Chapter 5 The Time Value of Money
Before You Go On Questions and Answers Section 5.1 1.
Why is a dollar today worth more than a dollar one year from now?
A dollar is worth more today than one year from now, due to its potential earning capacity. If you have the money in your hand today, you have the opportunity to invest it and earn interest or you can purchase goods and services for your immediate consumption. Given that people have a positive preference for consumption, time value of money holds true.
2.
What is a time line, and why is it important in financial analysis?
A time line is a horizontal line that starts at time zero (today) and shows cash flows as they occur over time. It is an important tool used to analyze cash flows over ov er certain time periods, as timing of each cash flow has a big impact on the final figure, and therefore on the resulting investment decision.
Section 5.2 1.
What is compounding, and how does it affect the future value of an investment?
Compounding is the process that refers to converting the initial (principal) amount into a future value. In order to obtain the future value of the principal amount, you calculate what the value at the end of the time period will be assuming the initial investment will earn interest, which is reinvested and will earn additional interest in the future periods.
1
2.
What is the difference between simple interest and compound interest?
The difference is the interest earned on interest.
3.
How does changing the compounding period affect the amount of interest earned on an investment?
The more frequent the compounding schedule, the higher the interest earned. For example, $100 invested for one year at 10 percent compounded annually will earn you $10 of interest at the end of the year, but if your bank compounded compoun ded interest quarterly, your earnings from interest would increase to $10.38.
Section 5.3 1.
What is the present value and when is it used?
Present value is the amount a future sum is worth today given a certain return rate. The present value concept should be used when calculating how much money you need toda y in order to reach your financial goal sometime in the future.
2.
What is the discount rate? How does the discount rate differ from the interest rate in the future value equation?
The discount rate is the compound interest rate used to determine the present value of future cash flows. Both discount and interest rates essentially represent the same conc ept. The only difference is the context in which they are used.
3.
What is the relation between the present value factor and the future value factor?
The present value factor is the reverse of the future value factor. To obtain the present value factor, you divide 1 by the future value factor (1 + i).
2
4.
Explain why you would expect the discount factor to become smaller the longer the time to payment.
The discount factor will become smaller the longer the time to payment due to time value of money. The longer you have to wait to obtain the money, mon ey, the less value it will have to you. Mathematically, the discount factor is calculated as 1 /(1 + i)n. The longer the time to payment, the larger n gets, which will make the discount factor smaller.
Section 5.4 1.
What is the difference between the interest rate (i) and the growth rate ( g g ) in the future value equation?
The interest rate and the growth rate in the future value equation essentially represent the same concept. The growth rate is used when we deal with numerical values value s such as sales or change over time. When referring to money mone y being invested, we use the term t erm interest rate.
Self Study Problems 5.1
Amit Patel is planning to invest $10,000 in a bank certificate of deposit (CD) for five
years. The CD will pay interest of 9 percent. What Wh at is the future value of Amit’s investment?
Solution:
Present value of the investment = PV = $10,000 Interest rate = i = 9% Number of years = n = 5.
3
0
1
2
3
4
5
├───┼───┼───┼────┼───┤ -$10,000
FV5=?
FVn PV(1 i )n FV5 $10 $10,000( ,000(1 1 0.09 0.09))5 $15, 386.24 386.24 $15,
5.2
Megan Gaumer expects to need $50,000 as a down payment on a house in six years. How much does she need to invest today in an account paying 7.25 percent?
Solution:
Amount Megan will need in six years = FV6 = $50,000 Number of years = n = 6 Interest rate on investment = i = 7.25% Amount needed to be invested now = PV = ?
0
1
2
3
4
5
6 Year
├───┼───┼───┼────┼───┼───┤ PV = ?
PV
FV6 = $50,000
FVn (1 i ) n $50,000 (1 0.0725) 6
$32,853.85
5.3
Kelly Martin has $10,000 that she can deposit into a savings account for five years. Bank A pays compounds interest annually, Bank B twice a year, and Bank C quarterly. Each
4
bank has a stated interest rate of 4 percent. What amount would Kelly have at the end of the fifth year if she left all the interest paid on the deposit in each bank?
Solution:
Present value = PV = $10,000 Number of years = n = 5 Interest rate = i = 4% Compound period m: A=1 B=2 C=4 Amount at the end of 5 years = FV5 = ?
0
1
2
3
4
5 Year
├───┼───┼───┼────┼───┤ -$10,000
A:
FV5 = ?
FVn
= PV × (1 + i/m)m x n
FV5
= 10,000 × (1 + 0.04/1)1x5 = $12,166.53
B:
FV5
= 10,000 × (1 + 0.04/2)2x5 = $12,189.94
C:
FV5
= 10,000 × (1 + 0.04/4)4x5 = $12,201.90
5.4
You have an opportunity to invest $2,500 today and receive $3,000 in three years. What will be the return on your investment?
Solution:
5
Your investment today = PV = $2,500 Amount to be received = FV3= $3,000 Time of investment = n = 3 Return on the investment = i = ?
0
1
2
3 Year
├───┼────┼───┤ -$2,500
$3,000
FVn PV (1 (1 i ) n $3,00 ,000 $2,50 ,500 (1 i )n $3,000 (1 i )n $2,500 i = 6.27%
5.5
Emily Smith deposits $1,200 in her bank today. toda y. If the bank pays 4 percent pe rcent simple interest, how much money will she have at the end of five years? What if the bank pays compound interest? How much of the earnings will be interest on interest?
Solution:
Deposit today = PV = $1,200 Interest rate = i = 4% Number of years = n = 5 Amount to be received back = FV5 = ?
0
1
2
3
4
5 Year
├───┼───┼───┼────┼───┤ -$1.200
FV5 = ?
a. Future value with simple interest
6
Simple interest per year = $1,200 × (0.04) = $48 Simple interest for 5 years = $48 × 5 = $240 FV5 = $1,200 + $240 = $1,440
b. Future value with compound interest
FV5
= $1,200 × (1 + 0.04)5 = $1,459.98
Simple interest = ($1,440 – $1,200) × 5 = $240 Interest on interest = $1,459.98 – $1,200 – $240 = $19.98
Critical Thinking Questions
5.1
Explain the phrase ―a dollar today toda y is worth more than a dollar tomorrow.‖ The implication is that if one was to receive a dollar today instead of in the future, the dollar could be invested and will be worth more than a dollar tomorrow because of the interest earned during that one day. This makes mak es it more valuable than receiving a dollar tomorrow.
5.2
Explain the importance of a time line.
Time lines are important tools used to analyze investments inv estments that involve cash flow streams over a period of time. They are horizontal lines that start at time zero (today) and show cash flows as they occur over time. Because Be cause of time value of money, mone y, it is crucial to keep track of not only the size, but also the timing of the cash flows.
5.3
What are the two factors to be considered in time value of money?
7
The factors that are critical in time value of money are the size of the cash flows and the timing of the cash flows.
5.4
Differentiate future value from present value.
F uture value value measures what one or more cash flows are worth at the end of a specified
period, while prese present val ue measures what one or more cash flows that are to be received in the future will be worth today (at t = 0).
5.5
Differentiate between compounding and discounting.
The process of converting an amount given giv en at the present time into a future value is called compounding . It is the process of earning interest over time. Discounting is the process of converting future cash flows to what its present present value is. In other words, present value is the current value of the future cash flows that are discounted discounted at an appropriate interest rate.
5.6
Explain how compound interest differs from simple interest.
Suppose I invest $100 for three years at a rate of 10 percent. Simple interest would imply that I will earn $10 for each of the three years for a total of $30 interest. At the end of three years I would have $130. Compound interest recognizes that the interest earned in years 1 and 2 will also earn interest over the remaining period. Thus, the $10 earned in the first year would earn interest at 10 percent for the next two years, and the t he $10 earned in the second year would earn interest for the third year. Thus the total amount that I would have at the end of three years would be: $100(1.10)3 $133.10 . By compounding, I have earned an additional interest of $3.10. The total interest or compound compoun d interest is the $33.10 earned on the $100 invested, while the simple interest earned is equal to $30.
8
5.7
If you were given a choice of investing in an account that paid quarterly interest and one that paid monthly interest, which one should you choose if they both offer the same stated interest rate and why?
The impact of compounding really dictates that one should pick the account that pays interest more frequently (as long as the interest rates are the same). This allows for the interest earned in the earlier periods to earn interest and the investment to grow more.
5.8
Compound rates are exponential over time. Explain.
Growth rates, as well as interest rates, are not linear, but rather expo nential over time. In other words, the growth rate of the invested funds is accelerated by the compounding of interest. Over time, the principal amount you receive interest on will get larger with compounding, thus generating higher interest payments.
5.9
What is the Rule of 72?
This is a rule of thumb to determine de termine how fast an investment can double. It is a rule that allows you to closely approximate the time that it would take to double your money. It works well with interest rates between 5 and 20 percent, but varies more with higher h igher rates. The Rule of 72 says that the time to double your money (TDM) approximately equals 72/i, where i is expressed as a percentage:
5.10
You are planning to take a spring break trip to Cancun your senior year. The trip is exactly two years away, but you want to be prepared and have enough money when the time comes. Explain how you would determine the amount of money you will have to save in order to pay for the trip.
First, determine how much money you will need for the trip. Second, check how much you already have and how ho w it translates into future value cash — how how much it will be worth
in two years. Next, determine how much you will have to deposit today, given the bank’s 9
offered interest rate, to ensure that you will have saved up the difference when the time for your senior spring break comes.
Questions and Problems BASIC
5.1
Future value: Chuck Tomkovick is planning to invest $25,000 today in a mutual fund
that will provide a return of 8 percent each year. What will be the value of the investment in 10 years?
LO 2 Solution:
0
5 years
├────────────────────┤ PV = -$25,000
FV5 = ?
Amount invested today = PV = $25,000 Return expected from investment = i = 8% Duration of investment = n = 10 years Value of investment after 10 years
= FV10
FV10 PV (1 i ) n $25,000 (1.08)10
$53,973.12
5.2
Future value: Ted Rogers is investing $7,500 in a bank CD that pays a 6 percent annual
interest. How much will the CD be worth at the end of five years?
LO 2
Solution:
0
5 years
10
├────────────────────┤ PV = $7,500
FV5 = ?
Amount invested today = PV = $7,500 Return expected from investment = i = 6% Duration of investment = n = 5 years Value of investment after 5 years = FV5
FV5 PV (1 i ) n $7,500 (1.06) 5
$10,036.69
5.3
Future value: Your aunt is planning to invest in a bank deposit that will pay 7.5 percent
interest semiannually. If she has $5,000 to invest, how much will she have at the end en d of four years?
LO 2 Solution:
0
4 years
├────────────────────┤ PV = $5,000
FV4 = ?
Amount invested today = PV = $5,000 Return expected from investment = i = 7.5% Duration of investment = n = 4 years Frequency of compounding = m = 2 Value of investment after 4 years = FV4 i FV4 PV 1 m
mn
0.075 $5, 000 1 2
$5, 000 (1.0375 1.0375))8 $5,
$6,712.35
11
2 4
5.4
Future value : Kate Eden received a graduation present of $2,000 that she is planning on
investing in a mutual fund that earns 8.5 percent each year. How much money can she collect in three years?
LO 2 Solution:
0
3 years
├────────────────────┤ PV = $2,000
FV3 = ?
