This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 132 C H A P T E R 3 Mathematics of Finance Now using formula (2) with A = 2,539.62,P = 2,443.02, and t = £02 2 we
h 360 9
ave
A = P(1 + rt)
2,539.62 = 2,443.02(1 + gr)
= 2,443.02 + 1,357.23r
96.60 = 1,357.23r
_ 96.60 N o '* r — 1,357.23 ~ 0.07117 or 7.117/0 v, I :2;
Repeat Example 5 if 500 shares of stock were purchased for $17.64 per share and sold 270 days later for $22.36 per share. ‘ (é? lNSlGHT The commission schedule in Table 1 specifies a piecewise deﬁned function C with inde
pendent variable p, the principal (See Section 22). Thus, 29 + 0.016;; ifo s p < 2,500
C = 49 + 0.008p if2,500 s p < 10,000
99 + 0.003;: if 10,000 5 p Explore 8! Dis C USS 2 (A) Starting with formula (2), derive each of the following formulas: A r __ A  P t _ A  P
1 + rt Pt Pr
(B) Explain why it is unnecessary to memorize the formulas above for P, r, and t if you know formula (2). . P: Answers to Matched Problems 1. $650 2. $4,761.90 3. 3.485%. 4. 15.0% 5. 31.43994:
’ E x e rc i s e 3  1 A In Problems 1—8, make the indicated conversions assuminga 12 I = $15;r = 8703‘ = 3 quarters;P = ? 1
360day year 13. I = $48; P = $600;t = 240 days;r = ? 1
1. 7.25% = ? (decimal); 13 weeks = ?(year) 14. I = $28. P = $7001 = 13 weeks. , = ? 2. 6.15% = ? (decimal); 8months = ?(year) 15_ , = $60.}, 2 $2 400. , = 5%., = 9 l
 ’ I 7 ' 3. 0.21 = ?(percentage); 2quarters —— ? (year) 16. I = $96; P = $3,200; r = 4%; t = 7 t
4. 0.035 = 7 (percentage); 48 weeks = 7 (year) 1
5_ 042% = 7 (decimal); 120 days = ? (year) ' B In Problems 1724, use formula (2) for the amount to ﬁnd each of the indicated quantities r
17. P = $4,500;r = 10%;: = 1quarter;A = ? l
18. P = $3,000;r = 45%;: = 30days;A = 7 19. A = $910;r = 16%;t = 13 weeks;P = ‘l 20. A = $6,608;r = 24%;t = 3quarters;P = ? 6. 0.87% = ? (decimal); 3quarters = ? (year)
7. 1.1 = ? (percentage); 10 months = ?(year)
8. 2.09 = ? (percentage); 60days = ? (year) In Problems 9—16, use formula (1) for simple interest to ﬁnd each ofthe indicated quantities 21. A = $14,560;P = $110003 = 4months; , = 7
9. P = $300;r = 7%;t = 2years;I = ? 22. A = $22,135;P = $19,000;t = 39weeks;r = ?
10. P=$950;r=9%;t=1year;1=? ‘ 23. A=$736;P=$640;r=15%;t=? 11. I=$36gr=4%;t=6months;P=? 24. A=$410;P=$400;'r=10%; =‘? c [,1 Problems 25—30, solve each formula for the indicated 32.
variable. ‘
_ 25. I = Prt‘,forr 26. 1 = Prt;forP
27. A = P + Prt;forP ’ 28. A = P + Prt;forr
29. A = P(1 + rt);f0rt 30. I = Prt;f01't
31. Discuss the similarities and differences in the graphs of In all problems involving days, a 360day year is assumed. Applications" future value A as a function of time I if $1,000 is invested at simple interest at rates of 4%, 8%, and 12%, respec~ ‘
tively (see the figure). A = 1.000(1 +0120 A = 1.000(l + 0.08!)
A = l,000(l + 0.04!) Time (years) Figure for 31 41. When annual rates are requested as an answer, express the rate as a percentage, correct to three decimal places. 33. 34. 35. 37. 39. 40. "‘ The authors wish to thank Professor Roy Luke of Pierce College and
Professor Dennis Pence of Western Michigan University for their many
Useful suggestions of applications for this chapter. , 42.
If $3,000 is loaned for 4 months at a 4.5% annual rate, how f much interest is earned? 43.
If $5,000 is loaned for 9 months at a 6.2% annual rate, how 1 much interest is earned? How much interest will you have to pay for a credit card balance of $554 that is 1 month overdue, if a 20% annual : 44_
rate is charged? A department store charges an 18% annual rate for over due accounts. How much interest will be owed on an $835 account that is 2 months overdue? A loan of $7,260 was repaid at the end of 8 months What size repayment check (principal and interest) was written, ?
if an 8% annual rate of interest was charged? 4 A loan of $10,000 was repaid at the end of 6 months What
amount (principal and interest) was repaid, if a 6.5% an
nual rate of interest was charged? A loan of $4,000 was repaid at the end of 10 months with
a check for $4,270. What annual rate of interest was
charged? F A check for $3,097.50 was used to retire a 5month $3,000
loan. What annual rate of interest was charged? 49. ‘3 e c t i o n 3 ~ 1 Simple Interest 133 Discuss the similarities and differences in the graphs of
future value A as a function of time t for loans of $400, $800, and $1,200, respectively, each at 7.5% simple inter—
est (see the figure). 2 3 4 5 6 7 8
Time(years) Figure for 32 If you paid $30 to a loan company for the use of $1,000 for
60 days, what annual rate of interest did they charge? If you paid $120 to a loan company for the use of $2,000 for
90 days, what annual rate of interest did they charge? A radio commercial for a loan company states: “You only
pay 29¢ a day for each $500 borrowed." If you borrow
$1,500 for 120 days, what amount will you repay. and what
annual interest rate is the company actually charging? l George ﬁnds a company that charges 59¢ per day for each
$1,000 borrowed. If he borrows $3,000 for 60 days, what
amount will he repay, and what annual interest rate will
he be paying the company? i What annual interest rate is earned by a 13—week Tbill 1
with a maturity value of $1,000 that sells for $989.37? i i What annual interest rate is earned by a 33day T—bill with
a maturity value of $1,000 that sells for $996.16? What is the purchase price of a 50day Tbill with a matu—
rity value of $1,000 that earns an annual interest rate of g
5.53%? 7‘ What is the purchase price of a 26week Tbill with a ma turity value of $1,000 that earns an annual interest rate of
4.903 % ? For services rendered, an attorney accepts a 90—day note
for $5,500 at 8% simple interest from a client. (Both in
terest and principal will be repaid at the end of 90 days.) 5:
Wishing to be able to use her money sooner, the attorney ‘5,
sells the note to a third party for $5,560 after 30 days. What
annual interest rate will the third party receive for the
investment? it. 134 C H A P T E R 3 Mathematics of Finance 50. To complete the sale of a house, the seller accepts a ISOday ; 53.
note for $10,000 at 7% simple interest. (Both interest and '
principal will be repaid at the end of 180 days.) Wishing to be able to use the money sooner for the purchase of an— 5
other house, the seller sells the note to a third party for E An investor purchases 215 shares at $45.75 a share. holr
the stock for 300 days, and then sells the stock for $51.!
a share. 54. An investor purchases 75 shares at $37.90 a share, hoh $10,124 after 60 days. What annual interest rate will the me Stock for 150 days’ and me“ sens the “Mk for $4": third party receive for the investment? a share. Many tax preparation firms offer their clients a refund an
» ticipation loan (RA L). For a fee, the firm will give a client
2 his refund when the return is filed. The loan is repaid when
the Internal Revenue Service sends the refund directly to th
firm. Thus, the RAL fee 'is equivalent to the interest charge
for a loan. The schedule in Table 4 is from a major RAL
lender. Use this schedule to ﬁnd the annual rate of interest
for the RA Ls in Problems 55—58. TABLE 4 RAL Fee i RAL Amount 1
Use the commission schedule from Company A shown in $0_$500 $2900 ‘
Table 2 to find the annual rate of interest earned by each §
investment in Problems 51 and 52. $501‘m’000 $3900 stem—$1,500 $49.00 $1,501$2,000 $69.00
TABLE 2 COH‘DCH'W A Principal Commission ‘
. + .8 ' ' 1 i . . . . .
