MfB Ch6

7. A homeowner planning a kitchen remodeling can SECTION 6.5 Loans and Amortization 419
afford a $200 monthly payment. How much can the 20. A woman buys an apartment house for $1,250,000 by
homeowner borrow for 5 years at 6%, compounded
monthly, and still stay within the budget? making a down payment of $250,000 and amortizing
the rest of the debt with semiannual payments over
8. AdriAnne and Anna’s Auto Repair wants to add a new the next 10 years. The interest rate on the debt is 8%,
service bay. How much can they borrow at 5%, com- compounded semiannually.
pounded quarterly for 412 years, if the desired quarterly 21. A man buys a car for $36,000. If the interest rate on the
payment is $6000? loan is 12%, compounded monthly, and if he wants to
make monthly payments of $900 for 36 months, how
In Problems 9–12, develop an amortization schedule for much must he put down?
22. A woman buys a car for $40,000. If the interest rate on
the loan described. the loan is 12%, compounded monthly, and if she wants
9. $100,000 for 3 years at 9% compounded annually to make monthly payments of $700 for 3 years, how
10. $30,000 for 5 years at 7% compounded annually much must she have for a down payment?
11. $20,000 for 1 year at 12% compounded quarterly 23. A couple purchasing a home budget $1800 per month
12. A$5$01,000,000f0orlo2a21nyeisartos at 10% compounded semiannually for their loan payment. If they have $20,000 available
13. be amortized for 10 years with for a down payment and are considering a 25-year loan,
quarterly payments of $334.27. If the interest rate is how much can they spend on the home at each of the
6%, compounded quarterly, what is the unpaid balance following rates?
immediately after the sixth payment? (a) 6.9% compounded monthly
14. A debt of $8000 is to be amortized with 8 equal (b) 7.5% compounded monthly
semiannual payments of $1288.29. If the interest rate 24. A developer wants to buy a certain parcel of land. The
is 12%, compounded semiannually, find the unpaid developer feels she can afford payments of $44,000
balance immediately after the fifth payment. each half-year for the next 7 years. How much can she
15. When Maria Acosta bought a car 212 years ago, she borrow and hold to this budget at each of the following
borrowed $14,000 for 48 months at 8.1% compounded interest rates?
monthly. Her monthly payments are $342.44, but she’d (a) 8.9% compounded semiannually
like to pay off the loan early. How much will she owe (b) 7.3% compounded semiannually
just after her payment at the 212-year mark? 25. A couple who borrow $90,000 for 30 years at 7.2%,
16. Six and a half years ago, a small business borrowed compounded monthly, must make monthly payments
$50,000 for 10 years at 9%, compounded semiannually, of $610.91.
in order to update some equipment. Now the company (a) Find their unpaid balance after 1 year.
would like to pay off this loan. Find the payoff amount (b) During that first year, how much interest do
just after the company makes the 14th semiannual
payment of $3843.81. they pay?
26. A company that purchases a piece of equipment by
Problems 17–20 describe a debt to be amortized. In each
problem, find: borrowing $250,000 for 10 years at 6%, compounded
(a) the size of each payment. monthly, has monthly payments of $2775.51.
(b) the total amount paid over the life of the loan. (a) Find the unpaid balance on this loan after 1 year.
(c) the total interest paid over the life of the loan. (b) During that first year, how much interest does the
17. A man buys a house for $350,000. He makes a $150,000
company pay?
down payment and amortizes the rest of the debt with 27. A recent college graduate buys a new car by borrowing
semiannual payments over the next 10 years. The inter-
est rate on the debt is 12%, compounded semiannually. $18,000 at 8.4%, compounded monthly, for 5 years. She
18. Sean Lee purchases $20,000 worth of supplies for his decides to pay an extra $15 per payment.
restaurant by making a $3000 down payment and (a) What is the monthly payment required by the loan,
amortizing the rest with quarterly payments over the
next 5 years. The interest rate on the debt is 16%, com- and how much does she decide to pay each month?
pounded quarterly. (b) How many payments (that include the extra $15)
19. John Fare purchased $10,000 worth of equipment by
making a $2000 down payment and promising to pay will she make?
the remainder of the cost in semiannual payments over (c) How much will she save by paying the extra $15?
the next 4 years. The interest rate on the debt is 10%, 28. A young couple buying their first home borrow $85,000
compounded semiannually. for 30 years at 7.2%, compounded monthly, and make
payments of $576.97. After 3 years, they are able to
make a one-time payment of $2000 along with their
36th payment.
(a) Find the unpaid balance immediately after they

