Mathematics of Finances
1. Simple Interest- is calculated by the equation I=Prt
where;
I = dollar amount of interest earned
P= principal amount in dollars
r = annual interest rate in decimals
t = time in years
EXAMPLE: If $8000 is invested for 2 years at an annual interest rate of 9%, how much interest will be
received at the end of the 2-year period?
Solution: The interest is I= Prt = 8000 (0.09)(2) = $1440.
2. Future value of an investment or a loan is calculated by: S= P + I
where;
or S = P + Prt
S = the future value
I = interest
P = principal
r = annual interest rate in decimals
t = time in years
EXAMPLE: If $6000 is invested for 4 years at an annual interest rate of 8%, how much interest will be
received at the end of the 4-year period and what will be the future value of the investment?
Solution:
The simple interest is I =Prt = 6000 (0.08 )(4) = $1920
The future value is S=P+I= 6000 + 1920 = $7920.
3. Annual Compounding- The interest for each period is added to the principal before interest is calculated for the
next year.
S = P ∗ (1 + r ) n
where;
S = future value
P= principal in dollars
r = interest rate in decimal
n= number of years
EXAMPLE: If $3000 is invested for 4 years at 9% compounded annually, what is the future value of
investment?
Solution:
The future value is
the
4
S=$3000 ∗ (1+0.09)
=$3000 ∗ (1.4115816)
=$4234.7448
=$4234.74
NOTE: Future Value is a dollar amount. Therefore, it must be rounded to the nearest cent.
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4. Periodic Compounding-
is used when the interest on an investment or a loan is compounded more than once a
year.
Future Value:
r⎞
⎛
S = P ∗ (1 + i ) = P ∗ ⎜1 + ⎟
⎝ m⎠
( m∗t )
n
S = future value
P = principal or present value
i = interest rate per compounding period,
n- number of compounding periods
r
, where r is the nominal interest rate
m
and m is the number of times compounded per year. This is expressed as a
decimal.
- n is found by the equation n = m ∗ t where m is the number of times compounded per year and t is the
number of years invested.
-This method can be used when compounding semi- annually (2 times per year), quarterly (4 times per
year), monthly (12 times per year) or daily (365 times per year).
- interest(i) is found by the equation i =
EXAMPLE:
What amount must be invested now in order to have $12,000 after 3 years if money is worth
6% compounded semiannually?
Solution:
Find the preset value P. The future value is
0.06
S=$12,000. Use i=
=0.03 and n=3(2)=6.
2
n
S=P ∗ (1+i)
$12,000=P ∗ (1+0.03) 6
= P ∗ (1.03) 6
= P ∗ (1.1940523)
12,000
= $10,049.81, to the nearest cent.
P=
1.1940523
5. Continuous Compounding- is used to determine the interest that results from compounding continuously.
Future Value:
S = P ∗ e (r ∗t )
P = principal
r = interest rate,
t = number of years
Example:
Find the future value if $1000 is invested for 20 years at 8%,
compounded continuously.
Solution:
The future value is
(0.08 )∗( 20 )
S=$1000*e
=$1000 ∗ e 1.6
=$1000*(4.95303)
=$4953.03
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