# Bonds - Excel

Document Sample

```					              A                  B                C                   D                 E                  F                  G
1                                                                                                                           3/19/2009
2
3                                          Chapter 4. Tool Kit for Bond Valuation
4
5   The value of any financial asset is the present value of the asset's expected future cash flows. The key inputs are (1) the
6   expected cash flows and (2) the appropriate discount rate, given the bond's risk, maturity, and other characteristics. The
7   model developed here analyzes bonds in various ways.
8
9   BOND VALUATION (Section 4.3)
10
11   A bond has a 15-year maturity, a 10% annual coupon, and a \$1,000 par value. The required rate of return (or the yield to
12   maturity) on the bond is 10%, given its risk, maturity, liquidity, and other rates in the economy. What is a fair value for the
13   bond, i.e., its market price?
14
15   First, we list the key features of the bond as "model inputs":
16   Years to Mat:                                        15
17   Coupon rate:                                       10%
18   Annual Pmt:                                        \$100
19   Par value = FV:                                  \$1,000
20   Required return, rd:                               10%
21
22   The easiest way to solve this problem is to use Excel's PV function. Click fx, then financial, then PV. Then fill in
23   the menu items as shown in our snapshot in the screen shown just below.
24
25
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30
31
32
33
34
35
36
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38
39
40
41   Value of bond =           \$1,000.00 Thus, this bond sells at its par value. That situation always exists if the going
42                                       rate is equal to the coupon rate.
43
44
45   The PV function can only be used if the payments are constant, but that is normally the case for bonds.
46
A                 B                  C                 D                 E                 F                G
47   Bond Prices on Actual Dates
48
49
50   Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for
51   new issues, but it is generally not correct for outstanding bonds. However, Excel has several date and time functions, and a
52   bond valuation function that uses the calendar, so we can get exact valuations on any given date.
53
54   Here is the data for MicroDrive's bond as of the day it was issued.
55
56 Settlement date (day on which you find bond price) =                                  1/5/2010
57 Maturity date =                                                                       1/5/2025
58 Coupon rate =                                                                          10.00%
59 Required return, rd =                                                                  10.00%
Redemption (100 means the bond pays 100% of its
60 face value at maturity) =                                                                  100
61 Frequency (# payments per year) =                                                             1
62 Basis (1 is for actual number of days in month and year)                                      1
63
64 Click on fx on the formula bar (or click Insert and then Function). This gives you the "Insert Function" dialog box. To find
65 a bond's price, use the PRICE function (found in the "Financial" category of the "Insert Function dialog box). The PRICE
66 function returns the price per \$100 dollars of face value.
67
68 Using PRICE function with inputs that are cell references:
69 Value of bond based on \$100 face value =                                               \$100.00
70 Value of bond in dollars based on \$1,000 face value =                                \$1,000.00
71
72 Using the PRICE function with inputs that are not cell references:
73 Value of bond based on \$100 face value =                                    =PRICE(DATE(2009,1,5),DATE(2024,1,5),10%,10%,100,1,1)
74 Value of bond based on \$100 face value =                                              100.0000
75 Value of bond in dollars based on \$1,000 face value =                                \$1,000.00
76
77
78 Interest Rate Changes and Bond Prices
79
80 Suppose the going interest rate changed from 10%, falling to 5% or rising to 15%. How would those changes affect the
81 value of the bond?
82
83
84 We could simply go to the input data section shown above, change the value for r from 10% to 5% and then 15%, and
85 observe the changed values. An alternative is to set up a data table to show the bond's value at a range of rates, i.e., to show
86 the bond's sensitivity to changes in interest rates. This is done below, and the values at 5% and 15% are boldfaced.
87                           Bond Value
88    Going rate, r:              \$1,000                     To make the data table, first type the headings, then type the rates in
89          0%                 \$2,500.00                     cells A87:A91, and then put the formula =B41 in cell B86, then select
90          5%                 \$1,518.98                     the range A86:B912. Then click Data and then Table to get the
91         10%                 \$1,000.00                     menu. The input data are in a column, so put the cursor on column
92         15%                   \$707.63                     and enter C20 the place where the going rate is inputted. Click OK
93         20%                   \$532.45                     to complete the operation and get the table.
94
95 We can use the data table to construct a graph that shows the bond's
96 sensitivity to changing rates.
97
A                  B                     C              D                  E                  F                G
98
99                                Interest Rate Sensitivity
100
101        \$3,000
102        \$2,500
\$2,000
103
\$1,500
104        \$1,000
105         \$500
106            \$0
107                 0%           5%               10%          15%            20%
108
109
110
111   BOND YIELDS (Section 4.4)
112
113   Yield to Maturity
114
115   The YTM is defined as the rate of return that will be earned if a bond makes all scheduled payments and is held to
116   maturity. The YTM is the same as the total rate of return discussed in the chapter, and it can also be interpreted as the
117   "promised rate of return," or the return to investors if all promised payments are made. The YTM for a bond that sells at
118   par consists entirely of an interest yield. However, if the bond sells at any price other than its par value, the YTM consists
119   of the interest yield together with a positive or negative capital gains yield. The YTM can be determined by solving the
120   bond value formula for I. However, an easier method for finding it is to use Excel's Rate function. Since the price of a bond
121   is simply the sum of the present values of its cash flows, so we can use the time value of money techniques to solve these
122   problems.
123
124   Problem: Suppose that you are offered a 14-year, 10% annual coupon, \$1,000 par value bond at a price of \$1,494.93. What
125   is the Yield to Maturity of the bond?
126
127   Use the Rate function to solve the problem.
128
129   Years to Mat:                     14
130   Coupon rate:                   10%
131   Annual Pmt:                  \$100.00                     Going rate, r =YTM:                        5.00%
132   Current price:             \$1,494.93
133   Par value = FV:            \$1,000.00
134
135   The yield-to-maturity is the same as the expected rate of return only if (1) the probability of default is zero, and (2) the bond
136   can not be called. If there is any chance of default, then there is a chance some payments may not be made. In this case, the
137   expected rate of return will be less than the promised yield-to-maturity.
138
139   Finding the Yield to Maturity on Actual Dates
140
141   Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for
142   new issues, but it is generally not correct for outstanding bonds. However, Excel has a function that uses the actual
143   calendar when finding yields. Consider the bond above, with 14 years until maturity. Suppose the actual current date is
144   1/5/2011, so the bond matures on 1/5/2025.
145
146   Here is the data for the bond.
147
148   Settlement date (day on which you find bond price) =                                  01/05/11
149   Maturity date =                                                                       01/05/25
A                  B                 C                   D                   E                  F                 G
150 Coupon rate =                                                                             10.00%
151 Price = bond price per \$100 par value =                                                   \$149.49
Redemption (100 means the bond pays 100% of its
152 face value at maturity) =                                                                     100
153 Frequency (# payments per year) =                                                                1
154 Basis (1 is for actual number of days in month and year)                                         0
155
156 Using the YIELD function with inputs that are cell references:
157 Yield to maturity =                                                                         5.0%
158
159
160 Yield to Call
161 The yield to call is the rate of return investors will receive if their bonds are called. If the issuer has the right to call the
162 bonds, and if interest rates fall, then it would be logical for the issuer to call the bonds and replace them with new bonds
163 that carry a lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to
164 maturity is replaced with years to call, and the maturity value is replaced with the call price.
165
166 Problem: Suppose you purchase a 15-year, 10% annual coupon, \$1,000 par value bond with a call provision after 10 years
167 at a call price of \$1,100. One year later, interest rates have fallen from 10% to 5% causing the value of the bond to rise to
168 \$1,494.93. What is the bond's YTC? Note that this is the same bond as in the previous question, but now we assume it can
169 be called.
170
171 Use the Rate function to solve the problem.
172
173 Years to call:                        9
174 Coupon rate:                      10%
175 Annual Pmt:                    \$100.00                     Rate = I = YTC =                             4.21%
176 Current price:               \$1,494.93
177 Call price = FV              \$1,100.00
178 Par value                    \$1,000.00
179
180 This bond's YTM is 5%, but its YTC is only 4.21%. Which would an investor be more likely to actually earn?
181
182
183 This company could call the old bonds, which pay \$100 per year, and replace them with bonds that pay somewhere in the
184 vicinity of \$50 (or maybe even only \$42.10) per year. It would want to save that money, so it would in all likelihood call the
185 bonds. In that case, investors would earn the YTC, so the YTC is the expected return on the bonds.