pay the extra $2000 and their 36th payment.

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420 CHAPTER 6 Mathematics of Finance borrowed and is paid before the loan is made as part
of the closing costs of buying a home (closing costs
(b) How many regular payments of $576.97 will include points, title fees, attorney’s fees, assessor’s fees,
amortize the unpaid balance from part (a)? and so on).
(a) If $100,000 is borrowed for 25 years, for each of
(c) How much will the couple save over the life of the the following, find the payment size and the total
loan by paying the extra $2000? paid over the life of the loan (including points).
(i) 771214%%,, compounded monthly, with 0 points
A debt of $100,000 is amortized at 6%, compounded (ii) compounded monthly, with 1 point
monthly, over 25 years with 300 monthly payments of (iii) 7%, compounded monthly, with 2 points
$644.30 each. Figure 6.12 includes two graphs: one shows (b) Which loan in part (a) has the lowest total cost
the total amount paid (in monthly payments) as a func- over the life of the loan?
tion of time (in months), and the other shows the amount 34. Time-share sales provide an opportunity for vacation-
paid toward the principal of the debt as a function of ers to own a resort condo for 1 week (or more) each
time. Use this figure to complete Problems 29 and 30.

$

150 year forever. The owners may use their week at their
own condo or trade the week and vacation elsewhere.
In thousands Time-share vacation sales usually require payment in
100 full or financing through the time-share company, and

50 interest rates are usually in the 13% to 18% range.
Suppose the cost to buy a 1-week time share in a
3-bedroom condo is $21,833. Also suppose a 10%
time

50 100 150 200 250 300 down payment is required, with the balance financed
for 15 years at 16.5%, compounded monthly.
Months (a) Find the monthly payment.
Figure 6.12

29. (a) Correctly label each graph. (b) Determine the total cost over the life of the loan.
(b) Draw a vertical segment whose length represents (c) Suppose maintenance fees for this condo are $400
the total amount of interest paid on the debt after per year. Find the annual cost of the condo over
250 months. the life of the loan. Assume that the annual main-
tenance fees remain constant.
30. Draw a vertical segment whose length represents the (d) Use part (c) and the 10% down payment to
outstanding principal of the debt (or the payoff amount determine the average annual cost for having this
of the loan) after 150 months. vacation condo for 1 week over the life of the loan.
35. During four years of college, Nolan MacGregor’s
31. What difference does 0.5% make on a loan? To answer student loans are $4000, $3500, $4400, and $5000 for
this question, find (to the nearest dollar) the monthly freshman year through senior year, respectively. Each
payment and total interest paid over the life of the loan loan amount gathers interest of 1%, compounded
for each of the following. quarterly, while Nolan is in school and 3%, com-
(a) An auto loan of $15,000 at 8.0% versus 8.5%, com- pounded quarterly, during a 6-month grace period
pounded monthly, for 4 years. after graduation.
(b) A mortgage loan of $80,000 at 6.75% versus 7.25%, (a) What is the loan balance after the grace period?
compounded monthly, for 30 years. Assume the freshman year loan earns 1% interest
(c) In each of these 0.5% differences, what seems to for 3y4 year during the first year, then for 3 full
have the greatest effect on the borrower: amount years until graduation. Make similar assumptions
borrowed, interest rate, or duration of the loan? for the loans for the other years.
Explain. (b) After the grace period, the loan is amortized over
the next 10 years at 3%, compounded quarterly.
32. Some banks now have biweekly mortgages (that is, Find the quarterly payment.
with payments every other week). Compare a 20-year, (c) If Nolan decides to pay an additional $90 per
$100,000 loan at 8.1% by finding the payment size and payment, how many payments of this size will
the total interest paid over the life of the loan under amortize the debt?
each of the following conditions. (d) How much will Nolan save by paying the extra $90
(a) Payments are monthly, and the rate is 8.1%, with each payment?
compounded monthly.
(b) Payments are biweekly, and the rate is 8.1%, com-
pounded biweekly.

