Compound Interest
As described by the animation above, the comparison between the simple interest and the compound
interest has this distinct difference regarding the payment of the interest.
The simple interest is straight forward and linear in character. The interest is the same all throughout
the loan period. Unlike with the compound interest, the interest is described as accumulating as time
passes by. It is described as the interest of the previous period will be added to the next period as
part of the succeeding principal. In effect, the interest will include the amount of the previous interest
period and so on and so forth until reaching the agreed time frame of the loan or the investment.
Let us derived the equation,
Future Worth of Compound Interest Derivation Table
Period Principal, P Interest, I
P+I
F=P+I
1
P
Pi
P + Pi
P(1+i)
2
P ( 1 + i ) P (1 + i) i P (1+i) (1+i) P ( 1 + i )2
3
P (1+i)2
P (1+i)2 i P (1+i)2 (1+i) P ( 1 + i )3
n
P (1+i)n-1
P (1+i)n-1 i P (1+i)n-1 (1+i) P ( 1 + i )n
Looking at the table, we can see that as the period progresses the amount of the future worth
increases exponentially. Comparing this data to see the comparison.
In conclusion regarding derivation table above, F = P (1+i) n
Compound Interest Sample Problem
1. If I invest an amount of P50,000.00 to a corporation and we agreed on a compounding rate
of return of 6%. They promised me to give back my investment after the 3rd year with all its
profit. How much must I receive after that 3rd year?
Analysis:
Given,
P = P50,000.00
i = 0.06
n = 3 years
F =?
The value that is being asked is the total amount of gross profit I should be receiving after the 3rd
year of my investment. We just simply substitute for our future worth equation.
F = P(1+i)n
F = P50,000.00 (1.06)3
F = P59,550.80
(the future worth based on the principal, number of years and the rate of return)
2. Considering the first problem, how long will I wait in order to have 2.75 times of my
investment?
Analysis
Given,
P = P50,000.00
i = 0.06
n = ? years
F = 2.75 P
Since the number of years is in question, we are going to use the natural logarithm of the equation,
F = P(1+i)n , F = 2.75 P
then,
2.75 P = P (1+0.06)n
2.75 = 1.06n
utilizing the natural log of both sides of the equation,
ln 2.75 = ln 1.06n
ln 2.75 = n (ln 1.06)
n = ln 2.75/ln 1.06
n = 17.36 years
(the number of years required for my investment to have 2.75 times the original principal)
Practice problems
1. If I loaned a certain amount and I need to pay them back in full, with the principal and
compounded interest, after 4.5 years. If the bank requires me to pay it 1.75 times the
principal, what is the interest rate we originally agreed, in percent? (ans. 13.24%)
2. Consider practice prob. 1, what is the exact amount that I need to pay if I loaned
P150,000.00 in pesos? (ans. P262,500.00)
3. How much will I need to invest now if I want to have P1M payback just after 5 years with a
rate of return of 9%? (ans. P649,931.39)
4. Using the principal from practice prob. 3, how many years will I need to wait if I want a
payback amount of P1,500,000.00, in years? (ans. 9.7050 years)
5. In an investment, I have a full payback with an amount of P500,000.00 after 20.9123 years.
If we initially agreed on an 8% rate of return, compared with the principal, how many times
did my investment had after the mentioned period? (ans. 5 times)
6. Consider the data from practice prob. 5, what is the amount of the principal? (ans.
P100,000.00)
Compound Interest with compounding period more than once a year
We already discussed the behavior of interest when it is compounded compared with the simple
interest. Now let us imagine this, all the while our analysis is only compounded on a yearly basis,
which you should have noticed in the first place. Now let us analyze if we are going to have a
compounding period more than once a year, say twice, which we call semi-annually. Or we can have
a compounding period of four times in a year, which we call quarterly. And also, how about we
compute a certain loan that compounds on a monthly basis. How can we analyze this particular
financial set up or agreement between the lender and borrower or vice versa?
We will just modify our compounding equation from this form,
F = P (1+ i)n
To this form. Now we will consider the r as the nominal interest rate and the interest i will be equal to
the ratio of r with m. And the number of periods will become n multiplied by m in which the equation
will be like this,
F = P (1+r/m) n m
F = future worth
P = present worth/ principal
n = number of years
r = is the nominal interest rate/ rate per year
m = the number of periods compounded in a year
i = r/m
Compound Interest with compounding period more than once a year sample problem
1. Bernard borrowed P45,000.00 in a bank with an interest rate of 8.5% compounded
quarterly. How much will he pay after 2 years? And also compare if it is only compounded
yearly, does the amount increase or decrease by how much?
Analysis:
Given,
P = P45,000.00
r = 8.5%
m = 4 compounding periods per year
n = 2 years
F=?
Let us just substitute all the given data to our compound interest equation,
F = P (1 + r/m )mn
F = P45,000.00 (1+0.085/4) (4)(2)
F = P53,243.80
If we are to compare it with just compounded yearly we have,
F = P ( 1 + i )n
F = P45,000.00 (1+0.085)(2)
F = P52,975.13
let us compare and see the difference,
P53,243.80 - P52,975.13
= P268.68 decreased from compounded quarterly comparing it with compounded yearly
2. If I borrowed a certain amount of money from a bank and I committed to return it with its
value doubled within 5 years compounded semi annually. What is the nominal rate of
interest in percent?
Analysis:
Given,
F = 2P
m = 2 (semi-annually)
n = 5 years
Let us start with the working equation,
F = P ( 1 + r/m )mn
2P = P (1+r/2)(2)(5)
2 = (1+r/2)10
2(1/10) = (1+r/2)(10/10)
2(1/10) = 1+r/2
(100%)(2(1/10) - 1)(2) = r
r = 14.35%
(nominal interest means, 14.35 %, compounded yearly)
Continuous Compounding
Discount
Discount
Discount is related to interest since they are both the difference between the future worth of money
and its present worth. The difference is the point of view of both financial parameters.
The interest's concern is how much money will a principal earns in the future. The principal or the
original amount of money is usually the given data or the reference of the financial calculations.
On the other hand, discount's concern is how much money it needs to have this amount of future
worth of money. Usually the future amount of money is the given data and the reference of the
calculation of discount.
Summary: