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A B C D E F G 1 4/11/2010 2 3 Chapter 5. Tool Kit for Bonds, Bond Valuation, and Interest Rates 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 5.3) 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 Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for 50 new issues, but it is generally not correct for outstanding bonds. However, Excel has several date and time functions, and a 51 bond valuation function that uses the calendar, so we can get exact valuations on any given date. 52 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) = 1/5/2011 57 Maturity date = 1/5/2026 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(2011,1,5),DATE(2026,1,5),10%,10%,100,1,1) 74 Value of bond based on $100 face value = 100.000 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 A89:A93, and then put the formula =B41 in cell B88, then select 90 5% $1,518.98 the range A88:B93. Then click Data, What-IF-Analysis, and then 91 10% $1,000.00 Table to get the menu. The input data are in a column, so put the 92 15% $707.63 cursor on column and enter C20 the place where the going rate is 93 20% $532.45 inputted. Click OK 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 CHANGES IN BOND VALUES OVER TIME (Section 5.4) 112 113 What happens to a bond price over time? To set up this problem, we will enter the different interest rates, and use the array 114 of cash flows above. The following example operates under the precept that the bond is issued at par ($1,000) in year 0. 115 From this point, the example sets three conditions for interest rates to follow: interest rates stay constant at 10%, interest 116 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 117 each of the scenarios. 118 119 Suppose interest rates rose to 15% or fell to 5% immediately after the bond was issued, and they remained at the new level 120 for the next 15 years. What would happen to the price of the bond over time? 121 122 We could set up data tables to get the data for this problem, but instead we simply inserted the PV formula into the 123 following matrix to calculate the value of the bond over time. Note that the formula takes the interest rate from the column 124 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 125 results in the data table above. We can also plot the data, as shown in the graph below. 126 A B C D E F G 127 Value of Bond in Given Year: 128 N 5% 10% 15% 129 0 $1,519 $1,000 $708 130 1 $1,495 $1,000 $714 131 2 $1,470 $1,000 $721 132 3 $1,443 $1,000 $729 133 4 $1,415 $1,000 $738 134 5 $1,386 $1,000 $749 135 6 $1,355 $1,000 $761 136 7 $1,323 $1,000 $776 137 8 $1,289 $1,000 $792 138 9 $1,254 $1,000 $811 139 10 $1,216 $1,000 $832 140 11 $1,177 $1,000 $857 141 12 $1,136 $1,000 $886 142 13 $1,093 $1,000 $919 143 14 $1,048 $1,000 $957 144 15 $1,000 $1,000 $1,000 145 146 147 Price of Bond Over Time 148 149 $1,600 150 151 $1,400 152 $1,200 153 $1,000 Rate Drops to 5% 154 $800 Rate Stays at 10% 155 $600 Rate Rises to 15% 156 $400 157 158 $200 159 $0 160 0 5 10 15 161 162 163 If rates fall, the bond goes to a premium, but it moves toward par as maturity approaches. The reverse hold if rates rise 164 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. 165 Note that the above graph assumes that interest rates stay constant after the initial change. That is most unlikely--interest 166 rates fluctuate, and so do the prices of outstanding bonds. 