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Ch.5 13ed Bond MinicMaster

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Ch.5 13ed Bond MinicMaster Powered By Docstoc
					            A               B              C              D              E             F              G             H
 1
 2
 3                          Chapter 5. Mini Case: Bonds & Interest Rates
 4
 5   Situation
 6   Sam Strother and Shawna Tibbs are vice-presidents of Mutual of Seattle Insurance Company and co-directors of the
 7   company's pension fund management division. A major new client, the Northwestern Municipal Alliance, has requested that
 8   Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs, who will make
 9   the actual presentation, have asked you to help them by answering the following questions. Because the Boeing Company
10   operates in one of the league's cities, you are to work Boeing into the presentation.
11
12   a. What are the key features of a bond? Answer: See Chapter 5 Mini Case Ppt Show
13
14   b. What are call provisions and sinking fund provisions? Do these provisions make bonds more or less risky?
15
16   Call Provisions and Sinking Funds
17   A call provision that allows the issuer to redeem the bond at a specified time before the maturity date. If interest rates fall, the
18   issuer can refund the bonds and issue new bonds at a lower rate. Because of this, borrowers are willing to pay more and
19   lenders require more on callable bonds.
20
21   In a sinking fund provision, the issuer pays off the loan over its life rather than all at the maturity date. A sinking fund reduces
22   the risk to the investor and shortens the maturity. This is not good for investors if rates fall after issuance.
23
24   c. How is the value of any asset whose value is based on expected future cash flows determined? Answer: See Chapter 5 Mini
25   Case Ppt Show
26
27   d. How is the value of a bond determined? What is the value of a 10-year, $1,000 par value bond with a 10 percent annual
28   coupon if its required rate of return is 10 percent?
29
30   Finding the "Fair Value" of a Bond
31
32   First, we list the key features of the bond as "model inputs":
33   Years to Mat:                 10
34   Coupon rate:                10%
35   Annual Pmt:                 $100
36   Par value = FV:           $1,000
37   Going rate, rd:             10%
38
39   The easiest way to solve this problem is to use Excel's PV function. Click fx, then financial, then PV. Then fill in the menu
40   items as shown in our snapshot in the screen shown just below.
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            A               B              C             D              E              F             G              H
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60   Value of bond =     $1,000.00 Thus, this bond sells at its par value. That situation always exists if the going rate is equal to the
61                                 coupon rate.
62
63   The PV function can only be used if the payments are constant, but that is normally the case for bonds.
64
65   e. (1.) What would be the value of the bond described in Part d if, just after it had been issued, the expected inflation rate rose
66   by 3 percentage points, causing investors to require a 13 percent return? Would we now have a discount or a premium bond?
67
68
69   We could simply go to the input data section shown above, change the value for r from 10% to 13%. You can set up a data table
70   to show the bond's value at a range of rates, i.e., to show the bond's sensitivity to changes in interest rates. This is done below.
71
                                                  To make the data table, first type the headings, then type the rates in cells in the
72                      Bond Value                left column. Since the input values are listed down a column, type the formula in
73    Going rate, r:          $1,000              the row above the first value and one cell to the right of the column of values (this
74          0%             $2,000.00              is B73; note that the formula in B73 actually just refers to the bond pricing
75          7%             $1,210.71              formula above in B58). Select the range of cells that contains the formulas and
76         10%             $1,000.00              values you want to substitute (A73:B78). Then click Data, What-If-Analysis, and
77         13%               $837.21              then Data Table to get the menu. The input data are in a column, so put the cursor
78         20%               $580.75              on "column input cell" and enter the cell with the value for r (B37), then Click OK
79                                                to complete the operation and get the table.
80   We can use the data table to construct a graph that shows
81   the bond's sensitivity to changing rates.
82
83
84                 Interest Rate Sensitivity of a 10-Year Bond
85                            Value at 7%                 Value at 13%
86         $2,500
87         $2,000
88         $1,500
89         $1,000
90          $500
91             $0                                                            Put B37 here.
