TABLE OF CONTENTS
1.
Interest Rates
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
2.
1
5
9
17
21
27
33
39
47
51
Level Annuities
A.
B.
C.
D.
E.
3.
Simple and Compound Interest
Present Value and the Effective Rate of Discount
Nominal Rates of Interest and Discount
Force of Interest: Calculating Accumulated Values
Force of Interest: Calculating Times
Force of Interest: Calculating Forces of Interest
Force of Interest: Other Problems
Equations of Value: Calculating Amounts
Equations of Value: Calculating Times
Equations of Value: Calculating Rates
Annuities: Calculating Present and Accumulated Values
Annuities: Calculating Payment Amounts
Annuities: Other Problems
Perpetuities
Annuities: Calculating Rates
55
67
77
81
87
Different Periods
A.
Annuities Payable More Frequently Than Interest Is Convertible: Calculating Present
and Accumulated Values
B.
Annuities Payable More Frequently Than Interest Is Convertible: Calculating Payment
Amounts
C.
Annuities Payable Less Frequently Than Interest Is Convertible: Calculating Present
and Accumulated Values
D.
Annuities Payable Less Frequently Than Interest Is Convertible: Calculating Payment
Amounts
E.
Annuities Payable at a Different Frequency Than Interest Is Convertible: Calculating
Interest Rates
F.ContinuousAnnuities
4.
Varying Annuities with Payments in an Arithmetic Progression: Basic Problems
Varying Annuities with Payments in an Arithmetic Progression: Advanced Problems
Varying Annuities with Payments in a Geometric Progression: Present and Accumulated
Values
D.
Varying Annuities with Payments in a Geometric Progression: Calculating Payment
Amounts
E.
Varying Annuities: Calculating Interest Rates
F.Other Varying Annuities
G.
Continuous Varying Annuities
109
113
117
119
121
129
135
141
147
151
159
Yield Rates
A.
B.
C.
6.
101
Varying Annuities
A.
B.
C.
5.
97
Loans
Yield Rates
Reinvestment Rates
Dollar-Weighted and Time-Weighted Rates of Interest and Allocation Methods
163
173
183
A.
Calculating of Outstanding Balances
B.
Other Problems Involving Outstanding Balances
C.
Amortization: Calculating Interest Paid
D.
Amortization: Calculating Principal Payments
E.
Amortization: Other Problems
F.Sinking Funds: Calculating Payments
G.
Sinking Funds: Internet Rates
H.
Sinking Funds: Other Problems
I.
Varying Series of Payments
7.
Financial Instruments
A.
Types of Financial Institutions
B.
Price of a Bond: calculating Purchase Prices
C.
Price of a Bond: Other Problems
D.
Premium Discount
E.
Interim Prices
F.Bond Yields
G.
Callable Bonds
H.
Common Stock
8.
257
263
275
281
289
291
297
301
Term Structure
A.
B.
C.
9.
195
203
213
223
229
235
243
249
253
Recognition of Inflation
Calculation of Spot and Forward Rates
Other Problems Related to Yield Curves
309
311
319
Duration
Convexity
Matching Assets and Liabilities
Immunization
323
331
335
339
Duration
A.
B.
C.
D.
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
iii
10.
McDonald 1
341
11.
McDonald 2
343
A.
B.
12.
Call Options and Put Options
Other Topics
McDonald 3
A.
B.
Put-Call Parity
Other Topics
13.
McDonald 4
14.
McDonald 5
A.
B.
15.
Forward Contracts
Future Contracts
McDonald 8
353
361
377
379
385
391
NOTES
Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial
Society and the Society of Actuaries. They are reproduced in this study manual with the permission of the CAS
and SoA solely to aid students studying for the actuarial exams. Some editing of questions has been done.
Students may also request past exams directly from both societies. I am very grateful to these organizations for
their cooperation and permission to use this material. They are, of course, in no way responsible for the structure
or accuracy of the manual.
Exam questions are identified by numbers in parentheses at the end of each question. CAS questions have four
numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the
points assigned. SoA or joint exam questions usually lack the number for points assigned. W indicates a written
answer question; for questions of this type, the number of points assigned are also given. A indicates a question
from the afternoon part of an exam. MC indicates that a multiple choice question has been converted into a
true/false question. Page numbers (p.) with solutions refer to the reading to which the question has been assigned
unless otherwise noted.