33. Many banks charge points on mortgage loans. Each
point is the equivalent of a 1% charge on the amount

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36. Clark and Lana take a 30-year home mortgage of Key Terms and Formulas 421
$121,000 at 7.8%, compounded monthly. They make
their regular monthly payments for 5 years, then decide Use a spreadsheet or financial program on a calculator
to pay $1000 per month. or computer to complete Problems 37 and 38.
(a) Find their regular monthly payment. 37. Develop an amortization schedule for a 4-year car
(b) Find the unpaid balance when they begin paying
the $1000. loan if $16,700 is borrowed at 8.2%, compounded
(c) How many payments of $1000 will it take to pay monthly.
off the loan? 38. Develop an amortization schedule for a 10-year mort-
(d) How much interest will they save by paying the gage loan of $80,000 at 7.2%, compounded monthly.
loan in this way?

KEY TERMS
AND FORMULAS

Section Key Terms Formulas
I 5 Prt
6.1 Simple interest
Future value of a simple interest S5P1I
investment or loan
Sequence function an 5 an21 1 d 1n . 12
Arithmetic sequence
Common difference d
nth term of an 5 a1 1 1 n 2 1 2 d
Sum of first n terms n
sn 5 2 1 a1 1 an 2

6.2 Future value of compound interest P11 i2n Pa1 r mt
investment Pert m
Periodic compounding S 5 1 5 1 b
Continuous compounding S 5

Annual percentage yield APY 5 a1 1 r m 2 1
Periodic compounding m
b

5 11 1 i2m 2 1
Continuous compounding APY 5 er 2 1
Geometric sequence 1n . 12
an 5 ran21
Common ratio
nth term of r
Sum of first n terms an 5 a1r n21

6.3 Future values of annuities sn 5 a1 1 1 2 r n 2 1 if r 2 12
Ordinary annuity 12r
Required payment into an
ordinary annuity S 5 R B 11 1 i2n 2 1 R
Sinking fund i
Annuity due
Sdue 5 R B 11 1 i2n 2 1 R 11 1 i2
i

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422 CHAPTER 6 Mathematics of Finance

Section Key Terms Formulas
6.4 Present values of annuities
An 5 RB1 2 11 1 i 2 2n R
6.5 Ordinary annuity i
Bond pricing
A1n,due2 5 RB1 2 11 1 i 2 2n R 11 1 i2
Annuity due i

Deferred annuity A1n,k2 5 RB1 2 11 1 i 2 2n R 11 1 i 2 2k
i
True annual percentage
rate (APR) R 5 An B 1 2 i i 2 2n R
11 1
Amortization (Loan amount is the
same as present value of an An 2 k 5 RB1 2 11 1 i 2 21n2k2 R
ordinary annuity) i

Unpaid balance
(n 2 k payments left)

Amortization schedule

REVIEW EXERCISES

1. Find the first 4 terms of the sequence with nth term 9. The future value of a series of payments, with interest
compounded when the payments are made
an 5 1
n2 10. The simple interest earned on an investment
11. The present value of a series of payments, with interest
2. Identify any arithmetic sequences and find the com-
mon differences. compounded when the payments are made
(a) 12, 7, 2, 23, . . . (b) 1, 3, 6, 10, . . . 12. The future value of an investment that earns interest
(c) 16, 31, 12, 23. . . .
3. Find the 80th term of the arithmetic sequence with first compounded periodically
term 22 and common difference 3. 13. The regular payment size needed to amortize a debt, when
4. Find the 36th term of the arithmetic sequence with
third term 10 and eighth term 25. interest is compounded when the payments are made
5. Find the sum of the first 60 terms of the arithmetic 14. The future value of an investment that earns interest
6. sIdeqenuteinfycean31,y12,g23e,o.m. .etric sequences and their common
ratios. compounded continuously
(a) 14, 2, 16, 128, . . . (b) 16, 212, 9, 2247, . . . 15.  Loans  If $8000 is borrowed at 12% simple interest for
(c) 4, 16, 36, 64, . . .
7. Find the fourth term of the geometric sequence with 3 years, what is the future value of the loan at the end of
first term 64 and eighth term 21. the 3 years?
8. Find the sum of the first 16 terms of the geometric 16.  Loan rate  Mary Toy borrowed $2000 from her parents
sequence 91, 13, 1, . . . . and repaid them $2100 after 9 months. What simple
interest rate did she pay?
APPLICATIONS 17.  Tuition  How much summer earnings must a college
student deposit on August 31 in order to have $3000 for
Finance  Identify which of the following formulas applies tuition and fees on December 31 of the same year, if the
to each situation in Problems 9–14. investment earns 6% simple interest?
18.  Contest payments  Suppose the winner of a contest
I 5 Prt S 5 P 1 1 1 i 2 n S 5 Pert receives $10 on the first day of the month, $20 on the
second day, $30 on the third day, and so on for a 30-day
S 5 R c 11 1 i2n 2 1 d An 5 1 2 11 1 i 2 2n d month. What is the total won?
i Rc i 19.  Salaries  Suppose you are offered two identical jobs:
one paying a starting salary of $40,000 with yearly
raises of $2000 and one paying a starting salary of
$36,000 with yearly raises of $2500. Which job will pay
more money over a 10-year period?