gnﬁiim + 14::0ffprincipal 55. A client receives a $475 RAL, which is paid back i
’ ‘ ‘ 0 Purim?“ 20 days. What is the annual rate of interest for tl
Over $10,000 $107 + 0.7% of pnnctpal loan? 51. An investor purchases 200 shares 'at $14.20 a share, holds
the stock for 39 weeks, and then sells the stock for $15.75
a share. 52. An investor purchases 450 shares at $21.40 a share, holds
the stock for 26 weeks, and then sells the stock for $24.60
a share. Use the commission schedule from Company B shown in
Table 3 to ﬁnd the annual rate of interest earned by each invest
ment in Problems 53 and 54. TABLE 3 Company B 56. 57. A client receives a $1,100 RAL, which is paid back in
days What is the annual rate of interest for this loan? A client receives a $1,900 RAL, which is paid ba
in 15 days. What is the annual rate of interest for I]
loan? A client receives a $3,000 RAL, which is paid back in
days What is the annual rate of interest for this loan? Write a piecewise deﬁnition of the commission C as a fur tion of the principal p for the schedule in Table 2. Gra
CforO S p 5 $15,000. Write a piecewise deﬁnition of the commission C as a fur
tion of the principal p for the schedule in Table 3. Gra
C for 0 S p S. 315,000. Principal Commission Under $3,000 332 + 13% of principal 61. Refer to Problems 59 and 60. From the investor’s viewpoi
$3 0004“) 000 $56 + 1% of principal when would it be better to place an order through Compa
0",” $10 $106 + 0 5% of principal A and when would it be better to place the order throu Company B? Section 32 COMPOUND AND CONTINUOUS COMPOUND INTEREST I Compound Interest
l Continuous Compound Interest
l Growth and Time
I Annual Percentage Yield 146 C H A P T E R 3 Mathematics of Finance Find all dollar amounts to the nearest cent. When an interest rate is
requested as an answer, express the rate as apercentage correct [0
two decimal places, unless directed otherwise. In all problems in—
volving days, use a 365day year. A In Problems 1—8, use compound interest formula (1) to ﬁnd
each of the indicated values. 1. the?!» up 8. P = $100;i = 0.01;n =12;A =? P = $1,000;i = 0.015;n = 20;A = 7 P = $800;i = 0.06m = 25;A = ? P = $10,000;i = 0.08m = 30,11 = 7 A = $10,000;i = 0.03m = 48;P = ‘7 A = $1,000;i = 0.015;n = 60;P = ? A = $18,000;i = 0.01m = 90;P = ? A = $50,000;i = 0.005;n = 70;P = ? In Problems 9—16, use the continuous compound interest
formula (2) to ﬁnd each of the indicated values. 9.
10.
11.
12.
13.
14.
15.
16. P = $2,450;r = 8.12%;t = 3years;A = ? P = $995;r = 22%;t = 2years;A = ? A = $6,300;r = 9.45%;t = 8years;P = ? A = $19,000;r = 7.69%;t = Syears;P = ? A = $88,000;P = $71,153;r = 8.5%;t = ? A = $32,982;P = $27,200;r = 5.93%;t = ? A = $15,875;P = $12,100;t = 48months;r = ?
A = $23,600;P = $19,150;t = 60months;r = ? Given the annual rate and the compounding period in Prob
lems 1 7—24, ﬁnd i, the interest rate per compounding period. 17.
18.
19.
20.
21.
22.
23.
24. 15% compounded monthly
7% compounded semiannually
6% compounded quarterly
10.95% compounded daily
9% compounded semiannually
21% compounded monthly 7.3% compounded daily
5% compounded quarterly Given the rate per compounding period in Problems 25—32,
ﬁnd r, the annual rate. 25.
27.
29.
31. 33. 2.5% per halfyear 26. 1.4% per month 2.2% per quarter 28. 6.75% per year
0.018% per day 30. 9.65% per halfyear
1.5% per month 32. 3.25% per quarter
If $100 is invested at 6% compounded (A) annually (B) quarterly (C) monthly what is the amount after 4 years? How much interest is
earned? 35. 37. 38. 39. A
12:
25900 A =12,000(1+ 0137—5) r
22,500 12
20,000 ,
a 17,500 0.075 '2‘
3 15,000 A: 8,000(1+ ————12 )
5 12.500
0 10,000 m
g 7,500 A =4,000(1 + u. 12
5,000
2500
o t If $2,000 is invested at 7% compounded (C) monthly what is the amount after 5 years? How much interest is
earned? (A) annually (B) quarterly If $5,000 is invested at 5% compounded monthly, what is _ ’
the amount after ' (A) 2 years? (B) 4 years? If $20,000 is invested at 4% compounded monthly, what
is the amount after (A) 5 years? (B) 8 years? If $8,000 is invested at 7% compounded continuously,
what is the amount after 6 years? If $23,000 is invested at 13.5% compounded continu
ously, what is the amount after 15 years? Discuss the similarities and the differences in the graphs
of future value A as a function of time t it $1,000 is in
vested for 8 years and interest is compounded monthly at annual rates of 4%, 8%, and 12%, respectively (see
the ﬁgure). II
III
III
III
ﬁll I
I
I
I
I
I
I
I
'4 “HIIIII IIEIII HNIHII
!MIIIII I
I
I
I
E
I
I
. Time (years) Figure for 39 Discuss the similarities and differences in the graphs of
future value A as a function of time tfor loans of $4,000, “
$8,000, and $12,000, respectively, each at 7.5% com
pounded monthly for 8 years (see the ﬁgure). 2 3 4 5 6 7 8
Time(years) Figure for 40 :
.
.
i
;
E
i 41. 42. 43. 45. 47. 49. Applications 61. 62. 63. 65. ‘ : r 1 o n 3  2 Compound and Continuous Compound Interest lf$1,000 is invested in an account that earns 9.75% com
pounded annually for 6 years, find the interest earned
during each year and the amount in the account at the
end of each year. Organize your results in a table. If $2,000 is invested in an account that earns 8.25% com I pounded annually for 5 years. find the interest earned
during each year and the amount in the account at the
end of each year. Organize your results in a table. If an investment company pays 6% compounded semian nually, how much should you deposit now to have $10,000
(A) 5 years from now? (B) 10 years from now? If an investment company pays 8% compounded quar terly. how much should you deposit now to have $6,000
(A) 3 years from now? (B) 6 years from now? If an investment earns 9% compounded continuously, how much should you deposit now to have $25,000
(A) 36 months from now? (B) 9 years from now? If an investment earns 12% compounded continuously, how much should you deposit now to have $4,800
(A) 48 months from now? (B) 7 years from now? What is the annual percentage yield (APY) for money
invested at (A) 4.5% compounded monthly?
(B) 5.8% compounded quarterly? What is the APY for money invested at (A) 6.2% compounded semiannually?
(B) 711% compounded monthly? What is the APY for money invested at (A) 10% compounded semiannually?
(B) 9% compounded continuously? A newborn child receives a $20,000 gift toward a college education from her grandparents. How much will the ' $20,000 be worth in 17 years if it is invested at 7% com
pounded quarterly? A person with $14,000 is trying to decide whether to pur i
chase a car now, or to invest the money at 6.5% com I pounded semiannually and then buy a more expensive car. How much will be available for the purchase of a car at the end of 3 years? What will a $210,000 house cost 10 years from now if the in— ﬂation rate over that period averages 3% compounded an nually? If the inﬂation rate averages 4% per year compounded an— nually for the next 5 years, what will a car costing $17,000 ; now cost 5 years from now? Rental costs for office space have been going up at 4.8% per year compounded annually for the past 5 years If office i
space rent is now $25 per square foot per month, what were the rental rates 5 years ago? 50. 51. 52. 53. 54. 147 What is the APY for money invested at
(A) 18.75% compounded daily?
(B) 15.25% compounded continuously? How long will it take $4,000 to grow to $9,000 if it is
invested at 7% compounded monthly? How long will it take $5,000 to grow to $7,000 if it is
invested at 6% compounded quarterly? How long will it take $6,000 to grow to $8,600 if it is
inVested at 9.6% compounded continuously? How long will it take $42,000 to grow to $60,276 if it is
invested at 4.25% compounded continuously? C In Problems 55 and 56, use the compound interest formula (1)
to ﬁnd n to the nearest larger integer value. 55. 56.
57. 58. 59. 66. 67. 68. 69. 70. 71. A = 2P;i = 0.06;n = ?
A = 2P;i = 0.05;n =?
How long will it take money to double if it is invested at (A) 10% compounded quarterly?
(B) 12% compounded quarterly? How long will it take money to double if it is invested at (A) 8% compounded semiannually?
(B) 7% compounded semiannually? How long will it take money to double if it is invested at (A) 9% compounded continuously?
(B) 11% compounded continuously? How long will it take money to double if it is invested at (A) 21% compounded continuously?
(B) 33% compounded continuously? In a suburb of a city, housing costs have been increasing at
5.2% per year compounded annually for the past 8 yearsA
house with a $260,000 value now would have had what value
8 years ago? If the population in a particular country is growing at 1.7%
compounded continuously, how long will it take the popu— lation to double? (Round up to the nexthigher year if not
exact.) If the world population is now about 6.5 billion people and
is growing at 1.14% compounded continuously, how long
will it take the population to grow to 10 billion people?
(Round up to the nexthigher year if not exact.) Which is the better investment and why: 9% compounded
monthly or 9.3% compounded annually? Which is the better investment and why: 8% compounded
quarterly or 8.3% compounded annually? (A) If an investment of $100 were made in the year the
Declaration of Independence was signed, and if it earned?) % compounded quarterly, how much would it
be worth in 2016? 148 C H A PT E R 3 Mathematics of Finance 0 72. 73.
74. 75. 76. 78. (B) Discuss the effect of compounding interest monthly, daily, and continuously (rather than quarterly) on the ;
$100 investment. 5 (C) Use a graphing calculator to graph the growth of the .
investment of part (A). (A) Starting with formula (1), derive each of the following formulas:
P_ A i_(£)”"_1n_lnA—lnP
(1 + i)”’ P ’ ln(l + i) (B) Explain why it is unnecessary to memorize the formu
las above for P, i, and n if you know formula (1). A promissory note will pay $50,000 at maturity 6 years from
now. If you pay $28,000 for the note now, what rate com
pounded continuously would you earn? If you deposit $10,000 in a savings account now, what rate
compounded continuously would be required for you to
withdraw $12,500 at the end of 4 years? You have saved $7 ,000 toward the purchase of a car costing
$9,000. How long will the $7,000 have to be invested at 9%
compounded monthly to grow to $9,000? (Round up to the
nexthigher month if not exact.) A newly married couple has $15,000 toward the purchase of
a house. For the type of house the couple is interested in
buying, an estimated down payment of $20,000 will be nec
essary. How long will the money have to be invested at 7%
compounded quarterly to grow to $20,000? (Round up to
the nexthigher quarter if not exact.) An Individual Retirement Account (IRA) has $20,000 in it, and the owner decides not to add any more money to the
account other than interest earned at 6% compounded
daily. How much will be in the account 35 years from now
when the owner reaches retirement age? If $1 had been placed in a bank account in the year 1066
and forgotten until now, how much would be in the aceount
at the end of 2016 if the money earned 2% interest com
pounded annually? 2% simple interest? (Now you can see
the power of compounding and see why inactive accounts
are closed after a relatively short period of time.) How long will it take money to double if it is invested at
7% compounded daily? 8.2% compounded continuously? How long will it take money to triple if it is invested at 5%
compounded daily? 6% compounded continuously? In a conversation with a friend, you mention that you have
two real estate investments, one that has doubled in value
in the past 9 years and another that has doubled in value in
the past 12 years. Your friend replies immediately that the
first investment has been growing at approximately 8%
compounded annually and the second at 6% compounded
annually. How did your friend make these estimates? The
rule of 72 states that the annual compound rate of growth
r of an investment that doubles in n years can be approxi
mated by r = 72/n. Construct a table comparing the exact . rate of growth and the approximate rate provided by the 82. rule of 72 for doubling times of n = 6,7,... ,12 years.
Round both rates to one decimal place. Refer to Problem 81. Show that the exact annual compound
rate of growth of an investment that doubles in n years is l % Solve Problems 8386 using graphical approximation tech given by r = 100(2”"  1). Graph this equation and the,"
rule of 72 on a graphing calculator for 5 s n s 20. " mques on a graphing calculator. 83. How long does it take for a $2,400 investment at 13% com " pounded quarterly to be worth more than a $3,000 invest
ment at 6% compounded quarterly? 84. How long does it take for a $4,800 investment at 8% com
pounded monthly to be worth more than a $5,000 invest r
ment at 5% compounded monthly? 85. One investment pays 10% simple interest and another pays ;
7% compounded annually. Which investment would you!
choose? Why? 86. One investment pays 9% simple interest and another pays ;
6% compounded monthly. Which investment would you. ‘L
choose? Why? 87. What is the annual nominal rate compounded daily for a.,
bond that has an annual percentage yield of 6.8%? 88. What is the annual nominal rate compounded monthly for‘
a CD that has an annual percentage yield of 5.9%? “ 89. What annual nominal rate compounded monthly has th 90. What annual nominal rate compounded continuously h
the same annual percentage yield as 6% compounde
monthly? Problems 91 —94 refer to zero coupon bonds. A zero coupon
bond is a bond that is sold now at a discount and will pay its
face value at some time in the future when it matures—no
interest payments are made. 91. A zero coupon bond with a face value of $30,000 matures?
in 15 ypars. What should the bond be sold for now if its rate}
of return is to be 4.348% compounded annually? .. 3...... 92. A zero coupon bond with a face value of $20,000 mature
in 10 years. What should the bond be sold for now if its rate.
of return is to be 4.194% compounded annually? 93. If you pay $4,126 for a 20year zero coupon bond with a'
face value of $10,000, what is your annual compound rat
of return? ' 94. If you pay $32,000 for a 5year zero coupon bond with a5
face value of $40,000, what is your annual compound rate ’
of return? 95. An online ﬁnancial service recently listed the following
money market accounts: 5 (A) Republic Bank: 4.31% compounded continuously
(B) Chase Bank: 4.35% compounded daily (C) BankFirst: 4.36% compounded monthly What is the annual percentage yield of each? 96. An online ﬁnancial service recently listed the following
1year CD accounts: ' (A) Banking for CDs: 4.5% compounded quarterly
(B) Wingspan Bank: 4.6% compounded monthly
(C) Discover Bank: 4.6% compounded continuously
What is the annual percentage yield of each? S e ct i o n 3 « 3 Future Value of an Annuity; Sinking Funds 149 The buying and selling commission schedule shown below is from 97. An investor purchases 100 shares of stock at $65 per share,
” an online discount brokerage ﬁrm. Taking into consideration the holds the stock for 5 years, and then sells the stock for $125
' buying and selling commissions in this schedule, ﬁnd the annual a share.
compound rate of interest earned by each investment in Problems 98. An investor purchases 300 shares of stock at $95 per share,
holds the stock for 3 years, and then sells the stock for $156 a share.
