Corporate Finance
Lecture 2. Introduction to Financial
Mathematics

Learning Objectives
• Calculate future values
• Calculate present values
• Use the FV and/or PV formulate to know how to solve for time or interest rate
• Know how to compute the PV and FV of both an ordinary annuity and an annuity due
• Know how to calculate the PV of a perpetuity
• Know how to adjust for compounding periods

2

Time Lines
• The first step in time value analysis is to understand a time line.
Time 0

1

2

3

4

5

Today End of period
1, start of
Period 2
– Time 0 is today, Time 1 is one period from today or the end of the first period (year, month, etc), or the beginning of the second period, and so on.
3

Time Lines (cont.)
• Information on a time line is usually written as: – Cash flows are written immediately below the tick marks – Unknown cash flows are denoted with a question mark – Interest rates are written above the line and between the tick marks.

4

Time Lines (cont.)
• Here, there is a cash outflow of $100 at time 0
(hence the minus sign), the interest rate is
10% and the cash flow at Time 3 is unknown

Time 0
10
%
Cash
flows
100

1

2

3
?

5

Time Value of Money
• A dollar received today is worth more than a dollar received in the future.
– You could invest the dollar you receive today and it would generate a return meaning that you would get more than a dollar in the future.

6

Time Value of Money- An Example
• Imagine that you have $100 today and you invest it in the bank at a yearly interest rate of
10%.
– In one years time you will have $110

• So, in this example, $100 today is worth $110 in one year’s time.

7

Time Value of Money (cont.)
• This introduces some important concepts:
• Present Value (PV). This is the amount you have today, or earlier money on a time line.
• Future Value (FV). This is the amount you will have in the future, or later money a time line.

8

Time Value of Money (cont.)
• Interest rate (i ). This is the amount the bank pays on money invested each period.
Or, “Exchange rate” between earlier money and later money. - Discount rate
- Cost of capital
- Opportunity cost of capital
- Required return

9

Time Value of Money (cont.)
• Interest received (INT). The amount that you receive during the period.
• Number of periods involved (n). How many periods in the analysis.

10

Future Value
• Using these concepts the future value one period ahead can be estimated as:
FVn PV  INT PV  ( PV * i )

1 present i – Where 
PVPV
is the value, – INT is the interest received and
– i is interest rate
11

Future Value
• In the example above, n=1, i=10%, PV=$100. The future value (FV) is:

FVn PV  INT PV  ( PV * i )
PV 1  i 
FV1 100  (100 *10%) $110
• Here the interest is calculated on the principal amount only. This is simple interest.
12

Future Value Over Multiple Periods
(cont.)
Time 0
Cash
flows

100

10
%

1

2

3

?

?

?

13

Future Value Over Multiple Periods
(cont.)
Time 0
Cash
flows
Interest
earned

100

10
%

1

2

3

?

?

?

100*10
% =10

14

Future Value Over Multiple Periods
(cont.)
Time 0
Cash
flows
Interest
earned

100

Amount at end of each period,
FVn

10
%

1

2

3

?

?

?

100*10
% =10
100+1
0 =110

15

Future Value Over Multiple Periods
(cont.)
Time 0
Cash
flows
Interest
earned

100

1

2

3

?

?

?

100*10
% =10

100*10
% =10

10
%

Amount at end
100+1
of each period,
0 =110
FVn – Note that the interest in each period is calculated using the principal at the beginning. This is an example of simple interest 16

Future Value Over Multiple Periods
(cont.)
Time 0
Cash
flows
Interest
earned

100

1

2

3

?

?

?

100*10
% =10

100*10
% =10

10
%

Amount at end
100+1
110+1 of each period,
0 =110
0 =120
FVn – Note that the interest in each period is calculated using the principal at the beginning. This is an example of simple interest 17

Future Value Over Multiple Periods
(cont.)
Time 0
Cash
flows
Interest
earned

100

1

2

3

?

?

?

100*10
% =10

100*10
% =10