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20. Investments  An investment is made at 8%, com- Review Exercises 423
pounded quarterly, for 10 years. 35. Construction fund  A company wants to have $250,000
(a) Find the number of periods.
(b) Find the interest rate per period. available in 421 years for new construction. How much
21. Future value  Write the formula for the future value must be deposited at the beginning of each quarter to
of a compound interest investment, if interest is reach this goal if the investment earns 10.2%, com-
compounded pounded quarterly?
(a) periodically (b) continuously. 36. Time to reach a goal  If $1200 is deposited at the end
22. Comparing rates  Which compounding method, at 6%, of each quarter in an account that earns 7.2%, com-
earns more money? pounded quarterly, how long will it be until the account
(a) semiannually (b) monthly is worth $60,000?
23. Interest  If $1000 is invested for 4 years at 8%, com- 37. Annuity  What lump sum would have to be invested at
pounded quarterly, how much interest will be earned? 9%, compounded semiannually, to provide an annuity
24. Savings goal  How much must one invest now in order of $10,000 at the end of each half-year for 10 years?
to have $18,000 in 4 years if the investment earns 5.4%, 38. Cruise fund  A couple wish to set up an annuity that
compounded monthly? will provide 6 monthly payments of $3000 while they
25. Future value  What is the future value if $1000 is take an extended cruise. How much of a $15,000 inheri-
invested for 6 years at 8%, compounded continuously? tance must be set aside if they plan to leave in 5 years
26. College fund  A couple received an inheritance and and want the first payment before they leave? Assume
plan to invest some of it for their grandchild’s college that money is worth 7.8%, compounded monthly.
education. How much must they invest if they would 39. Powerball lottery  Winners of lotteries receive the
like the fund to have $100,000 after 15 years, and their jackpot distributed over a period of years, usually 20 or
investment earns 10.31%, compounded continuously? 25 years. The winners of the Powerball lottery on July
27. Investments  If $15,000 is invested at 6%, compounded 29, 1998, elected to take a one-time cash payout rather
quarterly, how long will it be before it grows to than receive the $295.7 million jackpot in 25 annual
$25,000? payments beginning on the date the lottery was won.
28. Investment rates  (a) If an initial investment of $35,000 (a) How much money would the winners have
grows to $257,000 in 15 years, what annual interest
rate, continuously compounded, was earned? (b) What received at the beginning of each of the 25 years?
is the annual percentage yield on this investment? (b) If the value of money was 5.91%, compounded
29. Comparing yields  Find the annual percentage yield
equivalent to a nominal rate of 7.2% (a) compounded annually, what one-time payout did they receive in
quarterly and (b) compounded continuously. lieu of the annual payments?
30. Chess legend  Legend has it that when the king of 40. Payment from an annuity  A recent college gradu-
Persia wanted to reward the inventor of chess with ate’s gift from her grandparents is $20,000. How much
whatever he desired, the inventor asked for one grain will this provide at the end of each month for the next
of wheat on the first square of the chessboard, with the 12 months while the graduate travels? Assume that
number of grains doubled on each square thereafter for money is worth 6.6%, compounded monthly.
the remaining 63 squares. Find the number of grains on 41. Payment from an annuity  An IRA of $250,000 is
the 64th square. (The sum of all the grains of wheat on rolled into an annuity that pays a retired couple at the
all the squares would cover Alaska more than 3 inches beginning of each quarter for the next 20 years. If the
deep with wheat!) annuity earns 6.2%, compounded quarterly, how much
31. Chess legend  If, in Problem 30, the king granted the will the couple receive each quarter?
inventor’s wish for half a chessboard, how many grains 42. Duration of an annuity  A retirement account that
would the inventor receive? earns 6.8%, compounded semiannually, contains
32. Annuity  Find the future value of an ordinary annu- $488,000. How long can $40,000 be withdrawn at the
ity of $800 paid at the end of every 6-month period end of each half-year until the account balance is $0?
for 10 years, if it earns interest at 12%, compounded 43. Amortization  A debt of $1000 with interest at 12%,
semiannually. compounded monthly, is amortized by 12 monthly
33. Sinking fund  How much would have to be invested at payments (of equal size). What is the size of each
the end of each year at 6%, compounded annually, to payment?
pay off a debt of $80,000 in 10 years? 44. Loan pay-off  A debt of $8000 is amortized with eight
34. Annuity  Find the future value of an annuity due of semiannual payments of $1288.29 each. If money is
$800 paid at the beginning of every 6-month period worth 12%, compounded semiannually, find the unpaid
for 10 years, if it earns interest at 12%, compounded balance after five payments have been made.
semiannually. 45. Cash value  A woman paid $90,000 down for a house
and agreed to pay 18 quarterly payments of $4500 each.
If money is worth 4%, compounded quarterly, how
much would the house have cost if she had paid cash?

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424 CHAPTER 6 Mathematics of Finance

46. Amortization schedule  Complete the next two lines 55. Kevin Patrick paid off the loan he took out to buy
of the amortization schedule for a $100,000 loan for his car, but once the loan was paid, he continued to
30 years at 7.5%, compounded monthly, with monthly deposit $400 on the first of each month in an account
payments of $699.22. that paid 5.4%, compounded monthly. After four
years of making these deposits, Kevin was ready to
Payment Payment Balance Unpaid buy a new car. How much did he have in the
Number Amount Interest Reduction Balance account?
$104.55 $95,042.20 56. A young couple receive an inheritance of $72,000 that
56 $699.22 $594.67 they want to set aside for a college fund for their two

