MCS107
Mathematics of Finance; REVIEW
Simple Interest - interest that is computed on the original principal only,
Notation
P = the principal (i.e., the amount of the original investment)
I = the amount of interest earned on a principal
r = interest rate , written as a decimal number
t = term of the investment (i.e., the length of time in YEARS that the money is invested)
A = the future amount (i.e., the sum of the principal and interest after t years)
Simple Interest Formulas
𝐼 = 𝑃.𝑟.𝑡
𝐴=𝑃 +𝐼
Example 1.
Find the simple interest on a $1500 investment made for 5 years at an interest rate of 3.5%
yearly. What is the future amount?
Solution: The interest is
𝐼 = 𝑃.𝑟.𝑡 = (1500)(0.035)(5) = $262.5
and the amount due at the end of 5 years is
𝐴 = 𝑃 + 𝐼 =$ (1500+ 262.5) = $ 1762.5
Example 2.
Find the total amount due on a loan of $1000 at 6% simple interest at the end of 4 months.
Solution: To find the future amount 𝐴 due in 4 months, we use the simple interest formula
with;
𝑃 = 1000, 𝑟 = 6% = 0.06, 𝑡 =
4
12
=
1
3
year.
Therefore,
𝐴 = 𝑃 + 𝐼 = (1000) + [(1000).(0.06).( 13 )] = $1020.
Example 3.
If you buy a 180-day treasury bill with a maturity value of $10000 for $9800, what annual
simple interest rate will you earn?
Solution: Again we use the simple interest formula, but this time we are interested in finding
180
= 12 = 0.5 year.
𝑟, given 𝑃 = 9800, 𝐴 = 10000, 𝑡 = 360
Therefore,
𝐴 = 𝑃 + 𝐼 = 𝑃 + 𝑃.𝑟.𝑡 = 𝑃 (1 + 𝑟.𝑡) = (9800)[1 + (0.5)𝑟].
1
Thus,
10000 = (9800)[1 + (0.5)𝑟] implies
10000
9800
= 1 + (0.5)𝑟
or
2
98
= 0.5𝑟, 𝑟 =
4
98
∼
= 0.04
Matched Problems:
Q1. Use formula for simple interest to find each of the indicated quantities:
a) 𝑃 = $300, 𝑟 = 6%, 𝑡 = 3years ; 𝐼 =?
b) 𝐼 = $36, 𝑟 = 4%, 𝑡 = 6 months ; 𝑃 =?
c) 𝐼 = $48, 𝑃 = $600, 𝑡 = 240days ; 𝑟 =?
d) 𝑃 = $2400, 𝑟 = 5%, 𝐼 = $60 ; 𝑡 =?
Q2. Use formula for simple interest to find each of the indicated quantities:
a) 𝑃 = $3000, 𝑟 = 4.5%, 𝑡 = 30 days ; 𝐴 =?
b) 𝐴 = $910, 𝑟 = 16%, 𝑡 = 13 weeks ; 𝑃 =?
c) 𝐴 = $14560, 𝑃 = $13000, 𝑡 = 4 months ; 𝑟 =?
Compound Interest - earned interest that is periodically added to the principal and
thereafter itself earns interest at the same rate. In other words, for compound interest, interest is earned on both the principal and also on the interest that has already accumulated.
Conversion Period - the interval of time between successive interest calculations. Most
often, interest is compounded annually, semiannually, quarterly, monthly, or daily.
Compound Interest Formula:
𝐴 = 𝑃 (1 + 𝑖)𝑛
where
𝑖 = rate of interest per convension period
𝐴 = Accumulated amount at the end of 𝑛 conversion periods
𝑃 = Principal
Example 4.
If $2000 is invested at 6% compounded
a) annually
b) semiannually
c) quarterly
d) monthly
What is the future amount after 2 years?
Solution:
a) Compounded annually means that there is one interest payment period per year. So,
𝑛 = 2, 𝑖 = 0.06, 𝑃 = 2000.
2
𝐴 = 𝑃 (1 + 𝑖)𝑛 = (2000)(1 + 0.06)2 ∼
= $2247.2
b) Compounded semiannually means that there are two interest payment periods per
year. So, the number of payment periods in 2 years is 𝑛 = (2)(2) = 4 and 𝑖 = ( 12 )(0.06) =
0.03, 𝑃 = 2000.
𝐴 = 𝑃 (1 + 𝑖)𝑛 = (2000)(1 + 0.03)4 = (2000)(1.1255) ∼
= $2251
c) Compounded quarterly means that there are four interest payment periods per year.
So, the number of payment periods in 2 years is 𝑛 = (2)(4) = 8 and 𝑖 = ( 14 )(0.06) = 0.015, 𝑃 =
2000.
𝐴 = 𝑃 (1 + 𝑖)𝑛 = (2000)(1 + 0.015)8 = (2000)(1.0934) ∼
= $2157
d) Compounded monthly means that there are twelve interest payment periods per year.
1
So, the number of payment periods in 2 years is 𝑛 = (2)(12) = 24 and 𝑖 = ( 12
)(0.06) =
0.005, 𝑃 = 2000.
𝐴 = 𝑃 (1 + 𝑖)𝑛 = (2000)(1 + 0.005)24 = (2000)(1.083) ∼
= $2158
Continuous Compound Interest
From the previous example note that, if the number of compounding periods increases
infinitely then the amount will approach to some limiting value.
Therefore, if a principal,𝑃 , is invested at an annual rate,𝑟, (expressed as a decimal number)
compounded continuously then the amount , 𝐴, in the account at the end of ,𝑡 years is given
by
𝐴 = 𝑃.𝑒𝑟𝑡
Example 5.
What amount will an account have after 3 years if $4000 is invested at an annual rate of 7%
compounded
a) dailly
b) continuously
Solution.
1
a) Using the compounded interest formula with 𝑃 = 4000, 𝑖 = ( 365
)(0.07), 𝑛 = (365)(3) is
𝐴 = 𝑃 (1 + 𝑖)𝑛
𝐴 = (4000)(1 + 0.00019)1095
b) Using the continuous compound interest formula 𝐴 = 𝑃 𝑒𝑟𝑡 , with 𝑃 = 4000, 𝑟 = 0.07
and 𝑡 = 3; we get
𝐴 = (4000)𝑒(0.079)(3) = (4000)𝑒(0.237)
3
Example 6.
How much should you invest now at 6% to have $14000 toward the puchase of a car in 3
years if interest is compounded
a) quarterly
b) continuously
Solution.
a) We are given a future value 𝐴 = 14000 for a compound interest investment, and we need
to find the present value 𝑃 given 𝑖 = ( 41 )(0.06) = 0.015, 𝑛 = (4)(3) = 12 is
𝐴 = 𝑃 (1 + 𝑖)𝑛
14000 = 𝑃 (1 + 0.015)12
𝑃 =
14000
(1+0.015)12
b) We are given a future value 𝐴 = 14000 for a continuous compound interest investment,and we need to find the present value 𝑃 given 𝑟 = 0.06 and 𝑡 = 3; we get
14000 = 𝑃 𝑒(0.06)(3)
𝑃 =
14000
𝑒(0.06)(3)
4