Topic: Compound Interest Level of difficulty: Easy Solution: The payment of compound interest means that we must compound or find the future value of the amount invested the present valu
Trang 1Chapter 5: Time Value of Money
Multiple Choice Questions
1 What is the total amount accumulated after three years if someone invests $1,000 today with a simple annual interest rate of 5 percent? With a compound annual interest rate of 5 percent?
Simple interest rate: $1,000 + ($1,000)(5%)(3) = $1,150
Compound interest rate: $1,000(1.05)3 = $1,158
2 Which of the following has the largest future value if $1,000 is invested today?
A Five years with a simple annual interest rate of 10 percent
B 10 years with a simple annual interest rate of 8 percent
C Eight years with a compound annual interest rate of 8 percent
D Eight years with a compound annual interest rate of 7 percent
Level of difficulty: Easy
Therefore, C is the largest
Interest rates in the following questions are compound rates unless otherwise stated
3 Suppose an investor wants to have $10 million to retire 45 years from now How much would she have to invest today with an annual rate of return equal to 15 percent?
Trang 24 Which of the following is false?
A The longer the time period, the smaller the present value, given a $100 future value and
holding the interest rate constant
B The greater the interest rate, the greater the present value, given a $100 future value and holding the time period constant
C A future dollar is always less valuable than a dollar today if interest rates are positive
D The discount factor is the reciprocal of the compound factor
Level of difficulty: Medium
Solution: B The greater the interest rate, the smaller the present value, given a $100 future value and holding time period constant
5 Maggie deposits $10,000 today and is promised a return of $17,000 in eight years What is the implied annual rate of return?
Trang 37 Which of the following concepts is incorrect?
A An ordinary annuity has payments at the end of each year
B An annuity due has payments at the beginning of each year
C A perpetuity is considered a perpetual annuity
D An ordinary annuity has a greater PV than an annuity due, if they both have the same periodic payments, discount rate and time period
Level of difficulty: Medium
Solution: D The annuity due has a greater PV because it pays one year earlier than ordinary
annuity
8 Jan plans to invest an equal amount of $2,000 in an equity fund every year-end beginning this year The expected annual return on the fund is 15 percent She plans to invest for 20 years How much could she expect to have at the end of 20 years?
20
15
1)15.1(000,2
PV
n
)1(
11
15
)15.1(
11000,2
Trang 4expected annual rate of return equal to 12 percent?
A How much interest will he owe his parents after one year?
B How much will he owe, in total, after three years?
Topic: Simple Interest
Level of difficulty: Easy
Solution:
A In one year you will own P x k = $1500 x 6% = $90 of interest
B After three years, the total (principal and interest) owing will be: P + (n x P x k) = $1500 + (3 x $1500 x 6%) = $1770
12 Your sister has been forced to borrow money to pay her tuition this year If she makes annual payments on the loan at year end for the next three years, and the loan is for $2,500 at a simple interest rate of 6 percent, how much will she pay each year?
Topic: Simple Interest
Level of difficulty: Easy
Solution:
As the exact amount of interest owing each year will be paid, there is no “compounding.” The amount of each annual payment will be P x k = $2500 x 6% = $150 Unfortunately, these payments never reduce the principal owing, so the loan will never be paid off!
13 Khalil’s summer job has given him $1,200 more than he needs for tuition this year The local bank pays simple interest at a rate of 0.5 percent per month How much interest will he earn
in one year?
Topic: Simple Interest
Level of difficulty: Easy
Solution:
Khalil will be paid interest each month for 12 months, but without compounding The total interest earned is (n x P x k) = (12 x $1200 x 0.5%) = $72
Trang 514 A new Internet bank pays compound interest of 0.5 percent per month on deposits How much interest will Khalil’s summer savings of $1,200 earn in one year with this online bank account?
Topic: Compound Interest
Level of difficulty: Easy
Solution:
The payment of compound interest means that we must compound (or find the future value)
of the amount invested (the present value):
01.1274
$)005.01(1200
15 History tells us that a group of Dutch colonists purchased the island of Manhattan from the
Native American residents in 1626 Payment was made with wampum (likely glass beads and
trinkets), which had an estimated value of $24 Suppose the Dutch had invested this money back home in Europe and earned an average return of 5 percent per year How much would this investment be worth today, 380 years later, using:
A Simple interest?
