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…