Compound Interest and Geometric Progression

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ARMY INSTITUTE OF BUSINESS ADMINISTRATION
(AIBA)
TERM PAPER ON
Compound interest and geometric progression
Course Name: Business Mathematics
Course Code: BUS 1205
Date of Submission: 27th October, 2016
Prepared by
Tuhin Parves
ID-B3160B005
BBA 3
Supervised By
Abul Kalam Azad
Assistant Professor
AIBA, Savar

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DECLARATION
I hereby declare this term report was prepared by myself and was not previously submitted to
any other organizations.
The work does not breach any copyright.
______________________________________
Date and Signature
MD.Tuhin parves
ID-B3160B005
BBA-3(A)

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ACKNOWLEDGEMENT
I am fully grateful to those people who helped me to prepare this precious term report. I would
like to thank Director of AIBA and Abul Kalam Azad who is our honorable faculty member of
AIBA. I would like to thank him for his outstanding inspiration, dedication and supervision to
prepare this term report.
______________
Date and Signature
MD.Tuhin Parves
ID –B3160B005
BBA-3(A)

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Table of Content
1. Cover Page -------------------------------------------
2. Declaration -------------------------------------------
3. Acknowledgement ----------------------------------
4. Abstract -----------------------------------------------5
5. Chapter 1: Introduction -----------------------------6
Objectives -------------------------------7
6. Chapter 3:Methodology ----------------------------8
7. Chapter 4: Research work --------------------------8-23
8. Chapter 7:Conclusion -------------------------------24
9. Reference --------------------------------------------- 25

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ABSTRACT
Compound interest was once regarded as the worst kind of usury and was severely condemned
by Roman law and the common laws of many other countries.
Jacob Bernoulli discovered the constant “e” in 1683 by studying a question about compound
interest.
Richard Witt's book Arithmetical Questions, published in 1613, was a landmark in the history of
compound interest. It was wholly devoted to the subject (previously called anatocism), whereas
previous writers had usually treated compound interest briefly in just one chapter in a
mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest
allowable on loans) and on other rates for different purposes, such as the valuation of property
leases. Witt was a London mathematical practitioner and his book is notable for its clarity of
expression, depth of insight and accuracy of calculation, with 124 worked examples.

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Chapter 1
Introduction
The addition of interest to the principal sum of a loan or deposit is called compounding.
Compound interest is interest on interest. It is the result of reinvesting interest, rather than paying
it out, so that interest in the next period is then earned on the principal sum plus previously-
accumulated interest. Compound interest is standard in finance and economics.
Compound interest may be contrasted with simple interest, where interest is not added to the
principal, so there is no compounding. The simple annual interest rate is the interest amount per
period, multiplied by the number of periods per year. The simple annual interest rate is also
known as the nominal interest rate.

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Objectives:
 To know about compound interest.
 To know about the formulas of compound interest.
 To know the use the formula in mathematics field.
 To know the effects and overall uses of compound interest.
 To know the advantages and disadvantages of compound interest.
 To know the use of compound interest in business field.

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CHAPTER 2: METHODOLOGY
All the data in this presentation are secondary data. We also some support from our supervisor.
Mainly information’s are collected from many sources like internet, journal, and papers. We get
also support from our supervisor. The processes of collecting data are given below:
2.1 Data collection: First author has collected data from various external sources. The sources
are -
1. Wikipedia
2. Google,
3. Research paper
4. Books
5. Journal etc. . . .
2.2 Section: Then author has selected those data which s/he needs.
2.3 Characteristic: Then find out the characteristic of those data. Here s/he made an
introduction and objectives of this paper.
2.4 Analyzed process: S/he has analyzed the data and made a research work of this paper.
2.5 Ideas and solutions: S/he got some ideas. Using those ideas and made the research work
she made data findings.
2.6 Report: At last s/he summarized the paper by providing limitations, recommendations.
S/he also gives references from where s/he had collected the data. Then s/he made the final
report.