92                0%            5%             10%           15%       20%
93
94
95
96     (2.) What would happen to the value of the 10-year bond over time if the required rate of return remained at 13 percent, or
97 if it remained at 7 percent? Would we now have a premium or a discount bond in either situation? You pick a rate.
98
                A              B            C             D              E                   F        G             H
99                             Value of Bond in Given Year:
100             N             7%          10%           13%
101             0           $1,211       $1,000         $837
102             1           $1,195       $1,000         $846
103             2           $1,179       $1,000         $856
104             3           $1,162       $1,000         $867
105             4           $1,143       $1,000         $880
106             5           $1,123       $1,000         $894
107             6           $1,102       $1,000         $911
108             7           $1,079       $1,000         $929
109             8           $1,054       $1,000         $950
110             9           $1,028       $1,000         $973
111             10          $1,000       $1,000        $1,000
112                                                                                                   You pick the rate for a bond:
113                                                                  Rates fall to 7%                        Your choice:
         Value of the bond over time
114                                                                  Rates stay the same                          20%
                                                                     Rates increase to 13%
115
116                                                                  Your choice
                $1,400
117                                                                                                       Resulting bond prices
118             $1,200                                                                                            $581
119                                                                                                               $597
120             $1,000                                                                                            $616
121                                                                                                               $640
                     $800
        Price




122                                                                                                               $667
123                  $600                                                                                         $701
124                                                                                                               $741
125                  $400                                                                                         $789
126                                                                                                               $847
127                  $200                                                                                         $917
128                                                                                                              $1,000
                       $0
129
                             1       3       5      7      9           11
130                                      Years to maturity
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133
134   If rates fall, the bond goes to a premium, but it moves towards par as maturity approaches. The reverse hold if rates rise and
135   the bond sells at a discount. If the going rate remains equal to the coupon rate, the bond will continue to sell at par. Note that
136   the above graph assumes that interest rates stay constant after the initial change. That is most unlikely--interest rates fluctuate,
137   and so do the prices of outstanding bonds.
138
139   Yield to Maturity (YTM)
140
141   f. (1.) What is the yield to maturity on a 10-year, 9 percent annual coupon, $1,000 par value bond that sells for $887.00? That
142   sells for $1,134.20? What does the fact that a bond sells at a discount or at a premium tell you about the relationship between r d
143   and the bond's coupon rate? What is the yield-to-maturity of the bond?
144
145   Use the Rate function to solve the problem.
146
147   Years to Mat:                10
148   Coupon rate:               9%
149   Annual Pmt:              $90.00               Going rate, r =YTM:                 10.91%               See RATE function at right.
150   Current price:         $887.00
151   Par value = FV:       $1,000.00
             A                B            C              D                E           F             G              H
152
153      (2.) What are the total return, the current yield, and the capital gains yield for the discount bond? (Assume the bond is held
154   to maturity and the company does not default on the bond.)
155
156   Current and Capital Gains Yields
157   The current yield is the annual interest payment divided by the bond's current price. The current yield provides information
158   regarding the amount of cash income that a bond will generate in a given year. However, it does not account for any capital
159   gains or losses that will be realized if the bond is held to maturity or call.
160
161   Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we would
162   still use the annual interest.
163
164   Par value            $1,000.00
165   Coupon rate:              9%                 Current Yield =    10.15%
166   Annual Pmt:             $90.00
167   Current price:        $887.00
168   YTM:                  10.91%
169
170   The current yield provides information on a bond's cash return, but it gives no indication of the bond's total return. To see this,
171   consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no interest income.
172   However, the zero appreciates through time, and its total return clearly exceeds zero.
173
174       YTM =        Current Yield           +     Capital Gains Yield
175
176
177   Capital Gains Yield =              YTM                   -   Current Yield
178
179   Capital Gains Yield =             10.91%             -          10.15%
180
181   Capital Gains Yield =                 0.76%
182
183
184   g. How does the equation for valuing a bond change if semiannual payments are made? Find the value of a 10-year, semiannual
185   payment, 10 percent coupon bond if nominal rd = 13%.
186
187   Bonds with Semiannual Coupons
188   Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three
189   modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by 2, and
190   (3) divide the nominal interest rate by 2.
