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
c. How is the value of any asset whose value is based on expected future cash flows determined? Answer: See Mini
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 coupon if its required rate of return is 10 percent?
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
74
75
76
77
78
79
and enter the cell with the value for r (B37), then Click OK to complete the
operation and get the table.
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
(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 if it remained at 7 percent? Would we now have a premium or a discount bond in either situation? You pick
98
A B C D E F G H I J K L M N O P Q R S T
11/20/2018
Situation
Call Provisions and Sinking Funds
Value of bond = $1,000.00
We can use the data table to construct a graph that
shows the bond’s sensitivity to changing rates.
Chapter 5 Mini Case
Sam Strother and Shawna Tibbs are vice-presidents of Mutual of Seattle Insurance Company and co-directors of the
company’s pension fund management division. A major new client, the Northwestern Municipal Alliance, has requested
that Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs,
who will make the actual presentation, have asked you to help them by answering the following questions. Because the
Boeing Company operates in one of the league’s cities, you are to work Boeing into the presentation.
a. What are the key features of a bond? Answer: See Mini Case Show
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 items as shown in our snapshot in the screen shown just below.
Thus, this bond sells at its par value. That situation always exists if the going rate is
equal to the coupon rate.
The PV function can only be used if the payments are constant, but that is normally the case for bonds.
A call provision that allows the issuer to redeem the bond at a specified time before the maturity date. If interest rates
fall, the issuer can refund the bonds and issue new bonds at a lower rate. Because of this, borrowers are willing to pay
more and lenders require more on callable bonds.
b. What are call provisions and sinking fund provisions? Do these provisions make bonds more or less risky?
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 by 3 percentage points, causing investors to require a 13 percent return? Would we now have a discount or a
premium bond?
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 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.
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 the risk to the investor and shortens the maturity. This is not good for investors if rates fall after issuance.
Value at 7%
Value at 13%
99
100
101
102
103
104
105
106
107
108
109
110
111
3$1,162 $1,000 $867
4$1,143 $1,000 $880
5$1,123 $1,000 $894
6$1,102 $1,000 $911
7$1,079 $1,000 $929
8$1,054 $1,000 $950
9$1,028 $1,000 $973
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
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
Current and Capital Gains Yields
(2.) What are the total return, the current yield, and the capital gains yield for the discount bond? (Assume the bond
is held to maturity and the company does not default on the bond.)
The current yield is the annual interest payment divided by the bond’s current price. The current yield provides
information regarding the amount of cash income that a bond will generate in a given year. However, it does not
account for any capital gains or losses that will be realized if the bond is held to maturity or call.
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
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
Periods to maturity = 10*2 = 20
Coupon rate: 10%
Semiannual pmt = $100/2 = $50.00 PV = $834.72
Future Value: $1,000.00
Periodic rate = 13%/2 = 6.5%
Note that the bond is now more valuable, because interest payments come in faster.
Excel Bond Functions
2, and (3) divide the nominal interest rate by 2.
Use the Rate function with adjusted data to solve the problem.
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
Settlement (today) 3/25/2018
Maturity 12/31/2027
Coupon rate 10.00%
Going rate, r 13.00%
Redemption (par value) 100
Frequency (for semiannual) 2
Basis (360 or 365 day year) 0
A B C D E F G H I J K L M N O P Q R S T
N7% 10% 13%
0$1,211 $1,000 $837
1$1,195 $1,000 $846
2$1,179 $1,000 $856
You pick the rate for a bond:
Your choice:
20%
Resulting bond prices
$581
$597
$616
$640
$667
$701
$741
$789
$847
$917
$1,000
Yield to Maturity (YTM)
Use the Rate function to solve the problem.
Years to Mat: 10
Coupon rate: 9%
Annual Pmt: $90.00 Going rate, r =YTM: 10.91% See RATE function at right.
Current price: $887.00
Par value = FV: $1,000.00
Par value $1,000.00
Coupon rate: 9%
Current Yield =
10.15%
Annual Pmt: $90.00
Current price: $887.00
YTM: 10.91%
YTM =
Current Yield
+ Capital Gains Yield
Capital Gains Yield = YTM – Current Yield
Capital Gains Yield = 10.91% – 10.15%
Capital Gains Yield = 0.76%
Bonds with Semiannual Coupons
Suppose today’s date is January 1, 2018, and the bond matures on December 31, 2027
Settlement (today) 1/1/2018
Maturity 12/31/2027
Coupon rate 10.00%
Going rate, r 13.00%
Redemption (par value) 100
Frequency (for semiannual) 2
Basis (360 or 365 day year) 0
Value of bond = $83.4737 or $834.74
Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three
modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by
Notice that you could choose a current date that is between coupon payments, and the PRICE function will calculate the
correct price. See the example below.