Amount Kate invested today = PV = $2,000 Return expected from investment = i = 8.5% Duration of investment = n = 3 years Value of investment after 3 years = FV3
FV3 PV (1 i ) n $2,000 (1.085) 3
$2,554.58
5.5
Future value: Your bank pays 5 percent interest semiannually on your savings account.
You don’t expect the current balance of $2,700 to change over the next four years. How much money can you expect to have at the end of this period?
LO 2
Solution:
0
4 years
├────────────────────┤ PV = -$2,700
FV4 = ?
Amount invested today = PV = $2,700 Return expected from investment = i = 5% Duration of investment = n = 4 years Frequency of compounding = m = 2 Value of investment after 4 years = FV4
12
mn
i 0.05 FV4 PV 1 $2,700 1 2 m $2,700 (1.025)8 $3,289.69
5.6
24
Future value : Your birthday is coming up, and instead of other presents, your parents
promised to give you $1,000 in cash. Since you have a part time job and thus don’t need the cash immediately, you decide to invest the money in a bank CD that pays 5.2 percent quarterly for the next two years. How much money can you expect to gain in this period of time?
LO 2
Solution:
0
2 years
├────────────────────┤ PV = -$1,000
FV2 = ?
Amount invested today = PV = $1,000 Return expected from investment = i = 5.2% Duration of investment = n = 2 years Frequency of compounding = m = 4 Value of investment after 2 years = FV2 mn
i 0.052 FV2 PV 1 $1,000 1 4 m $1,000 (1.013) 8 $1,108.86
5.7
42
Multiple compounding periods : Find the future value of an investment of $100,000
made today for five years and paying 8.75 percent for the following compounding periods: a.
Quarterly
b.
Monthly
13
c.
Daily
d.
Continuous
LO 2
Solution:
0
5 years
├────────────────────┤ PV = -$100,000
FV5 = ?
Amount invested today = PV = $100,000 Return expected from investment = i = 8.75% Duration of investment = n = 5 years
a.
Frequency of compounding = m = 4 Value of investment after 5 years = FV5 mn
45
mn
125
mn
36 55
i 0.0875 FV5 PV 1 $100,000 1 4 m $100,000 (1.021875) 20 $154,154.2 4
b.
Frequency of compounding = m = 12 Value of investment after 5 years = FV5 i 0.0875 FV5 PV 1 $100,000 1 12 m $100,000 (1.00729) 60 $154,637.3 7
c.
Frequency of compounding = m = 365 Value of investment after 5 years = FV5 i 0.0875 FV5 PV 1 $100,000 1 365 m $100,000 (1.00024)1825 $154,874.9 1
d.
Frequency of compounding = m = Continuous
14
Value of investment after 5 years = FV5
FV5 PV e in $100,000 e 0.08755
$100,000 1.5488303 $154,883.03
5.8
Growth rates: Joe Mauer, a catcher for the Minnesota Twins, is expected to hit 15 home
runs in 2012. If his home run hitting ability is expected to grow by 12 percent every year for the next five years, how many home runs is he expected to hit in 2017?
LO 4
Solution:
0
5 years
├────────────────────┤ PV = -15
FV5 = ?
Number of home runs hit in 2012 = PV = 15 Expected annual increase in home runs hit = i = 12% Growth period = n = 5 years Expected home runs in 2017 = FV5
FV5 PV (1 g)n 15 (1 (1.12)5
26.4 26.4 26 home home runs runs 5.9
Present value: Roy Gross is considering an investment that pays 7.6 percent. How much
will he have to invest today so that the investment will be worth $25,000 in six years?
LO3 Solution:
0
6 years
├────────────────────┤ PV = ?
FV6 = $25,000
15
Value of investment after 6 years = FV5 = $25,000 Return expected from investment = i = 7.6% Duration of investment = n = 6 years Amount to be invested today = PV
PV
FVn $25,000 n (1 i ) (1.076) 6
$16,108.92
5.10
o ffered a future payment of $750 two years from Present value : Maria Addai has been offered now. If she can earn 6.5 percent compounded annually on her investment, what should she pay for this investment today?
LO3
Solution:
0
2 years
├────────────────────┤ PV = ?
FV2 = $750
Value of investment after 2 years = FV2 = $750 Return expected from investment = i = 6.5% Duration of investment = n = 2 years Amount to be invested today = PV
PV
FVn
1 i
n
$750 (1.065) 2
$661.24
5.11
Present value: Your brother has asked you for a loan and has promised to pay back
$7,750 at the end of three years. If you normally invest to earn 6 percent per year, how much will you be willing to lend to your brother?
LO3
16
Solution:
0
3 years
├────────────────────┤ PV = ?
FV3 = $7,750
Loan repayment amount after 3 years = FV3 = $7,750 Return expected from investment = i = 6% Duration of investment = n = 3 years Amount to be invested today = PV PV
FVn
1 i
n
$7,750 (1.06) 3
$6,507.05
5.12
Present value: Tracy Chapman is saving to buy a house in five years time. She plans to
put down 20 percent down at that time, and she believes that she will need $35,000 for the down payment. If Tracy can invest in a fund that pays 9.25 percent annually, how much will she need to invest today?
LO3 Solution:
0
5 years
├────────────────────┤ PV = ?
FV5 = $35,000
Amount needed for down payment after 5 years = FV5 = $35,000 Return expected from investment = i = 9.25% Duration of investment = n = 5 years Amount to be invested today = PV
17
PV
FVn n
1 i
$35,000 (1.0925) 5
$22,488.52
5.13
Present value : You want to buy some bonds that will have a value of $1,000 at the end
of seven years. The bonds pay 4.5 percent interest annually. How much should you pay for them today?
LO3
Solution:
0
7 years
├────────────────────┤ PV = ?
FV7 = $1,000
Face value of bond at maturity = FV7 = $1,000 Appropriate discount rate = i = 4.5% Number of years to maturity = n = 7 years. Present value of bond = PV
PV
FVn n
1 i
$1,000 (1.045) 7
$734.83 5.14
Present value: Elizabeth Sweeney wants to accumulate $12,000 by the end of 12 years.
If the annual interest rate is 7 percent, how much will she have to invest today tod ay to achieve her goal?
LO3 Solution:
0
12 years
├────────────────────┤ PV = ?
FV12 = $12,000
18
Amount Ms. Sweeney wants at end of o f 12 years = FV12 = $12,000 Interest rate on investment = i = 7% Duration of investment = n = 12 years. Present value of investment = PV PV
FVn
1 i n
$12,000 (1.07)12
$5,328.14
5.15
Interest rate: You are in desperate need of cash and turn to your uncle who has offered
to lend you some money. You decide to borrow $1,300 and agree to pay back $1,500 in two years. Alternatively, you could borrow from your bank that is charging 6.5 percent interest annually. Should you go with your uncle or the bank?
LO 2, LO3
Solution:
0
2 years
├────────────────────┤ PV = -$1,300
FV2 = $1,500
Amount to be borrowed = PV = $1,300 Amount to be paid back after 2 years = FV2 = $1,500 Interest rate on investment = i = ? Duration of investment = n = 2 years. Present value of investment = PV
19
PV $1,300 (1 i) 2
FVn
1 i n $1,500 (1 i ) 2 $1,500
1.1538
$1,300
1.1538 1 i 7.42% i
You should go with the bank borrowing.
5.16
Number of periods : You invest $150 in a mutual fund today that pays 9 percent interest
annually. How long will it take to double do uble your money?
LO 2, LO3
Solution:
0
n years
├────────────────────┤ PV = -$150
FVn = $300
Value of investment today = PV = $150 $1 50 Interest on investment = n = 9% Future value of investment = FV = $300 $3 00 Number of years to double investment = n FVn PV (1 (1 i ) n $300 $300 $150 $150 (1.09 1.09))n (1.09 1.09)) n $300 $300 150 150 2.00 2.00 n ln( ln(1.09 1.09)) ln(2 ln(2.0 .00) 0) n
ln(2.00) 8 years ln(1.09)
20
INTERMEDIATE 5.17
53,2 50 copies in its first year. The publishing Growth rate: Your finance textbook sold 53,250 company expects the sales to grow at a rate of 20 percent each year for the next three years and by 10 percent in the fourth year. Calculate the total number numb er of copies that the publisher expects to sell in years 3 and 4. Draw a time line to show the sales level for each of the next four years.
LO 4 Solution:
Number of copies sold in its first year year = PV = 53,250 Expected annual growth in the next 3 years = g = 20% Number of copies sold after 3 years = FV3 = FVn = PV (1 g )n 53, 250 (1.20) 1.20)3 53,
92,016 copies Number of copies sold in the fourth year = FV4
FVn PV (1 (1 g )n 92, 01 016 (1 (1.10)
101, 218 copies 0
3
4 years
├───────────┼────────┤ PV = -53,250
5.18
92,016
10,218 copies
Growth rate: CelebNav, Inc., had sales last year of $700,000, and the analysts are
predicting a good year for the start up, with sales growing 20 percent a year for the next three years. After that, the sales should grow 11 percent p ercent per year for another two years, at which time the owners are planning to sell the company. What are the projected sales for the last year before the sale?
LO 4
21
Solution:
0
1
2
3
4
5 years
├───────┼────────┼───────┼────────┼───────┤ g1 = 20%
g2 = 11%
PV = -$700,000
FV5=?
Sales of CelebNav last year = PV = $700,000 Expected annual growth in the next 3 years = g 1 = 20% Expected annual growth in years 4 and 5 = g 2= 11% Sales in year 5 = FV5
FV5 PV (1 g1 )3 (1 g2 )2 $700,00 ,000 (1.20)3 (1.11)2 $1, 490,, 348.16 348.16 $1,490
5.19
Growth rate: You decide to take advantage of the current online dating craze and start
your own Web site. You know that you have 450 people who will sign up immediately, and through a careful marketing research and analysis you determine that membership can grow by 27 percent in the first two years, 22 percent in year 3, and 18 percent in year 4. How many members do you expect to have at the end of four years?
LO 4 Solution:
0
1
2
3
4 years
├───────┼────────┼───────┼────────┤ g 1-2 1-2=27%
g 3=22%
PV = -450
g 4=18%
FV4 = ?
Number of Web site memberships at t = 0 = PV = 450 Expected annual growth in the next 2 years = g 1-2 1-2 = 27% Expected annual growth in years 3 = g 3= 22% Expected annual growth in years 4 = g 4= 18% Number of members in year 4 = FV4
22
FV4 PV(1 g1 )2 (1 g3 ) (1 g4 ) 450 (1.27)2 (1.22) (1.18) members 1, 045 members
5.20
Multiple compounding periods : Find the future value of an investment of $2,500 made
today for the following rates and periods: a.
6.25 percent compounded semiannually for 12 years
b.
7.63 percent compounded quarterly for 6 years
c.
8.9 percent compounded monthly for 10 years
d.
10 percent compounded daily for 3 years
e.
8 percent compounded continuously for 2 years
LO 2
Solution:
a.
0.0625 FV12 PV 1 2 $5,232.09
b.
0.0763 FV12 PV 1 4 $3,934.48
c.
0.089 FV12 PV 1 12 $6,067.86
d.
0.010 FV12 PV 1 365 $3,374.51
212
$2,500 (2.0928)
46
$2,500 ( 2.4768)
1210
e.
$2,500 (2.4271)
36 53
$2,500 (1.3498)
FV3 PV e in $3,000 e 0.082
$2,500 1.1735 $2,933.78
23
5.21
Multiple compounding periods : Find the present value of $3,500 under each of the
following rates and periods. a.