167 168 169 Market rate = 5% Return Due to Return Due to 170 N Bond Price Coupon Payment Price Change Total Return 171 0 $1,518.98 172 1 $1,494.93 6.58% -1.58% 5.00% 173 2 $1,469.68 6.69% -1.69% 5.00% 174 3 $1,443.16 6.80% -1.80% 5.00% 175 4 $1,415.32 6.93% -1.93% 5.00% 176 5 $1,386.09 7.07% -2.07% 5.00% 177 6 $1,355.39 7.21% -2.21% 5.00% 178 7 $1,323.16 7.38% -2.38% 5.00% A B C D E F G 179 8 $1,289.32 7.56% -2.56% 5.00% 180 9 $1,253.78 7.76% -2.76% 5.00% 181 10 $1,216.47 7.98% -2.98% 5.00% 182 11 $1,177.30 8.22% -3.22% 5.00% 183 12 $1,136.16 8.49% -3.49% 5.00% 184 13 $1,092.97 8.80% -3.80% 5.00% 185 14 $1,047.62 9.15% -4.15% 5.00% 186 15 $1,000.00 9.55% -4.55% 5.00% 187 188 Market rate = 10% Return Due to Return Due to 189 N Bond Price Coupon Payment Price Change Total Return 190 0 $1,000 191 1 $1,000 10.00% 0.00% 10.00% 192 2 $1,000 10.00% 0.00% 10.00% 193 3 $1,000 10.00% 0.00% 10.00% 194 4 $1,000 10.00% 0.00% 10.00% 195 5 $1,000 10.00% 0.00% 10.00% 196 6 $1,000 10.00% 0.00% 10.00% 197 7 $1,000 10.00% 0.00% 10.00% 198 8 $1,000 10.00% 0.00% 10.00% 199 9 $1,000 10.00% 0.00% 10.00% 200 10 $1,000 10.00% 0.00% 10.00% 201 11 $1,000 10.00% 0.00% 10.00% 202 12 $1,000 10.00% 0.00% 10.00% 203 13 $1,000 10.00% 0.00% 10.00% 204 14 $1,000 10.00% 0.00% 10.00% 205 15 $1,000 10.00% 0.00% 10.00% 206 207 208 Market rate = 15% Return Due to Return Due to 209 N Bond Price Coupon Payment Price Change Total Return 210 0 $707.63 211 1 $713.78 14.13% 0.87% 15.00% 212 2 $720.84 14.01% 0.99% 15.00% 213 3 $728.97 13.87% 1.13% 15.00% 214 4 $738.31 13.72% 1.28% 15.00% 215 5 $749.06 13.54% 1.46% 15.00% 216 6 $761.42 13.35% 1.65% 15.00% 217 7 $775.63 13.13% 1.87% 15.00% 218 8 $791.98 12.89% 2.11% 15.00% 219 9 $810.78 12.63% 2.37% 15.00% 220 10 $832.39 12.33% 2.67% 15.00% 221 11 $857.25 12.01% 2.99% 15.00% 222 12 $885.84 11.67% 3.33% 15.00% 223 13 $918.71 11.29% 3.71% 15.00% 224 14 $956.52 10.88% 4.12% 15.00% 225 15 $1,000.00 10.45% 4.55% 15.00% 226 227 228 BONDS WITH SEMIANNUAL COUPONS (Section 5.5) 229 A B C D E F G 230 Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three 231 modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by 2, 232 and (3) divide the nominal interest rate by 2. 233 234 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 235 5%? The bond is not callable. 236 237 Use the Rate function with adjusted data to solve the problem. 238 239 Periods to maturity = 15*2 = 30 240 Coupon rate: 10% 241 Semiannual pmt = $100/2 = $50.00 PV = $1,523.26 242 Current price: $1,000.00 243 Periodic rate = 5%/2 = 2.5% 244 245 Note that the bond is now more valuable, because interest payments come in faster. 246 247 BOND YIELDS (Section 5.6) 248 249 Yield to Maturity 250 251 The YTM is defined as the rate of return that will be earned if a bond makes all scheduled payments and is held to 252 maturity. The YTM is the same as the total rate of return discussed in the chapter, and it can also be interpreted as the 253 "promised rate of return," or the return to investors if all promised payments are made. The YTM for a bond that sells at 254 par consists entirely of an interest yield. However, if the bond sells at any price other than its par value, the YTM consists 255 of the interest yield together with a positive or negative capital gains yield. The YTM can be determined by solving the 256 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 257 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 258 259 260 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 261 is the Yield to Maturity of the bond? 262 263 Use the Rate function to solve the problem. 264 265 Years to Mat: 14 266 Coupon rate: 10% 267 Annual Pmt: $100.00 Going rate, r =YTM: 5.00% 268 Current price: $1,494.93 269 Par value = FV: $1,000.00 270 271 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 272 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 273 expected rate of return will be less than the promised yield-to-maturity. 274 275 Finding the Yield to Maturity on Actual Dates 276 277 Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for 278 new issues, but it is generally not correct for outstanding bonds. However, Excel has a function that uses the actual 279 calendar when finding yields. Consider the bond above, with 14 years until maturity. Suppose the actual current date is 280 1/5/2012, so the bond matures on 1/5/2026. 