Page references refer to Samuel A. Broverman Mathematics of Investment and Credit (2008); James W. Daniel
and Leslie Jane Federer Vaaler, Mathematical Interest Theory (2009); Stephen G. Kellison, The Theory of Interest
(2008); Robert L. McDonald, Derivatives Markets, (2006); and Chris Ruckman and Joe Francis, Financial
Mathematics (2005).
Although I have made a conscientious effort to eliminate mistakes and incorrect answers, I am certain some
remain. I am very grateful to students who discovered errors in the past and encourage those of you who find
others to bring them to my attention. Please check our web site for corrections subsequent to publication. I would
also like to thank Graham Lord for checking some of the solutions in the manual.
Hanover, NH 6/30/11
PJM
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
Financial Instruments  263
B.
Price of a Bond: Calculating Purchase Prices
B1.
A bond with a $100 par value has 5.25% annual coupons and is due to mature at the end of 16 years. The
bond will be redeemed at maturity for an amount equal to its par value less a service charge. The service
charge is equal to 25% of the excess (if any) of the par value over the purchase price. A prospective
purchaser offers a price that will produce a yield equivalent to a 7% annual effective interest rate, taking
into account the deduction of the service charge. It is noted that (1.07)16 = 3. In which of the following
ranges does this price lie?
A. < $65
B2.
B. ≥ $65 but < $71
C. ≥ $71 but < $77
B. $127
D. ≥ $1,050 but < $1,075
D. $131
E. $133
(83–4–5–2)
C. 1,120
D. 1,165
E. 1,215
(84S–4–11)
A bond of amount 1 sells for (1 + p) at a certain fixed yield rate. If the bond's coupon rate were halved,
the price would be (1 + q). What would be the price if the coupon rate were doubled? Throughout,
assume the bond is unchanged in all other respects.
B. (1 + p)2/(1 + q)
C. 1 + p + 2q
D. 1 + 2p − q
E. 1 + 4p − 4q
(84–4–9–2)
A $1,000 par value bond with 9% coupons payable semiannually is purchased for $1,300. The yield to the
purchaser is 6%, convertible semiannually. If the same bond were redeemable at 120% of par, what price
would have been paid to obtain the same yield? (Answer to nearest $10.)
A. $1,260
B7.
C. $129
B. 1,085
A. 1 + 3p − 2q
B6.
C. ≥ $1,025 but < $1,050
A 9% bond with a 1,000 par value and coupons payable semiannually is redeemable at maturity for 1,100.
At a purchase price of P, the bond yields a nominal annual interest rate of 8%, compounded semiannually,
and the present value of the redemption amount is 190. Determine P.
A. 1,050
B5.
(80S–4–12)
Two fifteen-year bonds with $100 redemption values are each purchased to yield an effective annual
interest rate of 4%. The first bond bears annual g% coupons and is purchased at a premium of $11.12.
The second bond bears annual (g + 2)% coupons. Which of the following is closest to the purchase price
of the second bond?
A. $125
B4.
E. ≥ $83
A $1,000 bond with quarterly coupons of $25 each will be redeemed in 3.5 years for $1,010. It is
purchased to yield a nominal annual rate of 8% compounded quarterly. In which of the following ranges
is the purchase price of this bond?
A. < $1,000
B. ≥ $1,000 but < $1,025
E. ≥ $1,075 (81S–4–13)
B3.
D. ≥ $77 but < $83
B. $1,320
C. $1,380
D. $1,440
E. $1,500
(84F–4–10)
A $1,000 bond bearing coupons at an annual rate of 5.5% payable semiannually and redeemable at $1,100
is bought to yield a nominal annual rate of 4% convertible semiannually. If the present value of the
redemption value at this yield is $140, what is the purchase price?
A. < $1,310
B. ≥ $1,310 but < $1,330
E. ≥ $1,370 (85–4–15–2)
© 2010 ACTEX Publications, Inc.
C. ≥ $1,330 but < $1,350
D. ≥ $1,350 but < $1,370
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
264  Financial Instruments
Solutions are based on Broverman, pp. 226–33; Daniel, pp. 256–66; Kellison, pp. 200–5; Ruckman, pp. 170–75.