8.9% compounded monthly for five years.
b.
6.6% compounded quarterly for eight years.
c.
4.3% compounded daily for four years.
d.
5.7% compounded continuously for three years.
LO3 Solution:
0
n years
├────────────────────┤ PV = ?
a.
FVn = $3,500
Return expected from investment = i = 8.9% Duration of investment = n = 5 years Frequency of compounding = m = 12 Present value of amount = PV PV
b.
FV5 i 1 m
mn
$3,500
$3,500 $2,246.57 1.5579
Return expected from investment = i = 6.6% Duration of investment = n = 8 years Frequency of compounding = m = 4 Present Value of amount = PV
24
125
0.089 1 12
PV
c.
FV8 i 1 m
mn
$3,500
0.066 1 4
48
$3,500 $2,073.16 1.6882
Return expected from investment = i = 4.3% Duration of investment = n = 4 years Frequency of compounding = m = 365 Present Value of amount = PV PV
d.
FV4 i 1 m
mn
$3,500
0.043 1 365
36 54
$3,500 $2,946.96 1.1877
Return expected from investment = i = 5.7% Duration of investment = n = 3 years Frequency of compounding = m = Continuous Present value of amount = PV PV
5.22
FV3 $3,50 ,500 0.057 0.0573 e e in
$3,500 $2, 949.88 .88 $2,949 1.1865
Multiple compounding periods : Samantha is looking to invest some money, so that she
has $5,500 at the end of three years. Which investment should she make given the following choices: a.
4.2% compounded daily
b.
4.9% compounded monthly
c.
5.2% compounded quarterly
d.
5.4% compounded annually
LO2 25
Solution:
0
3 years
├────────────────────┤ PV = ?
a.
FV3 = $5,500
Return expected from investment = i = 4.2% Duration of investment = n = 3 years Frequency of compounding = m = 12 Present value of amount = PV PV
FV3 i 1 m
mn
$5,500
0.042 1 365
36 53
$5,500 $4,848.92 1.1343
Samantha should invest $4,848.92 today to reach her target of $5,500 in three years.
b.
Return expected from investment = i = 4.9% Duration of investment = n = 5 years Frequency of compounding = m = 12 Present value of amount = PV PV
FV3 i 1 m
mn
$5,500 123
0.049 1 12
$5,500 $4,749.54 1.5579
Samantha should invest $4,749.54 today to reach her target of $5,500 in three years.
c.
Return expected from investment = i = 5.2% Duration of investment = n = 3 years Frequency of compounding = m = 4
26
Present Value of amount = PV PV
FV3 i 1 m
mn
$5,500
0.052 1 4
43
$5,500 $4,710.31 1.1677
Samantha should invest $4,710.31 today to reach her target of $5,500 in three years.
Return expected from investment = i = 5.4%
d.
Duration of investment = n = 3 years Frequency of compounding = m = 1 Present value of amount = PV
PV
FV3 $5,500 $4,697.22 3 (1 i ) (1.054) 3
Samantha should invest $4,697.22 today to reach her target of $5,500 in three years. Samantha should invest in choice D.
Time to grow: Zephyr Sales Company has sales of $1.125 million. If the company
5.23
expects its sales to grow at 6.5 percent annually, how long will it be before the company can double its sales? Use a financial calculator to solve this problem.
LO 4 Solution: Enter 6.5% N
Answer:
% g
PMT
11 years
27
-$1.125
$2.250
PV
FV
5.24
Time to grow: You are able to deposit $850 into a bank CD today, and you will
withdraw the money only once the balance is $1,000. If the bank pays 5 percent interest, how long will it take for the balance to reach $1,000?
LO 2, LO3 Solution:
Amount invested today = PV = $850 Expected amount in the future = FV = $1,000 Interest rate on CD = i = 5% To calculate the time needed to reach the target FV, we set up the future value equation. FVn PV (1 i ) n $1,000 $850 (1.05) n $1,000 1.1764 $850 n ln(1.05) ln(1.1764) (1.05) n
n
5.25
ln(1.1764) 3.3 years ln(1.05)
Time to grow: Neon Lights Company is a private company with sales of $1.3 million a
year. Management wants to go public but has to wait until the sales reach $2 million. If the sales are expected to grow 12 1 2 percent annually, when is the earliest that Neon Lights can go public?
LO 4 Solution:
Current level of sales = PV = $1,300,000 Target sales level in the future = FV = $2,000,000 $ 2,000,000 Projected growth rate = g = 12% To calculate the time needed to reach the target FV, we set up the future value equation.
28
FV3 PV (1 g) n $2,000,000 $1,300,000 (1.12) n (1.12) n
$2,000,000 1.5385 $1,300,00
n ln(1.12) ln(1.5385) n
5.26
ln(1.5385) ln(1.12)
3.8 years
Time to grow: You have just inherited $550,000. You plan to save this money and
continue to live off the money that you are earning in your current job. If the $550,000 is everything that you have other than an old car and some beat-up furniture, and you can invest the money in a bond that pays 4.6 percent interest annually. annuall y. How long will it be before you are a millionaire?
LO 2, LO3 Solution:
FVn PV (1 (1 i )n $1, $1, 000,000 000,000 $550,000 $550,000 (1.046) 1.046)n $1,000,000 (1.0046) n $550,000
$1,000,000 n ln(1.046) $550,000
ln
$1,000,000 $550,000
ln n
n
ln(1.046) 0.59784 0.04497
n 13.2 13.29 9 years ears
29
5.27
Growth rates: Xenix Corp had sales of $353,866 in 2011. If management expects its
sales to be $476,450 in three years, what is the rate at which the company’s compan y’s sales are expected to grow?
LO 4 Solution:
Sales in 2008 = PV = $353,866 Expected sales three years from now = $476,450 To calculate the expected sales growth rate, we set up the future value equation. FV3 PV (1 g )3 $476 $476,450 ,450 $353 $353,86 ,866 6 (1 g)3 (1 g )3
$476,450 1.3464 $353,866
g (1.3464)
1
3
1
10.42% 5.28
Growth rate: Infosys Technologies, Inc., an Indian technology company reported net
income of $419 million this year. Analysts expect the company’s earnings to be $1.468 $1.46 8 billion in five years. What is the expected growth rate in the company’s earnings? LO 4 Solution:
Earnings in current year = PV = $419,000,000 Expected earnings five years from now = $1,468,000,000 To calculate the expected earnings growth rate, we set up the future value equation. FV5 PV (1 g )5 $1, $1, 468 468,000,000 ,000,000 $419 $419,000 ,000,000 ,000 (1 g)5 (1 g )5
$1,468,000,000 $419,000,000
g ( 3 .50 36 )
1
5
3.5036
1
28.5%
30
5.29
Present value: Caroline Weslin needs to decide whether to accept a bonus of $1,820
today or wait two years and receive $2,100 then. She can invest at 6 percent. What should she do?
LO3 Solution:
0
2 years
├────────────────────┤ PV = -$1,820
FV2 = ?
Amount to be received in 2 years = FV2 = $2,100 Return expected from investment = i = 6% Duration of investment = n = 2 years Present value of amount today PV =
PV
FV2 $2,100 (1 i ) n (1.06) 2
$1,868.99
Since $1869 is greater than $1,820, $1,820 , Caroline should wait two years unless she needs the money before then.
5.30.
Present value: Congress and the President have decided to increase the Federal tax rate
in an effort to reduce the budget bu dget deficit. Suppose that Caroline Weslin will pay 35 percent of her bonus to the Federal government for taxes if she accepts the bonus today and 40 percent if she receives her bonus in two years. Will the increase in tax rates affect affect her decision?
LO3
31
Solution:
Yes. It will affect her decision.
If Caroline accepts the bonus today, after paying the taxes, she will have $1,820 × (1 - 0.35) = $1,183.00 left over.
If she waits two years and pays the high tax rate, the present value of what she will have left over is only $1,869 × (1 - 0.40) = $1,121.40.
ADVANCED 5.31
You have $2,500 you want to invest in your classmate’s start-up business. You believe the business idea to be great and hope to get $3,700 back at the end of three years. If all goes according to the plan, what will be your return on investment?
LO 2, LO3
Solution:
0
3 years
├────────────────────┤ PV = -$2,500
FV3 = $3,700
Amount invested in project = PV = $2,500 $2,5 00 Expected return three years from now = FV =$3,700 To calculate the expected rate of return, we set up the future value equation. FV3 PV (1 i ) 3 $3,700 $2,500(1 i ) 3 (1 i ) 3
$3,700 1.4800 $2,500
i (1.4800)
1
3
1 0.1396
13.96%
32
5.32
Patrick Seeley has $2,400 that he is looking to invest. His brother approached him with an investment opportunity that could double his money m oney in four years. What interest rate would the investment have to yield in order for Patrick’s brother to deliver d eliver on his promise?
LO 2, LO3
Solution:
0
4 years
├────────────────────┤ PV = -$2,400
FV4 = $4,800
Amount invested in project = PV = $2,400 $2,4 00 Expected return three years from now = FV =$4,800 Investment period = n = 4 years To calculate the expected rate of return, we set up the future value equation. FV4 PV (1 (1 i )4 $4,80 ,800 $2,40 ,400 (1 i )4 (1 i ) 4 i
$4,800 1.4800 $2,400 1
(2.000) 4 1 0.1892 (2.
18.92% 5.33
You have $12,000 in cash. You can deposit it today in a mutual fund earning 8.2 percent semiannually; or you can wait, enjoy some of it, and invest $11,000 in your brother’s business in two years. Your brother is promising promising you a return of at least 10 percent on your investment. Whichever alternative you choose, you will need to cash in at the end of 10 years. Assume your brother is trustworthy and that both investments carry the same risk. Which one will you choose?
33
LO 2
Solution:
Option A: Invest in account paying 8.2 percent semiannually for 10 years. 0
10 years
├────────────────────┤ PV = -$12,000
FV10 = ?
Amount invested in project = PV = $12,000 $12 ,000 Investment period = n = 10 years Interest earned on investment = i = 8.2% Frequency of compounding = m = 2 Value of investment after 10 years = FV10
0.082 FV10 PV 1 2 $26,803.77
210
$12,000 (2.23365)
Option B: Invest in brother’s business to earn 10 percent for eight years. 0
8 years
├────────────────────┤ PV = -$11,000
FV8 = ?
Amount invested in project = PV = $11,000 $11 ,000 Investment period = n = 8 years Interest earned on investment = i = 10% Frequency of compounding = m = 1 Value of investment after 8 years = FV10
FV8 PV 1 0.10 $11,000 ( 2.14359) 8
$23,579.48
34
You are better off investing today in the mutual fund and earn 8.2 percent semiannually for 10 years.
5.34
When you were born, your parents set up a bank account in your name with an initial investment of $5,000. You are turning 21 in a few days and will have access to all your funds. The account was earning 7.3 percent for the first seven years, and then the rates went down to 5.5 percent for six years. The economy was doing well in the early 2000s and your account earned 8.2 percent three years in a row. Unfortunately, the next two years you only earned 4.6 percent. Finally, as the economy recovered, your return jumped to 7.6 percent for the last three years. a.
How much money was in your account before the rates went down drastically (end of year 16)?
b.
How much money is in your account now, end of year 21?
c.
What would be the balance now if your parents made another deposit of $1,200 at the end of year 7?
