1
McGraw-Hill/Irwin
Copyright 4- 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
Time value of money
A dollar today is worth more than a dollar
tomorrow
Why?
Default risk
Inflation
Opportunity cost
Discount rates help us value future cash
flows today
Basic Definitions
Present Value
(PV)
The current value of future cash flows
discounted at the appropriate discount rate
Value at t=0 on a time line
Future Value
(FV)
The amount an investment is worth after one
or more periods.
Later money on a time line
4- 3
Basic Definitions
Interest rate (r)
Discount rate
Cost of capital
Opportunity cost of capital
Required return
Terminology depends on usage
4- 4
Time Line of Cash Flows
Tick marks at ends of periods
Time 0 is today;
Time 1 is the end of Period 1
0
1
2
3
CF1
CF2
CF3
r%
CF0
+CF = Cash INFLOW -CF = Cash OUTFLOW
PMT = Constant CF
4- 5
Time Line for a $100 Lump Sum
due at the End of Year 2.
0
r%
1
2 Year
100
4- 6
Future Values: General Formula
FV = PV(1 + r)t
FV = future value
PV = present value
r = period interest rate, expressed
as a decimal
t = number of periods
Future value interest factor = (1 + r)t
Note: yx key on your calculator
4- 7
Future Values Example 1
Suppose you invest $100 for one year
at 10% per year.
What is the future value in one year?
Interest = 100(.10) = 10
Value in one year
= Principal + interest
= 100 + 10 = 110
Future Value (FV)
= 100(1 + .10) = 110
4- 8
Future Values Example 1
Suppose you leave the money in for another
year. How much will you have two years
from now?
FV = 100(1.10)(1.10)
= 100(1.10)2 = 121.00
4- 9
Effects of Compounding
Simple interest
Interest earned only on the original
principal
Compound interest
Interest earned on principal and on
interest received
Interest on interest interest earned
on reinvestment of previous interest
payments
4-10
Effects of Compounding
Consider the previous example
FV w/simple interest
= 100 + 10 + 10 = 120
FV w/compound interest
=100(1.10)2 = 121.00
The extra 1.00 comes from the interest
of .10(10) = 1.00 earned on the first
interest payment
4-11
Texas Instruments BA-II Plus
= future value
One of these MUST
be negative
= present value
= period interest rate (r)
= number of periods
4-12
Texas Instruments BA-II Plus
= period interest rate (r)
must equal 1 for the to be the period rate
( = 1 = default on new BAII+)
Interest is entered as a percent, not a decimal
5% interest = 5, not .05
= 0 for this chapter only!
Clear the registers before each problem
Or reenter each field
4-13
Texas Instruments BA-II Plus
Set number of decimal places to display
Press key,
Press key (above .),
Enter desired decimal places (e.g., 4).
Press to set the displayed choice.
4-14
Texas Instruments BA-II Plus
Be sure payment per period is set to 1
Press key,
Press (above ),
Enter 1,
Press
Press
4-15
TI BAII+: Set Time Value
Parameters
Be sure calculator is set for cash flows at the
END of each period
To set END (for cash flows occurring at the end
of the period),
Press key,
Press (above ).
This is a toggle switch. The default is END.
To change to BEGIN, hit (above ) to go back and forth.
Press to set the displayed choice.
4-16
Future Values Example 2
Suppose you invest the $100 from the previous
example for 5 years. How much would you
have?
Formula Solution:
FV
=PV(1+r)t
=100(1.10)5
=100(1.6105)
=161.05
Keystrokes: 1.1 5 100 161.0510
4-17
Table 4.1
4-18
Figure 4.1
4-19
Texas Instruments BA-II Plus
To calculate FV: 10% 5 years PV=$100
Key Entry
Display
0.00
5 5.00
10 10.00
100 -100.00
0 0.00
161.05
4-20
Excel Spreadsheet Functions
Excel TVM functions:
=FV(rate,nper,pmt,pv)
=PV(rate,nper,pmt,fv)
=RATE(nper,pmt,pv,fv)
=NPER(rate,pmt,pv,fv)
Use the formula icon (x) when you cant
remember the exact formula
4-21
Future Values Example 3
Suppose you had a relative deposit $10 at 5.5%
interest 200 years ago. How much would the
investment be worth today?
Formula Solution:
FV = PV(1+r)t
= 10(1.055)200
=10(44718.984)
=447,189.84
CalculatorSolution
200
5.5
10
0
= 447,189.84
Excel Solution: =FV(Rate,Nper,Pmt,PV)
=FV(0.055,200,0,-10) = 447189.84
NOTE: Rate = decimal
4-22
Future Value: General Growth
Formula
Suppose your company expects to increase unit
sales of widgets by 15% per year for the next 5
years. If you currently sell 3 million widgets in
one year, how many widgets do you expect to
sell in 5 years?
Formula Solution:
FV
=PV(1+r)t
=3(1.15)5
=3(2.0114)
=6.0341 million
Calculator Solution
5
15
3
0
= -6.0341
Excel Solution: =FV(Rate,Nper,Pmt,PV)
=FV(0.15,5,0,3) = -6.0341
4-23
Future Value:
Important Relationship I
For a given interest rate:
The longer the time period,
The higher the future value
FV = PV(1 + r)t
For a given r, as t increases, FV increases
4-24
Future Value
Important Relationship II
For a given time period:
The higher the interest rate,
The larger the future value
FV = PV(1 + r)t
For a given t, as r increases, FV increases
4-25
Figure 4.2
4-26
Quick Quiz: Part 1
What is the difference between simple
interest and compound interest?
(Slide 4.10)
Suppose you have $500 to invest and
you believe that you can earn 8% per
year over the next 15 years.
How much would you have at the end of 15
years using compound interest?
How much would you have using simple
interest?
4-27
Present Values
The current value of future cash flows
discounted at the appropriate discount rate
Value at t=0 on a time line
Answers the questions:
How much do I have to invest today to have
some amount in the future?
What is the current value of an amount to be
received in the future?
4-28
Present Values
Present Value = the current value of
an amount to be received in the future
Why is it worth less than face value?
Opportunity cost
Risk & Uncertainty
Discount Rate = (time, risk)
4-29
Time Line of Cash Flows
Tick marks at ends of periods
Time 0 is today;
Time 1 is the end of Period 1
0
1
2
3
CF1
CF2
CF3
r%
CF0
+CF = Cash INFLOW -CF = Cash OUTFLOW
PMT = Constant CF
4-30
Present Values
FV = PV(1 + r)t
Rearrange to solve for PV
PV = FV / (1+r)t
PV = FV(1+r)-t
Discounting = finding the present value
of one or more future amounts.
4-31
Whats the PV of $100 due in 3
Years if r = 10%?
Finding PVs is discounting, and its
the reverse of compounding.
0
10%
1
2
PV = ?
3
100
Formula: PV = FV(1+r)-t = 100(1.10)-3 = $75.13
Calculator: 3 10 0 100 = -75.13
Excel: =PV(.10,3,0,100) = -75.13
4-32
Present Value: Example 1
Single Period
Suppose you need $10,000 in one year for the down
payment on a new car. If you can earn 7% annually,
how much do you need to invest today?
Formula Solution:
PV
=FV(1+r)-t
=10,000(1.07)-1
=10,000/1.07
=9,345.79
Calculator Solution
1
7
0
10000
= -9345.79
Excel Solution: =PV(Rate,Nper,Pmt,FV)
=PV(0.07,1,0,10000) = -9345.79
4-33
Present Values: Example 2
Multi-Periods
You want to begin saving for your daughters college
education and you estimate that she will need
$150,000 in 17 years. If you feel confident that you
can earn 8% per year, how much do you need to
invest today?
Formula Solution:
PV
=FV(1+r)-t
=150,000(1.08)-17
=150,000/(1.08)17
=-40,540.34
Calculator Solution:
17
8
0
150000
= -40540.34
Excel =PV(Rate,Nper,Pmt,FV)
=PV(0.08,17,0,150000) Solution: = -40540.34
4-34
Present Values: Example 3
Multi-Periods
Your parents set up a trust fund for you 10
years ago that is now worth $19,671.51. If the
fund earned 7% per year, how much did your
parents invest?
Formula Solution:
PV
=FV(1+r)-t
=19,671.51(1.07)-10
=19.671.51/(1.07)10
=-10,000
Calculator Solution:
10
7
0
19671.51
= -10000
Excel Solution: =PV(Rate,Nper,Pmt,FV)
=PV(0.07,10,0,19671.51) = -10000
4-35
Present Value:
Important Relationship I
For a given interest rate:
The longer the time period,
The lower the present value
FV
PV =
t
(1 + r )
For a given r, as t increases, PV decreases
4-36
Present Value:
Important Relationship I
What is the present value of $500 to be
received in 5 years? 10 years? The
discount rate is 10%
0
-310.46
-192.77
PV?
r=10%
5
10
500
PV?
500
5 yrs: PV = 500/(1.10)5 = -310.46
(1.10)5 =1.6105
10 yrs: PV = 500/(1.10)10= -192.77
(1.10)10 = 2.5937
4-37
Present Value
Important Relationship II
For a given time period:
The higher the interest rate,
The smaller the present value
FV
PV =
t
(1 + r )
For a given t, as r increases, PV decreases
4-38
Present Value:
Important Relationship II
What is the present value of $500 received
in 5 years if the interest rate is 10%? 15%?
Rate = 10%
Calculator Solution:
5
10
0
500
= -310.46
Rate = 15%
Calculator Solution:
5
15
0
500
= -248.59
4-39
Figure 4.3
4-40
Quick Quiz: Part 2
What is the relationship between present
value and future value?
Suppose you need $15,000 in 3 years. If
you can earn 6% annually, how much do
you need to invest today?
If you could invest the money at 8%,
would you have to invest more or less
than at 6%? How much?
4-41
The Basic PV Equation - Refresher
PV = FV / (1 + r)t
There are four parts to this equation
PV, FV, r and t
Know any three, solve for the fourth
Be sure and remember the sign convention
+CF = Cash INFLOW -CF = Cash OUTFLOW
4-42
Discount Rate
To find the implied interest rate,
rearrange the basic PV equation and
solve for r:
FV = PV(1 + r)t
r = (FV / PV)1/t 1
If using formulas with a calculator, make
use of both the yx and the 1/x keys
4-43
Discount Rate Example 1
You are looking at an investment that will pay
$1200 in 5 years if you invest $1000 today.
What is the implied rate of interest?
Formula:
r = (1200 / 1000)1/5 1 = .03714 = 3.714%
Calculator the sign convention matters!!!
5
1000
(you pay $1,000 today)
0
1200
(you receive $1,200 in 5 years)
= 3.714%
Excel: =RATE(5,0,-1000,1200) = 0.03714
4-44
Discount Rate Example 2
Suppose you are offered an investment that will
allow you to double your money in 6 years. You
have $10,000 to invest. What is the implied rate of
interest?
6
10000
0
20000
= 12.25%
Excel: =RATE(6,0,-10000,20000) = 0.1225
4-45
Discount Rate Example 3
Suppose you have a 1-year old son and you want
to provide $75,000 in 17 years towards his college
education. You currently have $5,000 to invest.
What interest rate must you earn to have the
$75,000 when you need it?
Calculator:
17 ;5000 ;0 ; 75000 = 17.27%
Excel: =RATE(17,0,-5000,75000) = 0.1727
4-46
Quick Quiz: Part 3
What are some situations in which you
might want to compute the implied interest
rate?
Suppose you are offered the following
investment choices:
You can invest $500 today and receive $600 in
5 years. The investment is considered low risk.
You can invest the $500 in a bank account
paying 4% annually.
What is the implied interest rate for the first
choice and which investment should you
choose?
4-47
Finding the Number of Periods
Start with basic equation and solve for t:
FV = PV(1 + r)t
FV
ln
PV
t=
ln(1 + r )
Calculator:
Excel:
=NPER(Rate, Pmt, PV, FV)
4-48
Number of Periods Example
You want to purchase a new car and you are
willing to pay $20,000. If you can invest at 10%
per year and you currently have $15,000, how
long will it be before you have enough money to
pay cash for the car?
Calculator Solution:
10 ; 15000 ;20000 ;
= 3.02 years
Excel: =NPER(0.10,0,-15000,20000)= 3.02
4-49
Number of Periods Example
FV
ln
PV
t=
ln(1 + r )
Formula Solution:
FV/PV = 20,000/15,000 = 1.333
ln(1.333) = 0.2877
ln(1.10) = 0.0953
t = 0.2877/0.0953 = 3.0189
4-50
Quick Quiz: Part 4
When might you want to compute the
number of periods?
Suppose you want to buy some new
furniture for your family room. You
currently have $500 and the furniture
you want costs $600. If you can earn
6%, how long will you have to wait if
you dont add any additional money?
4-51
Table 4.4
4-52
Quick Quiz 1 Solution
Invest $500 at 8% per year over 15 years.
How much would you have at the end of 15
years using compound interest?
15 8 500 0 1586.08
500(1.08)15 = 1586.08
=FV(.08, 15, 0, -500)
How much would you have using simple
interest?
500 + 15(500)(.08) = 1,100
Return
to Quiz 4-53
Quick Quiz 2 Solution
You need $15,000 in 3 years. You can earn
6% annually, how much do you need to
invest today?
3 6 15000 0 -12594.29
PV= 15000/(1.06)3 = 15000/(1.191016) =
= 15000 x 0.83962) = 12594.29
=PV(.06, 3, 0, 15000)
Return
to Quiz
4-54
Quick Quiz 2 Solution
You need $15,000 in 3 years. If you could invest
the money at 8%, would you have to invest more
or less than at 6%? How much?
3 15000 0 -11907.48
PV= 15000/(1.08)3 = 15000/(1.125971)
= 15000 x (0.79383) = 11907.48
=PV(.08, 3, 0, 15000)
Difference = $686.81
Return
to Quiz 4-55
Quick Quiz 3 Solution
Investment choices:
Invest $500 today and receive $600 in 5 years. The
investment is considered low risk.
Invest the $500 in a bank account paying 4%
annually.
What is the implied interest rate for the first choice
and which investment should you choose?
5 500 0 600 3.714%
r = (600/500)1/5 - 1 = 3.714%
=RATE(5, 0, -500, 600)
The bank account pays a higher rate.
Return
to Quiz 4-56
Quick Quiz 4 Solution
Suppose you want to buy some new furniture for
Your family room. You currently have $500 and
the furniture you want costs $600. If you can earn
6%, how long will you have to wait if you dont
add any additional money?
6 500 0 600 3.13 years
t = ln(600/500) / ln(1.06) = 3.13 years
=NPER(.06, 0, -500, 600)
Return
to Quiz 4-57
Practice Problems Ch. 4
1.
Elaine has just received an insurance settlement of $25,000. She wants to
save this money until her daughter goes to college. If she can earn an
average of 6.5 percent, compounded annually, how much will she have
saved when her daughter enters college 8 years from now?
2.
Skyline Industries will need $1.8 million 5 years from now to replace some
equipment. Currently, the firm has some extra cash and would like to
establish a savings account for this purpose. The account pays 5.25
percent interest, compounded annually. How much money must the
company deposit today to fully fund the equipment purchase?
4-58
Practice Problems Ch. 4
3.
You have been told that you need $21,600 today in order to have
$100,000 when you retire 42 years from now. What rate of interest was
used in the present value computation? Assume interest is compounded
annually.
4.
You have $2,158 today in your savings account. How long must you wait
for your savings to be worth $4,000 if you are earning 2.1 percent interest,
compounded annually?
4-59
Chapter 4
END