Thusaction Size Commission Rate ‘ , r _ . ‘ $29 + 2.5%” ofprincipal , ,
501.$5.099_ ‘ R a $57 + 0.6%inprinciparZ '6 ‘
..$6m1—$22.0m_ ‘ v $75 + 0.30% otpriltci‘ ‘ ‘32,},m1—550000; , ' sen + 0.20% amine“; , ‘ ,
:‘ssohmssoomo , , $147 + 0.10% ofpn‘ncipal ;. , ' .,
'5500,001+ , _ _ ‘ ‘ $247 + 0.08%;0f panelist, #1 99. An investor purchases 200 shares of stock at $28 per share,
holds the stock for 4 years, and then sells the stock for $55
a share. 100. An investor purchases 400 shares of stock at $48 per share,
holds the stock for 6 years, and then sells the stock for $147
a share. Section 33 FUTURE VALUE OF AN ANNUITY: SINKING FUNDS I Future Value of an Annuity
I Sinking Funds
% I Approximating Interest Rates I Future Value of an Annuity An annuity is any sequence of equal periodic payments. If payments are made at the
end of each time interval, then the annuity is called an ordinary annuity. We consider
only ordinary annuities in this book. The amount, or future value, of an annuity is
the sum of all payments plus all interest earned. Suppose you decide to deposit $100 every 6 months into an account that pays 6%
compounded semiannually. If you make six deposits, one at the end of each interest
payment period, over 3 years, how much money will be in the account after the last
deposit is made? To solve this problem, let us look at it in terms of a time line. Using
the compound amount formula A = P(1 + i)", we can find the value of each de
posit after it has earned compound interest up through the sixth deposit, as shown in
Figure 1. l 1—. $1000.03)
$1000 .03)2 $l00(l.03)3 Future value
$100( I .03)‘
$1000 .03)5 FIGURE 1 We could, of course, evaluate each of the future values in Figure 1 using a calcu
lator and then add the results to ﬁnd the amount in the account at the time of the sixth
deposite tedious project at best. Instead, we take another approach, which leads
directly to a formula that will produce the same result in a few steps (even when the number of deposits is very large).We start by writing the total amount in the account
after the sixth deposit in the form 156 C H A PT E R 3 Mathematics of Finance fullfeatured financial equation solver suitable for this text and higherlevel courses in
ﬁnance. Figure 5 shows a solution to Example 4 on this ﬁnancial solver. Note that there
are some differences in the variables in Figure 4C and Figure 5. If you own a “TI84 Plus
and are planning to take additional courses in finance, it will be well worth your effort to
learn to use this financial solver. iamer 7P 93513 
PMT“: “5235131”: . FIGURE 5 Financial equation solver on
aTI84 Plus graphing calculator Answers to Matched Problems 1. Value: $29,778.08; interest: $9,778.08
2. (A) $95,094.67 (B) $248,628.89
3. $2,322.73 4. 9.64% ' A In Problems [—8, ﬁnd i (the rate per period) and n (the number 9 ﬂ = 20;i = 003; PMT = $500; F V = ?
ofperiods) for each annuity. 10. n = 25; i = 0.04; PMT = $100; FV = ?
1. Quarterly deposits of $500 are made for 20 years into an 11. n = 401 = 002. PMT = $1000. FV = ? annuity that pays 8% compounded quarterly.
_ , 12. n = 30;i = 0.01;PMT = $50;FV = ‘?
2. Monthly deposrts of $350 are made for 6 years into an annuity that pays 6% compounded monthly.  B
3. Semiannualéle‘posits 701; 3900 are mageéor 12 years lilnto 13. FV = $3,000;n = 20;i = 0.02;PMT = ?
a . .
4. :lanmln‘?’ tp 2 so; comPOZH: :Zm‘annf’at 3’ 14. FV<= $8,000;n = 30;i = 0.03; PMT = ?
nnua eposr so , are ma e or years in can _ _ __ . , _ ‘ _
annuity that pays 6.25% compounded annually. 15' FV "' ss’ooo’" " 15" _ 0'01’ PMT _ ?
5. Monthly deposits of $235 are made for4years into an 16° FV = 32500;" = 10;l = 0.08; PMT = ? annuity that pays 9% compounded monthly.
6. Semiannual deposits of $1,900 are made for 7 years into C an annuity that pays 8.5% compounded semiannually. 17‘ FV = $4’000"i = 0'02; P M T = 200; n = ?
7. Annual deposits of $3,100 are made for 12 years into an 18' FV = $8900“ 2 0'04; PMT = 500’ " = 7 annuity that pays 5.95% compounded annually. % 19. F V = $7,600; PMT = $500;n = 10;i = ?
8. Quarterly deposits of $1,200 are made for 18 years into (Round answer to two deCImal Places) an annuity that pays 7.6% compounded quarterly. % 20. F V = $4,100, PMT = $100; n = 20;i = ’2 (Round answer to two decimal places.)
In Problems 9—20, use future value formula (6) to find each
of the indicated values ' Applications 21. Recently, Guaranty Income Life offered an annuity that 22. Recently, USG Annuity and Life offered an annuity that .
pays 6.65% compounded monthly. If $500 is deposited into pays 7.25% compounded monthly. If $1,000 is deposited s *
this annuity every month, how much is in the account after into this annuity every month, how much is in the account ‘75
10 years? How much of this is interest? after 15 years? How much of this is interest? 1.. rt, 23. 25. 26. 27. 29. 30. 31. 32. in order to accumulate enough money for a down payment
on a house. a couple deposits $300 per month into an ac
count paying 6% compounded monthly. If payments are
made at the end of each period,how much money will be in
the account in 5 years? A self—employed person has a Keogh retirement plan. (This
type of plan is free of taxes until money is withdrawn.) If de
posits of $7.500 are made each year into an account paying
8% compounded annually. how much will be in the account
after 20 years? Sun America recently offered an annuity that pays 6.35%
compounded monthly. What equal monthly deposit should be
made into this annuity in order to have $200,000 in 15 years? Recently, The Hartford offered an annuity that pays 5.5%
compounded monthly. What equal monthly deposit should be
made into this annuity in order to have $100,000 in 10 years? A company estimates that it will need $100,000 in 8 years to
replace a computer. If it establishes a sinking fund by mak
ing fixed monthly payments into an account paying 7.5%
compounded monthly. how much should each payment be? Parents have set up a sinking fund in order to have $120,000
in 15 years for their children’s college education. How much
should be paid semiannually into an account paying 6.8%
compounded semiannually? 1f $1.000 is deposited at the end of each year for 5 years
into an ordinary annuity earning 8.32% compounded an
nually. construct a balance sheet showing the interest earned
during each year and the balance at the end of each year. If $2,000 is deposited at the end of each quarter for 2 years
into an ordinary annuity earning 7.9% compounded quar
terly, construct a balance sheet showing the interest earned
during each quarter and the balance at the end of each
quarter. Beginning in January, a person plans to deposit $100 at the
end of each month into an account earning 6% com
pounded monthly. Each year taxes must be paid on the in
terest earned during that year. Find the interest earned
during each year for the ﬁrst 3 years. If $500 is deposited each quarter into an account paying
8% compounded quarterly for 3 years, find the interest
earned during each of the 3 years. ~. Bob makes his first $1,000 deposit into an IRA earning 6.4% compounded annually on his 24th birthday and his
last $1,000 deposit on his 35th birthday (12 equal deposits
in all). With no additional deposits, the money in the IRA
continues to earn 6.4% interest compounded annually until
Bob retires on his 65th birthday. How much is in the IRA
when Bob retires? Refer to Problem 33. John procrastinates and does not make his ﬁrst $1,000 deposit into an IRA until he is 36, but then he continues to deposit $1,000 each year until he is 65 ' (30 deposits in all). If John’s IRA also earns 6.4% com
pounded annually, how much is in his IRA when he makes
his last deposit on his 65th birthday? Refer to Problems 33 and 34. How much would John have
to deposit each year in order to have the same amount at re
tirement as Bob has? 36. 37. 39. Future Value of an Annuity: Sinking Funds 157 Refer to Problems 33 and 34. Suppose that Bob decides to
continue to make $1,000 deposits into his IRA every year
until his 65th birthday. If John still waits until he is 36 to
start his IRA, how much must he deposit each year in order
to have the same amount at age 65 as Bob has? Compubank, an online banking service, offered a money
market account with an APY of 4.86%. (A) If interest is compounded monthly. what is the equiv