186
187 Current Yield
188 The current yield is the annual interest payment divided by the bond's current price. The current yield provides
189 information regarding the amount of cash income that a bond will generate in a given year. However, it does not account
190 for any capital gains or losses that will be realized fi the bond is held to maturity or call.
191
192 Problem: What is the current yield on a \$1,000 par value, 10% annual coupon bond that is currently selling for
193 \$985?
194
195 Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we would
196 still use the annual interest.
197
198 Par value                    \$1,000.00
199 Coupon rate:                      10%                       Current Yield =          10.15%
200 Annual Pmt:                    \$100.00
201 Current price:                 \$985.00
A                  B                C                  D                  E                  F                G
202
203   The current yield provides information on a bond's cash return, but it gives no indication of the bond's total return. To see
204   this, consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no interest income.
205   However, the zero appreciates through time, and its total return clearly exceeds zero.
206
207
208   CHANGES IN BOND VALUES OVER TIME (Section 4.4)
209
210   What happens to a bond price over time? To set up this problem, we will enter the different interest rates, and use the array
211   of cash flows above. The following example operates under the precept that the bond is issued at par (\$1,000) in year 0.
212   From this point, the example sets three conditions for interest rates to follow: interest rates stay constant at 10%, interest
213   rates fall to 5%, or interest rates rise to 15%. Then the price of the bond over the fifteen years of its life is determined for
214   each of the scenarios.
215
216   Suppose interest rates rose to 15% or fell to 5% immediately after the bond was issued, and they remained at the new level
217   for the next 15 years. What would happen to the price of the bond over time?
218
219   We could set up data tables to get the data for this problem, but instead we simply inserted the PV formula into the
220   following matrix to calculate the value of the bond over time. Note that the formula takes the interest rate from the column
221   heads, and the value of N from the left column. Note that the N = 0 values for the 5% and 15% rates are consistent with the
222   results in the data table above. We can also plot the data, as shown in the graph below.
223
A                 B                C                D                     E                       F          G
224                                  Value of Bond in Given Year:
225           N                 5%              10%               15%
226            0              \$1,519           \$1,000            \$708
227            1              \$1,495           \$1,000            \$714
228            2              \$1,470           \$1,000            \$721
229            3              \$1,443           \$1,000            \$729
230            4              \$1,415           \$1,000            \$738
231            5              \$1,386           \$1,000            \$749
232            6              \$1,355           \$1,000            \$761
233            7              \$1,323           \$1,000            \$776
234            8              \$1,289           \$1,000            \$792
235            9              \$1,254           \$1,000            \$811
236           10              \$1,216           \$1,000            \$832
237           11              \$1,177           \$1,000            \$857
238           12              \$1,136           \$1,000            \$886
239           13              \$1,093           \$1,000            \$919
240           14              \$1,048           \$1,000            \$957
241           15              \$1,000           \$1,000           \$1,000
242
243
244                                         Price of Bond Over Time
245
246
\$1,600
247
248        \$1,400
249        \$1,200
250        \$1,000                                                                            Rate Drops to 5%
251         \$800                                                                             Rate Stays at 10%
252         \$600                                                                             Rate Rises to 15%
253
\$400
254
255         \$200
256            \$0
257                 0                  5                  10                  15
258
259
260   If rates fall, the bond goes to a premium, but it moves toward par as maturity approaches. The reverse hold if rates rise
261   and the bond sells at a discount. If the going rate remains equal to the coupon rate, the bond will continue to sell at par.
262   Note that the above graph assumes that interest rates stay constant after the initial change. That is most unlikely--interest
263   rates fluctuate, and so do the prices of outstanding bonds.
264
265
266
267   BONDS WITH SEMIANNUAL COUPONS (Section 4.6)
268
269   Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three
270   modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by 2,
271   and (3) divide the nominal interest rate by 2.
272
273   Problem: What is the price of a 15-year, 10% semi-annual coupon, \$1,000 par value bond if the nominal rate (the YTM) is
274   5%? The bond is not callable.
275
276   Use the Rate function with adjusted data to solve the problem.
A                 B                  C                   D                 E             F               G
277
278   Periods to maturity = 15*2 =                        30
279   Coupon rate:                                      10%
280   Semiannual pmt = \$100/2 =                       \$50.00         PV =            \$1,523.26
281   Current price:                               \$1,000.00
282   Periodic rate = 5%/2 =                           2.5%
283
284   Note that the bond is now more valuable, because interest payments come in faster.
285
286
287   THE DETERMINANTS OF MARKET INTEREST RATES (Section 4.7)
288
289   Quoted market interest rate = rd = r* + IP + DRP + LP + MRP
290
291   r* =               Real risk-free rate of interest
293   DRP =              Default risk premium
295   MRP =              Maturity risk premium
296
297
298   THE REAL RISK-FREE RATE OF INTEREST, r* (Section 4.8)
299
300                 r* = Real risk-free rate of interest
301                 r* = Yield on short-term U.S. Treasury Inflation-Protected Security (TIPS)
302                 r* =         -0.05% (March 2008)
303
304
305   THE INFLATION PREMIUM (IP) (Section 4.9)
306                                                                         Maturity
307                                                                 5 Years          30 Years
308                       Non-indexed U.S. Treasury Bond                 2.46%             4.40%
309                                                 TIPS                -0.05%             1.63%
311
312
313
314   THE NOMINAL, OR QUOTED, RISK-FREE RATE OF INTEREST, rRF (Section 4.10)
315
316   Nominal, or quoted, rate = rd = rRF + DRP + LP + MRP
317
318
319   THE DEFAULT RISK PREMIUM (DRP) (Section 4.11)
320
321   Bond spreads are the difference between the yield on a bond and the yield on some other bond of the same maturity.
322
323
Yield            T-Bond                AAA               BBB                      Source: see
325 Long-term Bonds            (1)               (2)                  (3)               (4)                     comment.
326 U.S. Treasury                 4.40%
A                  B                 C                  D                  E                  F                G
327   AAA                             5.52%             1.12%
328   AA                              5.90%             1.50%              0.38%
329   A                               6.14%             1.74%              0.62%
330   BBB                             6.51%             2.11%              0.99%
331   BB                              7.09%             2.69%              1.57%              0.58%
332   B                               7.61%             3.21%              2.09%              1.10%
333   CCC                             9.01%             4.61%              3.49%              2.50%
334
335   Note: The spreads in Column (2) are found by taking the yields in Column (1) and subtracting the yield on the U.S. Treasury Bond.
336         The spreads in Column (3) are found by taking the yields in Column (1) and subtracting the yield on the AAA bond.
337         The spreads in Column (4) are found by taking the yields in Column (1) and subtracting the yield on the BBB bond.
338
339   For a bond with good liquidity, its spread relative to a T-bond of similar maturity is a good estmat of the default risk premium.
340
341
342   THE MATURITY RISK PREMIUM (MRP) (Section 4.12)
343
344   Bonds are exposed to interest rate risk and reinvestment rate risk. The net effect is the maturity risk premium.
345
346   Interest Rate Risk
347
348   Interest Rate Risk is the risk of a decline in a bond's price due to an increase in interest rates. Price sensitivity to interest
349   rates is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the same
350   coupon, the bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same maturity,
351   the one with the smaller coupon payment will have more interest rate sensitivity.
352
353   Compare the interest rate risk of two bonds, both of which have a 10% annual coupon and a \$1,000 face value. The first
354   bond matures in 1 year, the second in 25 years.