191
192   Use the Rate function with adjusted data to solve the problem.
193
194   Periods to maturity = 10*2 =              20
195   Coupon rate:                           10%
196   Semiannual pmt = $100/2 =             $50.00       PV =        $834.72
197   Future Value:                      $1,000.00
198   Periodic rate = 13%/2 =                6.5%
199
200   Note that the bond is now more valuable, because interest payments come in faster.
201
202
203   Excel Bond Functions
204   Supose today's date is January 1, 2010, and the bond matures on December 31, 2019
             A              B             C             D             E             F             G             H
205
206   Settlement (today)                 1/1/2010
207   Maturity                         12/31/2019
208   Coupon rate                         10.00%
209   Going rate, r                       13.00%
210   Redemption (par value)                 100
211   Frequency (for semiannual)                2
212   Basis (360 or 365 day year)               0
213
214
215   Value of bond =                   $83.4737        or            $834.74
216
217   Notice that you could choose a current date that is between coupon payments, and the PRICE function will calculate the correct
218   price. See the example below.
219
220
221   Settlement (today)                3/25/2010
222   Maturity                         12/31/2019
223   Coupon rate                         10.00%
224   Going rate, r                       13.00%
225   Redemption (par value)                 100
226   Frequency (for semiannual)                2
227   Basis (360 or 365 day year)               0
228
229   Value of bond =                   $83.6307        or            $836.31
230
231   This is the value of the bond, but it does not include the accrued interest you would pay. The ACCRINT function will calculate
232   accrued interest, as shown below.
233
234   Issue date                         1/1/2010
235   First interest date               6/30/2010
236   Settlement (today)                3/25/2010
237   Maturity                         12/31/2019
238   Coupon rate                         10.00%
239   Going rate, r                       13.00%
240   Redemption (par value)                 100
241   Frequency (for semiannual)                2
242   Basis (360 or 365 day year)               0
243
244   Accrued interest =                  $2.3333       or             $23.33
245
246
247   Suppose the bond's price is $1,150. You can also calculate the yield using the YIELD function, as shown below.
248
249   Curent price                   $ 1,150.00
250   Settlement (today)                1/1/2010
251   Maturity                        12/31/2019
252   Coupon rate                        10.00%
253   Redemption (par value)                100
254   Frequency (for semiannual)               2
255   Basis (360 or 365 day year)              0
256
              A               B              C              D               E              F             G              H
257   Yield                                  7.81%
258
259
260   h. Suppose a 10-year, 10 percent, semiannual coupon bond with a par value of $1,000 is currently selling for $1,135.90,
261   producing a nominal yield to maturity of 8 percent. However, the bond can be called after 5 years for a price of $1,050.
262      (1.) What is the bond's nominal yield to call (YTC)?
263      (2.) If you bought this bond, do you think you would be more likely to earn the YTM or the YTC? Why?
264
265   Yield to Call
266   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 bonds,
267   and if interest rates fall, then it would be logical for the issuer to call the bonds and replace them with new bonds that carry a
268   lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to maturity is
269   replaced with years to call, and the maturity value is replaced with the call price.
270
271   Use the Rate function to solve the problem.
272
273   Number of semiannual periods to call:                      10
274   Seminannual coupon rate:                                 5%               Semiannual Rate = I = YTC =           3.77%
275   Seminannual Pmt:                                       $50.00                    Annual nominal rate =          7.53%
276   Current price:                                      $1,135.90
277   Call price = FV                                     $1,050.00
278   Par value                                           $1,000.00
279
280
281   i. Write a general expression for the yield on any debt security (rd) and define these terms: real risk-free rate of interest (r*),
282   inflation premium (IP), default risk premium (DRP), liquidity premium (LP), and maturity risk premium (MRP). Answer: See
283   Chapter 5 Mini CasePpt Show.
284
285
286   j. Define the nominal risk-free rate (r RF). What security can be used as an estimate of rRF? Answer: See Chapter 5 Mini Case
287   Ppt Show.
288
289
290   k. Describe a way to estimate the inflation premium (IP) for a T-Year bond. Answer: See Chapter 5 Mini Case Ppt Show.
291
292
293   l. What is a bond spread and how is it related to the default risk premium? How are bond ratings related to default risk? What
294   factors affect a company’s bond rating? Answer: See Chapter 5 Mini CasePpt Show.
295
296
297   m. What is interest rate (or price) risk? Which bond has more interest rate risk, an annual payment 1-year bond or a 30-year
298   bond? Why?
299
300
301   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 rates
302   is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the same coupon, the
303   bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same maturity, the one with
304   the smaller coupon payment will have more interest rate sensitivity.
305
306
307
308                                                   Your Choice of Maturity             10-Yr Maturity                1-Yr Maturity
309   Years to Mat:                10                    Rate         Price              Rate       Price              Rate
             A                       B        C                D               E              F                 G              H
310   Coupon rate:                     9%                                      $929.60                          $887.63
311   Annual Pmt:                   $90.00                      5.0%      1173.179067              5.0%       $1,308.87         5.0%
312   Current price:              $887.63                       7.0%      1082.003949              7.0%       $1,140.47         7.0%
313   Par value = FV:            $1,000.00                      9.0%              1000             9.0%       $1,000.00         9.0%
314   YTM =                         10.9%                      11.0%      926.0820596             11.0%         $882.22        11.0%
315                                                            13.0%      859.3107495             13.0%         $782.95        13.0%
316   Years to Mat:                      1
317   Coupon rate:                     9%
318   Annual Pmt:                   $90.00                                          10 Yr. versus 1 Yr.
319   Current price:              $982.87         $1,400.00                                                                           Your Choice

320   Par value = FV:            $1,000.00        $1,300.00
321   YTM =                         10.9%
322                                               $1,200.00

323                                               $1,100.00
324      Enter your                               $1,000.00
325   choice for years
                                                     $900.00
326     to maturity:                     5
327                                                  $800.00
328
                                                     $700.00
329
330                                                                5.0%            7.0%            9.0%            11.0%           13.0%
                                                                                                    YTM
331
332
333
334
335   n. What is reinvestment rate risk? Which has more reinvestment rate risk, a 1-year bond or a 10-year bond? Answer: See
336   Chapter 5 Mini Case Ppt Show.
337
338
339   o. How are interest rate risk and reinvestment rate risk related to the maturity risk premium? Answer: See Chapter 5 Mini
340   Case Ppt Show.
341
342
343   p. What is the term structure of interest rates? What is a yield curve?
344
345   The term structure describes the relationship between long-term and short-term interest rates. Graphically, this relationship
346   can be shown in what is known as the yield curve. See the hypothetical curve below.
347
348
349   Hypothetical Inputs                                                                                 See to right for actual date used in graph.
350   Real risk free rate                    3.00%
351   Expected inflation of                    5%       for the next           1         years.
352   Expected inflation of                    6%       for the next           1         years.
353   Expected inflation of                    8%       thereafter.
354
355                                  Hypothetical Treasury Yield Curve
356
357
358                         14.00%
359
360                         12.00%                                     MRP

361
                            10.00%
            Interest Rate
           A               B             C              D             E             F            G           H
362              10.00%
          Interest Rate                                     Inflation
363                                                         Premium
364               8.00%
365
366
                  6.00%
367                                          Real Risk
368
                  4.00%                      Free Rate

369
370
                  2.00%
371
372
                  0.00%
373                          1    3      5      7      9 11 13 15 17 19
374                                                    Maturity
375
376
377
378 The yield is upward sloping due to increasing expected inflation and an increasing maturity risk premium
379
380
381 q. Briefly describe bankruptcy law. If a firm were to default on the bonds, would the company be immediately liquidated?
382 Would the bondholders be assured of receiving all of their promised payments? Answer: See Chapter 5 Mini Case Ppt Show.
383
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                        I      J   K   L   M   N   O   P
              1       1/1/2012
              2
              3
              4
              5
 -directors of the
              6
 ance, has requested that
              7
r and Tibbs, who will make
              8
              9
the Boeing Company
             10
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ess risky? 14
             15
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e. If interest rates fall, the
             17
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 ng to pay more and
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 te. A sinking fund reduces
uance.       22
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 wer: See Chapter 5 Mini
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th a 10 percent annual
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  Then fill in the menu
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                        I     J   K   L   M   N   O   P
            54
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 e going rate is equal to the
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ds.         63
            64
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xpected inflation rate rose
            66
ount or a premium bond?
            67
            68
            69
 You can set up a data table
rates. This70 done below.
             is
            71
 e the rates in cells in the
            72
umn, type the formula in
            73
 the column of values (this
            74
 to the bond pricing
            75
tains the formulas and
            76
 a, What-If-Analysis, and
            77
 a column, so put the cursor
            78
 for r (B37), then Click OK
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 emained at 13 percent, or
You pick a 97
            rate.
            98
                        I       J   K   L   M   N   O   P
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 k the rate112 a bond:
              for
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 ulting bond prices
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erse hold if rates rise and
             134
              par.
 e to sell at135 Note that
             136
ely--interest rates fluctuate,
             137
             138
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              $887.00? That
at sells for141
 the relationship between r d
             142
             143
             144
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             146
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             148
See RATE149  function at right.
             150
             151
                       I         J   K   L   M   N   O   P
            152
  (Assume 153bond is held
            the
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 eld provides information
            157
            158
 account for any capital
            159
            160
            161
ual payments, we would
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            169
d's total return. To see this,
            170
            171
o interest income.
            172
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            178
            179
            180
            181
            182
            183
            184
ue of a 10-year, semiannual
            185
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            187
must make three
            188
            189
ears to maturity by 2, and
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on will calculate the correct
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INT function will calculate
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                         I     J   K   L   M   N   O   P
            257
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 ling for $1,135.90,
            261
r a price of $1,050.
            262
  Why? 263
            264
            265
the right to call the bonds,
            266
h new bonds that carry a
            267
            268
 years to maturity is
            269
            270
            271
            272
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            275
            276
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            278
            279
            280
-free rate of interest (r*),
            281
            282
mium (MRP). Answer: See
            283
            284
            285
   See Chapter 5 Mini Case
            286
            287
            288
            289
            290
5 Mini Case Ppt Show.
            291
            292
            293
  lated to default risk? What
            294
            295
            296
            297
  1-year bond or a 30-year
            298
            299
            300
             to interest rates
sensitivity 301
ve the same coupon, the
            302
            303
 e maturity, the one with
            304
            305
            306
            307
            308
         1-Yr Maturity
            309       Price
                         I     J   K          L                M                N                O                P
          310          $982.87
          311       $1,038.10
          312       $1,018.69
          313       $1,000.00
          314          $981.98
          315          $964.60
          316                          Scratch sheet for Your Choice
          317                          Years to Mat:                 5
          318                          Coupon rate:                9%
          319Your Choice               Annual Pmt:              $90.00
          320                          Current price:          $929.60
          321                          Par value = FV:       $1,000.00
          322                          YTM =                    10.9%
           323
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           13.0%
           330
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 ar bond? Answer: See
           336
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wer: See Chapter 5 Mini
           340
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           344
           345
phically, this relationship
           346
           347
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           349
r actual date used in graph.
           350
           351                         Suppose most investors expect the inflation rate to be 5 percent next year, 6 percent the following year
           352                         rate is 3 percent. The maturity risk premium is zero for securities that mature in 1 year or less, 0.1 pe
           353                         increases by 0.1 percent per year thereafter for 20 years, after which it is stable. What is the interest
           354                         securities? Draw a yield curve with these data. What factors can explain why this constructed yield c
