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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).