children. How much will this provide at the end of
MISCELLANEOUS FINANCIAL PROBLEMS each half-year for a period of 9 years if it is deferred
47. If an initial investment of $2500 grows to $38,000 in for 11 years and can be invested at 7.3%, compounded
semiannually?
18 years, what annual interest rate, compounded annu- 57. A bank is trying to decide whether to advertise some
ally, did this investment earn? new 18-month certificates of deposit (CDs) at 6.52%,
48. How much must be deposited at the end of each month compounded quarterly, or at 6.48%, compounded con-
in an account that earns 8.4%, compounded monthly, if tinuously. Which rate is a better investment for the
the goal is to have $40,000 after 10 years? consumer who buys such a CD? Which rate is better
49. What is the future value if $1000 is invested for for the bank?
6 years at 8% (a) simple interest and (b) compounded 58. A divorce settlement of $40,000 is paid in $1000
semiannually? payments at the end of each of 40 months. What is
50. Quarterly payments of $500 are deposited in an the present value of this settlement if money is worth
account that pays 8%, compounded quarterly. How 12%, compounded monthly?
much will have accrued in the account at the end of 59. If $8000 is invested at 12%, compounded continuously,
4 years if each payment is made at the end of each for 3 years, what is the total interest earned at the end
quarter? of the 3 years?
51. If $8000 is invested at 7%, compounded continuously, 60. A couple borrowed $184,000 to buy a condominium.
how long will it be before it grows to $22,000? Their loan was for 25 years and money is worth 6%,
52. An investment broker bought some stock at $87.89 per compounded monthly.
share and sold it after 3 months for $105.34 per share. (a) Find their monthly payment size.
What was the annual simple interest rate earned on this (b) Find the total amount they would pay over
transaction? 25 years.
53. Suppose that a 30-year municipal bond has a maturity (c) Find the total interest they would pay over
value of $5000 and a coupon rate of 8%, with coupons 25 years.
paid semiannually. Find the market price of the bond if (d) Find the unpaid balance after 7 years.
the current yield rate is 10% compounded semiannu- 61. Suppose a salesman invests his $12,500 bonus in a fund
ally. Is this bond selling at a discount or at a premium? that earns 10.8%, compounded monthly. Suppose also
54. A $5000 Mauranol, Inc. corporate bond matures that he makes contributions of $150 at the end of each
in 10 years and has a coupon rate of 7.2% paid month to this fund.
semiannually. (a) IFfinafdtetrhtehfeu1tu2r21eyveaalruse, tahfetefru1n2d12 years.
(a) If the yield rate is 8% compounded semiannually, (b) is used to set up an
annuity, how much will it pay at the end of each
find the bond’s market price. month for the next 10 years?
(b) Three years later the yield rate is 5.8%. If the bond 62. Three years from now, a couple plan to spend
4 months traveling in China, Japan, and Southeast
holder wishes to sell at this time, what is the bond’s Asia. When they take their trip, they would like to
current market price? withdraw $10,000 at the beginning of each month to
(c) If you bought a $5000 Mauranol, Inc. bond at the cover their expenses for that month. Starting now,
price in part (a), collected 3 years of coupons, and how much must they deposit at the beginning of each
sold at the price in part (b), what amount would month for the next 3 years so that the account will
you have earned on your original investment? provide the money they want while they are
(d) Suppose you made a single investment with traveling? Assume that such an account pays 6.6%,
principal equal to the amount paid in part (a), compounded monthly.
and after 3 years you received a single lump sum
return equal to the amount from part (c). What
interest rate compounded semiannually would you
have earned?

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63. At age 22, Aruam Sdlonyer receives a $4000 IRA from Chapter Test 425
her parents. At age 30, she decides that at age 67 she’d 64. A company borrows $2.6 million for 15 years at 5.6%,
like to have a retirement fund that would pay $20,000 at
the end of each month for 20 years. Suppose all invest- compounded quarterly. After 2 years of regular pay-
ments earn 8.4%, compounded monthly. How much ments, the company’s profits are such that manage-
does Aruam need to deposit at the end of each month ment feels they can increase the quarterly payments to
from ages 30 to 67 to realize her goal? $70,000 each. How long will it take to pay off the loan,
and how much interest will be saved?

