Financial mathematics

1. 1
Chapter 1: Simple Interest
I - Overview
A - Definition
1) Example
We invest $10,000 for 3 years at an annual rate of 8%.
- the interest produced each year is equal to:
10 000 x 8/100 = $800
- After 3 years, the placement will increase by:
$800 x 3 = $2 400
- At the end of the 3 years, the placement will amount to:
10 000 + 2400 = $12 400
2) Remark:
In simple interest, interest is not added to the capital at the end of the capitalization
period to produce interest.
3) Definition:
Interest represents the rent or the remuneration of the money loaned. This money
is called "capital or investment"
Simple interest generally applies to short-term loans or investments (less than a
year).
B – Formula of the simple interest
C : Invested capital
t : Interest rate for $100 of capital
n : Duration of placement
Va : Acquired value
I : Interest
I= C x t /100 x n
Va = C + C x t/100 x n
Va = C + I
If we invest some money today, how much will we receive tomorrow? "This is called
compounding.

2. 2
Verification
I = 10 000 x 8/100 x 3 = $2 400
Va = 10 000 x 10 000 + 8/100 x 3 or 10 000+ 2 400
Va = 12 400
C - Study of each of the elements of the simple interest formula
1 - Capital
This is the amount of money invested on a specific date
Example:
A capital placed at an annual rate of 3% for 3 years has yielded $18 of interest.
Determine the capital initially invested.
Solution:
I = C x t/100 x n
I x 100 = C x t x n
C = I x 100
t x n
C= 18 x 100 = $200
3 x 3
2 - Duration
The duration of the loan (or investment) can be expressed in years, months or days.
Example 1: duration expressed in years
A capital of $5,000 is placed at a simple interest at 11.5% for 2 years
I = C x t/100 x n
I = 5 000 x 11,5/100 x 2 = $1 150
Ex 2: Duration expressed in months:
A capital of $3,600 is placed at a simple interest for 9 months at 6%
I = C x t/100 x n/12(months)

3. 3
I = C x t x n
1 200
I = 3 600 x 6 x 9 = $162
1 200
Remark
In Lebanon, the financial year is 360 days.
When the placement period is between 2 dates, the months are counted for their actual
duration, even if the year is reduced to 360 days.
For the calculation of the number of days, we do not count the day of departure, but the
day of arrival.
Ex :
A capital of $4,800 is placed at an annual rate of 4.75% from March 12 to July 17.
Placement period is:
March - April - May - June - July
(31d-12) - 30d - 31d - 30d - 17d
19d + 30d + 31d + 30d + 17d = 127 days
I = 4800 x 4,75 x 127 = $80,43
36 000
3 – Interest rate
It is expressed in% (percentage) and indicates the amount of money brought back by $100
in a determined period (in principle a year).
Ex :
A capital of $5200 with simple interest yielded $52 of interest for 36 days.
Determine the interest rate.
I = C x t x n
36000
t = I x 36000
C x n
T= 52 x 36000 = 10%
52000 x 36

4. 4
Verification
I = 52000 x 10 x 36 = $52
36000
Remark
The rate used must correspond to the placement period chosen (yearly, semi-annually,
quarterly and monthly)
Corresponding proportional rates
Semi-annually quarterly monthly
Annual rate (ta) ts = ta /2 tt = ta /4 tm = ta /12
Ex :
At what semi-annual rate was a capital of $5000 which, in 102d, earned $85 of interest?
Solution
Let t be the annual rate
5000 x t x 102 = $85
360
t = 0,06
so the semi-annual rate is:
ts = ta /2 = 0,06 = 0,03 soit 3% .
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