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Chapter 3: Discussion
3.1 Compound Interest
Compound interest is interest calculated on the initial principal and also on the accumulated
interest of previous periods of a deposit or loan. Compound interest can be thought of as “interest
on interest,” and will make a deposit or loan grow at a faster rate than simple interest, which is
interest calculated only on the principal amount. The rate at which compound interest accrues
depends on the frequency of compounding; the higher the number of compounding periods, the
greater the compound interest. Thus, the amount of compound interest accrued on $100
compounded at 10% annually will be lower than that on $100 compounded at 5% semi-annually
over the same time period. Compound interest is also known as compounding.
3.1.1 History
Compound interest was once regarded as the worst kind of usury and was severely condemned
by Roman law and the common laws of many other countries.
Jacob Bernoulli discovered the constant in 1683 by studying a question about compound interest.
Richard Witt's book Arithmetical Questions, published in 1613, was a landmark in the history of
compound interest. It was wholly devoted to the subject (previously called anatocism), whereas
previous writers had usually treated compound interest briefly in just one chapter in a
mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest
allowable on loans) and on other rates for different purposes, such as the valuation of property
leases. Witt was a London mathematical practitioner and his book is notable for its clarity of
expression, depth of insight and accuracy of calculation, with 124 worked examples.[4][5]

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3.1.2 Compounding frequency
The compounding frequency is the number of times per year (or other unit of time) the
accumulated interest is paid out, or capitalized (credited to the account), on a regular basis. The
frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily (or not at all, until
maturity).For example, monthly capitalization with annual rate of interest means that the
compounding frequency is 12, with time periods measured in years.
The effect of compounding depends on:
1. The nominal interest rate which is applied and
2. The frequency interest is compounded.
3.2 Annual compound interest formula
The formula for annual compound interest is A = P (1 + r/n) ^ nt:
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
Example:
Annual Compound Interest Formula:
A = P(1+r/n)^ nt

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If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%,
compounded monthly, the value of the investment after 10 years can be calculated as follows...
P = 5000. r = 5/100 = 0.05 (decimal). n = 12. t = 10.
If we plug those figures into the formula, we get:
A = 5000 (1 + 0.05 / 12) ^ 12(10) = 8235.05.
So, the investment balance after 10 years is $8,235.05.
You may have seen some examples giving a formula of A = P ( 1+r ) ^ t . This simplified
formula assumes that interest is compounded once per period, rather than multiple times per
period.
3.2.1 Other formulas
If you want to work backwards and find out how much you would need to start with in order to
achieve a chosen future value, try the following version of the formula: P = A / ( 1 + r/n ) ^ nt.
Let's say your goal is to end up with $10,000 in 5 years, and you can get an 8% interest rate on
your savings, compounded monthly. Your calculation would be: P = 10000 / (1 + 0.08/12) ^ 1
3.3 Continuous Compounding
Continuous compounding is the mathematical limit that compound interest can reach. It is an
extreme case of compounding since most interest is compounded on a monthly, quarterly
or semiannual basis. Hypothetically, with continuous compounding, interest is calculated and
added to the account's balance every infinitesimally small instant. While this is not possible in
practice, the concept of continuously compounded interest is important in finance.
3.3.1 Continuous Compounding Formula and Calculation

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A=Pert
Example: Find future value if $1000 is invested for 20 years at 8% continually.
Solution:
P=1000
r=8%
t=20
A=Pert
.
. . A=1000xe0.08x20
= 4053.03 (Answer)
3.4 Effects of compound interest
Suppose that one cent had been invested at year 0 at a constant annual interest rate of 2%. After
the first year, this interest rate was applied to the initial principal of one cent and the capital grew
to 1.02 cent. In the second year, the interest earned was again 2%. However, from that time
onwards, it was not applied to the principal only but to the compound capital value (i.e., 1.02
cent). Thus, after the second year, the capital increased to 1.02×1.02 cent. After the third year,
the capital grew to 1.023 cent. After 2015 years, the capital has eventually grown to 1.022015 cent,
which is roughly equal to 2.13x1017 cent or, more precisely, 213,474,546,813,926,768.7 cent.
Compare this figure to a similar investment using simple interest rather than compound interest.
Suppose again that 1 cent is invested for a period of 2015 years at a constant annual interest rate
of 2%. In this case, after 2015 years, the final capital is only 41.3 cent. This comparison
highlights the effect of compounding, especially for long-term investments.

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3.5 Geometric sequence or progression
About Geometric Sequence
A geometric sequence is a sequence such that any element after the first is obtained by
multiplying the preceding element by a constant called the common ratio which is denoted by r.
The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,
where r common ratio
a1 first term
a2 second term
a3 third term
an-1 the term before the n th term
an the n th term
The geometric sequence is sometimes called the geometric progression.
What is Geometric sequence or progression?
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of
numbers where each term after the first is found by multiplying the previous one by a fixed, non-
zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric
progression with common ratio 3.
3.5.1 Formulas used with geometric sequences and geometric
series:

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To find any term of a geometric sequence:
where a1 is the first term of the sequence,
r is the common ratio, n is the number of the term to find.
To find the sum of a certain number of terms of a geometric sequence:
where Sn is the sum of n terms (nth partial sum),
a1 is the first term, r is the common ration.
NOTE THAT: a1 is often simply referred to as a
Find the common ratio for the sequence 1. The common ratio, r, can be found
by dividing the second term by the first
term, which in this problem yields -
1/2. Checking shows that multiplying
each entry by -1/2 yields the next entry.
2. Find the common ratio for the sequence given
by the formula
2. The formula indicates that 3 is the
common ratio by its position in the
formula. A listing of the terms will also
show what is happening in the sequence
(start with n = 1).
5, 15, 45, 135, ...
The list also shows the common ratio to
be 3.

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3. Find the 7th term of the sequence
2, 6, 18, 54, ...
3. n = 7; a1 = 2, r = 3
The seventh term is 1458.
4. Find the 11th term of the sequence 4. n = 11; a1 = 1, r = -1/2
5. Find a8 for the sequence
0.5, 3.5, 24.5, 171.5, ...
5. n = 8; a1 = 0.5, r = 7
6. Evaluate using a formula: 6. Examine the summation
This is a geometric series with a
common ratio of 3.
n = 5; a1 = 3, r = 3
7. Find the sum of the first 8 terms of the
sequence
-5, 15, -45, 135, ...
7. The word "sum" indicates a need for
the sum formula.
n = 8; a1 = -5, r= -3
8. The third term of a geometric sequence is 3
and the sixth term is 1/9. Find the first term.
8. Think of the sequence as "starting
with" 3, until you find the common
ratio.
For this modified sequence: a1 =
3, a4 = 1/9,
n = 4

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Now, work backward multiplying by 3
(or dividing by 1/3) to find the actual
first term.
a1 = 27
8. The third term of a geometric sequence is 3
and the sixth term is 1/9. Find the first term.
8. Think of the sequence as "starting
with" 3, until you find the common
ratio.
For this modified sequence: a1 =
3, a4 = 1/9,
n = 4
Now, work backward multiplying by 3
(or dividing by 1/3) to find the actual
first term.
a1 = 27
9. A ball is dropped from a height of 8 feet. The
ball bounces to 80% of its previous height with
each bounce. How high (to the nearest tenth of a
foot) does the ball bounce on the fifth bounce?
9. Set up a model drawing for each
"bounce".
6.4, 5.12, ___, ___, ___
The common ratio is 0.8.

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Answer: 2.6 feet
3.6 Uses of compound interest
Owners of small businesses often have limited sources of income and are further burdened by
expenses, making it extremely difficult to contribute generous sums to saving accounts. Even in
money-tight situations, you need to set a portion of your earnings aside for savings and pave a
pathway to a secured future. If you want maximum returns on your savings and investments,
investments that offer compound interest calculations can be viable options.
Calculation
Compound interest is one of two ways to calculate interest -- the other is simple interest. While
simple interest calculates interest on the initial principal only, compound interest lets you enjoy
higher returns by paying you interest on the principal amount plus the interest you previously
earned. If, for example, you invested $1,000 and earned $50 in interest at the end of the earning
period, your new principal becomes $1,050. The interest rate will be applied to $1,050, not to
your original $1,000, the next time interest is calculated.
Generating Profits
Compound interest opens doors to sources of profits for a company. For example, businesses can
please investors by earning them higher profits than expected. Financial managers are expected
to give dividends to investors. If these dividends are accumulated, or more precisely
compounded, and reinvested in the business, higher dividends may be payable the next year.

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Compound interest is a means for profit growth if used wisely. It works as a return multiplier,
and with each passing year, the interest that investors receive grows because they earn interest on
interest.
Ensuring Pension Payments
Various companies seek assistance of investment accounts to pay pensions. Typically, employers
exclude a fixed amount of their employees’ salaries and contribute it to their pension fund. The
amount accumulates for years until the employees reach retirement age, when the entire amount
is provided as pension. Businesses use the money in pension funds to invest in financial
instruments that pay guaranteed return rates. This helps to generate a smooth cycle of pension
payments, while earning consistent returns on the investment saved for so many years. The
retirement trust fund maintains its core essence of timely pension payment to its retired
employees, a process that is effectively enhanced by the concept of compound interest.
3.7 Advantages of compoundinterest
If you want to get the most return on money you save or invest, you want compound interest.
The two types of interest are simple and compound. Simple interest is paid only on the money
you save or invest (the principle), while compound interest is paid on your principle plus on the
interest you have already earned.
Example
Let's say you invest $100 at 10 percent annual interest. At the end of one year, you have $110
with either simple or compound interest. At the end of the second year, you have $120 with
simple interest or $121 with compound interest. After five years, the amounts are $150 for
simple interest and $161 for compound, and after 10 years, the amounts are $200 and $259
respectively. The difference then really starts to accelerate: At the 20-year point, the amounts are
$300 for simple interest and $672 for compound. And, after 50 years, you could have either $600
with simple interest or $11,739 with compound interest.
Start Early
The power of compound interest is the reason that financial planning and retirement experts
recommend starting a retirement plan early. A 20-year-old who places $5,000 in a one-time
investment that earns an average 8-percent annual return would have $160,000 at age 65, while a

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39-year-old who makes a one-time $5,000 investment at that rate of return would have only
$40,000 at age 65, according to Get Rich Slowly. The advantages are even greater for someone
making regular contributions to the investment.
Importance of Interest Rate
How much difference does a 1 percent change in the interest rate matter to how much you earn
with compound interest? For a $1,000 initial investment at 4-percent interest, you would have
$2,222 after 20 years and $4,801 after 40 years, whereas a 5-percent rate would give you $2,712
after 20 years and $7,040 after 40 years, according to Gary Beene's Retirement Information
Center.
Considerations
A consideration is how often the interest is compounded; an investment with interest
compounded monthly will grow faster than an investment with interest compounded annually.
While compound interest is beneficial if you are the one receiving the interest, if you are the one
paying compound interest on a loan or credit card, then it's costing you a lot of money, as interest
is charged on interest.
Benefit and Liability
The idea of compound interest is appealing only when you are on the earning side of the
financial balance. Banks typically pay compounded interest on deposits, a benefit for depositors.
If you are a credit card holder, knowledge of the workings of compound interest calculations
may be incentive to pay off your balances quickly. Credit card companies charge interest on the
principal amount and the accumulated interest. If you prolong paying off your credit card debt,
your principal will grow, because compound interest calculations reset your initial principal to
include previously earned interest. The benefit of compound interest is dependent on your
financial perspective. If you are a borrower, compound interest calculations translate to growth
in the amount you owe and the lender reaps the benefit. If you are the investor, you reap the
benefit as your money grows.

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If there's any magic in saving, this is it - the power of compound interest. When you're saving,
the bank (or financial institution) adds interest to your savings at regular intervals (for example,
every month). If you don't touch the interest, but let it add to your lump sum, then you start to
earn interest on your interest, as well as on the original amount you saved. This is called
compound interest.
The longer you leave your money, the more powerful the compound interest effect. So the earlier
you start saving, the more you make from compound interest. The same applies to other
investments such as shares, where you regularly reinvest dividends, or the company reinvests its
profits.
Consider the example of Viv. Viv's a 20 year old who decides to start saving a regular amount
each week. The tables on the next screen show how much she will end up with if she keeps up
her saving (either $10 per week or $50 per week) until she is 60. We've based Viv's results on an
interest rate of 2.5% after tax and allowing for inflation.
We've also assumed that Viv will increase the amount she saves each week to account for
inflation.
If inflation is 2% this year, Viv will increase her weekly savings by 2% from next year (from $50
to $51)
Look at the first five years and the last five years of the $10 table. In the first five years Viv
saves $2,600 and earns $170 in interest. In the last five years, she's still saving only $2,600, but
she earns a massive $3,970 in interest - far more than she saves. That's the power of compound
interest!
3.8 Disadvantages
The disadvantage is that the practice of compounding in credit card. As we all know, interest charged to
outstanding balances can be very high. Not only that, unless you read the small print you aren’t going to
be prepared for the way interest is charged on your card. For example, if you move a balance across from
one card to another that transfer will be at a very low rate. But if you go out and purchase something else
with that same card the new purchase will be at a very high rate and it won’t ever shift off your card until
the balance transfer sum has been completely paid off.
And so, as you whittle away each month at the balance, interest will be charged on top of interest
and that’s compound interest; which is a great thing when you are investing, but not when you
are trying to pay off a debt.

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3.9 Business uses of compound interest:
The way interest is calculated is an essential factor in generating favorable returns off of
financing agreements, and the true rate of interest often differs from the stated rate because of
compounding. Compound interest is used in a variety of financial instruments that are
commonplace in business. Although the purposes for business loans vary, the interest calculation
is similar.
Mortgage Loans
When purchasing a production facility or office building, many businesses choose to finance the
transaction and take out a mortgage. The financing agreement usually comes with a stated
interest rate and a loan term. However, the stated rate is not the effective interest rate because of
compounding. With compounding, interest is charged to the principal and any previous interest
already accrued. During the course of a year, mortgage interest is charged at varying intervals,
which raises the true rate paid. Mortgage terms significantly impact the amount of interest
lenders receive.
Hard-Money Loans
Sometimes businesses need to take out loans to keep operations going or to buy additional
inventory when cash flow is tight. Hard-money loans provide quick financing and are typically
secured using real estate as collateral. The application process and qualification criteria are
different compared with traditional business loans and are ideal for companies with poor credit
histories. Hard-money loans come with a stated loan term and interest rate, but the interest rate is
calculated using compounding. As a result, the true interest rate is far higher than the stated one.
Vehicle Loans
Freight companies or distributors use trucks and other vehicles in their operations to move
inventory and serve their customers. Some of the larger trucks require significant cash outlays,
making vehicle loans useful. These types of loans resemble personal auto loans with interest
compounding at regular intervals. The ultimate cost of a vehicle factors into business decision-
making and is an important consideration when choosing to buy a vehicle outright or finance it.
Equipment Loans
Production facilities and machine shops use heavy equipment to manufacture products.
Computer-controlled lathes, mills and other industrial machinery is expensive to purchase. These
pieces can be financed, and the interest paid is typically compounded over the loan term. The
interest expense can be written off against taxable income, which is an important consideration
when deciding to obtain a loan.

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Chapter4:Conclusion
Compound interest is interest calculated on the initial principal and also on the accumulated
interest of previous periods of a deposit or loan. Compound interest can be thought of as “interest
on interest.” The effect of compounding depends on:
3. The nominal interest rate which is applied and
4. The frequency interest is compounded.
Owners of small businesses often have limited sources of income and are further burdened by
expenses, making it extremely difficult to contribute generous sums to saving accounts. Even in
money-tight situations, you need to set a portion of your earnings aside for savings and pave a
pathway to a secured future. If you want maximum returns on your savings and investments,
investments that offer compound interest calculations can be viable options. The way interest is
calculated is an essential factor in generating favorable returns off of financing agreements, and
the true rate of interest often differs from the stated rate because of compounding. Compound
interest is used in a variety of financial instruments that are commonplace in business. Although
the purposes for business loans vary, the interest calculation is similar.

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References:
 https://en.wikipedia.org/wiki/Compound_interest
 http://www.investopedia.com/terms/c/compoundinterest.asp
 https://twitter.com/compoundchem?ref_src=twsrc%5Egoogle%7Ctwcamp%5Eserp%7Ct
wgr%5Eauthor
 http://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php
 http://smallbusiness.chron.com/uses-compound-interest-business-49709.html
 https://answers.yahoo.com/question/index?qid=20071201204953AAEOVEw
 http://www.sorted.org.nz/home/sorted-sections/saving/the-power-of-compound-interest
 http://www.ukpersonalloanstore.co.uk/financial/disadvantages_credit_card.html
 http://laws.justice.gc.ca/en/showdoc/cs/I-15/bo-ga:s_6//en#anchorbo-ga:s_6 Interest Act
(Canada), Department of Justice. The Interest Act specifies that interest is not
recoverable unless the mortgage loan contains a statement showing the rate of interest
chargeable, "calculated yearly or half-yearly, not in advance." In practice, banks use the
half-yearly rate.
 Jump up^ Munshi, Jamal. "A New Discounting Model". ssrn.com.
 Jump up^ This article incorporates text from a publication now in the public
domain: Chambers, Ephraim, ed. (1728). "article nameneeded". Cyclopædia, or an Universal
Dictionary of Arts and Sciences(first ed.). James and John Knapton, et al.
 Jump up^ Lewin, C G (1970). "An Early Book on Compound Interest - Richard Witt's
Arithmeticall Questions". Journal of the Institute of Actuaries. 96 (1): 121–132.
 Jump up^ Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal
of the Institute of Actuaries. 108 (3): 423–442.
 Jump up^ http://quoteinvestigator.com/2011/10/31/compound-interest/