355
356   Use the PV function, along with a two variable Data Table, to show the bonds' price sensitivity.
357   Coupon rate:                                    10%
358   Payment                                       \$100.00
359   Par value                                   \$1,000.00
360   Maturity                                            1
361   Going rate = r = YTM                            10%
362
363   Value of bond:                               \$1,000.00
364
365
366                        Value of the Bond Under Different Conditions
367      Going rate, r            Years to Maturity
368            \$1,000.00                 1              25
369                  0%          \$1,100.00        \$3,500.00
370                  5%          \$1,047.62        \$1,704.70
371                 10%          \$1,000.00        \$1,000.00
372                 15%            \$956.52          \$676.79
373                 20%            \$916.67          \$505.24
374                 25%            \$880.00          \$402.27
375
376
377
Bond Value
378                    (\$)
379
1,800
A                B              C              D                E          F   G
380       1,800
381
382       1,600                 25-Year Bond
383
384       1,400
385
1,200
386
387
1,000                                          1-Year Bond
388
389
800
390
391
600
392
393        400
394
395        200
396
397          0
398               0%       5%         10%      15%         20%            25%
399
400                                                          Interest Rate, rd
401
402
403
fff54e20-0c4b-4184-8d6c-2a818615c266.xls                                                                  Web 4A

3/19/2009
Web Extension 4A: Zero Coupon Bonds

Vandenburg Corporation needs to issue \$50 million to finance a project, and it has decided to raise
the funds by issuing \$1,000 par value, zero coupon bonds. The going interest rate on such debt is 6%,
and the corporate tax rate is 40%. Find the issue price of Vandenburg's bonds, construct a table to
analyze the cash flows attributable to one of the bonds, and determine the after-tax cost of debt for
the issue. Then, indicate the total par value of the issue.

This example analyzes the after-tax cost of issuing zero coupon debt.

Table 4A-1
Input Data
Amount needed =                                           \$50,000,000
Maturity value=                                                \$1,000
Pre-tax market interest rate, rd =                                 6%
Maturity (in years) =                                               5
Corporate tax rate =                                             40%
Coupon rate =                                                      0%
Coupon payment (assuming annual payments) =                        \$0
Issue Price =   PV of payments at rd =                        \$747.26

Analysis:
Years                                0          1             2           3         4           5
(1) Remaining years                  5          4             3           2         1           0
(2) Year-end accrued value         \$747.26     \$792.09       \$839.62    \$890.00   \$943.40   \$1,000.00
(3) Interest payment                             \$0.00         \$0.00      \$0.00     \$0.00       \$0.00
(4) Implied interest
deduction on discount                         \$44.84       \$47.53     \$50.38    \$53.40     \$56.60
(5) Tax savings                                  \$17.93       \$19.01     \$20.15    \$21.36     \$22.64
(6) Cash flow                      \$747.26       \$17.93       \$19.01     \$20.15    \$21.36   (\$977.36)

After-tax cost of debt =             3.60%

Number of \$1,000 zeros the
company must issue to raise \$50 million          =        Amount needed/Price per bond
=         66,911.279 bonds.
Face amount of bonds = # bonds x \$1,000          =        \$66,911,279

Michael C. Ehrhardt                                   Page 10                                            7/29/2011
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Web Extension 4C. Tool Kit for Duration

Duration is a measure of risk for bonds. The following example illustrates its calculation.
Figure 4C-1 Duration
Inputs
Years to maturity =                    20
Coupon rate =                      9.00%
Annual payment =                    \$90.0
Par value = FV =                   \$1,000
Going rate, r =                    9.00%

t                   CFt           PV of CFt                        t(PV of CFt)
(1)                   (2)              (3)                               (4)
1                   \$90             \$82.57                             82.57
2                   \$90             \$75.75                            151.50
3                   \$90             \$69.50                            208.49
4                   \$90             \$63.76                            255.03
5                   \$90             \$58.49                            292.47
6                   \$90             \$53.66                            321.98
7                   \$90             \$49.23                            344.63
8                   \$90             \$45.17                            361.34
9                   \$90             \$41.44                            372.95
10                   \$90             \$38.02                            380.17
11                   \$90             \$34.88                            383.66
12                   \$90             \$32.00                            383.98
13                   \$90             \$29.36                            381.63
14                   \$90             \$26.93                            377.05
15                   \$90             \$24.71                            370.63
16                   \$90             \$22.67                            362.69
17                   \$90             \$20.80                            353.54
18                   \$90             \$19.08                            343.43
19                   \$90             \$17.50                            332.58
20                  \$1,090          \$194.49                           3,889.79

Sum of
VB =     \$1,000.00      t(PV of CFt) =      \$9,950.11

Duration = Sum of t(PV of CFt) / VB=                 9.95

Finding Duration with the Excel Formula

Settlement date =                1/1/2010
Maturity                       12/31/2029
Coupon =                              9%
Yield =                               9%
Frequency =                             1

Michael C. Ehrhardt                                   Page 11                                    7/29/2011
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Duration =                            9.95

Consider the amount that would accumulate during the first 10 years, if all coupons are reinvested at the original interest
rate of 9%. To do this, first find the amount that would be in the account at 10 years (including the 10-year coupon). Then
we find the value of the bond at year 10 based on the payments from 11 and on.

