 266 267 BONDS WITH SEMIANNUAL COUPONS (Section 4.6) 268 269 Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three 270 modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by 2, 271 and (3) divide the nominal interest rate by 2. 272 273 Problem: What is the price of a 15-year, 10% semi-annual coupon, $1,000 par value bond if the nominal rate (the YTM) is 274 5%? The bond is not callable. 275 276 Use the Rate function with adjusted data to solve the problem. A B C D E F G 277 278 Periods to maturity = 15*2 = 30 279 Coupon rate: 10% 280 Semiannual pmt = $100/2 = $50.00 PV = $1,523.26 281 Current price: $1,000.00 282 Periodic rate = 5%/2 = 2.5% 283 284 Note that the bond is now more valuable, because interest payments come in faster. 285 286 287 THE DETERMINANTS OF MARKET INTEREST RATES (Section 4.7) 288 289 Quoted market interest rate = rd = r* + IP + DRP + LP + MRP 290 291 r* = Real risk-free rate of interest 292 IP = Inflation premium 293 DRP = Default risk premium 294 LP = Liquidity premium 295 MRP = Maturity risk premium 296 297 298 THE REAL RISK-FREE RATE OF INTEREST, r* (Section 4.8) 299 300 r* = Real risk-free rate of interest 301 r* = Yield on short-term U.S. Treasury Inflation-Protected Security (TIPS) 302 r* = -0.05% (March 2008) 303 304 305 THE INFLATION PREMIUM (IP) (Section 4.9) 306 Maturity 307 5 Years 30 Years 308 Non-indexed U.S. Treasury Bond 2.46% 4.40% 309 TIPS -0.05% 1.63% 310 Inflation premium 2.51% 2.77% 311 312 313 314 THE NOMINAL, OR QUOTED, RISK-FREE RATE OF INTEREST, rRF (Section 4.10) 315 316 Nominal, or quoted, rate = rd = rRF + DRP + LP + MRP 317 318 319 THE DEFAULT RISK PREMIUM (DRP) (Section 4.11) 320 321 Bond spreads are the difference between the yield on a bond and the yield on some other bond of the same maturity. 322 323 324 Spread relative to: Yield T-Bond AAA BBB Source: see 325 Long-term Bonds (1) (2) (3) (4) comment. 326 U.S. Treasury 4.40% A B C D E F G 327 AAA 5.52% 1.12% 328 AA 5.90% 1.50% 0.38% 329 A 6.14% 1.74% 0.62% 330 BBB 6.51% 2.11% 0.99% 331 BB 7.09% 2.69% 1.57% 0.58% 332 B 7.61% 3.21% 2.09% 1.10% 333 CCC 9.01% 4.61% 3.49% 2.50% 334 335 Note: The spreads in Column (2) are found by taking the yields in Column (1) and subtracting the yield on the U.S. Treasury Bond. 336 The spreads in Column (3) are found by taking the yields in Column (1) and subtracting the yield on the AAA bond. 337 The spreads in Column (4) are found by taking the yields in Column (1) and subtracting the yield on the BBB bond. 338 339 For a bond with good liquidity, its spread relative to a T-bond of similar maturity is a good estmat of the default risk premium. 340 341 342 THE MATURITY RISK PREMIUM (MRP) (Section 4.12) 343 344 Bonds are exposed to interest rate risk and reinvestment rate risk. The net effect is the maturity risk premium. 345 346 Interest Rate Risk 347 348 Interest Rate Risk is the risk of a decline in a bond's price due to an increase in interest rates. Price sensitivity to interest 349 rates is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the same 350 coupon, the bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same maturity, 351 the one with the smaller coupon payment will have more interest rate sensitivity. 352 353 Compare the interest rate risk of two bonds, both of which have a 10% annual coupon and a $1,000 face value. The first 354 bond matures in 1 year, the second in 25 years. 355 356 Use the PV function, along with a two variable Data Table, to show the bonds' price sensitivity. 