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1. Compound Interest
Compounding Period
Continuous Compounding
COMPOUND
INTEREST
INSPIRED BY

4. Present Value P at
Compound Interest
INSPIRED BY

7. Compounding
More than Once a
Year
INSPIRED BY

8. Definition of Terms:
Frequency of conversion (m) – number of
conversion periods in one year
Conversion or interest period– time between
successive conversions of interest
Total number of conversion periods n
n = mt = (frequency of conversion)×(time in
years)

9. Definition of Terms:
Frequency of conversion (m) – number of
conversion periods in one year
Conversion or interest period– time between
successive conversions of interest
Total number of conversion periods n
n = mt = (frequency of conversion)×(time in
years)

15. Finding Interest
Rate and Time in
Compound Interest
INSPIRED BY

20. Compound Interest
The compound
interest on P is the
amount of interest
charged to the amount
P and the interest
earned on previous
time periods.

21. Conversion per
year
Annually – 1
Semi-annually – 2
Quarterly – 4
Daily-365
Monthly – 12
Bi-monthly – 24
Weekly - 52

22. Future Value
under
Compound
Interest
The future value of P
borrowed (or
invested) at an
annual compound
interest rate r at time
t is given by
.

23. 1. Five years ago, Anna
put her savings worth
10,000 pesos in an
account providing a
compound interest rate
of 3.5% annually. Find
the value of her savings
today and the amount of
interest.
Given:
P= 10,000
r=0.035
t=5
The value of Anna’s savings today
is 11, 876.86 pesos. The amount
of interest earned is 1, 876.86.
11, 876.86-10,000 = 1, 876.86

24. Find the future of each of
the following at a given
annual rate compounded
annually and the given
time:
a. 1,000 at 2% after 3 years
b. 2, 500 at 2.27% after 5
years
c. 10, 000 at 1% after 4

25. At what interest rate
compounded annually
must 10,000 pesos be
invested in order to earn
an interest worth 2,000
in 5 years?
Given:
A=10,000+2000=12,000
P=10,000
T=5
=1.0371

26. Compounding
Periods
Interest can be compounded more
frequent than once per year.
Interest can be compounded
semiannually.
There are two compounding periods in a
year. Other common compounding
periods are four in a year (or quarterly)

27. 𝐴=𝑃 (1+
𝑟
𝑚 )
𝑚𝑡
Example 1
To have a capital for a
small food business,
Mang Nestor borrowed
10, 000 pesos at 2%
interest rate
compounded quarterly.
How much does Mang
Nestor need to pay after
2 years?
¿10,000(1+
0.02
4 )
4(2)
¿ 10,000 ( 1.005 )8
¿10, 407.07
Therefore, Mang Nestor needs
to pay 10, 407.07 after 2 years.

28. a. Quarterly
Example 2
Find the future value of
2, 000 pesos after one
year at 3% interest rate
compounded:
a. Quarterly
b. Monthly
c. Daily
¿2,000 (1+
0.03
4 )
4(1)
¿ 2,000 ( 1.0075 )4
¿2,060.68

29. b. Monthly
Example 2
Find the future value of
2, 000 pesos after one
year at 3% interest rate
compounded:
a. Quarterly
b. Monthly
c. Daily
¿2,000 (1+
0.03
12 )
12(1)
¿ 2,000 ( 1.0025 )12
¿2,060.83

30. Continuous
Compounding
The future value of P
compounded
continuously at annual
rate at time t is

31. P=2,000, r=0.03
and t=4.5
Example 1
What is the future value
of 2,000 pesos
compounded
continuously at 3%
interest after 4.5 years? ¿ 2,000 𝑒
0.03 ( 4.5)
¿2,289.07
𝐴= 𝑃𝑒𝑟𝑡
The future value is 2,
289.07.

32. 2 𝑃 = 𝑃𝑒0.03 𝑡
Example 1
How long will an
amount compounded
continuously at an
annual rate of 3%
double? How long will
the amount triple?
𝐼𝑛 2 =0.03 𝑡
𝑡=
𝐼𝑛 2
0.03
≈ 23.10
2 =𝑒0.03 𝑡
Thus, the amount will
double in 23.10 years.

33. 3 𝑃 = 𝑃𝑒0.03 𝑡
Example 1
How long will an
amount compounded
continuously at an
annual rate of 3%
double? How long will
the amount triple?
𝐼𝑛 3 = 0.03 𝑡
𝑡=
𝐼𝑛3
0.03
≈ 36.62
3 =𝑒0.03 𝑡
Hence, the amount will
triple in 36.62 years.

34. Present Value
under Compound
Interest

35. 1. The present
value of A at
an annual
compound
interest rate r
is:
𝑃=
𝐴
(1+𝑟 )𝑡

36. 2. The present
value of A at an
annual interest
rate of r
compounded m
times a year is:
𝑃=
𝐴
(1+
𝑟
𝑚 )
𝑚𝑡

37. 3. The present
value of A at an
annual interest
rate of r
compounded
continuously is:
𝑃=
𝐴
𝑒
𝑟𝑡
= 𝐴𝑒
− 𝑟𝑡

38. Three years from now, Mr. dela Cruz wants to start a
business. In order to do this, he estimated that he needs an
initial capital of 100,000 pesos. He can deposit an amount
today in one of the following accounts:
Account 1: offers 1.5% annual compound interest rate
Account 2: offers 1.4% annual interest rate compounded
monthly
Account 3: offers 1.1% annual interest rate compounded
monthly
Account 4: offers 1% annual interest rate compounded
continuously
Which of the accounts will require the least deposit?

39. Account 1:
Account 2:
Account 3:
Account 4:
Therefore, Account 1 require the smallest deposit today.

40. Check Yourself!
1. Complete the table
2. If 25,000 pesos is invested at an annual interest rate of 12%
compounded (a) annually, (b) quarterly, (c) monthly, what is the amount
after 2 years? How much interest is earned?
A P r t m
______ 10,000 2% 3 years 2
20,000 _____ 1.5% 1 year 12
15,000 10,000 2.5% 2 years ______

41. 3. If 100,000 is deposited at an account crediting a continuous
interest rate of 2%, how much will the deposit be worth after one
year?
4. A person has 100, 000 which he plans to put an investment for 3
years. He choosing between two investments. Investment A credits
5% interest compounded monthly while investment B credits 5.3%
interest compounded quarterly. Which investment is better? Why?
5. Mr. dela Paz is thinking of buying a new car in 5 years. He
estimated that he needs to have 700,000 in 5 years. In order to
achieve his goal, he decided to make a deposit today at an account
that gives 3% interest compounded semiannually. How much
deposit does Mr. Paz needed?