6 CHAPTER TEST 13. What amount must you invest in an account that earns
6.8%, compounded quarterly, if you want to have $9500
 1. If $8000 is invested at 7%, compounded continuously, after 5 years?
how long will it be before it grows to $47,000? 14. If $1500 is deposited at the end of each half-year in
a retirement account that earns 8.2%, compounded
 2. How much must you put aside at the end of each semiannually, how long will it be before the account
half-year if, in 6 years, you want to have $12,000, contains $500,000?
and the current interest rate is 6.2%, compounded 15. A couple would like to have $50,000 at the end of
semiannually? a4d21 dyietaiorsn.aTlhceoynctrainbuintivoensst $10,000 now and make
at the end of each quarter over
 3. To buy a municipal bond that matures to $10,000 in qthuear4t21eyrleya,rws.hIaftthsiezeinpvaeysmtmeenntst earns 7.7%, compounded
9 months, you must pay $9510. What simple interest their goal? will enable them to reach
rate is earned?
16. A woman makes $3000 contributions at the end of
 4. A debt of $280,000 is amortized with 40 equal each half-year to a retirement account for a period of
semiannual payments of $14,357.78. If interest is 8.2%, 8 years. For the next 20 years, she makes no additional
compounded semiannually, find the unpaid balance of contributions and no withdrawals.
the debt after 25 payments have been made. (a) If the account earns 7.5%, compounded semiannu-
ally, find the value of the account after the 28 years.
 5. If an investment of $1000 grew to $13,500 in 9 years, (b) If this account is used to set up an annuity that
what interest rate, compounded annually, did this pays her an amount at the beginning of each
investment earn? 6-month period for the next 20 years, how much
will each payment be?
6. A couple borrowed $97,000 at 7.2%, compounded 17. Maxine deposited $400 at the beginning of each month
monthly, for 25 years to purchase a condominium. for 15 years in an account that earned 6%, compounded
(a) Find their monthly payment. monthly. Find the value of the account after the
(b) Over the 25 years, how much interest will they pay? 15 years.
18. Grandparents plan to establish a college trust for their
7. If you borrow $2500 for 15 months at 4% simple youngest grandchild. How much is needed now so the
interest, how much money must you repay after the trust, which earns 6.4%, compounded quarterly, will
15 months? provide $4000 at the end of each quarter for 4 years
after being deferred for 10 years?
8. Suppose you invest $100 at the end of each month in an 19. The following sequence is arithmetic: 298.8, 293.3,
account that earns 6.9%, compounded monthly. What 287.8, 282.3, . . . .
is the future value of the account after 521 years? (a) If you were not told that this sequence was arith-
metic, how could you tell that it was?
9. Find the annual percentage yield (the effective (b) Find the 51st term of this sequence.
annual rate) for an investment that earns 8.4%, (c) Find the sum of the first 51 terms.
compounded monthly. 20. A 400-milligram dose of heart medicine is taken daily.
During each 24-hour period, the body eliminates 40%
10. An accountant wants to withdraw $3000 from an of this drug (so that 60% remains in the body). Thus
investment at the beginning of each quarter for the
next 15 years. How much must be deposited originally
if the investment earns 6%, compounded quarterly?
Assume that after the 15 years, the balance is zero.

11. If $10,000 is invested at 7%, compounded continuously,
what is the value of the investment after 20 years?

12. A collector made a $10,000 down payment for a classic
car and agreed to make 18 quarterly payments of $1500
at the end of each quarter. If money is worth 8%, com-
pounded quarterly, how much would the car have cost
if the collector had paid cash?

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

426 CHAPTER 6 Mathematics of Finance had an additional $2000. Would you save more over the
life of the loan by paying an extra $2000 with your 24th
the amount of drug in the bloodstream just after the payment or by paying $1160 per month beginning with
31st dose is given by the 25th payment? To help you answer this question,
400 1 400 1 0.6 2 1 400 1 0.6 2 2 1 400 1 0.6 2 3 1 . . . complete the following.
(a) Find the unpaid balance after 24 payments (both
1 400 1 0.6 2 29 1 400 1 0.6 2 30
Find the level of the drug in the bloodstream at this including and not including the $2000 payment).
time. (b) Determine how long it takes to pay off the loan
21. Suppose that a 10-year corporate bond has a matu-
rity value of $10,000 and a coupon rate of 10%, with with the regular $1142.76 payments and the
coupons paid semiannually. Find the market price of additional $2000 payment. Then find the total
the bond if the current yield rate is 6% compounded interest paid.
semiannually. Is this bond selling at a discount or at a (c) Determine how long it takes to pay off the loan
premium? without the additional $2000 payment, but with
22. If you took a home mortgage loan of $150,000 for monthly payments of $1160 from the 25th pay-
30 years at 8.4%, compounded monthly, your monthly ment. Then find the total interest paid.
payments would be $1142.76. Suppose after 2 years you (d) Which payment method costs less?