B1.
To earn 7% over 16 years, the AV of the coupons and the net redemption value must equal 3P.
[5.25][(1.07)16 − 1]
+
[100
−
(.25)(100
−
P)]
=
+ 75 + .25P
3P = 5.25s__
16|
.07
2.75P = (5.25)(2/.07) + 75 = 81.82
Answer: D
B2.
-14
-14 = [25][1 − (1.02) ] + (1,010)(.75788) = 1,068.12
+
(1,010)(1.02)
P = 25a__
14|
.02
Answer: D
B3.
-15
-15 = [g][1 − (1.04) ] + (100)(.55526)
+
(100)(1.04)
111.12 = ga__
|
15
.04
111.12 = 11.11839g + 55.526
g = 5.00018
g + 2 = 7.00018
__
P = 7.00018ga15| + (100)(1.04)-15 = (7.00018)(11.11839) + 55.526 = 133
Answer: E
B4.
g = Fr/C = (1,000)(.045)/1,100 = .04091
P = K + (g/i)(C − K) = 190 + (.04091/.04)(1,100 − 190) = 1,120.70
Answer: C
B5.
1 + q = (Fr/2)an_| + K
1 + p = Fr an_| + K
K = (2)(1 + q) − (1 + p) = 1 + 2q − p
Fr an_| = (2)(p − q)
2Fr an_| + K = (2)(2)(p − q) + (1 + 2q − p) = 1 + 3p − 2q
Answer: A
B6.
1,300 = P = K + (g/i)(C − K) = K + (.045/.03)(1,000 − K) = 1,500 − .5K
.5K = 200
K = 400
P' = P + .2K = 1,300 + (.2)(400) = 1,380
Answer: C
B7.
Solve for n and use to calculate the purchase price.
−ln (140/1,100)
2.06142
= .01980 = 104.1
ln 1.02
[27.5][1 − (1.02)-104.1]
+
K
=
+ 140 = (27.5)(43.63656) + 140 = 1,340.01
P = 27.5 a____
104.1|
.02
140 = Cvn = (1,100)(1.02)-n
n =
Answer: C
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
Financial Instruments  265
B8.
You are given the following information on a bond:
i)
ii)
iii)
Its par value and redemption value both equal 1,000.
Its coupon rate is 12% convertible semiannually.
It is priced to yield 10%, convertible semiannually.
The bond has a term of n years. If the term of the bond is doubled, the price will increase by 50. Calculate
the price of the n-year bond.
A. 1,050
B9.
B. 1,100
C. 1,150
D. 1,200
E. ≥ $930
B. 4P + 4Q
C. 4Q − 3P
D. 5P − 4Q
E. 5Q − 4P
(87S–140–10)
B. 1,375
C. 1,475
D. 2,100
E. 2,675
(88F–140–13)
A 100 par value 6% bond with semiannual coupons if purchased at 110 to yield a nominal rate of 4%
convertible semiannually. A similar 3% bond with semiannual coupons is purchased at P to provide the
buyer with the same yield. Calculate P.
A. 90
B13.
D. ≥ $920 but < $930
A 1,000 bond with coupon rate c convertible semiannually will be redeemed at par in n years. The
purchase price to yield 5% convertible semiannually is P. If the coupon rate were (c − .02), then the price
of the bond would be (P − 300). Another 1,000 bond is redeemable at par at the end of 2n years. It has a
coupon rate of 7% convertible semiannually and the yield rate is 5% convertible semiannually. Calculate
the price of this second bond.
A. 1,300
B12.
C. ≥ $910 but < $920
A bond with coupons of 40 sells for P. A second bond with the same maturity value and term has coupons
equal to 30 and sells for Q. A third bond with the same maturity value and term has coupons equal to 80.
All prices are based on the same yield rate, and all coupons are paid at the same frequency. Determine the
price of the third bond.
A. 4P − 4Q
B11.
(85F–4–6)
A $1,000 par value 20-year bond has coupons at 5% convertible semiannually. It has a redemption value
of $1,100. If the yield rate is 6% convertible semiannually, what is the bond's price?
A. < $900
B. ≥ $900 but < $910
(86–4–14–1)
B10.
E. 1,250
B. 95
C. 100
D. 105
E. 110
(88F–140–15)
You are given two n-year par value 1,000 bonds. Bond X has 14% semiannual coupons and a price of
1,407.70 to yield i, compounded semiannually. Bond Y has 12% semiannual coupons and a price of
1,271.80 to yield the same rate i compounded semiannually. Calculate the price of bond X to yield
(i − 1%).
A. 1,500
B. 1,550
© 2010 ACTEX Publications, Inc.
C. 1,600
D. 1,650
E. 1,700
(89S–140–13)
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
266  Financial Instruments
B8.
Calculate the PV of the redemption value and use to calculate the price of the bond:
P = K + (g/i)(C − K) = (1,000)(1.05)-2n + [.06/.05][1,000][1 − (1.05)-2n]
P = 1,200 − (200)(1.05)-2n
P' = P + 50 = 1,250 − (200)(1.05)-2n
P' = (1,000)(1.05)-4n + [.06/.05][1,000][1 − (1.05)-4n] = 1,200 − (200)(1.05)-4n
.25 = [1.05]-2n[1 − (1.05)-2n]
1,250 − (200)(1.05)-2n = 1,200 − (200)(1.05)-4n
-2n
(1.05)
= .5
P = 1,200 − (200)(.5) = 1,100
Answer: B
B9.
-40
-40 = [25][1 − (1.03) ] + (1,100)(.30656)
P = 25a__
40| + (1,100)(1.03)
.03
P = (25)(23.11477) + 337.22 = 915.09
Answer: C
B10.
P = K + 40an_|
Q = K + 30an_|
10an_| = P − Q
X = K + 80an_| = (4Q − 3P) + (8)(P − Q) = 5P − 4Q
Answer: D
B11.
Calculate the value of (1.025)-2n and use to calculate the price of the bond:
K = 4Q − 3P
P = (1,000)(1.025)-2n + (1,000c/2)a__
2n|
-2n
P − 300 = (1,000)(1.025) + [1,000][c − .02)/2]a__
2n|
-4n
[10][1 − (1.025)-2n]
-2n = .25
-4n + [35][1 − (1.025) ]
(1.025)
P'
=
(1,000)(1.025)
.025
.025
2]
[35][1
−
(.25)
= 62.50 + 1,312.50 = 1,375
P' = (1,000)(.25)2 +
.025
300 =
Answer: B
B12.
__
a__
110 = C + (Fr − Ci)a__
2n| = 100 + (100)(.03 − .02)a2n|
2n| = 10
P = C + (Fr − Ci)a__
2n| = 100 + (100)(.015 − .02)(10) = 95
Answer: B
B13.
-2n
-2n
1,271.80 = 60a__
1,407.70 = 70a__
2n| + 1,000v
2n| + 1,000v
a__
1,407.70 − (70)(13.59) = 1,000v-2n
135.90 = 10a__
2n|
2n| = 13.59
v-2n = .4564
i =
1 − v-2n
1 − .4564
= 13.59
= .04
__
a2n|
2i − 1 = (2)(.04) = .07
P =
(1.04)-2n = .4564
.78439
−ln .4564
n = 2 ln 1.04 = (2)(.03922) = 10
[70][1 − (1.035)-(2)(10)]]
+ (1,000)(1.035)-(2)(10) = (70)(14.21240) + (1,000)(.50257) = 1,497.34
.035
Answer: A
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
Financial Instruments  267
B14.
John buys a ten-year 1,000 par value bond with 8% semiannual coupons. The price of the bond to earn a
yield of 6% convertible semiannually is 1,204.15. The redemption value is more than the par value.
Calculate the price John would have to pay for the same bond to yield 10% convertible semiannually.
A. 875
B15.
B. 913
C. 951
D. 989
E. 1,027
(89F–140–13)
On June 1, 1990, an investor buys three fourteen-year bonds, each with a par value 1,000, to yield an
effective annual interest rate of i on each bond. Each bond is redeemable at par. You are given:
i)
ii)
iii)
The first bond is an accumulation bond priced at 195.63.
The second bond has 9.4% semiannual coupons and is priced at 825.72.
The third bond has 10% annual coupons and is priced at P.
Calculate P.
A. 825
B16.
B. 835
C. 845
D. 855
E. 865
(90S–140–14)
A ten-year bond with par value 1,000 and annual coupon rate r is redeemable at 1,100. You are given:
i)
ii)
iii)
The price to yield an effective annual interest rate of 4% is P.
The price to yield an effective annual interest rate of 5% is (P − 81.49).
The price to yield an effective annual interest rate of r is X.
Calculate X.
A. 1,061
B17.
D. 1,071
E. 1,075
(90S–140–15)
B. 14,100
C. 14,600
D. 15,400
E. 17,700
(90F–140–5)
Bart buys a 28-year bond with a par value of 1,200 and annual coupons. The bond is redeemable at par.
Bart pays 1,968 for the bond, assuming an annual effective yield rate of i. The coupon rate on the bond is
twice the yield rate. At the end of 7 years, Bart sells the bond for P, which produces the same annual
effective yield rate of i to the new buyer. Calculate P.
A. 1,470
B19.
C. 1,068
Jim buys a ten-year bond with par value of 10,000 and 8% semiannual coupons. The redemption value of
the bond at the end of ten years is 10,500. Calculate the purchase price to yield 6% convertible quarterly.
A. 11,700
B18.
B. 1,064
B. 1,620
C. 1,680
D. 1,840
E. 1,880
(90F–140–15)
A Treasury bond pays semiannual coupons at 8.8%, has a face value of $1,000, and a yield to maturity of
9.4%. The bond matures in two years. What is the price of the bond?
A. < $980 B. ≥ $980 but < $990
(91–5B–51–1)
© 2010 ACTEX Publications, Inc.
C. ≥ $990 but < $1,000
D. ≥ $1,000 but < $1,010
E. ≥ $1,010
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
268  Financial Instruments
B14.
Calculate the redemption value of the bond and use to calculate the price of the bond to yield 10%.
P − Fr a__
10|
1,204.15 − [40][1 − (1.03)-20]/.03
1,204.15 − (40)(14.87747)
C =
= 1,100
.55368
vn
(1.03)-20
-10
-20 = [40][1 − (1.05) ] + (1,100)(.37689)
+
(1,100)(1.05)
P' = 40a__
10|
.05
P' = (40)(12.46221) + 414.58 = 913
C =
=
Answer: B
B15.
(g)(C − K)
(.047)(1,000 − 195.63)
=
= .06
P−K
825.72 − 195.63
P = K + (g/i)(C − K) = 195.63 + (.10/.1236)(1,000 − 195.63) = 846.41
v14 = P/C = 195.63/1,000 = .19563
(1.06)2 = 1.1236
i =
Answer: C
B16.
-10
-10 = [1,000r][1 − (1.04) ] + (1,100)(.67556)
P = (1,000r)a__
10| + (1,100)(1.04)
.04
P = (1,000r)(8.11090) + 743.12 = 8,110.90r + 743.12
-10
-10 = [1,000r][1 − (1.05) ] + (1,100)(.61391)
+
(1,100)(1.05)
P − 81.49 = (1,000r)a__
10|
.05
P = 81.49 + (1,000r)(7.72173) + 675.30 = 756.79 + 7,721.73r = 8,110.90r + 743.12
r = .03513
-10
-10 = [35.13][1 − (1.03513) ] + 778.83
+
(1,100)(1.03513)
X = (1,000)(.03513)a__
10|
.03513
X = (35.13)(8.31116) + 778.83 = 1,070.80
Answer: D
B17.
(1.015)2 = 1.03023
-20
-20 = [400][1 − (1.03023) ] + (10,500)(.55121)
+
(10,500)(1.03023)
P = 400a__
|
20
.03023
P = (400)(14.84589) + 5,787.71 = 11,726.06
Answer: A
B18.
28 = (2,400)(1 − v28) + 1,200v28
1,968 = (1,200)(2i)a__
28| + 1,200v
21
P = (1,200)(2i)a__
v27 = .46479
21| + 1,200v
v28 = .36
P = (1,200)(2)(1 − .46479) + (1,200)(.46479) = 1,842.25
Answer: D
B19.
Assume the redemption amount equals the face amount.
1.094 − 1 = .04594
[44][1 − (1.04594)-4]
+ (1,000)(.83555)
P = 44a4_| + (1,000)(1.04594)-4 =
.04594
P = (44)(3.57965) + 835.55 = 993.05
Answer: C
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
Financial Instruments  269
B20.
An n-year zero-coupon bond with par value of 1,000 was purchased for 600. An n-year 1,000 par value
bond with semiannual coupons of X was purchased for 850. A 3n-year 1,000 par value bond with
semiannual coupons of X was purchased for P. All three bonds have the same yield rate. Calculate P.
A. 686
B21.
B. 696
C. 706
D. 716
E. 726
(93S–140–16)
You are given:
i)
ii)
iii)
iv)
A 10-year 8% semiannual coupon bond is purchased at a discount of X.
A 10-year 9% semiannual coupon bond is purchased at a premium of Y.
A 10-year 10% semiannual coupon bond is purchased at a premium of 2X.
All bonds were purchased at the same yield rate and have par values of 1,000.
Calculate Y.
A. X/3
B22.
B. 2X/5
B. 1,300
B. 618
C. 1,400
C. 718
(93F–140–14) (Sample1–2–10)
D. 1,500
E. 1,600
(95S–140–15)
D. 818
E. 918
(95S–140–17) (Sample1–2–12)
In July 1995, you purchase a July 1999 U.S. Treasury bond paying interest annually. Given the following,
what is the market value at the time of purchase (as a percentage of the face value)?
Face value
Coupon rate
A. < 90.0%
E. ≥ 102.0%
B25.
E. X
A 30-year bond has 10% annual coupons and a par value of 1,000. Coupons can be reinvested at a
nominal annual rate of 6% convertible semiannually. X is the highest price that an investor can pay for
the bond and obtain an effective yield of at least 10%. Calculate X.
A. 518
B24.
D. 2X/3
Two 1,000 par value bonds are purchased. The 2n-year bond costs 250 more than the n-year bond. Each
has 13% annual coupons and each is purchased to yield 6.5% annual effective. Calculate the price of the
n-year bond.
A. 1,200
B23.
C. X/2
$5,000
8%
B. ≥ 90.0% but < 94.0%
(95F–5B–1–1)
Yield to maturity
Expected inflation rate
C. ≥ 94.0% but < 98.0%
9.5%
6%
D. ≥ 98.0% but < 102.0%
On January 1, 1995, you purchased a bond with a December 31, 1999 maturity. At the time of purchase,
bonds with the same risk were yielding 7%. The purchased bond has a coupon rate of $80 paid annually
and a face value of $1,000. A year later, after you received the first $80 coupon, you sold the bond when
bonds with the same risk had a yield of 6.8%. What was the difference between your selling price and
buying price in nominal terms?
A.
B.
C.
D.
E.
The buying price was at least $25 more than the selling price.
The buying price was at least $15 more than the selling price, but less than $25 more than the
selling price.
The buying price was at least $5 more than the selling price, but less than $15 more than the
selling price.
The buying price was within $5 of the selling price.
The buying price was less than the selling price by at least $5. (98F–5B–1–1)
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
270  Financial Instruments
B20.
vn = P/C = 600/1,000 = .6
K' = 1,000v3n = (1,000)(.6)3 = 216
850 − 600
P−K
=
= .625
g/i =
1,000 − 600
C−K
P = K' + (g/i)(C − K') = 216 + (.625)(1,000 − 216) = 706
Answer: C
B21.
1,000 + 2X = K + 50a__
10a__
1,000 − X = K + 40a__
20|
20|
20| = 3X
K = (2,000 + X − 90a__
20| )/2 = 1,000 + X/2 − (45)(3/10)X = 1,000 − 13X
Y = K + 45a__
20| − 1,000 = (1,000 − 13X) + (45)(3X/10) − 1.000 = .5X
a__
20| = 3X/10
Answer: C
B22.
-2n
130an_| + (1,000)(1.065)-n + 250 = 130a__
2n| + (1,000)(1.065)
-2n
[130][1 − (1.065)-n]
-n + 250 = [130][1 − (1.065) ] + (1,000)(1.065)-2n
+
(1,000)(1.065)
.065
.065
-2n
-n
− (1,000)(1.065) + 250 = 0
(1,000)(1.065)
−(−1,000) ± (−1,000)2 − (4)(1,000)(250)
(1.065)-n =
= .5
(2)(1,000)
(130)(1 − .5)
+ (1,000)(.5) = 1,500
P =
.065
P + 250 = P'
Answer: D
B23.