357 Coupon rate: 10% 358 Payment $100.00 359 Par value $1,000.00 360 Maturity 1 361 Going rate = r = YTM 10% 362 363 Value of bond: $1,000.00 364 365 366 Value of the Bond Under Different Conditions 367 Going rate, r Years to Maturity 368 $1,000.00 1 25 369 0% $1,100.00 $3,500.00 370 5% $1,047.62 $1,704.70 371 10% $1,000.00 $1,000.00 372 15% $956.52 $676.79 373 20% $916.67 $505.24 374 25% $880.00 $402.27 375 376 377 Bond Value 378 ($) 379 1,800 A B C D E F G 380 1,800 381 382 1,600 25-Year Bond 383 384 1,400 385 1,200 386 387 1,000 1-Year Bond 388 389 800 390 391 600 392 393 400 394 395 200 396 397 0 398 0% 5% 10% 15% 20% 25% 399 400 Interest Rate, rd 401 402 403 fff54e20-0c4b-4184-8d6c-2a818615c266.xls Web 4A 3/19/2009 Web Extension 4A: Zero Coupon Bonds Vandenburg Corporation needs to issue $50 million to finance a project, and it has decided to raise the funds by issuing $1,000 par value, zero coupon bonds. The going interest rate on such debt is 6%, and the corporate tax rate is 40%. Find the issue price of Vandenburg's bonds, construct a table to analyze the cash flows attributable to one of the bonds, and determine the after-tax cost of debt for the issue. Then, indicate the total par value of the issue. This example analyzes the after-tax cost of issuing zero coupon debt. Table 4A-1 Input Data Amount needed = $50,000,000 Maturity value= $1,000 Pre-tax market interest rate, rd = 6% Maturity (in years) = 5 Corporate tax rate = 40% Coupon rate = 0% Coupon payment (assuming annual payments) = $0 Issue Price = PV of payments at rd = $747.26 Analysis: Years 0 1 2 3 4 5 (1) Remaining years 5 4 3 2 1 0 (2) Year-end accrued value $747.26 $792.09 $839.62 $890.00 $943.40 $1,000.00 (3) Interest payment $0.00 $0.00 $0.00 $0.00 $0.00 (4) Implied interest deduction on discount $44.84 $47.53 $50.38 $53.40 $56.60 (5) Tax savings $17.93 $19.01 $20.15 $21.36 $22.64 (6) Cash flow $747.26 $17.93 $19.01 $20.15 $21.36 ($977.36) After-tax cost of debt = 3.60% Number of $1,000 zeros the company must issue to raise $50 million = Amount needed/Price per bond = 66,911.279 bonds. Face amount of bonds = # bonds x $1,000 = $66,911,279 Michael C. Ehrhardt Page 10 7/29/2011 fff54e20-0c4b-4184-8d6c-2a818615c266.xls Web 4C Web Extension 4C. Tool Kit for Duration Duration is a measure of risk for bonds. The following example illustrates its calculation. Figure 4C-1 Duration Inputs Years to maturity = 20 Coupon rate = 9.00% Annual payment = $90.0 Par value = FV = $1,000 Going rate, r = 9.00% t CFt PV of CFt t(PV of CFt) (1) (2) (3) (4) 1 $90 $82.57 82.57 2 $90 $75.75 151.50 3 $90 $69.50 208.49 4 $90 $63.76 255.03 5 $90 $58.49 292.47 6 $90 $53.66 321.98 7 $90 $49.23 344.63 8 $90 $45.17 361.34 9 $90 $41.44 372.95 10 $90 $38.02 380.17 11 $90 $34.88 383.66 12 $90 $32.00 383.98 13 $90 $29.36 381.63 14 $90 $26.93 377.05 15 $90 $24.71 370.63 16 $90 $22.67 362.69 17 $90 $20.80 353.54 18 $90 $19.08 343.43 19 $90 $17.50 332.58 20 $1,090 $194.49 3,889.79 Sum of VB = $1,000.00 t(PV of CFt) = $9,950.11 Duration = Sum of t(PV of CFt) / VB= 9.95 Finding Duration with the Excel Formula Settlement date = 1/1/2010 Maturity 12/31/2029 Coupon = 9% Yield = 9% Frequency = 1 Michael C. Ehrhardt Page 11 7/29/2011 fff54e20-0c4b-4184-8d6c-2a818615c266.xls Web 4C Duration = 9.95 Consider the amount that would accumulate during the first 10 years, if all coupons are reinvested at the original interest rate of 9%. To do this, first find the amount that would