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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SIsouTthhweisst HaoGspiotalohdas aDn eopaerla?ting room used only for eye surgery. The annual cost of
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Sourcep:iDtaislcroevveireLwoaintsCednetceirsmioanilearbout the machine discussed in Question 3?

II. Profit Reinvestment

T. C. Hardware Store wants to construct a new building at a second location. If construction
could begin immediately, the cost would be $500,000. At this time, however, the owners do
not have the required 20% down payment, so they plan to invest $2000 per month of their
profits until they have the necessary amount. They can invest their money in an annuity
account that pays 6%, compounded monthly, but they are concerned about the 3% average
inflation rate in the construction industry. They would like you to give them some projec-
tions about how 3% inflation will affect the time required to accrue the down payment and
the building’s eventual cost. They would also like to know how their projected profits, after
the new building is complete, will affect their schedule for paying off the mortgage loan.

Specifically, the owners would like you to prepare a report that answers the following
questions.

1. If the 3% annual inflation rate is accurate, how long will it take to get the down payment?
Hint: Assume inflation is compounded monthly. Then the required down payment will
be 20% of

500,000 a 1 1 0.03 n
12 b

427

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

428 CHAPTER 6 Mathematics of Finance
where n is in months. Thus you must find n such that this amount equals the future
value of the owners’ annuity. The solution to the resulting equation is difficult to obtain
by algebraic methods but can be found with a graphing utility.

2. If the 3% annual inflation rate is accurate (compounded monthly), what will T. C.
Hardware’s projected construction costs be (to the nearest hundred dollars) when it
has its down payment?

3. If T. C. Hardware borrows 80% of its construction costs and amortizes that amount at
7.8%, compounded monthly, for 15 years, what will its monthly payment be?

4. Finally, the owners believe that 2 years after this new building is begun (that is, after
2 years of payments on the loan), their profits will increase enough so that they’ll be
able to make double mortgage payments each month until the loan is paid. Under these
assumptions, how long will it take T. C. Hardware to pay off the loan?

III. Purchasing a Home

Buying a home is the biggest single investment or purchase that most individuals make.
This project is designed to give you some insight into the home-buying process and the
associated costs.

Find all the costs associated with buying a home by making a 20% down payment and
borrowing 80% of the cost of the home.
1. Decide how much your home will cost (and how much you’ll have to put down and

how much to mortgage). Call a realtor to determine the cost of a home in your area
(this could be a typical starter home, an average home for the area, or your dream
house). Ask the realtor to identify and estimate all the closing costs associated with
buying your home. Closing costs include your down payment, attorney’s fees, title fees,
and so on.
2. Call at least three different banks to determine mortgage loan rates for 15 years,
25 years, and 30 years and the costs associated with obtaining your loan. These associ-
ated costs of a loan are paid at closing and include an application fee, an appraisal fee,
points, and so on.
3. For each bank and each different loan duration, develop a summary that contains
(a) an itemization and explanation of all closing costs and the total amount due at

closing
(b) the monthly payment
(c) the total amount paid over the life of the loan
(d) the amortization schedule for each loan (use a spreadsheet or financial software

package).
 4. For each duration (15 years, 25 years, and 30 years), identify which bank gives the best

loan rate and explain why you think it is best. Be sure to consider what is best for some-
one who is likely to remain in the home for 7 years or less compared with someone who
is likely to stay 20 years or more.

428

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.