Chapter 1
Time Value of Money
Time Value of Money
TVM explains the change in the amount of
money over time for funds owed by or
owned by a corporation (or individual)
• Purchasing power of money decreases over time
• Investments are expected to earn a return
Example
Exercise
You deposit $1,000 in a savings account
that pays interest at a rate of 10% per
year. How much money will you have after
one year?
Exercise
You deposit $1,000 in a savings account
that pays interest at a rate of 10% per
year. How much money will you have after
three years?
Solution
Total in Account at Amount Saved
Year Interest
Start of Year at End of Year
0 1,000 100 1,100
1 1,000
2
Total in Account at Amount Saved
Year Interest
Start of Year at End of Year
0 1,000 100 1,100
1 1,100
2
General Expressions
Simple Interest:
Interest = principal * number of periods * interest rate
I=Pni
Compound Interest for time period t:
I(t) = (principal + all accrued interest) * interest rate
# j=t−1 &
I t = %% P + ∑ I j (( (i)
$ j=1 '
Substitute
Simple Interest:
Interest = principal * number of periods * interest rate
I=Pni = (1,000) (3) (0.1) = 300
Compound Interest for time period t:
I(t) = (principal + all accrued interest) * interest rate
# j=t−1 & I3 = (1,000 + 100+110) (0.1) = 121
I t = %% P + ∑ I j (( (i)
$ j=1 ' Total I = 100 + 110 + 121 = 331
Compound Interest
$1,000 is lent for 3 years at i = 10%
per year compounded annually.
Total Owed at
Year Interest Amount Accumulated
Start of Year
0 P = 1,000 Pi = 100 F1 = 1,100
1 1,100 F1i = 110 F2 = F1 + F1i = 1,210
2 1,210 F2i = 121 F3 = F2 + F2i = 1,331
So, F3 = P(1 + i)3
Symbols
t = time, usually in periods such as years or
months
P = value or amount of money at a time t,
designated as present or time 0
F = value or amount of money at some future
time, such as at t = n periods in the future
n = number of interest periods
i = interest rate per time period
Exercise
If interest is compounded at 20% per
year, how long will it take for $50,000 to
accumulate to $86,400?
Exercise
What is the time it would take a given
sum of money to double at 4% simple
interest per year?
Cash Flow Diagram
Representing Money over Time
Cash Flow Diagram (CFD)
Up (+) for income
Down (-) for spending
End of period cash flow
Example
A company invests $500,000
to manufacture a new
product. The sale of this
product is expected to 0 1 2 3 4 5
provide a net income of
$70,000 a year for five
years, beginning at the end
of the first year.
Draw the CFD.
Example
A company plans expenditures
of $1 million now and each of
the next four years just for the
0 1 2 3 4
improvement of its products.
Construct the cash flow diagram
to find the equivalent value of
these expenditures at the end
of year 4, using a cost of capital
estimate for safety-related
funds of 12% per year.
Commonly used Symbols
t = time, usually in periods such as years or months
P = value or amount of money at a time t, designated as
present or time 0
F = value or amount of money at some future time, such
as at t = n periods in the future
A = series of consecutive, equal, end-of-period amounts
of money
n = number of interest periods; years, months, quarters
i = interest rate or rate of return per time period;
percent per year or month or any other period
Exercise
Suppose that you have a savings plan covering the next
10 years, according to which you put aside $600 today,
$500 at the end of the second and fourth years, and
$400 at the end of each year during the last five years.
As part of this plan, you expect to withdraw $300 at
the end of every year for the first three years and
$350 at the end of years 5, 7, and 9.
a) Tabulate your cash flows
b) Draw your cash flow diagram
Year Savings Withdrawals Cash Flows
0 600 0 - 600
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
$600
Reminder: Interest
Simple Interest:
Interest = principal * number of periods * interest rate
I=Pni
Compound Interest:
Interest = (principal + all accrued interest) * interest rate
# j=t−1 &
I t = %% P + ∑ I j (( (i)
$ j=1 '
Example
$1,000 is lent for 3 years at i = 10%
per year compounded annually.
Total Owed at
Year Interest Amount Accumulated
Start of Year
0 P = 1,000 Pi = 100 F1 = 1,100
1 1,100 F1i = 110 F2 = F1 + F1i = 1,210
2 1,210 F2i = 121 F3 = F2 + F2i = 1,331
So, F3 = P(1 + i)3
Compound Interest
Future Value,
n
F = P (1 + i)
Present Value,
1
P=F n
(1 + i)
Exercise
A person deposits $5,000 into an account
which pays interest at a rate of 8% per
year. Find the amount in the account
after 10 years.
Exercise
A small company wants to make a single
deposit now so it will have enough money
to purchase a tractor costing $50,000
five years from now. If the account will
earn interest of 10% per year, find the
amount that must be deposited now.