be in the account at 10 years (including the 10-year coupon). Then we find the value of the bond at year 10 based on the payments from 11 and on. Duration of Bond = 9.95011 Target value at year 10 = $10,000.00 FV of reinvested coupons at year 10 if no change in rates = $1,367.36 PV at year 10 of remaining payments if no change in rates = $1,000.00 Total value at year 10 if no change in rates = $2,367.36 Value of bonds to be purchased to provide target at 10 years = $4,224.11 Number of bonds purchased = 4.22 Now find the value at year 10 if the market interest rate (shown below) changes immediately after time zero, based on the total number of bonds that were purchased. Interest rate = 9.00% FV at year 10 = $5,775.89 PV of payments beyond year 10 discounted back to year 10 = $4,224.11 The total value of the position at time 9.95011 is the value of the reinvested coupon and the current value of the bond. Value of reinvested coupons: $5,775.89 Current value of bond: $4,224.11 Total value of position = $10,000.00 As the table below shows, the total value of a position at a future time equal to the orginal duration will not fall if interest rates change. For example, if rates go up, the value of reinvested coupons increases and the value of the bond at the future date (t=duration) falls, but the net affect is an increase in total value. If rates go down, the value of reinvested coupons goes down, but the future value of the bond goes up, for a net increase in value. Thus, if the desired time horizon is equal to the bond's duration, the value of the position will not fall if interest rates change. Change in Total Reinvested Current Price Value from Coupons at t=Duration Total Value Original Target $5,775.89 $4,224.11 $10,000.00 1% $3,977.42 $7,424.73 $11,402.15 $1,402.15 2% $4,162.75 $6,880.15 $11,042.90 $1,042.90 Michael C. Ehrhardt Page 12 7/29/2011 fff54e20-0c4b-4184-8d6c-2a818615c266.xls Web 4C 3% $4,358.22 $6,386.06 $10,744.28 $744.28 4% $4,564.36 $5,937.17 $10,501.53 $501.53 5% $4,781.73 $5,528.81 $10,310.54 $310.54 6% $5,010.94 $5,156.80 $10,167.74 $167.74 7% $5,252.60 $4,817.48 $10,070.07 $70.07 8% $5,507.35 $4,507.55 $10,014.90 $14.90 9% $5,775.89 $4,224.11 $10,000.00 $0.00 10% $6,058.93 $3,964.55 $10,023.48 $23.48 11% $6,357.20 $3,726.57 $10,083.77 $83.77 12% $6,671.50 $3,508.09 $10,179.59 $179.59 13% $7,002.63 $3,307.27 $10,309.90 $309.90 14% $7,351.45 $3,122.44 $10,473.89 $473.89 15% $7,718.86 $2,952.12 $10,670.98 $670.98 16% $8,105.78 $2,794.98 $10,900.76 $900.76 Michael C. Ehrhardt Page 13 7/29/2011 fff54e20-0c4b-4184-8d6c-2a818615c266.xls Web 4C 3/19/2009 xtension 4C. Tool Kit for Duration ollowing example illustrates its calculation. Michael C. Ehrhardt Page 14 7/29/2011 fff54e20-0c4b-4184-8d6c-2a818615c266.xls Web 4C during the first 10 years, if all coupons are reinvested at the original interest hat would be in the account at 10 years (including the 10-year coupon). Then on the payments from 11 and on. erest rate (shown below) changes immediately after time zero, based on the is the value of the reinvested coupon and the current value of the bond. position at a future time equal to the orginal duration will not fall if interest value of reinvested coupons increases and the value of the bond at the future increase in total value. If rates go down, the value of reinvested coupons goes p, for a net increase in value. Thus, if the desired time horizon is equal to the not fall if interest rates change. Michael