Duration of Bond =                 9.95011

Target value
at year 10 =                   \$10,000.00
FV of reinvested
coupons at year 10 if no
change in rates =               \$1,367.36
PV at year 10 of
remaining payments if
no change in rates =            \$1,000.00
Total value at year 10 if
no change in rates =            \$2,367.36
Value of bonds to be
purchased to provide
target at 10 years =            \$4,224.11
Number of bonds
purchased =                            4.22

Now find the value at year 10 if the market interest rate (shown below) changes immediately after time zero, based on the
total number of bonds that were purchased.

Interest rate =                     9.00%

FV at year 10 =             \$5,775.89
PV of payments beyond year 10 discounted back to year 10 =                            \$4,224.11

The total value of the position at time 9.95011 is the value of the reinvested coupon and the current value of the bond.

Value of reinvested coupons:                      \$5,775.89
Current value of bond:                            \$4,224.11
Total value of position =       \$10,000.00

As the table below shows, the total value of a position at a future time equal to the orginal duration will not fall if interest
rates change. For example, if rates go up, the value of reinvested coupons increases and the value of the bond at the future
date (t=duration) falls, but the net affect is an increase in total value. If rates go down, the value of reinvested coupons goes
down, but the future value of the bond goes up, for a net increase in value. Thus, if the desired time horizon is equal to the
bond's duration, the value of the position will not fall if interest rates change.

Change in Total
Reinvested       Current Price                      Value from
Coupons         at t=Duration    Total Value     Original Target
\$5,775.89           \$4,224.11     \$10,000.00
1%       \$3,977.42           \$7,424.73     \$11,402.15           \$1,402.15
2%       \$4,162.75           \$6,880.15     \$11,042.90           \$1,042.90

Michael C. Ehrhardt                                     Page 12                                                7/29/2011
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3%     \$4,358.22    \$6,386.06   \$10,744.28   \$744.28
4%     \$4,564.36    \$5,937.17   \$10,501.53   \$501.53
5%     \$4,781.73    \$5,528.81   \$10,310.54   \$310.54
6%     \$5,010.94    \$5,156.80   \$10,167.74   \$167.74
7%     \$5,252.60    \$4,817.48   \$10,070.07    \$70.07
8%     \$5,507.35    \$4,507.55   \$10,014.90    \$14.90
9%     \$5,775.89    \$4,224.11   \$10,000.00     \$0.00
10%     \$6,058.93    \$3,964.55   \$10,023.48    \$23.48
11%     \$6,357.20    \$3,726.57   \$10,083.77    \$83.77
12%     \$6,671.50    \$3,508.09   \$10,179.59   \$179.59
13%     \$7,002.63    \$3,307.27   \$10,309.90   \$309.90
14%     \$7,351.45    \$3,122.44   \$10,473.89   \$473.89
15%     \$7,718.86    \$2,952.12   \$10,670.98   \$670.98
16%     \$8,105.78    \$2,794.98   \$10,900.76   \$900.76

Michael C. Ehrhardt                          Page 13                          7/29/2011
fff54e20-0c4b-4184-8d6c-2a818615c266.xls                  Web 4C

3/19/2009

xtension 4C. Tool Kit for Duration

ollowing example illustrates its calculation.

Michael C. Ehrhardt                        Page 14       7/29/2011
fff54e20-0c4b-4184-8d6c-2a818615c266.xls                               Web 4C

during the first 10 years, if all coupons are reinvested at the original interest
hat would be in the account at 10 years (including the 10-year coupon). Then
on the payments from 11 and on.

erest rate (shown below) changes immediately after time zero, based on the

is the value of the reinvested coupon and the current value of the bond.