281 A B C D E F G 282 Here is the data for the bond. 283 284 Settlement date (day on which you find bond price) = 01/05/12 285 Maturity date = 01/05/26 286 Coupon rate = 10.00% 287 Price = bond price per $100 par value = $149.49 Redemption (100 means the bond pays 100% of its 288 face value at maturity) = 100 289 Frequency (# payments per year) = 1 290 Basis (1 is for actual number of days in month and year) 1 291 292 Using the YIELD function with inputs that are cell references: 293 Yield to maturity = 5.0% 294 295 296 Yield to Call 297 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 298 bonds, and if interest rates fall, then it would be logical for the issuer to call the bonds and replace them with new bonds 299 that carry a lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to 300 maturity is replaced with years to call, and the maturity value is replaced with the call price. 301 302 Problem: Suppose you purchase a 15-year, 10% annual coupon, $1,000 par value bond with a call provision after 10 years 303 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 304 $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 305 be called. 306 307 Use the Rate function to solve the problem. 308 309 Years to call: 9 310 Coupon rate: 10% 311 Annual Pmt: $100.00 Rate = I = YTC = 4.21% 312 Current price: $1,494.93 313 Call price = FV $1,100.00 314 Par value $1,000.00 315 316 This bond's YTM is 5%, but its YTC is only 4.21%. Which would an investor be more likely to actually earn? 317 318 This company could call the old bonds, which pay $100 per year, and replace them with bonds that pay somewhere in the 319 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 320 bonds. In that case, investors would earn the YTC, so the YTC is the expected return on the bonds. 321 322 323 Current Yield 324 The current yield is the annual interest payment divided by the bond's current price. The current yield provides 325 information regarding the amount of cash income that a bond will generate in a given year. However, it does not account 326 for any capital gains or losses that will be realized fi the bond is held to maturity or call. 327 328 Problem: What is the current yield on a $1,000 par value, 10% annual coupon bond that is currently selling for 329 $985? 330 331 Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we would 332 still use the annual interest. 333 A B C D E F G 334 Par value $1,000.00 335 Coupon rate: 10% Current Yield = 10.15% 336 Annual Pmt: $100.00 337 Current price: $985.00 338 339 The current yield provides information on a bond's cash return, but it gives no indication of the bond's total return. To see 340 this, consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no interest income. 341 However, the zero appreciates through time, and its total return clearly exceeds zero. 342 343 344 THE DETERMINANTS OF MARKET INTEREST RATES (Section 5.7) 345 346 Quoted market interest rate = rd = r* + IP + DRP + LP + MRP 347 348 r* = Real risk-free rate of interest 349 IP = Inflation premium 350 DRP = Default risk premium 351 LP = Liquidity premium 352 MRP = Maturity risk premium 353 354 355 THE REAL RISK-FREE RATE OF INTEREST, r* (Section 5.8) 356 357 r* = Real risk-free rate of interest 358 r* = Yield on short-term (1-year) U.S. Treasury Inflation-Protected Security (TIPS) 359 r* = 1.54% (March 2009) 360 361 362 THE INFLATION PREMIUM (IP) (Section 5.9) 363 Maturity 364 5 Years 20 Years 365 Non-indexed U.S. Treasury Bond 1.91% 3.93% 366 TIPS 1.41% 2.44% 367 Inflation premium 0.50% 1.49% 368 369 370 371 THE NOMINAL, OR QUOTED, RISK-FREE RATE OF INTEREST, rRF (Section 5.10) 372 373 Nominal, or quoted, rate = rd = rRF + DRP + LP + MRP 374 375 376 THE DEFAULT RISK PREMIUM (DRP) (Section 5.11) 377 378 Table 5-1 a b c 379 Rating Agency Percent defaulting within: Median Ratios Percent upgraded or downg 380 S&P and Fitch Moody’s 1 year 5 years Total Return on capital debt/Total capital Down 381 (1) (2) (3) (4) (5) (6) (7) 382 Investment grade bonds: 383 AAA Aaa 0.00 0.00 27.60 12.40 13.60 A B C D E F G 384 AA Aa 0 0.1 27 28.3 21.8 385 A A 0.1 0.6 17.5 37.5 8 386 BBB Baa 0.3 2.9 13.4 42.5 6.4 387 Junk bonds: 388 BB Ba 1.4 8.2 11.3 53.7 15.1 389 B B 1.8 9.2 8.7 75.9 10.8 390 CCC Caa 22.3 36.9 3.2 113.5 26.1 391 392 Notes: 393 a The ratings agencies also use “modifiers” for bonds rated below triple-A. S&P and Fitch use a plus and minus system; thus, A+ designa 394 rated bonds and A– the weakest. Moody’s uses a 1, 2, or 3 designation, with 1 denoting the strongest and 3 the weakest; thus, within the 395 Aa1 is the best, Aa2 is average, and Aa3 is the weakest. 