5
Introduction to Valuation: The Time Value of
Money
0
◦ Assume interest rates are 4.3884%.
◦ You have just won a lottery and must choose
between the following two options:
 Receive a cheque for $150,000 today.
 Receive $10,000 a year for the next 25
years.
KEY QUESTIONS FOR YOU:
Which option gives you the biggest “winnings”
How should you tackle this kind of problem?
1
1.
2.
3.
4.
5.
Be able to compute the future value of an investment
made today
Be able to compute the present value of cash to be
received at some future date
Be able to compute the return on an investment
Be able to compute the number of periods that equates a
present value and a future value given an interest rate
Be able to use a financial calculator and/or a spreadsheet
to solve time value of money problems
2

Consider the time line below:
0
1
2
3
……..
PV





t
FV
PV is the Present Value, that is, the value today.
FV is the Future Value, or the value at a future date.
The number of time periods between the Present Value and
the Future Value is represented by “t”.
The rate of interest is called “r”.
All time value questions involve the four values above: PV,
FV, r, and t. Given three of them, it is always possible to
calculate the fourth.
3



Present Value – earlier money on a time line
Future Value – later money on a time line
Interest rate – “exchange rate” between
earlier money and later money
◦
◦
◦
◦
Discount rate
Cost of capital
Opportunity cost of capital
Required return
4


Suppose you invest $1,000 for one year at 5% per
year. What is the future value in one year?
◦ Interest = 1,000(.05) = 50
◦ Value in one year = principal + interest = 1,000
+ 50 = 1,050
◦ Future Value (FV) = 1,000(1 + .05) = 1,050
Suppose you leave the money in for another year.
How much will you have two years from now?
◦ FV = 1,000(1.05)(1.05) = 1,000(1.05)2 =
1,102.50
5



Simple interest
Compound interest
Consider the previous example
◦ FV with simple interest = 1,000 + 50 + 50 =
1,100
◦ FV with compound interest = 1,102.50
◦ The extra 2.50 comes from the interest of .05(50)
= 2.50 earned on the first interest payment
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Beginning Simple Compound
Year Amount
Interest Interest
1
2
3
4
5
$100.00
110.00
121.00
133.10
146.41
Totals
$10.00
10.00
10.00
10.00
10.00
$50.00
$ 0.00
1.00
2.10
3.31
4.64
$ 11.05
Total
Interest
Total
Amount
$10.00
11.00
12.10
13.31
14.64
$ 61.05
$110.00
121.00
133.10
146.41
161.05
7

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
8

Suppose you invest the $1,000 from the
previous example for 5 years. How much
would you have?
◦ FV = 1,000(1.05)5 = 1,276.28

The effect of compounding is small for a
small number of periods, but increases as
the number of periods increases. (Simple
interest would have a future value of
$1,250, for a difference of $26.28.)
9

Suppose you had a relative deposit $10 at
5.5% interest 200 years ago. How much
would the investment be worth today?
◦ FV = 10(1.055)200 = 447,189.84

What is the effect of compounding?
◦ Simple interest = 10 + 200(10)(.055) = 120.00
◦ Compounding added $447,069.84 to the value of
the investment
10

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?
◦ FV = 3,000,000(1.15)5 = 6,034,072
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

What is the difference between simple
interest and compound interest?
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?
12
Q. You have just won a $1 million jackpot in the provincial lottery. You can
buy a ten year certificate of deposit which pays 6% compounded
annually. Alternatively, you can give the $1 million to your brother-in-law,
who promises to pay you 8% simple interest annually over the ten year
period. Which alternative will provide you with more money at the
end of ten years?
13

Deposit $5,000 today in an account paying
12%. How much will you have in 6 years? How
much is simple interest? How much is
compound interest?
14

How much do I have to invest today to have some
amount in the future?
◦ FV = PV(1 + r)t
◦ Rearrange to solve for PV = FV / (1 + r)t


When we talk about discounting, we mean finding
the present value of some future amount.
When we talk about the “value” of something, we
are talking about the present value unless we
specifically indicate that we want the future value.
15


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?
PV = 10,000 / (1.07)1 = 9,345.79
16

You want to begin saving for your daughter’s
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?
◦ PV = 150,000 / (1.08)17 = 40,540.34
17

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?
◦ PV = 19,671.51 / (1.07)10 = 10,000
18

For a given interest rate – the longer the
time period, the lower the present value
◦ What is the present value of $500 to be received
in 5 years? 10 years? The discount rate is 10%
◦ 5 years: PV = 500 / (1.1)5 = 310.46
◦ 10 years: PV = 500 / (1.1)10 = 192.77
19

For a given time period – the higher the
interest rate, the smaller the present value
◦ What is the present value of $500 received in 5
years if the interest rate is 10%? 15%?
 Rate = 10%: PV = 500 / (1.1)5 = 310.46
 Rate = 15%; PV = 500 / (1.15)5 = 248.59
20



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?
21


Suppose you need $3,000 next year to buy a
new computer. The interest rate is 8 percent
per year. How much money should you set
aside now in order to pay for the purchase?
What if you can wait 2 years till you purchase
the computer, how much money should you
set aside?
22

PV and FV are related!
◦ Have you noticed that $620,921 becomes $1 million (and that $1 million
requires $620,921) if you have a time period of 5 years and a discount
rate of 10%?
PV at 10%
$1,000,000
$620,921
FV at 10%
23


You should understand that a dollar received
today is worth more than a dollar received
tomorrow.
Lesson: The value of cash flows received at
different times can never be directly
compared. You must first discount all cash
flows to a common date and then compare
them.
24


Kangaroo Autos is offering free credit on a
$10,000 car. You pay $4,000 down and then
the balance at the end of 2 years. Turtle
Motors next door does not offer free credit
buy will give you $500 off the list price. If the
interest rate is 10 percent, which company is
offering the better deal?
Suppose your friend wants to buy a car but
does not like to be indebt to the car
company. What would you recommend?
25


PV = FV / (1 + r)t
There are four parts to this equation
◦ PV, FV, r and t
◦ If we know any three, we can solve for the
fourth
26


Often we will want to know what the
implied interest rate is in an investment
Rearrange the basic PV equation and solve
for r
◦ FV = PV(1 + r)t
◦ r = (FV / PV)1/t – 1
27

You are looking at an investment that will
pay $1,200 in 5 years if you invest $1,000
today. What is the implied rate of interest?
◦ r = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714%
28

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?
◦ r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%
29

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?
◦ r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27%
30


What are some situations in which you might want
to know the implied interest rate?
You are offered the following investments:
◦ 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%.
◦ What is the implied interest rate for the first choice and
which investment should you choose?
31

Benjamin Franklin died on April 17, 1790. In
his will, he gave 1,000 dollars to the city of
Boston and $1000 to the city of Philadelphia.
It was agreed that the money would be paid
out 200 years after Franklin’s death in 1990.
By that time, the Philadelphia bequest had
grown to about $2 million; the Boston
bequest had grown to $4.5 million. What was
the rate of return for the two states earn?
32

Suppose you deposit $5000 today in an
account paying r percent per year. If you will
get $10,000 in 10 years, what rate of return
are you being offered?
33

Start with basic equation and solve for t
(remember your logs)
◦ FV = PV(1 + r)t
◦ t = ln(FV / PV) / ln(1 + r)
34

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?
◦ t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
35

Suppose you want to buy a new house.
You currently have $15,000 and you figure
you need to have a 10% down payment
plus an additional 5% of the loan amount
for closing costs. Assume the type of
house you want will cost about $150,000
and you can earn 7.5% per year, how long
will it be before you have enough money
for the down payment and closing costs?
36


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 don’t add any additional money?
37

Suppose you need to come up with $40,000
for a world cruise. You have only $20,000.
You know you can earn 15% interest per year.
How many years do you have to wait?
38


Many financial calculators are available
online
Click on the web surfer to go to
Investopedia’s web site and work the
following example:
◦ You need $50,000 in 10 years. If you can earn 6%
interest, how much do you need to invest today?
◦ You should get $27,919.74
39

Use the following formulas for TVM calculations
◦
◦
◦
◦


FV(rate,nper,pmt,pv)
PV(rate,nper,pmt,fv)
RATE(nper,pmt,pv,fv)
NPER(rate,pmt,pv,fv)
The formula icon is very useful when you can’t
remember the exact formula
Click on the Excel icon to open a spreadsheet
containing four different examples.
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41




You have $10,000 to invest for five years.
How much additional interest will you earn
if the investment provides a 5% annual
return, when compared to a 4.5% annual
return?
How long will it take your $10,000 to
double in value if it earns 5% annually?
What annual rate has been earned if
$1,000 grows into $4,000 in 20 years?
42
6
Discounted Cash Flow Valuation
43






Be able to compute the future value of
multiple cash flows
Be able to compute the present value of
multiple cash flows
Be able to compute loan payments
Be able to find the interest rate on a loan
Understand how interest rates are quoted
Understand how loans are amortized or
paid off
44
Future Value Calculations
◦ So far, we have looked at problems involving
only a single cash flow.
◦ This is unrealistic – most business
investments will involve multiple cash flows
over time.
◦ We need a method for coping with such
streams of cash flows!
45
46

Assume interest rates are 8%.
You make 3 deposits to your bank account:
 $1,200 today
 $1,400 one year later.
 $1,000 two years later.
How much will you have in three years time?

47

Doing Future Value Calculations
◦ Calculate what each cash flow will be worth at the specified
future date and add up these future values.
$1,200
0
$1,400
1
$1,000
2
3
FV in Year 3:
$1,080.00 = $1,000 x 1.08
$1,632.96 = $1,400 x (1.08)2
$1,511.65 = $1,200 x (1.08)3
$4,224.61
48

Find the value at year 3 of the following
cash flows: Year 0 cash flow is 7000, year 1,
2, and 3 cash flows are 4000 each.
◦
◦
◦
◦
◦

Today (year 0): FV = 7000(1.08)3 = 8,817.98
Year 1: FV = 4,000(1.08)2 = 4,665.60
Year 2: FV = 4,000(1.08) = 4,320
Year 3: value = 4,000
Total value in 3 years = 8,817.98 + 4,665.60 +
4,320 + 4,000 = 21,803.58
Value at year 4 = 21,803.58(1.08) =
23,547.87
49

Suppose you invest $500 in a mutual fund
today and $600 in one year. If the fund
pays 9% annually, how much will you have in
two years?
◦ FV = 500(1.09)2 + 600(1.09) = 1,248.05
50


How much will you have in 5 years if you
make no further deposits?
First way:
◦ FV = 500(1.09)5 + 600(1.09)4 = 1,616.26

Second way – use value at year 2:
◦ FV = 1,248.05(1.09)3 = 1,616.26
51

Suppose you plan to deposit $100 into an
account in one year and $300 into the
account in three years. How much will be in
the account in five years if the interest rate
is 8%?
52
53
You wish to buy a car making three
installments:
$8,000 today
$4,000 one year later.
$4,000 two years later.
How much money would you have to place in
an account today to generate this stream of
cash flows?
54

Present Value Calculations
◦ You would need to place $15,133.06 in an
account today to generate the desired cash
flows:
-$8,000
PV today:
0
-$4,000
1
-$4,000
2
$8,000.00
$4,000 / (1.08) = $3,703.30
$4,000 / (1.08)2 = $3,429.36
$15,133.06
55
0
1
200
2
3
4
400
600
800
178.57
318.88
427.07
508.41
1,432.93
56


You can use the PV or FV functions in
Excel to find the present value or future
value of a set of cash flows
Click on the Excel icon for an example
57

You are considering an investment that will
pay you $1,000 in one year, $2,000 in two
years and $3000 in three years. If you want
to earn 10% on your money, how much
would you be willing to pay?
58

Another way to use the financial calculator for uneven cash
flows is to use the cash flow keys
◦ Usually you press CF and enter the cash flows beginning
with year 0.
◦ In Excel you can use the NPV function.
59




Suppose you are looking at the following possible
cash flows: Year 1 CF = $100; Years 2 and 3 CFs =
$200; Years 4 and 5 CFs = $300. The required
discount rate is 7%
What is the value of the cash flows at year 5?
What is the value of the cash flows today?
What is the value of the cash flows at year 3?
60