position at a future time equal to the orginal duration will not fall if interest
value of reinvested coupons increases and the value of the bond at the future
increase in total value. If rates go down, the value of reinvested coupons goes
p, for a net increase in value. Thus, if the desired time horizon is equal to the
not fall if interest rates change.

Michael C. Ehrhardt                                      Page 15      7/29/2011
3/19/2009

Web Extension 4D. The Pure Expectations Theory and Estimation of Forward Rates

The shape of the yield curve depends primarily on two key factors: (1) expectations about future inflation
and (2) perceptions about the relative riskiness of securities of different maturities. The first factor is the
basis for the Pure Expectations Hypothesis. If the relationship between expectations for future inflation
and bond yields is controlling, i. e., if no maturity premiums existed, then the pure expectations theory
posits that forward interest rates can be predicted by "backing them out of the yield curve." Essentially,
under the pure expectations theory, long-term security rates are a weighted average of the yields on all
the shorter maturities that make up the longer maturity. This calculation will hold true, providing that the
MRP=0 assumption is valid.

For instance, if the yield on a 1-year bond is 5% and that on a 2-year bond is 6%, the rate on a 1-year
bond one year from now should be 7%, because (1.06)2 = (1.05)(1.07).

Generally, r designates the rate, or yield, and our notation involves two subscripts. The first subscript
denotes when in the future we expect the yield to exist, and the second denotes the maturity of the
security. For instance, the rate expected 3 years from now on a 2-year bond would be denoted by 3r2.

Assuming that expectations theory holds, use the yield information below to back out the following

Expected forward rates, in words:                                 Symbol:
Yield on 1-year bond 1 year from now   =                             1r1
Yield on 1-year bond 2 years from now =                              2r1
Yield on 1-year bond 3 years from now =                              3r1
Yield on 1-year bond 4 years from now =                              4r1
Yield on 5-year bond 5 years from now =                              5r5
Yield on 10-year bond 10 years from now =                           10r10
Yield on 20-year bond 10 years from now =                           10r20
Yield on 10-year bond 20 years from now =                           20r10

Maturity         Maturity         Yield
1 year             1             5.02%
2 year             2             5.31%
3 year             3             5.48%
4 year             4             5.65%
5 year             5             5.73%
10 year            10             5.68%
20 year            20             6.01%
30 year            30             5.92%

(1+ r2)2           =      (      (1 + r1)          x           (1 + 1r1)
1.1090             =      (      1.0502            x           (1 + 1r1)
1r1              =             5.60%

(1+ r3)3           =      (     (1+ r2)2           x           (1 + 2r1)
1.1736            =     (      1.1090            x           (1 + 2r1)
2r1             =            5.82%

(1+ r4)4          =     (      (1+ r3)3          x           (1 + 3r1)
1.2459            =     (      1.1736            x           (1 + 3r1)
3r1             =             6.16%

(1+ r5)5          =     (      (1+ r4)4          x           (1 + 4r1)
1.3213            =     (      1.2459            x           (1 + 4r1)
4r1             =             6.05%

(1+ r10)10         =     (      (1+ r5)5          x          (1 + 5r5)5
1.7375            =     (      1.3213            x          (1 + 5r5)5
5r5             =             5.63%

(1+ r20)20         =     (     (1+ r10)10         x         (1 + 10r10)10
3.2132            =     (      1.7375            x         (1 + 10r10)10
10r10            =            6.34%

(1+ r30)30         =     (     (1+ r20)20         x         (1 + 20r10)10
5.6149            =     (      3.2132            x         (1 + 20r10)10
20r10            =            5.74%

The data used to construct the yield curve are readily available, and forward rates can be calculated as

SOLUTIONS TO SELF-TEST QUESTIONS

4a Assume the interest rate on a 1-year T-bond is currently 7% and the rate on a 2-year bond is 9%. If

1-year Treasury yield                   7.0%
2-year Treasury yield                   9.0%

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 4.3
SOLUTIONS TO SELF-TEST

2 A bond that matures in six years has a par value of \$1,000, an annual coupon payment of \$80, and a market
interest rate of 9%. What is its price?

