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CHAPTER 3Compound Interest
Recall…◦What can you say to the
amount of interest
earned in simple
interest?
Do you know?
◦An interest can also
earn an interest?
Compound Interest
◦Whenever a simple interest is
added to the principal at regular
intervals, and the sum become
the new principal, the interest is
said to be compounded.
Challenge:
◦Give me an example of a
business transaction which
involves compounding or
compound interest.
Do you know?
◦Most savings accounts
pay compounded
interest every three
months or quarterly.
Therefore,
◦Adding interest to the
principal gives more
interest.
Compounded
◦Whenever we encounter the
word “compounded” what we
mean is that “the interest is
added to the principal to have a
new principal.”
Something to think about…
◦How does the
interest become
compounded?
Compound Amount
◦The final amount at the
end of the term. We will
denote this using
majuscule letter S.
Compound Interest
◦It refers to the difference
between the compound
amount S and the original
Principal P.
Example 3.1:◦If P2,000.00 is invested at
an interest rate of 8%
compounded annually for 3
years, find the compound
amount and interest.
Final Answer:
◦The compound amount in
3 years is P2,519.42 and
the compound interest is
P519.42.
Example 3.2:◦Find the compound amount
and compound interest if
P5,000.00 is invested at 10%
compounded semi-annually
for 2 years.
Compounded Semi-annually
◦It means the interest earned
in 6 months is added to the
principal to earn additional
interest for the next 6
months.
Therefore…
◦Since we’re just finding for
the interest for 6 months we
will divide the rate of interest
by 2.
Final Answer:◦The compound amount semi-
annually for 2 years is
P6,077.53 and the
compound interest is
P1,077.53.
Reflect:◦How do you find the process
of computing for the
compound amount and
compound interest?
◦YAS! I feel
you! It’s very
TEDIOUS!!!!!!
LESSON 3.2Finding the Compound Amount and Compound Interest Using
the Formula
Example 3.3
◦Find the accumulated value
of P5,000.00 in 4 years if it
is invested at 12%
compounded quarterly.
Reflect:◦Most of the time, interest is
compounded into the
principal more than once a
year.
Conversion/Interest Period
◦The time between two
successive conversions
of interest.
Conversion:
◦Annually – 1 year
◦Semi-annually – 6 months
◦Quarterly – 3 months
◦Monthly – 1 month
Frequency of Conversion
◦The number of conversion
periods at a certain time.
To denote this we will use
minuscule letter m.
Frequency of Conversion:
◦Annually – 1
◦Semi-annually – 2
◦Quarterly – 4
◦Monthly – 12
◦Daily – 365/366
Nominal Rate
◦The rate of interest in
compound interest. To
denote this we will use the
minuscule letter j.
The rate of interest for each conversion period is denoted
by i.◦Formula:
𝑖 =𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑎𝑡𝑒
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛
Equation 3.1
The number of conversion periods in the term is denoted
by n.◦Formula:
𝒏 = 𝒕𝒊𝒎𝒆 𝒊𝒏 𝒚𝒆𝒂𝒓𝒔 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒔𝒊𝒐𝒏
𝒐𝒓 𝒏 = 𝒕 ∗ 𝒎Equation 3.2
Example 3.2
◦If money is invested at
8% compounded
quarterly for 3 years.
Formula for Compound Amount:
◦Let:
◦P = the original principal invested
◦i = rate of interest
◦S = compound amount of P
◦n = number of conversion periods
Formula for Compound Amount:
◦𝑺 = 𝑷(𝟏 + 𝒊)𝒏
Equation 3.3
Where:◦S = compound amount or accumulated value of P at the end
of n periods
◦P = the original principal invested
◦ i = rate of interest = 𝑗
𝑚
j = nominal rate of interest (annual rate)
m = frequency of conversion
◦n = number of conversion periods = 𝑡 ∗ 𝑚
t = term of investment
m = frequency of conversion
Accumulation Factor
◦In the compound
amount formula, it is
the factor (1 + 𝑖)𝑛.
Example 3.3
◦Find the accumulated value
of P5,000.00 in 4 years if it
is invested at 12%
compounded quarterly.
Final Answer:
◦The accumulated
value in 4 years is
P8,023.53.
Do you know?