396 bDefault data are from Fitch Ratings Global Corporate Finance 2008 Transition and Default Study, March 5, 2009: see 397 http://www.fitchratings.com/corporate/reports/report_frame.cfm?rpt_id=428182. c 398 Median ratios are from Standard & Poor’s 2006 Corporate Ratings Criteria, April 23, 2007: see 399 http://www2.standardandpoors.com/spf/pdf/fixedincome/Corporate_Ratings_2006.pdf. d 400 Composite yields for AAA, AA, and A bonds can be found at http://finance.yahoo.com/bonds/composite_bond_rates. Representative yie 401 and CCC bonds can be found using the bond screener at http://screen.yahoo.com/bonds. .html. 402 403 404 Bond spreads are the difference between the yield on a bond and the yield on some other bond of the same maturity. 405 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. 406 407 Figure 5-3: Bond Spreads 408 409 Data for chart to right 410 411 Spread 412 (%) 413 414 7.00 415 416 417 418 6.00 419 BAA − T-bond 420 421 5.00 422 423 424 4.00 425 426 427 3.00 428 429 430 2.00 431 432 433 1.00 434 AAA − T-bond A B C AAA − T-bond D E F G 435 436 0.00 437 1999-01 2001-07 2003-07 2004-01 2006-01 2008-07 1999-07 2000-01 2000-07 2001-01 2002-01 2002-07 2003-01 2004-07 2005-01 2005-07 2006-07 2007-01 2007-07 2008-01 2009-01 438 439 440 441 442 443 444 THE LIQUIDITY PREMIUM (LP) (Section 5.12) 445 446 A differential of at least 2 percentage points (and perhaps up to 4 or 5 percentage points) exists between the least liquid and 447 the most liquid financial assets of similar default risk and maturity. 448 449 450 THE MATURITY RISK PREMIUM (MRP) (Section 5.13) 451 452 Bonds are exposed to interest rate risk and reinvestment rate risk. The net effect is the maturity risk premium. 453 454 Interest Rate Risk 455 456 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 457 rates is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the same 458 coupon, the bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same maturity, 459 the one with the smaller coupon payment will have more interest rate sensitivity. 460 461 Compare the interest rate risk of two bonds, both of which have a 10% annual coupon and a $1,000 face value. The first 462 bond matures in 1 year, the second in 25 years. 463 464 Use the PV function, along with a two variable Data Table, to show the bonds' price sensitivity. 465 Coupon rate: 10% 466 Payment $100.00 467 Par value $1,000.00 468 Maturity 1 469 Going rate = r = YTM 10% 470 471 Value of bond: $1,000.00 472 473 474 Value of the Bond Under Different Conditions 475 Going rate, r Years to Maturity 476 $1,000.00 1 25 477 0% $1,100.00 $3,500.00 478 5% $1,047.62 $1,704.70 479 10% $1,000.00 $1,000.00 480 15% $956.52 $676.79 481 20% $916.67 $505.24 482 25% $880.00 $402.27 483 484 Figure 5-4 485 486 Bond Value A B C D E F G Bond Value 487 ($) 488 489 1,800 490 1,600 25-Year Bond 491 492 1,400 493 494 1,200 495 496 1,000 1-Year Bond 497 498 800 499 500 600 501 502 400 503 504 200 505 506 0 507 0% 5% 10% 15% 20% 25% 508 Interest Rate, rd 509 510 511 512 THE TERM STRUCTURE OF INTEREST RATES (Section 5.14) 513 514 515 The term structure describes the relationship between long-term and short-term interest rates. Graphically, this relationship can be sho 516 in what is known as the yield curve. In practice, the yield curve is relatively easy to obtain. It is published daily in the Wall Street Journ 517 and can be accessed through the internet, via www. bloomberg.com. However, the "building block approach" to generating a yield cur 518 more complicated. We will see that later when we build our own yield curve. 