           355
           356                         Now, we want to set up a table that encompasses all of the information for our yield curve.
           357
           358                         INPUT DATA
           359                         Real risk free rate                   3.00%
           360                         Expected inflation of                   5%        for the next             1
           361                         Expected inflation of                   6%        for the next             1
                     I     J   K         L                 M               N             O                  P
           362                     Expected inflation of                  8%      thereafter.
           363                        Years to      Real risk-free     Inflation   Maturity Risk        Treasury
           364                       Maturity          rate (r*)     Premium (IP) Premium (MRP)           Yield
           365                            1              3.00%          5.00%         0.00%              8.00%
           366                            2              3.00%          5.50%         0.10%              8.60%
           367                            3              3.00%          6.33%         0.20%              9.53%
           368                            4              3.00%          6.75%         0.30%              10.05%
           369                            5              3.00%          7.00%         0.40%              10.40%
           370                            6              3.00%          7.17%         0.50%              10.67%
           371                            7              3.00%          7.29%         0.60%              10.89%
           372                            8              3.00%          7.38%         0.70%              11.08%
           373                            9              3.00%          7.44%         0.80%              11.24%
           374                           10              3.00%          7.50%         0.90%              11.40%
           375                           11              3.00%          7.55%         1.00%              11.55%
           376                           12              3.00%          7.58%         1.10%              11.68%
           377                           13              3.00%          7.62%         1.20%              11.82%
           378                           14              3.00%          7.64%         1.30%              11.94%
           379                           15              3.00%          7.67%         1.40%              12.07%
           380                           16              3.00%          7.69%         1.50%              12.19%
           381
mmediately liquidated?                   17              3.00%          7.71%         1.60%              12.31%
 er 5 Mini 382 Ppt Show.
           Case                          18              3.00%          7.72%         1.70%              12.42%
           383                           19              3.00%          7.74%         1.80%              12.54%
           384                           20              3.00%          7.75%         1.90%              12.65%
           385                           21              3.00%          7.76%         2.00%              12.76%
           386                           22              3.00%          7.77%         2.10%              12.87%
           387                           23              3.00%          7.78%         2.20%              12.98%
           388                           24              3.00%          7.79%         2.30%              13.09%
           389                           25              3.00%          7.80%         2.40%              13.20%
           390                           26              3.00%          7.81%         2.50%              13.31%
           391                           27              3.00%          7.81%         2.60%              13.41%
           392                           28              3.00%          7.82%         2.70%              13.52%
           393                           29              3.00%          7.83%         2.80%              13.63%
           394                           30              3.00%          7.83%         2.90%              13.73%
           395
           396                     The table above gives us all of the components for our Treasury yield curve. Recall, we have said that
           397                     of risk premiums, the inflation premium and the maturity risk premium. Just as we "built" Treasury
           398                     curve based upon these expectations.
     Q   R   S   T
 1
 2
 3
 4
 5
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35
36
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39
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48
49
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51
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53
     Q   R   S   T
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
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72
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98
      Q   R   S   T
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
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131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
      Q   R   S   T
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
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188
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190
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      Q   R   S   T
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
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239
240
241
242
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248
249
250
251
252
253
254
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256
      Q   R   S   T
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
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282
283
284
285
286
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288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
                      Q                R                S                T
          310
          311
          312
          313
          314
          315
          316
          317
          318
          319
          320
          321
          322
            323
            324
            325
            326
            327
            328
            329
            330
            331
            332
            333
            334
            335
            336
            337
            338
            339
            340
            341
            342
            343
            344
            345
            346
            347
            348
            349
            350
  year, 6 percent the following year, and 8 percent thereafter. The real risk-free
            351
hat mature in 1 year or less, 0.1 percent for 2-year securities, and then the MRP
            352
            353
h it is stable. What is the interest rate on 1-year, 10-year, and 20-year Treasury
xplain why354 constructed yield curve is upward sloping?
             this
            355
 on for our356 curve.
             yield
            357
            358
            359
            360 years.
            361 years.
                     Q                R                S                T
          362
          363
          364
          365
          366
          367
          368
          369
          370
          371
          372
          373
          374
          375
          376
          377
          378
          379
          380
          381
          382
          383
          384
          385
          386
          387
          388
          389
          390
          391
          392
          393
          394
          395
d curve. Recall, we have said that Treasury securities are subject to two kinds
          396
ium. Just 397 "built" Treasury yields in the table, we can "build" a yield
          as we
          398

				
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