◦The value of (1 + 0.03)16can
be obtained using a scientific
calculator or by the use of
Table II.
Example 3.4
◦Find the compound amount
and the compound interest on
P10,000.00 for 9 ¼ years at
6% compounded quarterly.
Final Answer:
◦The compound amount
is P17,347.77 and the
compound interest is
P7,347.77.
Let’s Practice:◦Find the interest rate (i) for each period, the total
number of conversion periods (n) and the conversion
period (m) at the end of the indicated time. If
principal is P15,000.00, determine also the S and I.
(a) 8 years at 9% compounded semi-annually
(b) 12 years and 6 months at 10% compounded
monthly
Assignment:◦Find the interest rate (i) for each period, the total number
of conversion periods (n) and the conversion period (m) at
the end of the indicated time.
(c) 10 years and 9 months at 10.5% compounded quarterly
(d) From April 1, 2015 to December 31, 2007 at 12%
compounded quarterly
(e) From June 1, 2002 to May 31, 2008 at 11%
compounded annually
Something to think about…
◦How about when Present
ValueP is missing? What
should we do?
LESSON 3.3Finding the Present Value
at Compound Interest
Present Value◦It is an amount due in ninterest periods which is
invested at a given rate. We
denote this using majuscule
letter P.
Challenge:◦Derive the formula for finding
Present Value P using the
formula for computing for
Compound Amount S.
Formula for Present Value P:
◦𝑃 =𝑆
(1+𝑖)𝑛or
◦𝑃 = 𝑆(1 + 𝑖)−𝑛
Equations 3.4 and 3.5
Discount Factor◦This refers to the factor
(1 + 𝑖)−𝑛. “To discount an
amount S due in n periods”
means to find its present value
P at n periods before S is due.
Example 3.5◦Find the present value of
P18,500.00 due in 5 years if
money is worth 8%
compounded semi-annually.
Final Answer:
◦The present value
is P12,497.93.
Example 3.6◦A 60 square meter house and lot is
purchased on installment. The buyer
makes a P110,400.00 down-payment
and owes a balance of P257,600.00
payable in 5 years. Find the cash value
of the house and lot if money is worth
10% compounded quarterly.
Final Answer:
◦The cash value of
the house and lot is
P157,205.79.
Let’s Practice: Solve the following problems:1. Find the present value of P30,700.00 due in 6
years if money is worth 8% compounded quarterly.
2. On the birth of a son, a father wished to invest
sufficient money to accumulate P2,500,000.00 by the
time his son turns 21 years old. If the father invests at
a rate of 10% compounded semi-annually, how much
should the investment be?
LESSON 3.4Compound Amount at a
Fraction of a Period
Example 3.7◦Find the compound amount if
P10,000.00 is invested for 4 years
and 9 months at 10% compounded
semi-annually assuming simple
interest over the final fractional
part.
Something to think about…
◦In the formula 𝑆 =𝑃(1 + 𝑖)𝑛, what can you
say about the value of
𝑛?
◦Steps in Computing
𝑆 where 𝑛 is a
fraction
Step 1:◦Find the compound
amount at the end of the
largest number of whole
periods in the given time.
Step 2:◦Accumulate the result in the
first step for the remaining
time (which is less than a
period) at simple interest,
nominal rate.
Final Answer:
◦The compound
amount is
P15,901.11.
LESSON 3.5Present Value at a
Fraction of a Period
Example 3.8◦Find the present value of
P20,000.00 due in 5 years
and 4 months at 12%
compounded quarterly.
◦Steps in Computing
Present Value 𝑃 at a
Fraction of a Period
Step 1:◦Increase the number of whole
periods by one. Using this new
period, discount S. It means,
add a few months to compute
the fractional period.
Step 2:◦Accumulate the result in the
first step at simple interest,
at nominal rate for the
number of months added in
step number 1.
Final Answer:
◦The present
value is
P10,646.61.
Let’s Practice: Solve the following problems:1. Find the compound amount if
P26,000.00 is invested at 8%
compounded quarterly for 4 years and 5
months.
2. Find the present value of P19,200.00 due
in 3 years and 8 months if money is
worth 10% compounded semi-annually.