519 520 Before jumping into the creation of our own yield curve, let's look at some historical interest rate data and draw some historical yield 521 curves. 522 523 Maturity (yrs) Mar-80 Feb-00 Mar-09 524 0.5 15.0% 6.0% 0.4% 525 1 14.0% 6.2% 0.6% 526 5 13.5% 6.7% 1.7% 527 10 12.8% 6.7% 2.7% 528 30 12.3% 6.3% 3.7% 529 530 From this data, we can plot three line graphs. Each line graph represents the U.S. Treasury yield curve at a 531 different point in time. 532 533 Figure 5-5. U.S. Treasury Bond Interest Rates on Different Dates 534 535 536 537 538 Interest Rate (%) A B C D E F G (%) 539 540 541 16% 542 Yield Curve for March 1980 543 14% 544 545 12% 546 547 10% 548 549 550 8% Yield Curve for February 2000 551 552 6% 553 554 4% 555 556 2% Yield Curve for March 2009 557 558 559 0% 560 0 5 10 15 20 25 30 561 Years to Maturity 562 563 564 565 H I J K L M N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 H I J K L M N 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 6,1,5),10%,10%,100,1,1) 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 H I J K L M N 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 H I J K L M N 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 H I J K L M N 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 H I J K L M N 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 H I J K L M N 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 H I J K L M N 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 b 379 ercent upgraded or downgraded in 2008: 380 Up Yieldd 381 (8) (9) 382 383 NA 5.50 H I J K L M N 384 0 5.62 385 1.8 5.79 386 2.6 7.53 387 388 6.8 11.62 389 5.6 13.7 390 8.7 26.3 391 392 393 system; thus, A+ designates the strongest A- 394 weakest; thus, within the double-A category, 395 2009: see 396 397 398 399 400 _rates. Representative yields for BBB, BB, B, 401 402 403 404 405 lt risk premium. 406 407 408 409 410 DATE AAA - T-bond BAA - T-bond 411 2009-02 2.40 5.21 412 2009-01 2.53 5.62 413 2008-12 2.63 6.01 414 2008-11 2.59 5.68 415 2008-10 2.47 5.07 416 2008-09 1.96 3.62 417 2008-08 1.75 3.26 418 2008-07 1.66 3.15 419 2008-06 1.58 2.97 420 2008-05 1.69 3.05 421 2008-04 1.87 3.29 422 2008-03 2.00 3.38 423 2008-02 1.79 3.08 424 2008-01 1.59 2.80 425 2007-12 1.39 2.55 426 2007-11 1.29 2.25 427 2007-10 1.13 1.95 428 2007-09 1.22 2.07 429 2007-08 1.12 1.98 430 2007-07 0.73 1.65 431 2007-06 0.69 1.60 432 2007-05 0.72 1.64 433 2007-04 0.78 1.70 434 2007-03 0.74 1.71 H I J K L M N 435 2007-02 0.67 1.56 436 2007-01 0.64 1.58 437 2006-12 0.76 1.66 438 2006-11 0.73 1.60 439 2006-10 0.78 1.69 440 2006-09 0.79 1.71 441 2006-08 0.80 1.71 442 2006-07 0.76 1.67 443 2006-06 0.78 1.67 444 2006-05 0.84 1.64 445 2006-04 0.85 1.69 446 2006-03 0.81 1.69 447 2006-02 0.78 1.70 448 2006-01 0.87 1.82 449 2005-12 0.90 1.85 450 2005-11 0.88 1.85 451 2005-10 0.89 1.84 452 2005-09 0.93 1.83 453 2005-08 0.83 1.70 454 2005-07 0.88 1.77 455 2005-06 0.96 1.86 456 2005-05 1.01 1.87 457 2005-04 0.99 1.71 458 2005-03 0.90 1.56 459 2005-02 1.03 1.65 460 2005-01 1.14 1.80 461 2004-12 1.24 1.92 462 2004-11 1.33 2.01 463 2004-10 1.37 2.11 464 2004-09 1.33 2.14 465 2004-08 1.37 2.18 466 2004-07 1.32 2.12 467 2004-06 1.28 2.05 468 2004-05 1.32 2.03 469 2004-04 1.38 2.11 470 2004-03 1.50 2.28 471 2004-02 1.42 2.19 472 2004-01 1.39 2.29 473 2003-12 1.35 2.33 474 2003-11 1.35 2.36 475 2003-10 1.41 2.44 476 2003-09 1.45 2.52 477 2003-08 1.43 2.56 478 2003-07 1.51 2.64 479 2003-06 1.64 2.86 480 2003-05 1.65 2.81 481 2003-04 1.78 2.89 482 2003-03 2.08 3.14 483 2003-02 2.05 3.16 484 2003-01 2.12 3.30 485 2002-12 2.18 3.42 486 2002-11 2.26 3.57 H I J K L M N 487 2002-10 2.38 3.79 488 2002-09 2.28 3.53 489 2002-08 2.11 3.32 490 2002-07 1.88 3.25 491 2002-06 1.70 