B Compound interest?
Topic: Simple and Compound Interest
Level of difficulty: Easy
$)08.1(
12500
$)1(
1
1 1
PV
Or using a financial calculator (TI BAII Plus),
N=1, I/Y=8, PMT=0, FV= -2500, CPT PV=2314.81
Trang 617 Grace, a retired librarian, would like to donate some money to her alma mater to endow a
$4,000 annual scholarship The university will manage the funds, and expects to earn 7 percent per year How much will Grace have to donate so that the endowment fund never runs out?
142,
57
$07
0
14000
$
1
k PMT
PV
18 Grace decides that creating a perpetual scholarship is too costly (see Problem 17Error!
Reference source not found.) Instead, she would like to support the education of her
favourite grand-nephew, Stephen, who plans to begin university in three years How much will Grace have to invest today, at 7 percent, to be able to give Stephen $4,000 at the end of each year for four years?
Topic: Ordinary Annuities
Level of difficulty: Easy
Solution:
Find the present value of the four-year annuity at year 3:
85.548,13
$07
.0
)07.01(
11
4000
$)1(
11
PV
n
Now, find the present value of this amount today:
90.059,11
$)07.1(
185
.548,13
$)1(
1
3 3
PV
19 Bank A pays 7.25 percent interest compounded semi-annually, Bank B pays 7.20 percent compounded quarterly, and Bank C pays 7.15 percent compounded monthly Which bank pays the highest effective annual rate?
Topic: Effective Annual Rates
Level of difficulty: Easy
Solution:
2
0725
01
Trang 7For Bank B, 17.40%
4
0720
01
Bank B pays the highest effective annual rate
20 Jimmie is buying a new car His bank quotes a rate of 9.5 percent per year for a car loan Calculate the effective annual rate if the compounding occurs:
A Annually
B Quarterly
C Monthly
Topic: Determining Effective Annual Rates
Level of difficulty: Easy
Solution:
A For annual compounding, the effective annual rate will be the same as the quoted rate To check this:
%5.91
%5.9111
B With quarterly compounding, set m=4,
%84.914
%5.91
%5.91
Topic: Effective vs Quoted Rates
Level of difficulty: Easy
Solution:
A k Quoted Rate6%FV1year PV0(1k)$50,000(1.06)$53,000
Trang 8Level of difficulty: Easy
Solution:
50.780,885
$)10.01(000,550
$12.0
Each share is worth $16.67
24 Mary-Beth is planning to live in a university residence for four years while completing her degree The annual cost for food and lodging is $5,800 and must be paid at the start of each school year What is the total present value of Mary-Beth’s residence fees if the discount rate (interest rate) is 6 percent per year?
Topic: Annuities Due
Level of difficulty: Easy
Solution:
Because the fees are paid at the start of the year, this is not an ordinary annuity, but rather, an annuity due
Trang 9$)06.01(06
.0
)06.01(
11
25 Calculate the effective annual rates for the following:
A 24 percent, compounded daily
B 24 percent, compounded quarterly
C 24 percent, compounded every four months
D 24 percent, compounded semi-annually
E 24 percent, compounded continuously
F Calculate the effective monthly rate for A to D
Level of difficulty: Medium
Solution:
365
24 1
Trang 10A one year?
B five years?
C ten years?
Topic: Compound Interest
Level of difficulty: Medium
Topic: Compound Interest
Level of difficulty: Medium
Solution:
First find the value of the investment after each period of time, and then compare to the values from Problem 26 to determine how much difference a small change in the interest rate can make
A FV1year $20,000(10.105)1$22,100.00 You are ($22,100 – $22,000) = $100 better off after one year
B FV5years $20,000(10.105)5$32,948.94 You are ($32,948.94 – $32,210.20) = $738.74 better off after five years
C FV10years $20,000(10.105)10 $54,281.62 You are ($54,281.62 – $51,874.85) = $2,406.77 better off after 10 years
28 When Jon graduates in three years, he wants to throw a big party, which will cost $800 To have this amount available, how much does he have to invest today if he can earn a compound return of 5 percent per year?