alent annual nominal rate? (B) If you wish to have $10,000 in this account after 4 years,
what equal deposit should you make each month? American Express‘s online banking division offered a
money market account with an APY of 5.65%. (A) If interest is compounded monthly, what is the equiv
alent annual nominal rate? (B) If a company wishes to have $1,000,000 in this ac
count after 8 years, what equal deposit should be
made each month? You can afford monthly deposits of $200 into an account
that pays 5.7% compounded monthly. How long will it be
until you have $7,000 to buy a boat? (Round to the next
higher month if not exact.) A company establishes a sinking fund for upgrading office
equipment with monthly payments of $2,000 into an ac
count paying 6.6% compounded monthly. How long will it
be before the account has $100,000? (Round up to the next
higher month if not exact.) 9 In Problems 41—44, use graphical approximation techniques or an equation solver to approximate the desired interest rate. Express each answer as rt percentage, correct to two decimal
places. 41. 42. A person makes annual payments of $1,000 into an ordi
nary annuity. At the end of 5 years, the amount in the
annuity is $5,840. What annual nominal compounding rate
has this annuity earned? A person invests $2,000 annually in an IRA. At the end of
6 years, the amount in the fund is $14,000. What annual
nominal compounding rate has this fund earned? At the end of each month, an employee deposits $50 into a
Christmas club fund. At the end of the year, the fund con
tains $620. What annual nominal rate compounded monthly
has this fund earned? At the end of each month, an employee deposits $80 into a
credit union acoount.At the end of 2 years, the account con
tains $2,100. What annual nominal rate compounded
monthly has this account earned? In Problems 45 and 46, use graphical approximation tech niques to answer the questions. 45. When would an ordinary annuity consisting of quarterly pay
ments of $500 at 6% compounded quarterly be worth more
than a principal of $5,000 invested at 4% simple interest? . When would an ordinary annuity consisting of monthly pay ments of $200 at 5% compounded monthly be worth more
than a principal of $10000 invested at 7.5% compounded
monthly? —7 A [1; Problems I 8, ﬁnd i (the rate per period) and n (the number
ofpcriods) for each loan at the given annual rate. it. Monthly payments of $245.65 are made for 4 years to
repay a loan at 7.2% compounded monthly. 3. Semiannual payments of $3,200 are made for 12 years to
repay a loan at 9.9% compounded semiannually. Quarterly payments of $975 are made for 10 years to
repay a loan at 9.9% compounded quarterly. SJ 4. Annual payments of $1,045 are made for 5 years to
repay a loan at 4.75% compounded annually. 5. Semiannual payments of $4,500 are made for 16 years to
repay a loan at 5.05% compounded semiannually. 6. Quarterly payments of $610 are made for 6 years to
repay a loan at 8.24% compounded quarterly. 7. Annual payments of $5,195 are made for 9 years to
repay a loan at 5.48% compounded annually. 8. Monthly payments of $433 are made for 3 years to repay
a loan at 10.8% compounded monthly. 21. American General offers a lOyear ordinary annuity with a
guaranteed rate of 6.65% compounded annually. How much should you pay for one of these annuities if you want to re
ceive payments of $5,000 annually over the 10—year period? 22. American General also offers a 7year ordinary annuity
with a guaranteed rate of 6.35% compounded annually.
How much should you pay for one of these annuities if you
want to receive payments of $10,000 annually over the
7year period? 23. ELoan, an online lending service, recently offered 36
month auto loans at 7.56% compounded monthly to appli
cants with good credit ratings. If you have a good credit can you borrow from ELoan? What is the total interest
you will pay for this loan? 24. E—Loan recently offered 36month auto loans at 9.84%
compounded monthly to applicants with fair credit ratings
If you have a fair credit rating and can afford monthly pay
ments of $350, how much can you borrow from ELoan?
What is the total interest you will pay for this loan? be paid? $3  If you buy a computer directly from the manufacturer for
$3,500 and agree to repay it in 60 equal installments at 1.75% interest per month on the unpaid balance, how much ‘
are your monthly payments? How much total interest will ' be paid? rating and can afford monthly payments of $350, how much ' 25. If you buy a computer directly from the manufacturer for ; $2,500 and agree to repay it in 48 equal installments at ‘
’ 1.25% interest per month on the unpaid balance, how much "
are your monthly payments? How much total interest will I Present Value of an Annuity; Amortization 167 In Problems 9—20, use formula (5) or (6) to solve each problem
9. n = 30;i = 0.04;PMT = $200;PV = ? 10. n = 40;i = 0.01;PMT = $400;PV =? 11. n = 25;i = 0.025; PMT = $250; PV = ? 12. n = 60;i = 0.0075;PMT = $500; PV = ? B
13. PV = $6,000;n = 36;i = 0.01; PMT = ?
14. PV = $1,200;n = 40;i = 0.025;PMT = ?
15. PV = $40,000;n = 96;i = 0.0075; PMT = ?
16. PV = $14,000;n = 72;i = 0.005; PMT = ?
C 17. PV = $5,000;i = 0.01;PMT = $2m;n = ?
18. PV = $20,000;i = 0.0175; PMT = $500;n = ? "1*; 19. PV = $9,000; PMT = $600;n = 20;i = ?
(Round answer to three decimal places.) 20. PV = $12,000;PMT = $400;n = 40;i = ?
(Round answer to three decimal places.) Problems 27 and 28 refer to the following ads. 27. Use the information given in the Bison sedan ad to deter
mine if this is really 0% financing. If not, explain why and
determine what rate a consumer would be charged for
financing one of these sedans 2008 BISON SEDAN Zero down  0% ﬁnancing
3 1 79~ per month* 2008 BISON WAGON
Zero down  0% ﬁnancing 'Blsonsedan.msdown,0%br72nm 'Blsonmgm.0%down.0%br72mm 28. Use the information given in the Bison wagon ad to deter
mine if this is really 0% ﬁnancing. If not, explain why and
determine what rate a consumer would be charged for
ﬁnancing one of these wagons 29. You want to purchase an automobile for $27,300.'Ihe dealer
offers you 0% financing for 60 months or a $5,000 rebate.
You can obtain 6.3% ﬁnancing for 60 months at the local
bank. Which option should you choose? Explain. 30. You want to purchase an automobile for $28,500.The deal
er offers you 0% financing for 60 months or a $6,000 re
bate. You can obtain 6.2% ﬁnancing for 60 months at the
local bank. Which option should you choose? Explain. 31. A sailboat costs $35,000. You pay 20% down and amortize
the rest with equal monthly payments over a 12year period.
If you must pay 8.75% compounded monthly, what is your
monthly payment? How much interest will you pay? 168 C H A P T E R 3 Mathematics of Finance 32. A recreational vehicle costs $80,000. You pay 10% down
and amortize the rest with equal monthly payments over
a 7year period. If you must pay 9.25% compounded
monthly, what is your monthly payment? How much inter
est will you pay? 33. Construct the amortization schedule for a $5,000 debt that
is to be amortized in eight equal quarterly payments at 2.8%
interest per quarter on the unpaid balance. 34. Construct the amortization schedule for a $10,000 debt that
is to be amortized in six equal quarterly payments at 2.6%
interest per quarter on the unpaid balance. 35. A woman borrows $6,000 at 9% compounded monthly,
which is to be amortized over 3 years in equal monthly pay
ments. For tax purposes, she needs to know the amount of
interest paid during each year of the loan. Find the interest
paid during the first year, the second year, and the third
year of the loan. [Hint Find the unpaid balance after 12
payments and after 24 payments] Q6} man establishes an annuity for retirement by depositing
$50,000 into an account that pays 7.2% compounded monthly.
Equal monthly withdrawals will be made each month for 5
years, at which time the account will have a zero balance.
Each year taxes must be paid on the interest earned by the
account during that year. How much interest was earned
during the ﬁrst year? [Hint The amount in the account at the
end of the ﬁrst year is the present value of a 4year annuity] 37. Some friends tell you that they paid $25,000 down on a new
house and are to pay $525 per month for 30 years If inter
est is 7.8% compounded monthly, what was the selling price (3% the house? How much interest will they pay in 30 years?