3.02 492 2002-05 1.59 2.93 493 2002-04 1.55 2.82 494 2002-03 1.53 2.83 495 2002-02 1.60 2.98 496 2002-01 1.51 2.83 497 2001-12 1.68 2.96 498 2001-11 2.32 3.16 499 2001-10 2.46 3.34 500 2001-09 2.44 3.30 501 2001-08 2.05 2.88 502 2001-07 1.89 2.73 503 2001-06 1.90 2.69 504 2001-05 1.90 2.68 505 2001-04 2.06 2.93 506 2001-03 2.09 2.95 507 2001-02 2.00 2.77 508 2001-01 1.99 2.77 509 2000-12 1.97 2.78 510 2000-11 1.73 2.56 511 2000-10 1.81 2.60 512 2000-09 1.82 2.55 513 2000-08 1.72 2.43 514 2000-07 1.60 2.30 his relationship can be shown 515 2000-06 1.57 2.38 y in the Wall Street Journal 516 2000-05 1.55 2.46 517 to generating a yield curve is 2000-04 1.65 2.41 518 2000-03 1.42 2.11 519 2000-02 1.16 1.77 520 w some historical yield 2000-01 1.12 1.67 521 1999-12 1.27 1.91 522 1999-11 1.33 2.12 523 1999-10 1.44 2.27 524 1999-09 1.47 2.28 525 1999-08 1.46 2.21 526 1999-07 1.40 2.16 527 1999-06 1.33 2.12 528 1999-05 1.39 2.18 529 1999-04 1.46 2.30 530 1999-03 1.39 2.30 531 1999-02 1.40 2.39 532 1999-01 1.52 2.57 533 534 535 536 537 538 7e99da02-8d69-4f0c-834f-f55575846ed0.xlsx Web 5A 4/11/2010 Web Extension 5A: 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 5A-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 24 10/14/2011 7e99da02-8d69-4f0c-834f-f55575846ed0.xlsx Web 5C Web Extension 5C. Tool Kit for Duration Duration is a measure of risk for bonds. The following example illustrates its calculation. Figure 5C-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 = 1/1/2009 Maturity 12/31/2028 Coupon = 9% Yield = 9% Frequency = 1 Michael C. Ehrhardt Page 25 10/14/2011 7e99da02-8d69-4f0c-834f-f55575846ed0.xlsx Web 5C 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 26 10/14/2011 7e99da02-8d69-4f0c-834f-f55575846ed0.xlsx Web 5C 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 27 10/14/2011 7e99da02-8d69-4f0c-834f-f55575846ed0.xlsx Web 5C 4/24/2009 xtension 5C. Tool Kit for Duration ollowing example illustrates its calculation. Michael C. Ehrhardt Page 28 10/14/2011 7e99da02-8d69-4f0c-834f-f55575846ed0.xlsx Web 5C 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 29 10/14/2011 3/28/2009 Web Extension 5D. 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 5.3 SOLUTIONS TO SELF-TEST 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 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 and a market interest arket interest rate of SECTION 5.4 SOLUTIONS TO SELF-TEST 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 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 Suppose that one year later the ave 29 years to maturity? her than 6%). What is the price? SECTION 5.5 SOLUTIONS TO SELF-TEST A bond has a 25-year maturity, an 8% annual coupon paid semiannually, and a face value of $1,000. The going nomina (rd) is 6%. 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 alue of $1,000. The going nominal annual interest rate SECTION 5.6 SOLUTIONS TO SELF-TEST A bond currently sells for $850. It has an eight-year maturity, an annual coupon of $80, and a par value of $1,000. Wha 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% 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 ye 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. nd a par value of $1,000. What is ty, but it can be called in 5 years ange? 5 $110 $1,250 $1,110 6.85% SECTION 5.9 SOLUTIONS TO SELF-TEST 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% ercent. What is the inflation SECTION 5.11 SOLUTIONS TO SELF-TEST A 10-year T-bond has a yield of 4.5 percent. A corporate bond with a rating of AA has a yield of 6.0 percent. If the corp has excellent liquidty, what is an estimate of the corporate bond’s default risk premium? Yield on T-Bond 4.5% Yield on corporate bond 6.0% Default risk premium 1.5% yield of 6.0 percent. If the corporate bond ? SECTION 5.13 SOLUTIONS TO SELF-TEST QUESTIONS 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% 2.5% for the foreseeable future. nd’s yield?