Trang 11$)05.1(
1800
$)1(
1
3 3
A simple interest at 5 percent per year?
B compound interest at 5 percent per year?
Topic: Simple and Compound Interest
Level of difficulty: Medium
$)(
of the year
Topic: Ordinary Annuities
Level of difficulty: Medium
Solution:
The future value amount is $40,000 The amount to be saved each year is really the payment
on an ordinary annuity:
71.3898
$07
.0
1)07.01(000
Topic: Ordinary Annuities (Solving for IRR)
Trang 12Level of difficulty: Medium
Solution:
Solve for the interest rate (or internal rate of return) on an ordinary annuity This is quite
difficult to do algebraically, but is easily handled with a financial calculator (TI BAII Plus)
Note that we must use a negative sign for the annual payment (savings) or the future value amount, but not both
N=8, PMT = 3,000, PV=0, FV= –40,000, CPT I/Y=14.2067%
32 Shortly after John was born, his parents began to put money in a savings account to pay for his post-secondary education They save $1,000 each year, and earn a return of 9 percent per year However, the interest income is taxed each year at a rate of 30 percent How much will John’s account be worth after 17 years?
Topic: Ordinary Annuity (Future Value)
Level of difficulty: Medium
Solution:
Taxation of the interest income has the effect of reducing the rate of return In
effect:k9%(10.30)6.3% Using this rate of return we find the future value of John’s
063.0
1)063.01(1000
33 John’s parents used a regular savings account to save for his post-secondary education Based
on the amount accumulated (from your answer in Problem 320), how much can John withdraw from the account at the beginning of each year for his four years at university? The account will continue to earn 9 percent per year, but interest income is taxed at a rate of 30 percent
Topic: Annuities Due
Level of difficulty: Medium
Solution:
As in Problem 32, k9%(10.30)6.3% because of taxation John’s withdrawals at the beginning of each year are essentially “payments” on an annuity due:
95.7919
$)
063.01(063.0
)063.01(
11
11.973
Trang 13payable on the interest income This RESP account provides a return of 6 percent per year
A How much will Jane’s account be worth when she begins her university studies?
B As an incentive to save for higher education, the government will add 20 percent to any money contributed to an RESP each year Including these grants, how much will Jane have in her account?
Topic: Ordinary Annuity (Future Value)
Level of difficulty: Medium
Solution:
A The future value of Jane’s account will be:
88.212,28
$06
.0
1)06.01(1000
$06
.0
1)06.01(1200
Topic: Annuities Due
Level of difficulty: Medium
Solution:
Each year, Jane can withdraw:
36.9217
$)
06.01(06.0
)06.01(
11
46.855
$)15.01(
%
36
36 Stephen has learned that his great-aunt (see Problem 18Error! Reference source not found.)
intends to give him $4,000 each year he is studying at university Tuition must be paid in advance, so Stephen would like to receive his payments at the beginning of each school year
Trang 14How much will his great-aunt have to invest today at 7 percent, to make the four annual (start-of-year) payments?
Topic: Annuities Due
Level of difficulty: Medium
Solution:
Find the present value of the four-year annuity due:
26.497,14
$)07.01(07
.0
)07.01(
11
4000
$)1()1(
PV
n
Now, discount this amount back three years:
08.834,11
$)07.1(
126
.497,14
$)1(
1
3 3
PV
37 Rather than give her grand-nephew some money each year while he is studying, Stephen’s great-aunt has decided to save the money and pay off Stephen’s student loans when he finishes his degree The total amount owing at that time will be $16,000 How much will she have to save each year until that time if her investments earn a return of 7 percent per year? Topic: Ordinary Annuities
Level of difficulty: Medium
Solution:
Assuming the savings are invested at the end of each year, find the payment (amount to be saved) for an ordinary annuity with a future value of $16,000
85.848,107
.0
1)07.01(000
,16
Topic: Effective Interest Rates and Loan Arrangements
Level of difficulty: Medium
Solution:
First, find the effective interest corresponding to the frequency of Jimmie’s car payments (f
=12); with monthly compounding, set m=12,
%7083.0112
%5.8111
12 12
m
QR k
The 60 car payments form an “annuity” whose present value is the amount of the loan (the price of the car):
Trang 15$007083
.0
)007083
01(
11
39 Create an amortization schedule for Jimmie’s car loan (see Problem 38Error! Reference
source not found.) What portion of the first monthly payment goes towards repaying the
principal amount of the loan? What portion of the last monthly payment goes towards the principal?