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 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 rd and the bond’s coupon rate? What is the yield-to-maturity of the bond?
g. How does the equation for valuing a bond change if semiannual payments are made? Find the value of a 10-year,
semiannual payment, 10 percent coupon bond if nominal rd = 13%.
Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we
would still use the annual interest.
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, consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no
interest income. However, the zero appreciates through time, and its total return clearly exceeds zero.
If rates fall, the bond goes to a premium, but it moves towards par as maturity approaches. The reverse hold if rates
rise 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. Note that the above graph assumes that interest rates stay constant after the initial change. That is most unlikely–
interest rates fluctuate, and so do the prices of outstanding bonds.
Value of Bond in Given Year:
$0
$200
$400
$600
$800
$1,000
$1,200
$1,400
1 3 5 7 9 11
Price
Years to maturity
Value of the bond over time Rates fall to 7%
Rates stay the same
Rates increase to 13%
Your choice
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
Going rate, r 13.00%
Redemption (par value) 100
Frequency (for semiannual) 2
Basis (360 or 365 day year) 0
Accrued interest = $2.3333 or $23.33
Suppose the bond’s price is $1,150. You can also calculate the yield using the YIELD function, as shown below.
Curent price 1,150.00$
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
282
283
284
285
286
Current price: $1,135.90
Call price = FV $1,050.00
Par value $1,000.00
(r*), inflation premium (IP), default risk premium (DRP), liquidity premium (LP), and maturity risk premium (MRP).
Answer: See Mini Case Show.
j. Define the real risk-free rate (r*). What security can be used as an estimate of r*? What is the nominal risk-free rate
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
334
335
336
337
339
340
341
342
343
As the interst rate goes from 5% to 15%, the price changes are bigger for the 10-year bond.
344
345
346
347
348
349
350
351
352
353
354
A B C D E F G H I J K L M N O P Q R S T
Value of bond = $83.6307 or $836.31
Issue date 1/1/2018
First interest date 6/30/2018
Settlement (today) 3/25/2018
Maturity 12/31/2027
Coupon rate 10.00%
Settlement (today) 1/1/2018
Maturity 12/31/2027
Coupon rate 10.00%
Redemption (par value) 100
Frequency (for semiannual) 2
Basis (360 or 365 day year) 0
Yield 7.81%
Yield to Call
Use the Rate function to solve the problem.
Number of semiannual periods to call: 10
Seminannual coupon rate: 5% Semiannual Rate = I = YTC = 3.77%
Seminannual Pmt: $50.00 Annual nominal rate = 7.53%
Your Choice of Maturity
Years to Mat: 10 Rate Price Rate Price Rate Price
Coupon rate: 10% $966.65 $946.77 $991.88
Annual Pmt: $100.00 5.0% 1,216.47 5.0% $1,386.09 5.0% $1,047.62
Current price: $946.77 7.0% 1,123.01 7.0% $1,210.71 7.0% $1,028.04
Par value = FV: $1,000.00 10.0% 1,000.00 10.0% $1,000.00 10.0% $1,000.00
YTM = 10.9% 13.0% 894.48 13.0% $837.21 13.0% $973.45
15.0% 832.39 15.0% $749.06 15.0% $956.52
Years to Mat: 1Scratch sheet for Your Choice
Coupon rate: 10% Years to Mat: 5
Annual Pmt: $100.00 Coupon rate: 10%
Current price: $991.88 Annual Pmt: $100.00
Par value = FV: $1,000.00 Current price: $966.65
YTM = 10.9%
Par value = FV:
$1,000.00
YTM = 10.9%
5
p. What is the term structure of interest rates? What is a yield curve?
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 is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the
same coupon, the bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same
maturity, the one with the smaller coupon payment will have more interest rate sensitivity.
m. What is interest rate (or price) risk? Which bond has more interest rate risk, an annual payment 1-year bond or a 10-
year bond? Why?
h. Suppose a 10-year, 10 percent, semiannual coupon bond with a par value of $1,000 is currently selling for $1,135.90,
producing a nominal yield to maturity of 8 percent. However, the bond can be called after 5 years for a price of $1,050.