Equate the AV value of the investment at the specified yield and that of its components and solve for P:
[100][(1.0609)30 − 1]
+
1,000
=
P(1.10)30 = 17.44940P = 100s__
+ 1,000
30|
.0609
17.44940P = (100)(80.32189) + 1,000 = 9,032.19
P = 517.62
(1.03)2 = 1.0609
Answer: A
B24.
[400][1 − (1.095)-4]
P = 400a4_| + (5,000)(1.095)-4 =
+ (5,000)(.69557)
.095
P = (400)(3.20448) + 3,477.85 = 4,759.64
P/F = 4,758.64/5,000 = 95.2%
Answer: C
B25.
[80][1 − (1.07)-5]
+ (1,000)(.71299)
P = 80a5_| + (1,000)(1.07)-5 =
.07
P = (80)(4.10020) + 712.99 = 1,041.01
[80][1 − (1.068)-4]
P' = 80a4_| + (1,000)(1.068)-4 =
+ (1,000)(.76863) = 1,040.83
.068
P' = (80)(3.40256) + 768.63 = 1,040.83
P − P' = 1,041.01 − 1,040.83 = .18
Answer: D
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
Financial Instruments  271
B26. For an asset, you are given:
i)
ii)
iii)
The value of the asset at the beginning of year 1 is 1,000.
The asset returns 10% each year for five years.
The asset pays a dividend of 50 at the end of each year for five years.
Your opportunity cost of capital is 12%. Calculate the present value of the asset at the beginning of year 1.
A. 909
B. 921
C. 930
D. 939
E. 951
(Sample1–2–47)
B27. Consider the following bonds:
Bond A: a ten-year bond with 5% annual coupons and a face value of 1000
Bond B: a five-year bond with 10% annual coupons and a face value of 700
Bond C: a ten-year bond with 12% annual coupons and a face value of 600
Assuming a discount rate of 10%, rank the following:
I. Price of bond A
A. I > II > III
II. Price of bond B
B. I > III > II
C. II > I > III
III. Price of bond C
D. III > I > II
E. III > II > I
(Sample1–2–49)
B28. A firm has proposed the following restructuring for one of its 1,000 par value bonds. The bond presently
has ten years remaining until maturity. The coupon rate on the existing bond is 6.75% per annum paid
semiannually. The current nominal semiannual yield on the bond is 7.40%. The company proposes
suspending coupon payments for four years with the suspended coupon payments being repaid, with
accrued interest, when the bond comes due. Accrued interest is calculated using a nominal semiannual rate
of 7.40%. Calculate the market value of the restructured bond.
A. 755
B. 805
C. 855
D. 905
E. 955
(00S–2–29)
B29. You are given the following information about a bond:
i)
ii)
iii)
iv)
The term-to-maturity is two years.
The bond has a 9% annual coupon rate, paid semiannually.
The annual bond-equivalent yield-to-maturity is 8%.
The par value is $100.
Calculate the current price of the bond. (00–8–14a–.5)
B30. A 1,000 par value twenty-year bond with annual coupons and redeemable at maturity at 1,050 is purchased
for P to yield an annual effective rate of 8.25%. The first coupon is 75. Each subsequent coupon is 3%
greater than the preceding coupon. Determine P.
A. 985
B. 1,000
C. 1,050
© 2010 ACTEX Publications, Inc.
D. 1,075
E. 1,115
(00F–2–30)
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
272  Financial Instruments
B26.
VA1 = (1,000)(1.1) − 50 = 1,050
VA2 = (1,050)(1.1) − 50 = 1,105
VA3 = (1,105)(1.1) − 50 = 1,165.50
VA4 = (1,165.5)(1.1) − 50 = 1,232.05
VA5 = (1,232.05)(1.1) − 50 = 1,305.26
[50][1 − (1.12)-5]
PV = 50a5|_ + (1,305.26)(1.12)-5 =
+ (1,305.26)(.56743)
.12
PV = (50)(3.60478) + 740.64 = 921
Answer: B
B27.
-10
-10 = [50][1 − (1.10) ] + (1,000)(.38554)
PA = 50a__
10| + (1,000)(1.10)
.10
PA = (50)(6.14457) + 385.54 = 692.77
[70][1 − (1.10)-5]
+ (700)(.62092)
PB = 70a5_| + (700)(1.10)-5 =
.10
PB = (70)(3.79079) + 434.64 = 700.00
-10
-10 = [120][1 − (1.10) ] + (600)(.38554)
+
(600)(1.10)
PC = 120a__
10|
.10
PC = (72)(6.14457) + 231.32 = 673.73
Answer: C
B28.
Since interest is accrued at the yield rate, there is no effect on the bond's market value.
.0675/2 = .03375
.074/2 = .037
-20
-20 = [33.75][1 − (1.037) ] + (1,000)(.48353)
P = 33.75a__
20| + (1,000)(1.037)
.037
P = (33.75)(13.95861) + 483.53 = 955
Answer: E
[4.5][1 − (1.04)-4]
+ (100)(.85480)
.04
P = (4.5)(3.62990) + 85.48 = 101.81
B29.
_ + (100)(1.04)-4 =
P = 4.5a4|
B30.
1.0825/1.03 = 1.050971
-20
-20 = [72.81553][1 − (1.050971) ] + (1,050)(.20485)
P = (1.03)-175a__
20| + (1,050)(1.0825)
.050971
P = (72.81553)(12.36024) + 215.09 = 1,115
Answer: E
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
Financial Instruments  273
B31.
You have decided to invest in two bonds. Bond X is an n-year bond with semiannual coupons, while bond
Y is an accumulation bond redeemable in n/2 years. The desired yield rate is the same for both bonds.
You also have the following information:
i)
ii)
iii)
iv)
v)
Par value of bond X is 1,000.
For bond X, the ratio of the semiannual bond rate to the desired semiannual yield rate, (r/i), is
1.03125.
The present value of the redemption value of bond X is 381.50.
Redemption value of bond Y is the same as the redemption value of bond X.
Price to yield of bond Y is 647.80.
What is the price of bond X?
A. 1,019
B32.
B. 1,029
C. 1,050
D. 1,055
E. 1,072
(01F–2–31)
You have decided to invest in bond X, an n-year bond with semiannual coupons, with the following
characteristics:
i)
ii)
iii)
Par value of bond is 1,000.
For bond X, the ratio of the semiannual bond rate to the desired semiannual yield rate, (r/i), is
1.03125.
The present value of the redemption value X is 381.50.
Given vn = .5889, what is the price of bond X?
A. 1,019
B33.
B. 1,029
C. 1,050
D. 1,055
E. 1,072
(Sample–FM–22)
Susan can buy a zero-coupon bond that will pay 1,000 at the end of twelve years and is currently selling
for 624.60. Instead she purchases a 6% bond with coupons payable semiannually that will pay 1,000 at
the end of ten years. If she pays X, she will earn the same annual effective rate as the zero-coupon bond.
Calculate X.
A. 1,164
B. 1,167
© 2010 ACTEX Publications, Inc.
C. 1,170
D. 1,173
E. 1,176
(05S–FM–5)
SOA Exam FM and CAS Exam 2 – Peter J. Murdza
274  Financial Instruments
B31.
Use the PV of the two redemption values to calculate n and the redemption value. Then use the base
amount to calculate the price of bond X:
KX = 381.50 = Cvn
KY = 647.80 = Cvn/2
C = 1,100
vn = .34682
GX = F(r/i) = (1,000)(1.03125) = 1,031.25
PX = G + (C − G)vn = 1,031.25 + (1,100 − 1,031.25)(.34682) = 1,055.09
Answer: D
B32.
See B31.
Answer: D
B33.
i = (1,000/624.60)1/12 − 1 = .04
i' = (1.04).5 = 1.01980
-20
-20 = [30][1 − (1.0198) ] + (1,000)(.67556)
P = 60a__
10| + (1,000)(1.0198)
.01980
P = (30)(16.38304) + 675.62 = 1,167.11
Answer: B
© 2010 ACTEX Publications, Inc.
SOA Exam FM and CAS Exam 2 – Peter J. Murdza