Topic: Loan Arrangements
Level of difficulty: Medium
(3) Interest
=k*(1)
(4) Principal Repayment
= (2)-(3)
Ending Principal
35 14,083.18 594.98 99.76 495.22 13,587.95
36 13,587.95 594.98 96.25 498.73 13,089.22
37 13,089.22 594.98 92.72 502.26 12,586.96
59 1,177.43 594.98 8.34 586.64 590.79
60 590.79 594.98 4.18 590.79 0.00 The first monthly payment repays $389.56 of the principal amount of the loan and the last payment repays $590.79
40 Using the amortization schedule from Problem 39, determine how much Jimmie still owes on the car loan after three years of payments on the five-year loan What is the present value of this amount?
Trang 16Topic: Loan Arrangements and Discounting
Level of difficulty: Medium
Solution:
After three years, or 36 monthly payments, the principal outstanding is $13,089.22 (from the amortization table) The present value of this amount is:
30.152,10
$)007083
01(
122
.089,13
Topic: Loan Arrangements
Level of difficulty: Medium
Solution:
Use the effective monthly interest rate from Problem 38, k=0.7083%
Find the present value of Jimmie’s 36 payments:
95.847,18
$007083
.0
)007083
01(
11
98.594
$95.847,18
$00
42 Jimmie is offered another loan of $29,000 that requires 60 monthly payments of $588.02 (see
Problem 38Error! Reference source not found.) What is the effective annual interest rate
on this loan? What would the quoted rate be?
Topic: Loan arrangements and Effective Annual Rates
Level of difficulty: Medium
Solution:
The 60 monthly payments form an annuity whose present value is $29,000 Finding the
interest rate is most easily done with a financial calculator (TI BAII Plus):
N=60, PMT=588.027, PV= -29,000, CPT I/Y = 0.6667%
Note that we used N=60 months, so the solution is a monthly interest rate, however, the
problem asks for the effective annual rate
%30.81)006667
01(1)1
Trang 17nominal rate would the bank quote for this loan?
Topic: Determining Rates of Return and Effective Interest Rates
Level of difficulty: Medium
Solution:
Solve the annuity equation to find k, the interest rate:
?)
1(
1124.6935
$00.000
The calculations are most easily done with a financial calculator (TI BAII Plus),
A What is the effective monthly rate on this loan?
B With monthly compounding, what is the nominal (annual) interest rate on this loan? Topic: Determining Rates of Return and Effective Interest Rates
Level of difficulty: Medium
Solution:
A There will be 5 x 12 = 60 monthly payments The calculations are most easily done with a
financial calculator (TI BAII Plus),
%0.1
Trang 1845 Compare the loans in problems 043 and 440 Which is the better deal, and why?
Topic: Effective Interest Rates Level of difficulty: Medium Solution:
The two loans have the same principal amount; one requires monthly payments, the other annual, so they cannot be compared on that basis We need an effective rate for the same period of time for comparison In Problem 43, the effective annual rate was 12% For Problem 44, the effective annual rate is:
%7.121)01.01(1)1
46 After losing money playing on-line poker, Scott visits a loan shark for a $750 loan To avoid
a visit from the “collection agency,” he will have to repay $800 in just one week
A What is the nominal interest rate per week? Per year?
B What is the effective annual interest rate?
Topic: Effective Interest Rates Level of difficulty: Medium Solution:
A You will pay interest of (800–750) = $50 after one week This implies a nominal interest rate of 50/750 = 6.67% per week With 52 weeks in the year, the nominal rate per year is then
A What is the effective annual interest rate?
B What is the effective monthly interest rate?
C How much will Josephine’s monthly mortgage payments be?
Topic: Mortgage Loans and Effective Interest Rates Level of difficulty: Medium
Solution:
A In Canada, fixed-rate mortgages use semi-annual compounding of interest, so m=2 The effective annual rate is therefore: