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
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46
H I J K L M N
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73
6,1,5),10%,10%,100,1,1)
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H I J K L M N
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124
125
126
H I J K L M N
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133
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136
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142
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144
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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
381612d7-6cb9-4718-9499-6fe4e3f77b4a.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 11/22/2011
381612d7-6cb9-4718-9499-6fe4e3f77b4a.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 11/22/2011
381612d7-6cb9-4718-9499-6fe4e3f77b4a.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
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381612d7-6cb9-4718-9499-6fe4e3f77b4a.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 11/22/2011
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4/24/2009
xtension 5C. Tool Kit for Duration
ollowing example illustrates its calculation.
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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 11/22/2011
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?