Chapter 3 The Time Value of Money (Part 1) LEARNING OBJECTIVES (Slides 3-­‐1 to 3-­‐2) 1. Calculate future values and understand compounding.
2. Calculate present values and understand discounting.
3. Calculate implied interest rates and waiting time from the time value of money
equation.
4. Apply the time value of money equation using a formula, a calculator, and a
spreadsheet.
5. Explain the Rule of 72, a simple estimation of doubling values.
IN A NUTSHELL…. This chapter is the first in a two-part unit on the time value of money. It provides an
introduction to financial mathematics, a sound understanding of which is imperative for a
student to do well in the later, more complex topics of corporate finance. The students
should be well grounded in measuring future values, present values, interest rates, and
number of periods necessary to achieve various cash flow streams.
The author covers the compounding and discounting techniques associated with single
cash outflows and inflows over a single period of time such as a year as well as over
multiple periods of time. He shows how time value problems associated with single cash
flow streams are a function of four variables i.e. the present value (PV), future value
(FV), interest rate (i), and number of periods (n) involved, and can be solved by hand
using a formula, a financial calculator, a spreadsheet program such as Excel, or time
value of money tables. It is important to let students know that tables are fairly limited in
scope, since they only provide interest factors for discrete time and interest rate intervals,
and that proper knowledge of using a financial calculator or a spreadsheet is the key to
doing these problems quickly and accurately. The author ends with a discussion of some
common applications of time value functions, such as planning for retirement, and the
Rule of 72, which is used for estimating the number of years or the required rate of return
necessary to double a sum of money.
LECTURE OUTLINE 3.1 Future Value and Compounding Interest (Slides 3-­‐3 to 3-­‐11) When a sum of money is invested into an account which pays a rate of interest for the
period during which the money is invested, the value of money at the end of the stated
period is called the future or compound value of that sum of money. This type of
calculation is done when determining the attractiveness of alternative investments or
48 ©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 49 when one is trying to figure out the effect of inflation on the future cost of assets such as
a car or a house. In the latter case, we use the expected annual inflation rate instead of the
rate of interest in order to calculate the future cost of the asset.
The Single-Period Scenario: In problems involving a single period such a 1 year, the
future value of a sum of money would equal the original sum plus the amount of interest
earned over that period.
FV = PV + PV × interest rate or FV = PV(1+interest rate (in decimals)
Example 1 Let’s say John deposits $200 for a year in an account that pays 6% per year. At the end of
the year, he will have:
FV = $200 + $200 × .06 = $212 = $200(1.06) = $212
The Multiple-­‐Period Scenario: If the amount of money is left in the account for multiple
periods such as 2 years or more, then the future value of a sum of money would be equal
to the original sum plus interest over the multiple periods involved plus interest earned on
the interest as well.
FV = PV × (1 + i)n
Example 2 If John closes out his account after 3 years, how much money will he have accumulated?
How much of that is the interest on interest component? What about after 10 years?
FV3 = $200(1.06)3 = $200 × 1.191016 = $238.20
where, 6% interest per year for 3 years = $200 × .06 × 3=$36
Interest on interest = $238.20 – $200 – $36 = $2.20
FV10 = $200(1.06)10 = $200 × 1.790847 = $358.17
where, 6% interest per year for 10 years = $200 × .06 × 10 = $120
Interest on interest = $358.17 – $200 – $120 = $38.17
Methods of Solving Future Value Problems: include the use of a formula, a financial
calculator, a spreadsheet program such as Excel, and the use of time value tables.
Method 1: The formula method: is the most time-consuming, although the least
dependent method that one can use. It requires a thorough understanding of the relevant
mathematical functions. To calculate the future or compound value (FV) of a sum of
money, the following formula would apply
FV = PV × (1 + i)n
where: PV=present value of the lump sum; i = the periodic rate of interest; and n = the
number of periods for which interest is to be compounded.
©2013 Pearson Education, Inc. Publishing as Prentice Hall 50 Brooks n Financial Management: Core Concepts, 2e ·Method 2: The financial calculator approach: is quick and easy since all that has to be
done is to enter the values for the present value (PV), interest rate (I/Y), number of
periods (N), and annuity involved (PMT), and then solve for the future value (FV). It is
important to let students know that the PV should be entered as a negative (since it is an
outflow) and that PMT should be set to 0. Also, interest rates are to be entered in percent
form, i.e.12.13 for 12.13% and not in decimal form.
·Method 3: The spreadsheet method: like the calculator method is also very quick and
easy but is even more versatile, since it can be used to perform sensitivity analyses by
varying the input values, and allows for quick review of all the inputs used. Students
should be reminded that in the case of spreadsheet programs such as Excel, the default
format for entering the interest rate (Rate) is the decimal format, i.e. 0.1213 for 12.13%
and the variable name for the number of periods is “Nper.”
Example 3: Compounding of interest Let’s say you want to know how much money will have accumulated into your bank
account after 4 years, if you deposit all $5,000 of your graduation gifts into the account
which pays a fixed interest rate of 5% per year, and leave it there untouched for all four
of your college years….
Formula Method: The formula for solving this problem is as follows:
Future Value = Present Value × (1 + r)n
or
FV = PV × (1+r)n;
where: PV = $5,000; r = 5%; n=4; and
FV = $5000 × (1.05)(1.05)(1.05)(1.05) = $6077.53;
Note that the total amount of interest earned over the four year period = $1077.53, and
can be broken down as follows:
Year 1; 5% × $5,000.00 = $250.00
Year 2; 5% × $5,250.00 = $262.50
Year 3; 5% × $5,512.50 = $275.62
Year 4; 5% × $5,788.125= $289.41
Total interest = $1077.53
Calculator method: PV =-5,000; n=4; i=5; PMT=0; CPT FV=$6077.53
Spreadsheet method: Rate = .05; Nper = 4; Pmt=0; PV=-5,000; Type =0; FV=6077.31
Time value table method: FV = PV(FVIF, 5%, 4) = 5000 × (1.215506)=6077.53
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 51 where (FVIF, 5%,4) = Future value interest factor listed under the 5% column and in the
4 year row of the Future value of $1 table.
Example 4: Future cost due to inflation Let’s say that you have seen your dream house which is currently listed at $300,000, but
unfortunately, you are not in a position to buy it right away and will have to wait at least
another 5 years before you will be able to afford it. If house values are appreciating at the
average annual rate of inflation of say 5%, how much will a similar house cost after 5
years?
In this case, PV = current cost of the house = $300,000; n = 5 years; I = average annual
inflation rate = 5%; and we have to solve for FV….
FV = $300,000 × (1.05)(1.05)(1.05)(1.05)(1.05)
= $300,000 × (1.276282)
= $382,884.5
So the house will cost $382,884.5 after 5 years
Calculator method: PV =-300,000; n=5; i=5; PMT=0; CPT FV=$382,884.5
Spreadsheet method: Rate = .05; Nper = 5; Pmt=0; PV=-$300,000; Type =0; FV=$382,884.5
3.2 Present Value and Discounting (Slides 3-­‐12 to 3-­‐19) Calculating the present value (PV) of a single sum of money that will be received after a
number of periods involves discounting the amount of interest that would have been
earned over that period at a given rate of interest. It is therefore the exact opposite or
inverse of calculating the future value of a sum of money. Such calculations are useful for
determining today’s price or value of an asset or cash flow that will be received in the
future. The formula used for determining PV is as follows:
⎡ 1 ⎤
PV = FV × ⎢
n ⎥
⎣⎢ (1+ r ) ⎦⎥
where, the term in brackets is the present value interest factor for the relevant rate of
interest and number of periods involved, and is the reciprocal of the future value interest
factor (FVIF) i.e. (1 + r)n.
Example 5: Discounting interest Let’s say you just won a jackpot of $50,000 at the casino and would like to save a portion
of it so as to have $40,000 to put down on a house after 5 years. Your bank pays a 6%
rate of interest. How much money will you have to set aside from the jackpot winnings?
©2013 Pearson Education, Inc. Publishing as Prentice Hall 52 Brooks n Financial Management: Core Concepts, 2e Here, the future value (FV) is $40,000 i.e. the down payment needed; and since n=5; and
i=6%; we can solve for the present value of savings (PV) necessary by using the
following formula:
⎡ 1 ⎤
PV = FV × ⎢
n ⎥
⎣⎢ (1+ r ) ⎦⎥
⎡ 1 ⎤
PV = $40,000× ⎢
5 ⎥
⎣⎢ (1.06) ⎦⎥
PV = $40,000 × 0.747258
PV = $29,890.32
Note that $28,090.33 × (1.06)(1.06)(1.06)(1.06)(1,06) = $40,000
Thus, we have to discount interest of $10,110.67 off the FV of $40,000 to calculate the
PV or amount of saving needed.
Calculator method: FV 40,000; n = 5; i = 6%; PMT = 0; CPT PV = -$29,890.33
Note that the calculator would give PV = –29,890.33 to signify the outflow needed to get
an inflow of $40,000 after 5 years.
Spreadsheet method: Rate = .06; Nper = 5; Pmt=0; FV=$40,000; Type =0; PV=-$29,890.33
Time value table method: PV = FV(PVIF, 6%, 5) = 40,000 × (0.7473)=$29,892
where (PVIF, 6%,5) = Present value interest factor listed under the 6% column and in the
5 year row of the Present value of $1 table = 0.7473
Note: Since the table values are rounded to 4 decimals, the PV is slightly higher.
3.3 One Equation and Four Variables (Slide 3-­‐20) It can be extremely helpful to show students that any time value problem involving lump
sums i.e. a single outflow and a single inflow, requires the use of a single equation
consisting of 4 variables. These variables are the present value (PV), the future value
(FV), the rate of interest, growth or inflation (i), and the number of periods involved (n).
If 3 of the 4 variables are given, the fourth one can be easily determined by algebraic
manipulation of the single equation. The generic equation (stated in terms of the
unknown future value) is as follows:
FV = PV × (1+r)n
where, if we are given PV, r, and i we can solve for FV (as done in Example 1 above).
This equation can be restated with any one of the other 3 variables on the left hand side,
and used to solve for the unknown PV (given FV, n and i), the unknown growth (interest
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 53 rate) needed to compound a given sum of money into a given future value over a stated
number of years, or to solve the unknown number of periods that would be required for a
lump sum to grow into a future higher value at a given rate of interest. The other 3 forms
of the equation are as follows:
⎡ 1 ⎤
PV = FV × ⎢
n ⎥
⎢⎣ (1+ r ) ⎥⎦
1
⎛ FV ⎞ n
r = ⎜
⎟ − 1
⎝ PV ⎠
⎛ FV ⎞
ln⎜
⎟
PV ⎠
⎝
n=
ln(1+ r )
Example 6: Solving for an unknown interest rate Jim firmly believes that if he invests his graduation gifts of $5,000 he will be able to
triple its value by time he graduates with a college degree in 4 years. What rate of return
per year will Jim have to earn on his investment to make this belief come true?
PV = $5,000; FV = $15,000; n = 4; i=?
Using the equation listed above:
1
⎛ FV ⎞ n
r = ⎜
⎟ − 1r
⎝ PV ⎠
1
⎛ $15,000 ⎞ 4
r = ⎜
⎟ − 1
⎝ $5,000 ⎠
r = 0.31607 or 31.607% per year
Example 7: Solving for the number of periods involved Let’s say that Jim realizes that a rate of return of 31.607% per year is pretty high and
lowers his sights by settling for an investment that will pay 12% per year. If he still hopes
to triple his money, for how many years will he have to keep the $5,000 worth of
graduation gifts invested?
PV = $5,000; FV = $15,000; I = 12%; n = ?
The appropriate equation to use is as follows:
©2013 Pearson Education, Inc. Publishing as Prentice Hall 54 Brooks n Financial Management: Core Concepts, 2e ⎛ FV ⎞
ln⎜
⎟
PV ⎠
⎝
n=
ln(1+ r )
n = ln(3)/ln(1.12)
n = 9.89404 years i.e almost 10 years
3.4 Applications of the Time Value of Money Equation (Slides 3-­‐22 to 3-­‐36) In this section, the author presents various real-life applications of the time-value-ofmoney equation, in order to show how practical, useful, and necessary it is for students to
learn how to solve time value problems. Once again, it is important to stress the point that
these type of problems can be solved using any one of the four methods explained above,
i.e. by formula, by a financial calculator, by using a spreadsheet, or by referring to a time
value table. By drawing a time line and clearly indicating the values of the 3 variables
that are given in the problem, any one of the 4 variables can be easily solved for.
3.5 Doubling Your Money: The Rule of 72 (Slide 3-­‐37) Prior to the time when calculators were developed, the Rule of 72 was used to estimate
the number of years required to double a sum of money at a given rate of interest.
Basically, by dividing the rate of interest into 72, a fairly good estimate of the period
required to double a sum of money could be calculated. For example, if the rate of
interest was 9%, it would take approximately 8 years (i.e. 72/9 = 8) to double a sum of
money. This estimate was fairly accurate for interest rates ranging between 4% and 30%
with a difference of less than a third of a year as compared to the formula-based answer.
The rule could also be used to calculate the rate of interest needed to double a sum of
money by a certain number of years. For example, the rule estimates that to double a sum
of money in 4 years, the rate of return would have to be approximately 18% (i.e.
72/4=18).
Questions 1. What are the four basic parts (variables) of the time-value of money equation?
The four variables are present value (PV), time as stated as the number of periods (n),
interest rate (r), and future value (FV).
2. What does the term compounding mean?
Compounding means that interest earned in a prior period earns interest in the current
period.
3. Define a growth rate and a discount rate. What is the difference between them?
A growth rate is the annual percentage increase and the discount rate is the annual
reduction rate. The difference between a growth rate and a discount rate is the
direction of the problem. A growth rate implies going forward in time, a discount rate
implies going backwards in time.
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 55 4. What happens to a future value as you increase the interest (growth) rate?
The future value gets larger as you increase the interest rate.
5. What happens to a present value as you increase the discount rate?
The present value gets smaller as you increase the discount rate.
6. What happens to a future value as you increase the time to the future date?
The future value increases as you increase the time to the future.
7. What happens to the present value as the time to the future value increases?
The present value decreases as you increase the time between the future value date
and the present value date.
8. What is the Rule of 72?
The Rule of 72 is a simple estimation method for how long it takes something to
double in value or size or given a specific time period the required growth rate to
double in value or size.
9. Is the present value is always less than the future value?
Yes, as long as interest rates are positive—and interest rates are always positive—the
present value of a sum of money will always be less than its future value. One can get
confused with this concept if they think about the depreciation of an asset over time.
Present value and future value in the TVM equation refers to cash or an identical cash
flow at two different points in time.
10. When a lottery price is offered as $10,000,000 but will pay out a series of
$250,000 payments over forty years, is it really a $10,000,000 lottery prize?
The present value of the lottery is not worth $10,000,000. The total payments over
time are $10,000,000 but this is not a value of the lottery because these payments are
at different points in time and cash flows can only be added if they are at the same
point in time.
Prepping for Exams 1.
2.
3.
4.
5.
d.
b.
c.
c.
a.
6. c.
7. d.
8. b.
9. a.
10. b.
Problems 1. Future Values. Fill in the future values for the following table
a. Using the future value formula, FV = PV × (1+r)n.
b. Using the time value of money keys or function from a calculator or spreadsheet.
Present Value
Interest Rate
Number of Periods
©2013 Pearson Education, Inc. Publishing as Prentice Hall Future Value
56 Brooks n Financial Management: Core Concepts, 2e $400.00
5.0%
5
$17,411.00
6.0%
30
$35,000.00
10%
20
$26,981.75
16%
15
Present Value
Interest Rate
Number of Periods
Future Value
$400.00
5.0%
5
$510.51
$17,411.00
6.0%
30
$99,999.92
$35,000.00
10%
20
$235,462.50
$26,981.75
16%
15
$249,999.97
ANSWER a. With TVM formula (rounding to second decimal only for final answer)
FV = $400.00 × (1.05)5 = $400.00 × 1.2763 = $510.51
FV = $17,411.00 × (1.06)30 = $17,411.00 × 5.7435 = $99,999.92
FV = $35,000.00 × (1.10)20 = $35,000.00 × 6.7275 = $235,462.50
FV = $26,981.75 × (1.16)15 = $26,981.75 × 9.2655 = $249,999.97
b. Time Value of Money Keys or Spreadsheet
Input
Variable
Compute
5
N
5.0
I/Y
-400
PV
0
PMT
Input
Variable
Compute
30
N
6.0
I/Y
-17,411
PV
0
PMT
Input
Variable
Compute
20
N
10.0
I/Y
-35,000
PV
0
PMT
Input
Variable
Compute
15
N
16.0
I/Y
-26,981.75
PV
0
PMT
FV
510.51
FV
99,999.92
FV
235,462.50
FV
249,999.97
2. Future Value (with changing years). Dixie Bank offers a certificate of deposit with
an option to select your own investment period. Jonathan has $7,000 for his CD
investment. If the bank is offering a 6% interest rate, how much will the CD be worth
at maturity if Jonathan picks a a. two-year investment period?
b. five-year investment period?
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 57 c. eight-year investment period?
d. fifteen-year investment period?
ANSWER a. FV = $7,000 × (1.06)2 = $7,000 × 1.1236 = $7,865.20
b. FV = $7,000 × (1.06)5 = $7,000 × 1.3382 = $9,367.58
c. FV = $7,000 × (1.06)8 = $7,000 × 1.5938 = $11,156.94
d. FV = $7,000 × (1.06)15 = $7,000 × 2.3966 = $16,775.91
3. Future Value (with changing interest rates). Jose has $4,000 to invest for a two year
period. He is looking at four different investment choices. What will be the value of
his investment at the end of two years for each of the following potential
investments?
a. bank CD at 4%.
b. bond fund at 8%.
c. mutual stock fund at 12%.
d. new venture stock at 24%.
ANSWER a. FV = $4,000 × (1.04)2 = $4,000 × 1.0816 = $4,326.40
b. FV = $4,000 × (1.08)2 = $4,000 × 1.1664 = $4,665.60
c. FV = $4,000 × (1.12)2 = $4,000 × 1.2544 = $5,017.60
d. FV = $4,000 × (1.24)2 = $4,000 × 1.5376 = $6,150.40
4. Future Value. Grand Opening Bank is offering a one-time investment opportunity for
its new customers. A customer opening a new checking account can buy a special
savings bond for $100 today which the bank will compound at 7.5% for the next
twenty years. The savings bond must be held for at least five years but can then be
cashed in at the end of any year starting with year five. What is the value of the bond
at each cash-in date up through twenty years? (Use an Excel Spreadsheet to solve this
problem.)
Spreadsheet Solution In Cells A1 through A16 put in the Present Value of the savings bond, $100.00.
In Cells B1 through B16 put in the annual interest rate, 0.075, R.
In Cells C1 through C16 put in the waiting time in years, 5 through 20, N.
In Cell D1 put in the FV function using the row value in A1 for the PV, row value in B1
for the interest rate, row value in C1 for n. Copy the formula for cells D2 through D16.
The displayed value will be the future value of the $100 savings bond at 7.5% annual
interest.
©2013 Pearson Education, Inc. Publishing as Prentice Hall 58 Brooks n Financial Management: Core Concepts, 2e A
B
C
D
1
-100.00
0.075
5
FV (PV = A1, rate = B1, n = C1) = $143.56
2
-100.00
0.075
6
FV (PV = A2, rate = B2, n = C2) = $154.33
3
-100.00
0.075
7
FV (PV = A3, rate = B3, n = C3) = $165.90
4
-100.00
0.075
8
FV (PV = A4, rate = B4, n = C4) = $178.35
5
-100.00
0.075
9
FV (PV = A5, rate = B5, n = C5) = $191.72
6
-100.00
0.075
10
FV (PV = A6, rate = B6, n = C6) = $206.10
7
-100.00
0.075
11
FV (PV = A7, rate = B7, n = C7) = $221.56
8
-100.00
0.075
12
FV (PV = A8, rate = B8, n = C8) = $238.18
9
-100.00
0.075
13
FV (PV = A9, rate = B9, n = C9) = $256.04
10
-100.00
0.075
14
FV (PV = A10, rate = B10, n = C10) = $275.24
11
-100.00
0.075
15
FV (PV = A11, rate = B11, n = C11) = $295.89
12
-100.00
0.075
16
FV (PV = A12, rate = B12, n = C12) = $318.08
13
-100.00
0.075
17
FV (PV = A13, rate = B13, n = C13) = $341.94
14
-100.00
0.075
18
FV (PV = A14, rate = B14, n = C14) = $367.58
15
-100.00
0.075
19
FV (PV = A15, rate = B15, n = C15) = $395.15
16
-100.00
0.075
20
FV (PV = A16, rate = B16, n = C16) = $424.79
Note: The negative values in column A denote the cash outflow and produce a
positive cash inflow in column D.
5. Future Value. Jackson Enterprises just purchased some land for $230,000. The land
was purchased for a future beach front property development project that will include
rental cabins, lodge, and recreational facilities. Jackson Enterprises has not committed
to the development project, but will decide in five years whether to go forward with
the project or sell off the land. Real estate values increase annually at 4.5% for
unimproved property. For how much can Jackson Enterprises expect to sell the
property in five years if it chooses not to proceed with the beach front development
project? What if Jackson Enterprises holds the property for ten years and then sells?
ANSWER Holding Property Five Years:
FV= $230,000 × (1.045)5 = $230,000 × 1.2462 = $286,621.85
Holding Property Ten Years:
FV= $230,000 × (1.045)10 = $230,000 × 1.5530 = $357,182.97
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 59 6. Future Value. The Portland Stallions professional football team is looking at its
future revenue stream from ticket sales. Currently, a season package costs $325 per
seat. The Portland Stallions box office has promised season ticket holders this same
rate for the next three years. Four years from now, the organization will raise season
ticket prices based on the estimated inflation rate of 3.25%. What will the season
tickets sell for in four years?
ANSWER FV = $325 × (1.0325)4 = $325 × 1.136475928 = $369.35
7. Future Value. Upstate University charges $16,000 a year in graduate tuition. Tuition
rates are growing at 4.5% each year. You plan on enrolling in graduate school in five
years. What is your expected graduate tuition in five years?
ANSWER FV = $16,000 × (1.045)5 = $16,000 × 1.246181938 = $19,938.91
8. Present Values. Fill in the present value for the following table
a. Using the present value formula, PV = FV × [1/(1+r)n].
b. Using the time value of money keys or function from a calculator or spreadsheet.
Future Value
Interest Rate
Number of Periods
Present Value
$900.00
5%
5
?
$80,000.00
6%
30
?
$350,000.00
10%
20
?
$26,981.75
16%
15
?
Future Value
Interest Rate
Number of Periods
Present Value
$900.00
5.0%
5
$705.17
$80,000.00
6.0%
30
$13,928.81
$350,000.00
10%
20
$52,025.27
$26,981.75
16%
15
$2,912.06
ANSWER a. With TVM formula (rounding to second decimal only for final answer)
PV = $900.00 × 1/(1.05)5 = $400.00 × 0.7835 = $705.17
PV = $80,000.00 × 1/(1.06)30 = $80,000.00 × 0.1741 = $13,928.81
PV = $350,000.00 × 1/(1.10)20 = $350,000.00 × 0.1486 = $52,025.27
PV = $26,981.75 × 1/(1.16)15 = $26,981.75 × 0.1079 = $2,912.06
©2013 Pearson Education, Inc. Publishing as Prentice Hall 60 Brooks n Financial Management: Core Concepts, 2e b. Time Value of Money Keys or Spreadsheet
Input
5
5.0
0
Variable
N
I/Y
PV
Compute
-900
PMT
Input
Variable
Compute
30
N
6.0
I/Y
0
PV
-80,000
PMT
Input
Variable
Compute
20
N
10.0
I/Y
0
PV
-350,000
PMT
Input
Variable
Compute
15
N
16.0
I/Y
0
PV
-26,981.75
PMT
FV
705.17
FV
13,928.81
FV
52,025.27
FV
2,912.06
9. Present Value (with changing years). When they are first born, Grandma gives every
grandchild a $2,500 savings bond that matures in eighteen years. What is the present
value of each of these savings bonds if the current discount rate is 4%?
a. Seth turned sixteen years old today.
b. Shawn turned thirteen years old today.
c. Sherry turned nine years old today.
d. Sheila turned four years old today.
e. Shane was just born.
ANSWER a.
b.
c.
d.
e.
PV = $2,500 × 1/(1.04)2 = $2,500 × 0.9246 = $2,311.39
PV = $2,500 × 1/(1.04)5= $2,500 × 0.8219 = $2,054.82
PV = $2,500 × 1/(1.04)9 = $2,500 × 0.7026 = $1,756.47
PV = $2,500 × 1/(1.04)14 = $2,500 × 0.5775 = $1,443.69
PV = $2,500 × 1/(1.04)18 = $2,500 × 0.4936 = $1,234.07
10. Present Value (with changing interest rates). Marty has been offered an injury
settlement of $10,000 payable in three years. He wants to know what the present
value of the injury settlement is if his opportunity cost is 5%. (The opportunity cost is
the interest rate in this problem.) What if the opportunity cost is 8%? What if it is
12%?
ANSWER PV = $10,000 × 1/(1.05)3 = $10,000 × 0.8638 = $8,638.38
PV = $10,000 × 1/(1.08)3 = $10,000 × 0.7938 = $7,938.32
PV = $10,000 × 1/(1.12)3 = $10,000 × 0.7118 = $7,117.80
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 61 11. Present Value. The State of Confusion wants to change the current retirement policy
for state employees. To do so, however, the state must pay the current pension fund
members the present value of their promised future payments. There are 240,000
current employees in the state pension fund. The average employee is twenty-two
years away from retirement, and the average promised future retirement benefit is
$400,000 per employee. If the state has a discount rate of 5% on all its funds, how
much money will the state have to pay to the employees before it can start a new
pension plan?
ANSWER Present Value of Retirement Payments of the average employee:
PV = $400,000 × 1/(1.05)22 = $400,000 × 0.3418 = $136,739.95
With 240,000 Employees the total obligations are:
PV Obligation of Pension Fund = 240,000 × $136,739.95
= $32,817,590,000
Note: the PV was not rounded to nearest cent prior to multiplying it by the number
of employees.
12. Present Value. Two rival football fans have made the following wager: if their
college football team wins the conference title outright, the other fan will donate
$1,000 to the winning school. Both schools have had relatively unsuccessful teams
but are improving each season. If the two fans must put up their potential donation
today and the discount rate is 8% for the funds, what is the required upfront deposit if
a team is expected to win the conference title in five years? Ten years? Twenty years?
ANSWER Present Value of Payoff in Five Years:
PV = $1,000 × 1/(1.08)5 = $1,000 × 0.6806 = $680.58
Present Value of Payoff in Ten Years:
PV = $1,000 × 1/(1.08)10 = $1,000 × 0.4632 = $463.19
Present Value of Payoff in Twenty Years:
PV = $1,000 × 1/(1.08)20 = $1,000 × 0.2145 = $214.55
13. Present Value. Prestigious University is offering a new admission and tuition
payment plan for all alumni. On the birth of a child, parents can guarantee admission
to Prestigious University if they pay the first year’s tuition. The university will pay an
annual rate of return of 4.5% on the deposited tuition, and a full refund will be
available if the child chooses another university. The tuition is $12,000 per year at
Prestigious University and is frozen at that level for the next eighteen years. What
would parents pay today if they just gave birth to a new baby and the child will attend
©2013 Pearson Education, Inc. Publishing as Prentice Hall 62 Brooks n Financial Management: Core Concepts, 2e college in eighteen years? How much is the required payment to secure admission for
their child if the interest rate falls to 2.5%?
ANSWER PV = $12,000 × 1/(1.045)18 = $12,000 × 1/(2.2085) = $12,000 × 0.4528 = $5,433.60
PV = $12,000 × 1/(1.025)18 = $12,000 × 1/(1.5597) = $12,000 × 0.6412 = $7,693.99
14. Present Value. Standard Insurance is developing a long-life insurance policy for
people that outlive their retirement nest egg. The policy will pay out $250,000 on
your eighty-fifth birthday. You must buy the policy on your sixty-fifth birthday. The
insurance company can earn 7% on the purchase price of your policy. What is the
minimum purchase price the insurance company should charge for this policy?
ANSWER PV = $250,000 × (1/(1+0.07)20 = $250,000 × 0.258419 = $64,604.75
15. Present Value. You are currently in the job market. Your dream is to earn a six-figure
salary ($100,000). You hope to accomplish this goal within the next thirty years. In
your field, salaries grow at 3.75% per year. What starting salary do you need to reach
this goal?
ANSWER PV = $100,000 × (1/(1+0.0375)30 = $100,000 × 0.3314033 = $33,140.33
16. Interest rate or discount rate. Fill in the interest rate for the following table
a. Using the interest rate formula, r = (FV/PV)1/n – 1
b. Using the time value of money keys or function from a calculator or spreadsheet.
Present Value
Future Value
Number of Periods
Interest Rate
$500.00
$1,998.00
18
?
$17,335.36
$230,000.00
30
?
$35,000.00
$63,214.00
20
?
$27,651.26
$225,000.00
15
?
Present Value
Future Value
Number of
Periods
Interest Rate
$500.00
$1,998.00
18
8.00%
$17,335.36
$230,000.00
30
9.00%
ANSWER ©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 63 $35,000.00
$63,214.00
20
3.00%
$27,651.26
$225,000.00
15
15.00%
a. With TVM formula (rounding to second decimal only for final answer)
r = ($1998.00/$500.00)1/18 – 1 = 1.0800 – 1 = 8.00%
r = ($230,000/$17,335.36)1/30 – 1 = 1.0900 – 1 = 9.00%
r = ($63,214/$35,000)1/20 – 1 = 1.0300 – 1 = 3.00%
r = ($225,000/$27,651.26)1/15 – 1 = 1.1500 – 1 = 15.00%
b. Time Value of Money Keys or Spreadsheet
Input
18
500.00
0
-1,998.00
Variable
N
I/Y
PV
PMT
Compute
Input
Variable
Compute
30
N
17,335.36
I/Y
0
PV
-230,000
PMT
Input
Variable
Compute
20
N
35,000
I/Y
0
PV
-63,214
PMT
Input
Variable
Compute
15
N
27,651.26
I/Y
0
PV
-225,000
PMT
FV
8.00
FV
9.00
FV
3.00
FV
15.00
17. Interest rate (with changing years). Keiko is looking at the following investment
choices and wants to know what interest rate each choice produces.
a. Invest $400 and receive $786.86 in ten years.
b. Invest $3,000 and receive $10,927.45 in fifteen years.
c. Invest $31,180.47 and receive $100,000 in twenty years.
d. Invest $31,327.88 and receive $1,000,000 in forty-five years.
ANSWER a.
b.
c.
d.
r = ($786.86/$400)1/10 – 1 = 1.07 – 1 = 7.00%
r = ($10,927.45/$3,000)1/15 – 1 = 1.09 – 1 = 9.00%
r = ($100,000/$31,180.47)1/20 – 1 = 1.06 – 1 = 6.00%
r = ($1,000,000/$31,327.88)1/45 – 1 = 1.08 – 1 = 8.00%
18. Interest rate. Two mutual fund managers, Martha and David, have been bragging
about whose fund is the top performer. Martha states that investors bought shares in
her mutual fund ten years ago for $21.00, and those shares are now worth $65.00.
David states that investors bought shares in his mutual fund for only $3.00 six years
ago, and now they are worth $7.30. Which mutual fund manager has had the highest
©2013 Pearson Education, Inc. Publishing as Prentice Hall 64 Brooks n Financial Management: Core Concepts, 2e growth rate for the management period? Should this comparison be made over
different management periods? Why or why not?
ANSWER Martha’s Rate, r = ($65.00/$21.00)1/10 – 1 = 1.1196 – 1 = 11.96%
David’s Rate, r = ($7.30/$3.00)1/6 – 1 = 1.1598 – 1 = 15.98%
Fund Two has the higher annual return but for a shorter period of time. To be appropriate
for comparing performance the funds’ should be evaluated over the same period of time.
The first fund may have had a higher return over the last six years than the second fund
but low returns in the earlier years reduced the return below the 16% return rate of the
second fund.
19. Interest rate. In 1972, Bob purchased a new Datsun 240Z for $3,000. Datsun later
changed its name to Nissan, and the 1972 240Z became a classic. Bob kept his car in
excellent condition and in 2002 could sell the car for six times what he originally
paid. What was Bob’s return on owning this car? What if he keeps the car for another
thirty years and earns the same rate? What could he sell the car for in 2032?
ANSWER Thirty years’ return on car:
r = ($18,000/$3,000)1/30 – 1 = 1.0615 – 1 = 6.15%
Price Thirty Years from now:
FV = $18,000 × (1.0615)30 = $108,000
Or just increase the price six fold again, $18,000 × 6 = $108,000.
20. Interest rate. Upstate Bank is offering long-term certificates of deposit with a face
value of $100,000 (future value). Bank customers can buy these CDs today for
$67,000 and will receive the $100,000 in fifteen years. What interest rate is the bank
paying on these CDs?
ANSWER r = ($100,000/$67,000)1/15 – 1 = 1.027058 – 1 = 2.7058%
21. Discount rate. Future Bookstore sells books before they are published. Today they
offered the book Adventures in Finance for $14.20, but the book will not be published
for another two years. The retail price when the book is published will be $24.00.
What is the discount rate Future Bookstore is offering its customers for this book?
ANSWER r = ($24/$14.20)1/2 – 1 = 1.30 – 1 = 30%
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 65 22. Growth and future value. A famous disease control scientist is trying to determine
the potential infected population of the new West Columbia flu. Two weeks ago, the
first patient showed up with the disease. Four days later, the disease control center in
Atlanta had six confirmed cases. The scientist estimates that it will be another two
days before a cure will be ready, a total of 16 days from the first confirmed case. How
many patients will be infected two days from now?
ANSWER The disease is spreading at a growth rate of nearly 57% per day.
r = (6/1)1/4 – 1 = 1.3161 – 1 = 56.51%
Using the same growth rate for the 16 day period, (two weeks and two days) the number
of patients infected will be:
FV = 1 × (1.5156)16 = 1,296 or 1,296 patients
Or you could realize that every four days the number of people infected increases by six
times and with four periods of four days you have
1 × 6 × 6 × 6 × 6 = 1 × (6)4 = 1,296.
23. Waiting period. Fill in the number of periods for the following table.
a. Using the waiting period formula, n = ln(FV/PV) / ln(1+r).
b. Using the time value of money keys or function from a calculator or spreadsheet.
Present Value
Future Value
Interest Rate
Number of Periods
$800.00
$1,609.76
6%
?
$17,843.09
$100,000.00
9%
?
$35,000.00
$3,256,783.97
12%
?
$25,410.99
$300,000.00
28%
?
ANSWER Solutions with TVM Keys or Spreadsheet Present Value
Future Value
Interest Rate
Number of
Periods
$800.00
$1,609.76
6%
12
$17,843.09
$100,000.00
9%
20
$35,000.00
$3,256,783.97
12%
40
$25,410.99
$300,000.00
28%
10
a. With TVM formula (rounding to second decimal only for final answer)
n = ln ($1,609.76/$800.00) / ln (1.06) = 0.6992 / 0.0583 = 12
©2013 Pearson Education, Inc. Publishing as Prentice Hall 66 Brooks n Financial Management: Core Concepts, 2e n = ln ($100,000/$17,843.09) / ln (1.09) = 1.7236 / 0.0862 = 20
n = ln ($3,256,783.97/$35,000) / ln (1.12) = 4.5331 / 0.1133 = 40
n = ln ($300,000/$25,410.99) / ln (1.28) = 2.4686 / 0.2469 = 10
b. Time Value of Money Keys or Spreadsheet inputs
Input
Variable
Compute
6.00
N
800.00
I/Y
0
PV
-1,609.76
PMT
Input
Variable
Compute
9.00
N
17,843.09
I/Y
0
PV
-100,000
PMT
Input
Variable
Compute
12.0
N
35,000
I/Y
0
PV
-3,256,783.97
PMT
Input
Variable
Compute
28.0
N
25,410.99
I/Y
0
PV
-300,000
PMT
FV
12
FV
20
FV
40
FV
10
24. Waiting period (with changing years). Jamal is waiting to be a millionaire. He wants
to know how long he must wait if
a. he invests $24,465.28 at 16% today
b. he invests $47,101.95 at 13% today
c. he invests $115,967.84 at 9% today
d. he invests $295,302.77 at 5% today
ANSWER a.
b.
c.
d.
n = ln($1,000,000/$24,465.28) / ln (1.16) = 3.7105 / 0.1484 = 25
n = ln($1,000,000/$47,101.95) / ln (1.13) = 3.0554 / 0.1222 = 25
n = ln($1,000,000/$115,967.84) / ln (1.09) = 2.1544 / 0.0862 = 25
n = ln($1,000,000/$295,302.77) / ln (1.05) = 1.2198 / 0.0488 = 25
25. Waiting period. Jeff, a local traffic engineer, has designed a new pedestrian
footbridge. The bridge has been designed to handle 200 pedestrians daily. Once the
bridge reaches 1,000 pedestrians daily, however, it will require a new bracing system.
Jeff has estimated that traffic will increase annually at 5%. How long will the current
bridge system work before a new bracing system is required? What if the annual
traffic rate increases at 8% annually? At what traffic increase rate will the current
system last only ten years?
ANSWER For 5% growth rate in pedestrian traffic:
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 67 n = ln(1000/200) / ln (1.05) = 1.6904 / 0.0488 = 32.9869 years
For 8% growth rate in pedestrian traffic:
n = ln(1000/200) / ln (1.08) = 1.6904 / 0.0770 = 20.9124 years
Ten Years Maximum would require growth rate of:
r = (1000/200)1/10 – 1 = 1.1746 – 1 = 17.46%
26. Waiting period. Susan Norman seeks your financial advice. She wants to know how
long it will take for her to become a millionaire. She tells you that she has $1,330
today and wants to invest it in an aggressive stock portfolio. The historical return on
this type of investment is 18% per year. How long will she have to wait if the $1,330
is the only amount she invests and she never withdraws from the market until she
reaches her $1 million? (Assume no taxes on the earnings). What if the rate of return
is only 14% annually? What if the rate of return is only 10% annually?
ANSWER Waiting time if investment is earning 18%:
n = ln(1,000,000/1,330) / ln (1.18) = 6.6226 / 0.1655 = 40.0121 years
Waiting time if investment is earning 14%:
n = ln(1,000,000/1,330) / ln (1.14) = 6.6226 / 0.1310 = 50.5431 years
Waiting time if investment is earning 10%:
n = ln(1,000,000/1,330) / ln (1.10) = 6.6226 / 0.0953 = 69.4845 years
27. Waiting period. Upstate University currently has a 6,000 car parking capacity for
faculty, staff, and students. This year the university issued 4,356 parking passes.
Parking passes have been growing at a rate of 6% per year. How long will it be before
the university will need to add additional parking?
ANSWER Waiting time if parking growth rate is 6%:
n = ln(6,000/4,356) / ln (1.06) = 0.320305264 / 0.058268908 = 5.4953 years
28. Double your money. Approximately how long will it take to double your money if
you get 5.5%, 7.5%, or 9.5% annual return on your investment? Verify the
approximate doubling period with the time value of money equation.
ANSWER Approximating the double period with the Rule of 72s:
At 5.5%, 72/5.5 = 13.09 or a little over 13 years,
©2013 Pearson Education, Inc. Publishing as Prentice Hall 68 Brooks n Financial Management: Core Concepts, 2e At 7.5%, 72/7.5 = 9.60 or a little over 9 ½ years,
At 9.5%, 72/9.5 = 7.5789 or a little over 7 ½ years.
Verification:
n = ln 2 / ln 1.055 = 0.6931 / 0.0535 = 12.9462 or a little under 13 years
n = ln 2 / ln 1.075 = 0.6931 / 0.0723 = 9.5844 or a little over 9 ½ years
n = ln 2 / ln 1.095 = 0.6931 / 0.0908 = 7.6376 or a little over 7 ½ years
29. Double your wealth. Kant Miss Company is promising its investors that it will
double their money every three years. Is this promise too good to be true? What
annual rate is Kant Miss promising? If you invested $250 now and Kant Miss were
able to deliver on its promise, how long would it take before your investment reaches
$32,000?
ANSWER Using the rule of 72s the approximate rate is: 72/3 = 24%
The actual rate is: r = 21/3 – 1 = 25.99%
To calculate how long it would take for $250 to turn into $32,000:
n = ln (32,000/250) / ln (1.2599) = 21 years
30. Challenge question. In the chapter text, we dealt exclusively with a single lump sum,
but often we may be looking at several lump-sum values simultaneously. Let’s
consider the retirement plan of a couple. Currently, the couple has four different
investments: a 401(k) plan, two pension plans, and a personal portfolio. The couple is
five years away from retirement. They believe they have sufficient money in their
plans today so that they do not have to contribute to the plans over the next five years
and will still meet their $2 million retirement goal. Here are the current values and the
growth rate of each plan:
401(k): $88,000 growing at 6.5%
Pension plan 1: $304,000 growing at 7%
Pension plan 2: $214,000 growing at 7.25%
Personal portfolio: $149,000 growing at 8.5%
Does the couple have enough already invested to make their goal in five years?
Hint: View each payment as a separate problem and find the future value of each lump
sum. Then add up all the future values five years from now.
ANSWER: Future value of 401-K, $88,000 × (1.065)5 = $88,000 × 1.3701 = $120,567.63
Future value of Pension Plan 1, $304,000 × (1.07)5 = $304,000 × 1.4026 =$426,375.73
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 69 Future value of Pension Plan 2, $214,000 × (1.0725)5 = $214,000 × 1.4190 =
$303,668.87
Future value of Personal Portfolio, $149,000 × (1.085)5 = $149,000 × 1.5037 =
$224,044.85
Total = $120,567.63 + $426,375.73 + $303,668.87 + $224,044.85 = $1,074, 657.08
They are only half way to their $2 million dollar goal with their current savings,
pensions, and 401-K plan.
©2013 Pearson Education, Inc. Publishing as Prentice Hall 70 Brooks n Financial Management: Core Concepts, 2e Solutions to Advanced Problems for Spreadsheet Application 1. Future value of a portfolio. It would take 20 years to reach their $2,000,000 goal.
Money
Market
Government
Bonds
Large Cap
Small Cap
Real Estate
2.5%
5.5%
9.5%
12.0%
4.0%
0
$ 37,000.00
$ 140,000.00
$
107,000.00
$ 71,000.00
$ 87,000.00
$ 442,000.00
1
$ 37,925.00
$ 147,700.00
$
117,165.00
$ 79,520.00
$ 90,480.00
$ 472,790.00
2
$ 38,873.13
$ 155,823.50
$
128,295.68
$ 89,062.40
$ 94,099.20
$ 506,153.90
3
$ 39,844.95
$ 164,393.79
$
140,483.76
$ 99,749.89
$ 97,863.17
$ 542,335.57
4
$ 40,841.08
$ 173,435.45
$
153,829.72
$
111,719.87
$ 101,777.69
$ 581,603.82
5
$ 41,862.10
$ 182,974.40
$
168,443.55
$
125,126.26
$ 105,848.80
$ 624,255.11
6
$ 42,908.66
$ 193,037.99
$
184,445.68
$
140,141.41
$ 110,082.75
$ 670,616.50
7
$ 43,981.37
$ 203,655.08
$
201,968.02
$
156,958.38
$ 114,486.06
$ 721,048.92
8
$ 45,080.91
$ 214,856.11
$
221,154.98
$
175,793.39
$ 119,065.51
$ 775,950.90
9
$ 46,207.93
$ 226,673.20
$
242,164.71
$
196,888.59
$ 123,828.13
$ 835,762.56
10
$ 47,363.13
$ 239,140.22
$
265,170.35
$
220,515.22
$ 128,781.25
$ 900,970.18
11
$ 48,547.21
$ 252,292.94
$
290,361.54
$
246,977.05
$ 133,932.50
$ 972,111.23
12
$ 49,760.89
$ 266,169.05
$
317,945.88
$
276,614.30
$ 139,289.80
$
1,049,779.92
13
$ 51,004.91
$ 280,808.35
$
348,150.74
$
309,808.01
$ 144,861.40
$
1,134,633.40
14
$ 52,280.03
$ 296,252.80
$
381,225.06
$
346,984.97
$ 150,655.85
$
1,227,398.72
15
$ 53,587.03
$ 312,546.71
$
417,441.45
$
388,623.17
$ 156,682.08
$
1,328,880.44
16
$ 54,926.71
$ 329,736.78
$
457,098.38
$
435,257.95
$ 162,949.37
$
1,439,969.19
Growt
h Rates
Portfolio
Total
Year
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 71 17
$ 56,299.88
$ 347,872.30
$
500,522.73
$
487,488.90
$ 169,467.34
$
1,561,651.15
18
$ 57,707.37
$ 367,005.28
$
548,072.39
$
545,987.57
$ 176,246.04
$
1,695,018.65
19
$ 59,150.06
$ 387,190.57
$
600,139.26
$
611,506.08
$ 183,295.88
$
1,841,281.85
20
$ 60,628.81
$ 408,486.05
$
657,152.50
$
684,886.81
$ 190,627.71
$
2,001,781.88
2. Changing future value growth rates. The trees produce their highest gross profit at a
height of just over 11 feet six years after planting.
Years of Growth
Growth Rate
Height of
Tree
0
1.00
$5 per foot
30% increase
Revenue
Annual Cost
Total
Cost
Gross
Profit
1
0.6
1.60
$ 8.00
$ 3.00
$ 3.00
$ 5.00
2
0.6
2.56
$ 12.80
$ 3.90
$ 6.90
$ 5.90
3
0.6
4.10
$ 20.48
$ 5.07
$ 11.97
$ 8.51
4
0.4
5.73
$ 28.67
$ 6.59
$ 18.56
$ 10.11
5
0.4
8.03
$ 40.14
$ 8.57
$ 27.13
$ 13.01
6
0.4
11.24
$ 56.20
$ 11.14
$ 38.27
$ 17.93
7
0.2
13.49
$ 67.44
$ 14.48
$ 52.75
$ 14.69
8
0.2
16.18
$ 80.92
$ 18.82
$ 71.57
$ 9.35
9
0.2
19.42
$ 97.11
$ 24.47
$ 96.04
$ 1.06
10
0.2
23.31
$ 116.53
$ 31.81
$ 127.86
$ (11.33)
Solutions to Mini-­‐Case Richards Tree Farm, Inc.: The Continuing Saga
This mini-case illustrates the role that present and future value computations play in basic
business planning and price negotiation decisions. It introduces the idea of discount rates
as opportunity costs and present values as prices, preparing the way for concepts that will
be introduced in the chapters on valuation and capital budgeting.
1. Because of inflation, Jake expects the price at which he can sell the trees to
increase by 3% per year. What price does he expect to receive if he keeps the
trees until they reach 8 feet or 10 feet tall?
©2013 Pearson Education, Inc. Publishing as Prentice Hall 72 Brooks n Financial Management: Core Concepts, 2e The trees which are currently 6 feet tall will grow to 8 ft. in 3 years and 10 feet in 6
years from now. Hence the price that Jake can expect to receive at 8 and 10 ft. heights
can be calculated as follows:
Using formulas
FV = 34(1.03)3 = 37.15, FV = 34(1.03)6 = 47.76
Using a financial calculator:
N
i/y
PV
PMT
FV
3
3
-34
0
37.15
6
3
-40
0
47.76
The trees will be worth $37.15 in 3 years and $47.16 in 6 years.
2. If Jake discounts the future price of the trees at 10% per year, what is the
present value of their future prices?
Using formulas:
PV = 37.15/(1.10)3 = 27.91, PV = 47.76/(1.10)6 = 26.96
Using a financial calculator:
N
i/y
PV
PMT
FV
3
10
27.91
0
37.15
6
10
26.96
0
47.76
The present value of selling the trees now is $24.00, in 3 years $27.91, and in 6 years,
$26.96.
3. Using time value of money equation, compute the growth rate of the trees
between the third and the sixth year, and between the sixth and the ninth year.
Using formulas:
Rate= (8/6-1)1/3 =.101, rate = (10/6-1)1/6 =.077
Using a financial calculator:
N
i/y
PV
PMT
FV
3
10.1%
(6.00)
0
8.00
3
7.7%
(8.00)
0
10.00
The trees grow at the rate of 10.1% per year between the 3rd and 6th years and 7.7%
between the 6th and 9th years.
4. When should Jake sell the trees?
Richards should sell the trees after the 6th year when they are 8 feet tall and their
present value is highest.
5. Challenge Question: A major landscape contractor, who has bid successfully on
a large-scale Boston beautification and urban greening project, has offered to
©2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter 3 n The Time Value of Money (Part 1) 73 buy all 10,000 of the flowering dogwood trees at a price of $28,000, to be paid
immediately. The trees, however, will not be needed for three years. If Jake
accepts, he will be obliged to deliver 10,000 trees 3 years from today. If anything
should happen to his own crop, he would need to buy trees on the open market
at the prevailing price, which might be higher or lower than the price estimated
in question 1. Should Jake accept the offer if his required rate of return is 10%?
Hint: what is the present value of the price he expects to receive for the trees 3
years in the future? Discount the price at 10%.
If Richards waits 3 years to sell the trees, the present value of his planting is the
$27.91 per tree calculated in question 2 times 10,000 or $27,910. The present value of
the offer for future delivery is $28,000, so it should probably be accepted.
By accepting the offer, Richards avoids the risk that prices will fall due to market
conditions. On the other hand, if prices should be higher than expected, he loses out
on the opportunity to sell at the higher price. The biggest risk is that something will
happen to his crop. If the crop should fail, he would have to absorb all of his costs up
to that point, and he would also be obliged to buy trees at the market price, whatever
that may be, to meet his commitment to deliver 10,000 trees. The discount rate
implies that funds could be reinvested at 10%, so another way to look at the problem
is that the $28,000 is worth $28,000(1.10)3 or $37,268 three years from now, which is
slightly more than he expects to receive from selling the trees then.
Additional Problems with Solutions 1. Joanna’s dad is looking to deposit a sum of money immediately into an account that
pays an annual interest rate of 9% so that her first year college tuition costs are
provided for. Currently the average college tuition cost is $15,000 and is expected to
increase by 4% (the average annual inflation rate). Joanna just turned 5, and is
expected to start college when she turns 18. How much money will Joanna’s dad have
to deposit into the account?
ANSWER: (Slides 3-­‐38 to 3-­‐40) Step 1. Calculate the average annual college tuition cost when Joanna turns 18 i.e. the
future compounded value of the current tuition cost at an annual increase of 4%.
PV = –15,000; n= 13; i=4%; PMT=0; CPT FV=$24, 976.10 (rounded to 2 decimals)
OR
FV = $15,000 × (1.04)13 = $15,000 × 1.66507 = $24,976.10
Step 2. Calculate the present value of the annual tuition cost using an interest rate of 9%
per year.
FV = 24,976.10; n=13; i=9%; PMT = 0; CPT PV = $8,146.67 (rounded to 2 decimals)
OR
PV = $24,976.10 × (1/(1+0.09)13 = $24,976.10 × 0.32618 = $8,146.67
©2013 Pearson Education, Inc. Publishing as Prentice Hall 74 Brooks n Financial Management: Core Concepts, 2e So, Joanna’s dad will have to deposit $8,146.67 into the account today so that she will
have her first year tuition costs provided for when she starts college at the age of 18.
2. Bank A offers to pay you lump sum of $20,000 after 5 years, if you deposit $9,500
with them today. Bank B, on the other hand, says that they will pay you a lump sum
of $22,000 after 5 years, if you deposit $10,700 with them today. Which offer should
you accept and why?
ANSWER: (Slides 3-­‐41 to 3-­‐43) To answer this question, you have to calculate the rate of return that will be earned on
each investment and accept the one that has the higher rate of return.
Bank A’s Offer:
PV = -$9,500; n=5; FV =$20,000; PMT = 0; CPT I = 16.054%
OR
Rate = (FV/PV)1/n – 1 = ($20,000/$9,500)1/5 – 1 = 1.16054 – 1 = 16.054%
Bank B’s Offer:
PV = -$10,700; n=5; FV =$22,000; PMT = 0; CPT I = 15.507%
OR
Rate = (FV/PV)1/n – 1 = ($22,000/$10,700)1/5 – 1 = 1.15507 – 1 = 15.507%
You should accept Bank A’s offer, since it provides a higher annual rate of return.
3. You have decided that you will sell off your house, which is currently valued at
$300,000 when it appreciates in value to $450,000. If houses are appreciating at an
average annual rate of 4.5% in your neighborhood, for how approximately long will
you be staying in the house?
ANSWER: (Slides 3-­‐44 to 3-­‐45) PV = –300,000; FV = 450,000; I = 4.5%; PMT = 0; CPT n = 9.21 years or 9
and 3 months
OR
n = [ln(FV/PV)]/[ln(1+i)]
n = [ln(450,000/(300,000])/[ln(1.045)] = .40547 / .04402 = 9.21 years
©2013 Pearson Education, Inc. Publishing as Prentice Hall years
Chapter 3 n The Time Value of Money (Part 1) 75 4. Your arch nemesis, who happens to be an accounting major, makes the following
remark, “You finance types think you know it all…well, let’s see if you can tell me,
without using a financial calculator, what rate of return would an investor have to
earn in order to double $100 in 6 years?” How would you respond?
ANSWER (Slides 3-­‐46 to 3-­‐48) Use the rule of 72 to silence him once and for all and then prove the answer by
compounding a sum of money at that rate for 6 years to show him how close your
response was to the actual rate of return…Then ask him politely if he would like to
borrow your jar of beans!
Rate of return required to double a sum of money = 72/N = 72/6 = 12%
Verification: $100(1.12)6 = $197.38…which is pretty close to double
The accurate answer would be calculated as follows:
PV = –100; FV = 200; n = 6; PMT = 0; I = 12.246%
OR
r = (FV/PV)1/n – 1 = (200/100)1/ 6 – 1 = 1.12246 – 1 = .12246 or 12.246%
5. You want to save $25,000 for a down payment on a house. Bank A offers to pay
9.35% per year if you deposit $11,000 with them, while Bank B offers 8.25% per
year if you invest $10,000 with them. How long will you have to wait to have the
down payment accumulated under each option?
ANSWER (Slides 3-­‐49 to 3-­‐50) Bank A
FV = $25,000; I = 9.35%; PMT = 0; PV = –11,000; CPT N = 9.18 years
Bank B
FV = $25,000; I = 8.25%; PMT = 0; PV = –10,000; CPT N = 11.558 years
©2013 Pearson Education, Inc. Publishing as Prentice Hall