C. Ehrhardt Page 15 7/29/2011 3/19/2009 Web Extension 4D. The Pure Expectations Theory and Estimation of Forward Rates The shape of the yield curve depends primarily on two key factors: (1) expectations about future inflation and (2) perceptions about the relative riskiness of securities of different maturities. The first factor is the basis for the Pure Expectations Hypothesis. If the relationship between expectations for future inflation and bond yields is controlling, i. e., if no maturity premiums existed, then the pure expectations theory posits that forward interest rates can be predicted by "backing them out of the yield curve." Essentially, under the pure expectations theory, long-term security rates are a weighted average of the yields on all the shorter maturities that make up the longer maturity. This calculation will hold true, providing that the MRP=0 assumption is valid. For instance, if the yield on a 1-year bond is 5% and that on a 2-year bond is 6%, the rate on a 1-year bond one year from now should be 7%, because (1.06)2 = (1.05)(1.07). Generally, r designates the rate, or yield, and our notation involves two subscripts. The first subscript denotes when in the future we expect the yield to exist, and the second denotes the maturity of the security. For instance, the rate expected 3 years from now on a 2-year bond would be denoted by 3r2. Assuming that expectations theory holds, use the yield information below to back out the following Expected forward rates, in words: Symbol: Yield on 1-year bond 1 year from now = 1r1 Yield on 1-year bond 2 years from now = 2r1 Yield on 1-year bond 3 years from now = 3r1 Yield on 1-year bond 4 years from now = 4r1 Yield on 5-year bond 5 years from now = 5r5 Yield on 10-year bond 10 years from now = 10r10 Yield on 20-year bond 10 years from now = 10r20 Yield on 10-year bond 20 years from now = 20r10 Maturity Maturity Yield 1 year 1 5.02% 2 year 2 5.31% 3 year 3 5.48% 4 year 4 5.65% 5 year 5 5.73% 10 year 10 5.68% 20 year 20 6.01% 30 year 30 5.92% (1+ r2)2 = ( (1 + r1) x (1 + 1r1) 1.1090 = ( 1.0502 x (1 + 1r1) 1r1 = 5.60% (1+ r3)3 = ( (1+ r2)2 x (1 + 2r1) 1.1736 = ( 1.1090 x (1 + 2r1) 2r1 = 5.82% (1+ r4)4 = ( (1+ r3)3 x (1 + 3r1) 1.2459 = ( 1.1736 x (1 + 3r1) 3r1 = 6.16% (1+ r5)5 = ( (1+ r4)4 x (1 + 4r1) 1.3213 = ( 1.2459 x (1 + 4r1) 4r1 = 6.05% (1+ r10)10 = ( (1+ r5)5 x (1 + 5r5)5 1.7375 = ( 1.3213 x (1 + 5r5)5 5r5 = 5.63% (1+ r20)20 = ( (1+ r10)10 x (1 + 10r10)10 3.2132 = ( 1.7375 x (1 + 10r10)10 10r10 = 6.34% (1+ r30)30 = ( (1+ r20)20 x (1 + 20r10)10 5.6149 = ( 3.2132 x (1 + 20r10)10 20r10 = 5.74% The data used to construct the yield curve are readily available, and forward rates can be calculated as SOLUTIONS TO SELF-TEST QUESTIONS 4a Assume the interest rate on a 1-year T-bond is currently 7% and the rate on a 2-year bond is 9%. If 1-year Treasury yield 7.0% 2-year Treasury yield 9.0% Maturity Risk Premium 0.0% 1-year rate, 1 year from now 11.04% 4b What would the forecast be if the maturity risk premium on the 2-year bond were 0.5% and it was zero 1-year Treasury yield 7.0% 2-year Treasury yield 9.0% Maturity Risk Premium 0.5% 1-year rate, 1 year from now 10.02% SECTION 4.3 SOLUTIONS TO SELF-TEST 2 A bond that matures in six years has a par value of $1,000, an annual coupon payment of $80, and a market interest rate of 9%. What is its price? Years to Maturity 6 Annual Payment $80 Par value $1,000 Going rate, rd 9% Value of bond = $955.14 3 A bond that matures in 18 years