38 . . family is thinking about buying a new house costing
120,000. The family must pay 20% down, and the rest is to
be amortized over 30 years in equal monthly payments. If
money costs 7.5% compounded monthly, what will the fam
ily’s monthly payment be? How much total interest will be
paid over the 30 years? ' 39. A student receives a federally backed student loan of $6,000
at 3.5% interest compounded monthly. After finishing col—
lege in 2 years, the student must amortize the loan in the
next 4 years by making equal monthly payments What will
the payments be and what total interest will the student
pay? [Hint This is a twopart problem. ﬁrst ﬁnd the amount
of the debt at the end of the first 2 years; then amortize this
amount over the next 4 years] A person establishes a sinking fund for retirement by con ”/ tributing $7,500 per year at the end of each year for 20 years.
For the next 20 years, equal yearly payments are withdrawn,
at the end of which time the account will have a zero bal
ance. If money is worth 9% compounded annually, what i
i
l
i l yearly payments will the person receive for the last 20'
years? 41. A family has a $75,000, 30year mortgage at 8.1% com '1
a x 'i pounded monthly. Find the monthly payment.Also ﬁnd the L unpaid balance after
(A) 10 years
(B) 20 years
(C) 25 years 42. A family has a $50,000, 20year mortgage at 7.2% com '
pounded monthly. Find the monthly payment. Also ﬁnd the '« unpaid balance after
(A) 5 years (B) 10 years (C) 15 years 43. A family has a $80,000, 20year mortgage at 8%,; compounded monthly. (A) Find the monthly payment and the total interest paid. i
(B) Suppose the family decides to add an extra $100 to
its mortgage payment each month starting with the
very first payment. How long will it take the family
to pay off the mortgage? How much interest will the family save? ‘ At the time they retire, a couple has $200,000 in an account; N that pays 8.4% compounded monthly. (A) If the couple decides to withdraw equal monthly
payments for 10 years, at the end of which time the 'A
account will have a zero balance, how much should ;
the couple withdraw each month? i (B) If the couple decides to withdraw $3,000 a month
until the balance in the account is zero, how many
withdrawals can the couple make? ,, 45. An ordinary annuity that earns 7.5% compounded monthly
'has a current balance of $500,000. The owner of the account is about to retire and has to decide how much to withdraw
from the account each month. Find the number of with:
drawals under each of the following options: ’ (A) $5,000 monthly
(B) $4,000 monthly
(C) $3,000 monthly 46. Refer to Problem 45. If the account owner decides to with
draw $3,000 monthly, how much is in the account after 1(
years? After 20 years? After 30 years? 47. An ordinary annuity pays 7.44% compounded monthly. (A) A person deposits $100 monthly for 30 years and
then makes equal monthly withdrawals for the next
15 years, reducing the balance to zero. What are the
monthly withdrawals? How much interest is earned
during the entire 45year process? (B) If the person wants to make withdrawals of $2,000
per month for the last 15 years, how much must be
deposited monthly for the first 30 years? 48. An ordinary annuity pays 6.48% compounded monthly.
(A) A person wants to make equal monthly deposits int! the account for 15 years in order to then make equa
monthly withdrawals of $1,500 for the next 20 years i l
\ 5
a 49. 50. 51. 53. A person purchased a $200,000 home 20 years ago by pay S e ct i o n 3 ~ 4 Present Value of an Annuity; Amortization 169 reducing the balance to zero. How much should be
deposited each month for the first 15 years? What is V
the total interest earned during this 35year process? , (B) If the person makes monthly deposits of $1,000 for
the first 15 years, how much can be withdrawn
monthly for the next 20 years? A couple wishes to borrow money using the equity in its home for collateral. A loan company will loan the couple up to 70% of their equity. The couple purchased the home 12 years ago for $79,000. The home was financed by paying 20% down and signing a 30year mortgage at 12% on the f
unpaid balance. Equal monthly payments were made to i
amortize the loan over the 30year period.The net market value of the house is now $100,000.After making the 144th payment, the couple applied to the loan company for the maximum loan. How much (to the nearest dollar) will the 3
couple receive? ‘ A person purchased a house 10 years ago for $100,000. The house was financed by paying 20% clown and signing a E
30year mortgage at 9.6% on the unpaid balance. Equal monthly payments were made to amortize the loan over a 30—year period. The owner now (after the 120th payment) 3‘ wishes to refinance the house because of a need for addi % tional cash. If the loan company agrees to a new 30—year :
mortgage of 80% of the new appraised value of the house, which is $136,000, how much cash (to the nearest dollar) will the owner receive after repaying the balance of the orig if
inal mortgage? 5 A person purchased a $120,000 home 10 years ago by pay
ing 20% down and signing a 30year mortgage at 10.2%
compounded monthly. Interest rates have dropped and the
owner wants to refinance the unpaid balance by signing a
new 20—year mortgage at 7.5 % compounded monthly. How
much interest will refinancing save? ing 20% down and signing a 30year mortgage at 13.2%
compounded monthly. Interest rates have dropped and the
owner wants to refinance the unpaid balance by signing a new 10year mortgage at 8.2% compounded monthly. How
much interest will refinancing save? Discuss the similarities and differences in the graphs of un
paid balance as a function of time for 30year mortgages of
$50,000,375,000, and $100,000, respectively, each at 9% com
pounded monthly (see the ﬁgure). Include computations of
the monthly payment and total interest paid in each case. Unpaid balance (5) o 6 12 Is 24; 30 V
'l'ime(years) l Figure for 53. 54. Discuss the similarities and differences in the graphs of un
paid balance as a function of time for 30—year mortgages of
$60,000 at rates of 7% , 10%, and 13%, respectively (see the
figure). Include computations of the monthly payment and
total interest paid in each case. Unpaid balance (3)
“E 0 6 l2 I8 24 30
Time (years) Figure for 54 In Problems 55—58, use graphical approximation techniques or
an equation solver to approximate the desired interest rate.
Express each answer as a percentage, correct to two decimal
places. 55. S7. 58. A discount electronics store offers to let you pay for a
$1,000 stereo in 12 equal $90 installments. The store claims
that since you repay $1,080 in 1 year, the $80 finance
charge represents an 8% annual rate. This would be true
if you repaid the loan in a single payment at the end of
the year. But since you start repayment after 1 month, this
is an amortized loan, and 8% is not the correct rate. What
is the annual nominal compounding rate for this loan?
[Did you expect the rate to be this high? Loans of this type
are called addon interest loans and were very common
before Congress enacted the mm in Lending Act. Now
credit agreements must fully disclose all credit costs and
must express interest rates in terms of the rates used in
the amortization process] A $2,000 computer can be financed by paying $100 per
month for 2 years What is the annual nominal compound—
ing rate for this loan? The owner of a small business has received two offers of
purchase. The first prospective buyer offers to pay the
owner $100,000 in cash now. The second offers to pay the
owner $10,000 now and monthly payments of $1,200 for 10
years. In effect, the second buyer is asking the owner for a
$90,000 loan. If the owner accepts the second offer, what
annual nominal compounding rate will the owner receive
for ﬁnancing this purchase? At the time they retire, a couple has $200,000 invested in an
annuity. The couple can take the entire amount in a single
payment, or receive monthly payments of $2,000 for 15
years. If the couple elects to receive the monthly payments, what annual nominal compounding rate will the couple cam
on the money invested in the annuity? l
l
l
l 170 C H A P T E R 3 Mathematics of Finance I ,CHAPTERSHREVEESW‘ ,, ¥ ‘
' vi, anmnceptsr 31 Simple Interest Examples
' Interest is‘the fee paid for the use of a sum of money P, called the principal. Simple interest is
given by
I = Prt where
I = interest P, = principal _
r = annual simple interest rate (written as a decimal)
t = time in years ,
0 If a principal P‘(present value) is borrowed, then the amount A (future value) is the total of the prin Ex. 1, p. 129 cipal and the interest: Ex. 2, p. 129
Ex.3,p. 130 A=P+P" Ex.4,p.130 = P(1 + rt) Ex.5,p. 131 32 Compound and Continuous Compound Interest . ' Compound interest is interest paid on the principal plus reinvested interest. The future and present Ex. 1, p. 136
values are related by ‘ _ . A = P( 1 + i)"
wherei = r/m and
L A = amount Or future value
L P = principal or present value ‘
I r '= annual neminal rate (or just rate)
m = of compounding periods per year
i = rate per compounding period n =’ total number of compounding periods ' i“, _
r , 'If a principal P is invested at an annual rate r earning continuous compound interest, then the moon Ex. 2, p. 138
~ *A after t years is given by .  ‘ Ex. 3, p. 139
‘ _ ‘ A = Pa"? ‘ . a ‘ Ex. 4, p. 141 H L or 0 The growth time of an investment is the time it takes fora given principal to grow to a particular
‘ » amount. Three methods for ﬁnding the growth time are as follows: _ 1. Use logarithms and a calculator.