Solution:
0
1
7
13
14
15
16
21 years
├───┼∙∙∙∙∙∙∙∙∙∙┼∙∙∙∙∙∙∙∙∙∙∙∙────┼────┼───┼───∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙──┤ PV = -$5,000 i1 = 7.3%
a.
FV21 = ? i2 = 5.5%
i3 = 8.2%
i4 = 4.6%
i5 = 7.6%
Initial investment = PV = $5,000 Interest rate for first 7 years = i1 = 7.3% Interest rate for next 6 years = i2 = 5.5% Interest rate for next 3 years = i3 = 8.2% Investment value at age 16 years = FV16 FV16 PV (1 i1 ) 7 (1 i2 ) 6 (1 i3 ) 3 7
$5,000 1 0.073 (1.055) 6 (1.082) 3 $5,000 (1.6376) (1.3788) (1.2667) $14,300.55
35
b.
Interest rate for from age 17 to 18 = i4 = 4.6% Interest rate for next 3 years = i5 = 7.6% Investment at start of 16th year = PV = $14,300.55 Investment value at age 21 years = FV21 FV21 FV16 (1 i4 ) 2 (1 i5 ) 3 2
$14,300.55 1 0.046 (1.076) 3 $14,300.55 (1.0941) (1.2458)) $19,492.38
c.
Additional investment at start of 8th year = $1,200 Total investment for next 6 years = $8,187.82 + $1,200 = $9,387.82 Interest rate for next 6 years = i2 = 5.5% Interest rate for years 13 to 16 = i3 = 8.2% Interest rate for from age 17 to 18 = i4 = 4.6% Interest rate for next 3 years = i5 = 7.6% Investment value at age 21 = FV21 FV21 FV7 (1 (1 i2 )6 (1 i3 ) 3 (1 i4 )2 (1 i5 )3 2
$9,387. 7.82 82 (1.05 1.055 5)6 (1.08 1.082) 2)3 1.04 1.046 6 (1.07 1.076) 6)3 $9,38 $9387.82 (1.3788 1.3788)) (1.2667) 1.2667) (1.0941 1.0941)) (1.2458) 1.2458) $9387.82 349.16 $22, 349.16
LO 2 5.35
Sam Bradford, a number 1 draft pick of the St. Louis Rams, and his agent are evaluating three contract options. Each option offers a signing bonus and a series of payments over ove r the life of the contract. Bradford uses a 10.25 percent rate of return to evaluate ev aluate the contracts. Given the cash flows for each of the following options, which one should he choose?
Year
Cash Flow Type
0
Signing Bonus
Option A
$3,100,000
36
Option B
$4,000,000
Option C
$4,250,000
1
Annual Salary
$ 650,000
$ 825,000
$ 550,000
2
Annual Salary
$ 715,000
$ 850,000
$ 625,000
3
Annual Salary
$ 822,250
$ 925,000
$ 800,000
4
Annual Salary
$ 975,000
$1,250,000
$ 900,000
5
Annual Salary
$1,100,000
6
Annual Salary
$1,250,000
$1,000,000
LO3 Solution:
To decide on the best contract c ontract from Sam Bradford’s viewpoint, we need to find the present value of each option. The contract with the highest present value should be the one chosen.
Option A:
Discount rate to be used = i= 10.25% Present value of contract = PVA PVA $3,100,000
$650,000 $715,000 $822,250 $975,000 $1,100,000 $1,250,000 (1.1025)1 (1.1025) 2 (1.1025) 3 (1.1025) 4 (1.1025) 5 (1.1025) 6
$3,100,000 $589,569 $588,232 $613,576 $659,918 $675,305 $696,047 $6,922,647 Option B:
Discount rate to be used = i= 10.25% Present value of contract = PVB PVB $4,000,000
$825,000 $850,000 $925,000 $1,250,000 (1.1025)1 (1.1025) 2 (1.1025) 3 (1.1025) 4
$4,000,000 $748,299 $699,297 $690,249 $846,049 $6,983,894
Option C:
Discount rate to be used = i= 10.25% Present value of contract = PVC
37
PVC $4,250,000
$550,000 $625,000 $800,000 $900,000 $1,000,000 (1.1025)1 (1.1025) 2 (1.1025) 3 (1.1025) 4 (1.1025) 5
$4,250,000 $498,866 $514,189 $596,972 $609,155 $613,913 $7,083,096 Option C is the best choice for Sam Bradford.
5.36
Surmec, Inc., reported earnings of $2.1 million last year. The company’s compan y’s primary business line is the manufacture of nuts and bolts. Since this is a mature mature industry, the analysts are confident that the sales will grow at a steady rate of 7 percent a year for as far as they can tell. The company’s net income equals 23 percent of sales. Management would like to buy a new fleet of trucks but can only do so once the profit reaches $620,000 a year. At the end of what year will they be able to buy the trucks? What will sales and net income be in that year?
LO 4 Solution:
Current level of sales for Surmec = PV = $2,100,000 $2,100 ,000 Profit margin = 23% Net Income for the year = 0.23 x $2,100,000 = $483,000 Target profit level in the future = FV = $620,000 $620,00 0 Projected growth rate of sales = g = 7% To calculate the time needed to reach the target FV, we set up the future value equation. FVn PV (1 g) n $620,000 $483,000 (1.07) n (1.07) n
$620,000 1.2836 $483,00
n ln(1.07) ln(1.2836) n
ln(1.2836) 3.7 years ln(1.12)
The company achieves its profit target during the fourth year. Sales level at end of year 4 = FV4
FVn PV (1 g) n
$2,100,000 (1.07) 4 $2,752,671 .62
38
Profit for the year = $2,752,671.62 x 0.23 = $633,114.47
5.37
You are graduating in two years and you start thinking about your future. You know that you will want to buy a house ho use five years after you graduate and that you will want to put down $60,000. As of right now, you have $8,000 in your savings account. You are also fairly certain that once you graduate, you can work in the family business and earn $32,000 a year, with a 5 percent raise every year. You plan to live with your parents for the first two years after graduation, which will enable you to minimize your expenses and put away $10,000 each year. The next three years, you will have to live out on your own, as your younger sister will be graduating from college and has already announced her plan to move back into the family house. Thus, you will only be able to save 13 percent of your annual salary. Assume that you will be able to invest savings from your salary at 7.2 percent. What is the interest rate at which you need to invest the current savings account balance in order to achieve your goal? Hint: Draw a time line that shows all the cash flows for years 0 through 7. Remember, you want to buy a house h ouse seven years from now and your first salary will be in year 3.
LO 2 Solution:
0
1
2
3
4
5
6
7
├─────┼──────┼─────┼─────┼──────┼─────┼──────┤ $10,000
$10,000
$4,586.40
$4,815,72 $5,056.48
Starting salary in year 3 = $32,000 Annual pay increase = 5% Savings in first 2 years = $10,000 Savings rate for years 3 to 7 = 13% Year
1
2
3
4
5
6
7
Salary
$0
$0
$32,000
$33,600
$35,280
$37,044
$38,896
Savings
$0
$0
$10,000
$10,000
$4,586.40
$4,815.72
$5,056.48
39
Investment rate = i = 7.2% Future value of savings from salary = FV7 FV7 $0 $0 $10,000 (1.072) 4 $10,000 (1.072) 3 $4,586.40 (1.072) 2
$4,815.72 (1.072)1 $5,056.48 (1.072) 0 $13,206.24 $12,319.25 $5,267.86 $5,162.45 $5,056.48 $41,012.28 Target down payment = $60,000 Amount needed to reach target = $60,000 - $41,012.28 = FV = $18,987.72 Current savings balance = PV $8,000 Time to achieve target = n = 7 years. To solve for the investment rate needed to achieve target, we need to set up the future value equation: FV PV (1 i ) 7 $18,987.72 $8,000 (1 i ) 7 (1 i ) 7
$18,987.72 2.3735 $8,000
i ( 2.3735)
17
1
1.1314 1 13.14%
Sample Test Problems 5.1
Santiago Hernandez is planning to invest $25,000 in a money market account for two years. The account pays an interest of 5.75 percent compounded on a monthly basis. How much will Santiago Hernandez have at the end of two years?
LO 2 Solution:
0
2 years
├────────────────────┤ PV = -$25,000
FV = ?
40
Amount invested today = PV = $25,000 Return expected from investment = i = 5.75% Duration of investment = n = 2 years Frequency of compounding = m = 12 Value of investment after 2 years = FV2 mn
i 0.0575 FV2 PV 1 $25,000 1 12 m $25,000 (1.1216) 24 $28,039.13
5.2
122
Michael Carter is expecting an inheritance of $1.25 $1.2 5 million in four years. If he had the money today, he could earn interest at an annual rate of 7.35 percent. What is the present value of this inheritance?
LO3 Solution:
0
4 years
├────────────────────┤ PV = ?
FV = $1,250,000
Amount needed for down payment after 4 years = FV4 = $1,250,000 Return expected from investment = i = 7.35% Duration of investment = n = 4 years Amount to be invested today = PV PV
FVn
1 i n
$1,250,000 (1.0735) 4
$941,243.1 3
5.3
What is the future value of an investment of $3,000 after three years with compounding at the following rates and frequencies? a.
8.75% compounded monthly.
41
b.
8.625% compounded daily.
c.
8.5% compounded continuously.
LO 2
Solution: a.
Interest rate on investment = i = 8.75% Frequency of compounding = m = 12 Value of investment after 3 years = FV3 i FV3 PV 1 m
123
mn
0.0875 $3, 000 1 12
$3, $3, 000 (1.00729) 1.00729)36 $3,896.82
b.
Frequency of compounding = m = 365 Value of investment after 3 years = FV3 mn
i 0.08625 FV3 PV 1 $3,000 1 365 m $3,000 (1.000236)1095 $3,885.81
c.
36 53
Frequency of compounding = m = Continuous Value of investment after 3 years = FV3
FV3 PV e in $3,000 e 0.08 53
$3,000 1.29046 $3,871.38
5.4.
Twenty-five years ago, Amanda Cortez invested $10,000 in an account paying an annual interest rate of 5.75 percent. What is the value of the investment today? What is the interest-on-interest earned on this investment?
LO3 Solution:
0
25 years
├────────────────────┤
42
PV = -$10,000
FV = ?
Amount invested today = PV = $10,000 Return expected from investment = i = 5.75% Duration of investment = n = 25 years Frequency of compounding = m = 1 Value of investment after 25 years = FV25
FV25 PV (1 i ) 25 $10,000(1.0575) 25
$40,458.46 Simple interest on investment = $10,000 × 0.0575 × 25 = $14,375 FV25 = $24,375 Interest-on-interest = $40,458.46 – $24,375 = $16,083.46”
5.5
You just bought a corporate bond at $863.75 today. In five years the bond will mature and you will receive $1,000. What is the rate of return on this bond? bon d?
LO 2, LO3
Solution:
0
5 years
├────────────────────┤ PV = -$863.75
FV = $1,000
Amount to be borrowed = PV = $863.75 Amount to be paid back after 5 years = FV5 = $1,000 Years to maturity = n = 5 years. Interest rate on investment = i Present value of investment = PV
43
PV $863.75 (1 i ) 5
FVn
1 i n $1,000 (1 i ) 5 $1,000 $863.75
i (1.1577)
1.1577 15
1
i 2.97%
The rate of return on this bond is 2.97 percent
44