LESSON 3.6Finding the
Nominal Rate
Note:◦The nominal rate
𝒋 can be determined
if 𝑺, 𝑷 and 𝒏 are
given.
Example 3.9◦At what nominal rate
compounded semi-annually
will P50,000.00
accumulate to P85,000.00
in 12 years?
Final Answer:
◦The nominal
rate is 4.47%
Reflect:◦How do you find the
process for finding
nominal rate?
◦YAS! I feel
you! It’s very
TEDIOUS!!!!!!
Formula:Finding for j:
𝑗 = 𝑖𝑥𝑚
Equation 3.6
Formula:
𝑖 =𝑆
𝑃
1𝑛− 1
Equation 3.7
Example 3.10◦At what rate compounded
quarterly, will P16,000.00
amount to P20,000.00 in 5
years?
Final Answer:
◦The nominal
rate is 4.49%
Let’s Practice: Solve the following problems:1. At what nominal rate compounded
semi-annually will P18,000.00
amount to P25,000.00 in 5 years?
2. What rate compounded quarterly will
double any sum of money in 9 years?
LESSON 3.7Finding the Time
How can we find time?
◦Using Logarithm
◦Interpolation
Method
Recall…◦What is your idea
about logarithms?
Logarithms◦In mathematics, the logarithm
of a number is the exponent to
which another fixed value, the
base, must be raised to produce
that number.
Logarithms◦The idea of logarithms
is to reverse the
operation of
exponentiation.
Example 3.11◦How long will it take for
P6,500.00 to become
P9,800.00 if it is invested
at 8% compounded
quarterly?
Final Answer:◦It will take 5.18
years for P6,500.00
to become
P9,800.00.
Example 3.12◦How long will it take for
P6,000.00 to become
P11,300.00 at 6%
compounded semi-
annually?
Final Answer:◦It take 10.71 years
for P6,000.00 to
become P11,300.00.
Let’s Practice: Solve the following problems:1. How long will it take for P8,000.00 to
accumulate to P9,500.00 at 8%
compounded semi-annually?
2. After how many years will P21,000.00
accumulate to P42,000.00 if is its
invested at 12% compounded quarterly?
LESSON 3.7Nominal Rate and
Effective Rate
Example 3.13◦Find the compound amount of
P1,000.00 invested in one year:
(a) at 12% compounded semi-
annually; and
(b) at 12.36% compounded
annually.
Final Answer:
The compound amount at
12% compounded semi-
annually is P1,123.60 and
also at 12.36% compounded
annually is P1,123.60.
Something to think about…◦Two annual rates of interest
with different conversion
periods are said to be
equivalent if they earn the
same compound amount for
the same time.
Nominal Rate◦It is a rate wherein the
interest is compounded
more often than once a
year.
Something to think…
◦From the two rates
on the board, what
is the nominal rate?
Answer:
◦12% is the
nominal rate.
Effective Rate◦It is the rate that, when
compounded annually, produces
the same compound amount
each year as the nominal rate (j)
compounded (m) times a year.
Something to think…
◦From the two rates
on the board, what
is the effective rate?
Answer:
◦12.36% is the
nominal rate.
Effective rate
◦We will use miniscule
letter u to denote
effective rate.
Challenge:
◦Let’s derive the
formula for finding
effective rate u.
Representations…◦Let:
◦u be the effective rate
◦P be the principal
invested at two
investment rates.
Formula:◦Effective Rate:
𝑢 = [1 +𝑗
𝑚]𝑚−1
Equation 3.8
Something to think about…
◦What can you say if
nominal rate is
compounded
annually?
Note:
◦If the nominal rate
j is compounded
annually, then, u=j.
Example 3.14◦Find the effective
rate equivalent to 8%
compounded
quarterly.
Example 3.15◦What nominal rate
compounded semi-
annually is equivalent to
7% effective rate?
Formula to be used:◦Nominal Rate:
(1 +𝑗
𝑚)𝑚= 1 + 𝑢
Equation 3.9
Example 3.15◦The nominal rate
equivalent to 7%
effective rate is
6.88%.
Let’s Practice: Solve the following problems:
1. What nominal rate, compounded
semi-annually is equivalent to 8%
effective rate?
2. Find the effective rate equivalent
to 9% (m=4).