# Bonds

Document Sample

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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
350   DRP =               Default risk 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%
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)
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
408
409 Data for chart to right
410
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
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
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
521d38c9-0029-4c6e-afcb-8ee574999ba5.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                                            12/7/2011
521d38c9-0029-4c6e-afcb-8ee574999ba5.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                                    12/7/2011
521d38c9-0029-4c6e-afcb-8ee574999ba5.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                                                12/7/2011
521d38c9-0029-4c6e-afcb-8ee574999ba5.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                          12/7/2011
521d38c9-0029-4c6e-afcb-8ee574999ba5.xlsx                  Web 5C

4/24/2009

xtension 5C. Tool Kit for Duration

ollowing example illustrates its calculation.

Michael C. Ehrhardt                         Page 28       12/7/2011
521d38c9-0029-4c6e-afcb-8ee574999ba5.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      12/7/2011
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%

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%

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

Yield on T-Bond                             5%
Yield on TIPS                               3%

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%

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%