. _2. Use graphical approximation on a graphing calculator. Ex. 5, p. 142 L ' 3. Use equation solver on a graphing calculator or a computer. _ ' The annual percentage yield APY (also called the effective rate or the true interest rate) is the
simple interest rate that would earn the same amount as a given annual rate for which interest is
compounded. '5 If a principal is invested at the annual rate r compounded m times a year, then the annual percentage EX 6,13. 144
yield is given by ,, 7 * _ Ex. 7. p. 144 , I r m
APY = (1 + —) — 1 i if a principal is“ invested at the annual rate r compounded continuously, then the annual percentage
‘ ‘ yield is given by  'APY ='e' +1 V
s  A aero coupon bond is a bondthat is sold now at a discount and will pay its face Value at some time I
‘ in the future when it matures ’ ‘ ‘ ‘ ‘ '  g, am Valui ol’an .\ttnnit§: Sinking l‘mnls . . \n annuin is any sequence of equal periodic payments. If payments are made at the end of each
time interval. then the annuity is called an ordinary annuity. The amount. or future value, of an
maturity is the sum of all payments plus all interest earned and is given by (l + i)" 1
iv =~ PM'I‘ / r \ llc‘l’C
FV 7 future value (amount)
PMT : periodic payment 1‘ 2 rate per period u = number of payments (periods)
,\n account that is established to accumulate funds to meet future obligations or deth is called a
sinking fund. The sinking fund payment can be found by solving the future value formula for PM ‘I‘: i PMT= FV~——
(t + 1')” ~ I 3.  sent Value ol‘au Annuity: Amortization ’t it equal payments are made from an account until the amount in the account is l). the payment and
the present value are related by the following formula: Ml e (l + i)”’
PV = PM I —~i~~
where
PV = present value ofall payments
PMT = periodic payment
i I rate per period
it = number of periods * Amortizing a debt means that the debt is retired in a given length of time by equal periodic pay ments that include compound interest Solving the present value formula for the payment gives us
the amortization formula: t . _ _~1L
lMT PV1_(1+1)" An amortization schedule is a table that shows the interest due and the balance reduction for each
payment of a loan the equity in a property is the difference between the current net market value and the unpaid loan balance. The unpaid balance of a loan with a remaining payments is given by the present
value formula. ‘EXERClSE work through all the problems in this chapter review and
check your answers in the back of the book. Answers to all
review problems are there along with section numbers in italics
to indicate where each type of problem is dismissal. Where
weaknesses show up, review appropriate sections in the text. A = P(1+i’)“. In Problems 1—4, ﬁnd the indicated quantity, given
A = P(1 + rt). 1. A = ?;P = $100;r = 9%;t = 6months 2. A = $808:P = ?;r = 12%;t = lmonth 3. A = $212;P = $200;r = 8%;r = ? 4. A = $41sz = $4,000;r = ?;t = 6months A = P2". Chapter Al Review 171 l\ l_ti l‘l
l \J p lit
l'x lpd"; map 15.: F\_ R, p lit” iix. (x p. 10> l.\.~l.p. lo”)
limip. lo} 8 In Problems 5 and 6, find the indicated quantity, given 5. A = ?;P = $1,200;i = 0.005;n = 30
6 A = $5,000;P= ?;i = 040075;n = 60 In Problems 7 and 8, ﬁnd the indicated quantity, given ’7. A = ?;P = $4.750;r = 6.8%;t = 3years
8 A = $36,000;P = ?;r = 9.3%;t = 60months H. % 19. 172 C H A PT E R 3 Mathematics of Finance In Problems 9 and 10, ﬁnd the indicated quantity, given
(1 + i)" — 1 I
9. FV = ?;PMT = $1,000;i = 0.005;n = 60
10. FV = $8,000;PMT = ?;i = 0.015;n = 48 FV=PMT In Problems 11 and 12, ﬁnd the indicated quantity, given
1 — (1 + if"
i 11. PV = ?;PMT = $2,500;i = 0.02m = 16
12. PV = $8,000;PMT = ?',i = 0.0075;n = 60 PV = PMT APPLICATIONS Find all dollar amounts correct to the nearest cent. When an in
terest rate is requested as an answer, express the rate as a per
centage, correct to two decimal places 15. If you borrow $3,000 at 14% simple interest for 10 months,
how much will you owe in 10 months? How much interest
will you pay? 16. Grandparents deposited $6,000 into a grandchild’s account
toward a college education. How much money (to the near
est dollar) will be in the account 17 years from now if the ac
count eams 7% compounded monthly? 17. How much should you pay for a corporate bond paying
6.6% compounded monthly in order to have $25,000 in 10
years? 18. A savings account pays 5.4% compounded annually.
Construct a balance sheet showing the interest earned dur
ing each year and the balance at the end of each year for
4 years if
(A) A single deposit of $400 is made at the beginning of the first year. (B Four (18 OSltS Of are made at the end Of each
p
year. One investment pays 13% simple interest and another 9%
compounded annually. Which investment would you choose? Why? 20. A $10,000 retirement account is left to earn interest at 7%
compounded daily. How much money will be in the account
40 years from now when the owner reaches 65? (Use a 365—
day year and round answer to the nearest dollar.) 21. A couple wishes to have $40,000 in 6 years for the down
payment on a house. At what rate of interest compounded
continuously must $25,000 be invested now to accomplish
this goal? 22. Which is the better investment and why: 9% compounded,
quarterly or 9.25% compounded annually? 23. What is the value of an ordinary annuity at the end of
8 years if $200 per month is deposited into an account earn
ing 7.2% compounded monthly? How much of this value is
interest? C r'r’rrrﬂﬂﬁ 13. Solve the equation 2,500 = 1.000(106)" for n to the nearest
integer using: (A) Logarithms % (B) Graphical approximation techniques or an equation
solver on a graphing calculator 14. Solve the equation 5 000 _ 1000.01)" — l
’ ” 0.01
for n to the nearest integer using:
(A) Logarithnts % (B) Graphical approximation techniques or an equation solver on a graphing calculator. 24. A credit card company charges a 22% annual rate for over
due accounts. How much interest will be owed on a $635
account 1 month overdue? 25. What will a $15,000 car cost (to the nearest dollar) 5 years
from now if the inflation rate over that period averages 5%
compounded annually? 26. What would the $15,000 car in Problem 25 have cost (to the
nearest dollar) 5 years ago if the inﬂation rate over that pe
riod had averaged 5% compounded annually? 27. A loan of $2,500 was repaid at the end of 10 months with a
check for $2,812.50. What annual rate of interest was
charged? 28. You want to purchase an automobile for $21,600. The dealer
offers you 0% ﬁnancing for 48 months or a $3,000 rebate.
You can obtain 4.8% ﬁnancing for 48 months at the local
bank. Which option should you choose? Explain. 29. Fmd the annual percentage yield on a CD earning 6.25% if
interest is compounded (A) monthly.