This is the value of the bond, but it does not include the accrued interest you would pay. The ACCRINT function will
calculate accrued interest, as shown below.
(1.) What is the bond’s nominal yield to call (YTC)?
(2.) If you bought this bond, do you think you would be more likely to earn the YTM or the YTC? Why?
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 factors affect a company’s bond rating? Answer: See Mini Case Show.
k. Describe a way to estimate the inflation premium (IP) for a T-Year bond. Answer: See Mini Case Show.
1-Yr Maturity
10-Yr Maturity
o. How are interest rate risk and reinvestment rate risk related to the maturity risk premium? Answer: See Mini Case
Show.
n. What is reinvestment rate risk? Which has more reinvestment rate risk, a 1-year bond or a 10-year bond? Answer:
See Mini Case Show.
Enter your choice
for years to
maturity:
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, 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 lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to
maturity is replaced with years to call, and the maturity value is replaced with the call price.
(rRF)? What securities can be used as estimates of rRF? Answer: See Mini Case Show.
$800.00
$900.00
$1,000.00
$1,100.00
$1,200.00
$1,300.00
$1,400.00
$1,500.00
10 Yr. versus 1 Yr.
Your Choice
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
Real risk free rate 3.00%
Expected inflation of 5% for the next 1 years.
Expected inflation of 6% for the next 1 years.
Expected inflation of 8% thereafter.
INPUT DATA
Real risk free rate 3.00%
Expected inflation of 5% for the next 1 years.
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
15 3.00% 7.67% 1.40% 12.07%
16 3.00% 7.69% 1.50% 12.19%
17 3.00% 7.71% 1.60% 12.31%
18 3.00% 7.72% 1.70% 12.42%
19 3.00% 7.74% 1.80% 12.54%
20 3.00% 7.75% 1.90% 12.65%
21 3.00% 7.76% 2.00% 12.76%
22 3.00% 7.77% 2.10% 12.87%
23 3.00% 7.78% 2.20% 12.98%
24 3.00% 7.79% 2.30% 13.09%
25 3.00% 7.80% 2.40% 13.20%
26 3.00% 7.81% 2.50% 13.31%
402
403
404
405
406
407
408
409
A B C D E F G H I J K L M N O P Q R S T
Expected inflation of 6% for the next 1 years.
Expected inflation of 8% thereafter.
Years to Real risk-free Inflation Maturity Risk Treasury
Maturity rate (r*) Premium (IP)
Premium (MRP)
Yield
1 3.00% 5.00% 0.00% 8.00%
2 3.00% 5.50% 0.10% 8.60%
3 3.00% 6.33% 0.20% 9.53%
4 3.00% 6.75% 0.30% 10.05%
5 3.00% 7.00% 0.40% 10.40%
6 3.00% 7.17% 0.50% 10.67%
7 3.00% 7.29% 0.60% 10.89%
8 3.00% 7.38% 0.70% 11.08%
9 3.00% 7.44% 0.80% 11.24%
10 3.00% 7.50% 0.90% 11.40%
11 3.00% 7.55% 1.00% 11.55%
12 3.00% 7.58% 1.10% 11.68%
13 3.00% 7.62% 1.20% 11.82%
The yield is upward sloping due to increasing expected inflation and an increasing maturity risk premium 14 3.00% 7.64% 1.30% 11.94%
27 3.00% 7.81% 2.60% 13.41%
28 3.00% 7.82% 2.70% 13.52%
29 3.00% 7.83% 2.80% 13.63%
30 3.00% 7.83% 2.90% 13.73%
The table above gives us all of the components for our Treasury yield curve. Recall, we have said that Treasury securities are subject to
two kinds of risk premiums, the inflation premium and the maturity risk premium. Just as we “built” Treasury yields in the table, we can
“build” a yield curve based upon these expectations.
The term structure describes the relationship between long-term and short-term interest rates. Graphically, this
relationship can be shown in what is known as the yield curve. See the hypothetical curve below.
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
1 2 3 4 5 6 7 8 9 1011121314151617181920
Interest Rate
Maturity
Inflation
Premium
Real Risk
Free Rate