Years to Maturity                                           6
Annual Payment                                            \$80
Par value                                              \$1,000
Going rate, rd                                            9%

Value of bond =                                      \$955.14

3 A bond that matures in 18 years has a par value of \$1,000, an annual coupon of 10%, and a market interest rate
of 7%. What is its price?

Years to Maturity                                          18
Coupon rate                                              10%
Annual Payment                                           \$100
Par value                                              \$1,000
Going rate, rd                                            7%

Value of bond =                                    \$1,301.77
80, and a market

market interest rate
SECTION 4.4
SOLUTIONS TO SELF-TEST

4 A bond currently sells for \$850. It has an eight-year maturity, an annual coupon of \$80, and a par value of \$1,000. W
is its yield to maturity? What is its current yield?

Years to Maturity                                                  8
Annual Payment                                                \$80.00
Current price                                                \$850.00
Par value = FV                                             \$1,000.00

Going rate, rd =YTM:                                     10.90%

Annual Payment                                                \$80.00
Current price                                                \$850.00

Current yield:                                           9.41%

5 A bond currently sells for \$1,250. It pays a \$110 annual coupon and has a 20-year maturity, but it can be called in 5
years at \$1,110. What are its YTM and its YTC? Is it likely to be called if interest rates don't change?

Years to Maturity                                                 20             Years to Call
Annual Payment                                                  \$110             Annual Payment
Current price                                                 \$1,250             Current price
Par value = FV                                                \$1,000             Call price

YTM                                                      8.38%                   YTC

The company will probably call the bond, because the YTC is less than the YTM.
and a par value of \$1,000. What

urity, but it can be called in 5
n't change?

5
\$110
\$1,250
\$1,110

6.85%
SECTION 4.5
SOLUTIONS TO SELF-TEST

2a Last year a firm issued 30-year, 8% annual coupon bonds at a par value of \$1,000. (1) Suppose that one year later
the going rate drops to 6%. What is the new price of the bonds, assuming that they now have 29 years to maturity?

Years to Maturity                                              29
Coupon rate                                                   8%
Annual Payment                                                \$80
Par value                                                  \$1,000
Going rate, rd                                                6%

Value of bond =                                        \$1,271.81

2b Suppose instead that one year after issue the going interest rate increases to 10% (rather than 6%). What is the
price?

Years to Maturity                                              29
Coupon rate                                                   8%
Annual Payment                                                \$80
Par value                                                  \$1,000
Going rate, rd                                               10%

Value of bond =                                          \$812.61
(1) Suppose that one year later
ow have 29 years to maturity?

(rather than 6%). What is the
SECTION 4.6
SOLUTIONS TO SELF-TEST

2 A bond has a 25-year maturity, an 8% semiannual coupon, and a face value of \$1,000. The going nominal annual in
What is the bond's price?

Coupons per year                               2

Annual values         Semiannual Inputs

Years to Maturity                             25                     50
Coupon rate                                  8%                     4%
Annual Payment                               \$80                    \$40
Par value                                 \$1,000                 \$1,000
Going rate, rd                               6%                   3.0%

Value of bond =                       \$1,255.67               \$1,257.30
00. The going nominal annual interest rate (r d) is 6%.
SECTION 4.9
SOLUTIONS TO SELF-TEST

2 The yield on a 15-year TIPS is 3 percent and the yield on a 15-year Treasury bond is 5 percent. What is the inflation

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

5 percent. What is the inflation
SECTION 4.11
SOLUTIONS TO SELF-TEST

5 A 10-year T-bond has a yield of 6 percent. A corporate bond with a rating of AA has a yield of 4.5 percent. If the cor
has excellent liquidty, what is an estimate of the corporate bond’s default risk premium?

Yield on T-Bond                                      6.0%
Yield on corporate bond                              7.5%

a yield of 4.5 percent. If the corporate bond
?
SECTION 4.13
SOLUTIONS TO SELF-TEST QUESTIONS

3 Assume that the real risk-free rate is r* = 3% and the average expected inflation rate is 2.5% for the foreseeable
future. The DRP and LP for a bond are each 1%, and the applicable MRP is 2%. What is the bond’s yield?

r*                                                             3.0%