(B) continuously. 30. You have $2,500 toward the purchase of a boat that will
cost $3,000. How long will it take the $2,500 to grow to
$3,000 if it is invested at 9% compounded quarterly?
(Round up to the nexthigher quarter it not exact.) 31. How long will it take money to double if it is invested at
6% compounded monthly? 9% compounded monthly?
(Round up to the nexthigher month if not exact.) 32. Starting on his let birthday, and continuing on every birth
day up to and including his 65th,.lohn deposits $2,000 a year
into an IRA. How much (to the nearest dollar) will be in the
account on John’s 65th birthday, if the account earns: (A) 7% compounded annually?
(B) 11% compounded annually? 33. If you just sold a stock for $17,388.17 (net) that cost you
$12,903.28 (net) 3 years ago, what annual compound rate
of return did you make on your investment? 34 35. 37. 38. 39. 41. 42. 43. The table shows the fees for refund anticipation loans
(RALS) offered by an online tax preparation firm. Find the
annual rate of interest for each of the following loans. (A) A $400 RAL paid back in 15 days
(13) A $1,800 RAL paid back in 21 days RAL Amount RAL Fee
$100'5500 $29.00
$501—$i .000 $39.00 $1.00 I —$ l .500 $49.00 S l $01—$21“) $69.00 $2.(x)1~$5.000 $82.00 Recently Lincoln Benefit Life offered an annuity that pays ' 5.5% compounded monthly. What equal monthly deposit
should be made into this annuity in order to have $50,000
in 5 years? A person wants to establish an annuity for retirement pur
poses. He wants to make quarterly deposits for 20 years so
that he can then make quarterly withdrawals of $5,000 for
10 years. The annuity earns 7.32% interest compounded
quarterly (A) How much will have to be in the account at the time
he retires? (B) How much should be deposited each quarter for 20
years in order to accumulate the required amount? (C) What is the total amount of interest earned during
the 30year period? If you borrow $4,000 from an online lending firm for the purchase of a computer and agree to repay it in 48 equal _
installments at 0.9% interest per month on the unpaid bal _ ance, how much are your monthly payments? How much
total interest will be paid? A company decides to establish a sinking fund to replace a
piece of equipment in 6 years at an estimated cost of
$50,000. To accomplish this, they decide to make fixed monthly payments into an account that pays 6.12% oom , pounded monthly. How much should each payment be? How long will it take money to double if it is invested at ‘ 7.5% compounded daily? 7.5% compounded annually? A student receives a student loan for $8,000 at 5.5% inter— f
est compounded monthly to help her finish the last 1.5 years ;
of college. Starting 1 year after finishing college, the stu ;
dent must amortize the loan in the next 5 years by making equal monthly payments. What will the payments be and what total interest will the student pay? If you invest $5,650 in an account paying 865% com _
pounded continuously, how much money will be in the ac i count at the end of 10 years? A company makes a payment of $1,200 each month into a sinking fund that earns 6% compounded monthly. Use graphical approximation techniques on a graphing calcula tor to determine when the fund will be worth $100,000. A couple has a $50,000, 20year mortgage at 9% com '»
pounded monthly. Use graphical approximation techniques ' 44. 45. 47. 49. Chapter 3 Review 1 73 on a graphing calculator to determine when the unpaid bal—
ance will drop below $10,000. A loan company advertises in the paper that you will pay
only 8d a day for each $100 borrowed. What annual rate of
interest are they charging? (Use a 360day year.) Construct the amortization schedule for a $1,000 debt that
is to be amortized in four equal quarterly payments at 2.5%
interest per quarter on the unpaid balance. You can afford monthly deposits of only $200 into an ac—
count that pays 7.98% compounded monthly. How long will
it be until you will have $5,000 to purchase a used car?
(Round to the nexthigher month if not exact.) A company establishes a sinking fund for plant retooling
in 6 years at an estimated cost of $850,000. How much
should be invested semiannually into an account paying
8.76% compounded semiannually? How much interest will
the account earn in the 6 years? What is the annual nominal rate compounded monthly for
a CD that has an annual percentage yield of 6.48%? If you buy a 13week T—bill with a maturity value of $5,000
for $4,922.15 from the US. Treasury Department, what an
nual interest rate will you earn? 50. In order to save enough money for the down payment on a condominium, a young couple deposits $200 each month
into an account that pays 7.02% interest compounded
monthly. If the couple needs $10,000 for a down payment,
how many deposits will the couple have to make? 51. A business borrows $80,000 at 9.42% interest compounded 52. 53. 54. 55. 56. monthly for 8 years. (A) What is the monthly payment? (B) What is the unpaid balance at the end of the first
year? (C) How much interest was paid during the ﬁrst year? You unexpectedly inherit $10,000 just after you have made
the 72nd monthly payment on a 30—year mortgage of $60,000
at 8.2% compounded monthly. Discuss the relative merits of
using the inheritance to reduce the principal of the loan, or
to buy a certificate of deposit paying 7% compounded
monthly. Your parents are considering a $75,000, 30year mortgage to
purchase a new home. The bank at which they have done
business for many years offers a rate of 7.54% compounded
monthly. A competitor is offering 6.87% compounded monthly. Would it be worthwhile for your parents to switch
banks? How much should a $5,000 face value zero coupon bond, maturing in 5 years, he sold for now, if its rate of return is to
be 5.6% compounded annually? If you pay $5,695 for a $10,000 face value zero coupon bond
that matures in 10 years, what is your annual compound rate
of return? If an investor wants to earn an annual interest rate of 6.4%
on a 26week T—bill with a maturity value of $5,000, how much
should the investor pay for the Tbill? 174 C H A P T E R 3 Mathematics of Finance 57. TWO years ago you borrowed $10,000 at 12% interest com
pounded monthly, which was to be amortized over 5 years
Now you have acquired some additional funds and decide
that you want to pay off this loan.What is the unpaid bal
ance after making equal monthly payments for 2 years? 58. What annual nominal rate compounded monthly has the same annual percentage yield as 7.28% compounded quar terly? 59. (A) A man deposits $2,000 in an IRA on his 21st birthday
and on each subsequent birthday up to, and including,
his 29th (nine deposits in all).The account earns 8%
compounded annually. If he then leaves the money in
the account without making any more deposits, how
much will he have on his 65th birthday, assuming the
account continues to earn the same rate of interest? (B) How much would be in the account (to the nearest
dollar) on his 65th birthday if he had started the de
posits on his 30th birthday and continued making de posits on each birthday until (and including) his 65th
birthday? 60. A promissory note will pay $27,000 at maturity 10 years
from now. How much money should you be willing to pay
now if money is worth 5.5 % compounded continuously? 61. In a new housing development, the houses are selling for
$100,000 and require a 20% down payment. The buyer is
given a choice of 30—year or 15year ﬁnancing, both at 7.68%
compounded monthly. 62. 63. A $600 stereo is ﬁnanced for 6 months by making month
% payments of $110. What is the annual nominal compoun 64. A person deposits $2,000 each year for 25 years into a
% IRA. When she retires immediately after making the 25 (A) What is the monthly payment for the 30year
choice? For the 15year choice? (B) What is the unpaid balance after 10 years for the 30. 7
year choice? For the ISyear choice? A loan company will loan up to 60%.of the equity in
home. A family purchased their home 8 years ago fo $83,000. The home was financed by paying 20% down an
signing a 30year mortgage at 8.4% for the balance. Equ
monthly payments were made to amortize the loan ove
the 30year period. The market value of the house is no
$95 ,000. After making the 96th payment, the family appli
to the loan company for the maximum loan. How much (t
the nearest dollar) will the family receive? ing rate for this loan? deposit, the IRA is worth $220,000. (A) Find the interest rate earned by the IRA over the v
25year period leading up to retirement.
(B) Assume that the IRA continues to earn the interest
rate found in part (A). How long can the retiree withdraw $30,000 per year? How long can she with
draw $24,000 per year? ' ...
View
Full Document
 Fall '08
 LEDUC
 Interest Rates, Mathematics of Finance

Click to edit the document details