has a par value of $1,000, an annual coupon of 10%, and a market interest rate of 7%. What is its price? Years to Maturity 18 Coupon rate 10% Annual Payment $100 Par value $1,000 Going rate, rd 7% Value of bond = $1,301.77 80, and a market market interest rate SECTION 4.4 SOLUTIONS TO SELF-TEST 4 A bond currently sells for $850. It has an eight-year maturity, an annual coupon of $80, and a par value of $1,000. W is its yield to maturity? What is its current yield? Years to Maturity 8 Annual Payment $80.00 Current price $850.00 Par value = FV $1,000.00 Going rate, rd =YTM: 10.90% Annual Payment $80.00 Current price $850.00 Current yield: 9.41% 5 A bond currently sells for $1,250. It pays a $110 annual coupon and has a 20-year maturity, but it can be called in 5 years at $1,110. What are its YTM and its YTC? Is it likely to be called if interest rates don't change? Years to Maturity 20 Years to Call Annual Payment $110 Annual Payment Current price $1,250 Current price Par value = FV $1,000 Call price YTM 8.38% YTC The company will probably call the bond, because the YTC is less than the YTM. and a par value of $1,000. What urity, but it can be called in 5 n't change? 5 $110 $1,250 $1,110 6.85% SECTION 4.5 SOLUTIONS TO SELF-TEST 2a Last year a firm issued 30-year, 8% annual coupon bonds at a par value of $1,000. (1) Suppose that one year later the going rate drops to 6%. What is the new price of the bonds, assuming that they now have 29 years to maturity? Years to Maturity 29 Coupon rate 8% Annual Payment $80 Par value $1,000 Going rate, rd 6% Value of bond = $1,271.81 2b Suppose instead that one year after issue the going interest rate increases to 10% (rather than 6%). What is the price? Years to Maturity 29 Coupon rate 8% Annual Payment $80 Par value $1,000 Going rate, rd 10% Value of bond = $812.61 (1) Suppose that one year later ow have 29 years to maturity? (rather than 6%). What is the SECTION 4.6 SOLUTIONS TO SELF-TEST 2 A bond has a 25-year maturity, an 8% semiannual coupon, and a face value of $1,000. The going nominal annual in What is the bond's price? Coupons per year 2 Annual values Semiannual Inputs Years to Maturity 25 50 Coupon rate 8% 4% Annual Payment $80 $40 Par value $1,000 $1,000 Going rate, rd 6% 3.0% Value of bond = $1,255.67 $1,257.30 00. The going nominal annual interest rate (r d) is 6%. SECTION 4.9 SOLUTIONS TO SELF-TEST 2 The yield on a 15-year TIPS is 3 percent and the yield on a 15-year Treasury bond is 5 percent. What is the inflation premium for a 15-year security Yield on T-Bond 5% Yield on TIPS 3% Inflation premium 2% 5 percent. What is the inflation SECTION 4.11 SOLUTIONS TO SELF-TEST 5 A 10-year T-bond has a yield of 6 percent. A corporate bond with a rating of AA has a yield of 4.5 percent. If the cor has excellent liquidty, what is an estimate of the corporate bond’s default risk premium? Yield on T-Bond 6.0% Yield on corporate bond 7.5% Default risk premium 1.5% a yield of 4.5 percent. If the corporate bond ? SECTION 4.13 SOLUTIONS TO SELF-TEST QUESTIONS 3 Assume that the real risk-free rate is r* = 3% and the average expected inflation rate is 2.5% for the foreseeable future. The DRP and LP for a bond are each 1%, and the applicable MRP is 2%. What is the bond’s yield? r* 3.0% Inflation Premium 2.5% Default Risk Premium 1.0% Liquidity Premium 1.0% Maturity Risk Premium 2.0% Yield 9.5% s 2.5% for the foreseeable s the bond’s yield?

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