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CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES

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BUSINESS MATHEMATICS AND FINANCE

MODULE-A

ACCOUNTING & FINANCE FOR BANKERS - JAIIB



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CONTENTS

1. Simple Interest

2. Compound Interest

3. Calculation of Equated Monthly Installments

4. Fixed and Floating Interest Rates

5. Interest Calculation using Products/Balances

6. Calculation of Annuities

7. Types of Annuities

8. Calculation of Annuities

9. Sinking Fund



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CALCULATION OF INTREST & ANNUITIES

INTRODUCTION

Interest is the cost of using somebody else’s

money. When you borrow money, you pay

interest. When you lend money, you earn

interest.

When borrowing: In order to borrow money,

you’ll need to repay what you borrow. In

addition, to compensate the lender for the risk of

lending to you (and their inability to use the

money anywhere else while you had it), you

need to repay more than you borrowed.

When lending: If you have extra money

available, you can lend it out yourself or deposit

it in a savings account and let the bank lend it

out. In exchange, you’ll expect to earn interest –

otherwise, you might be tempted to spend the

money today because there’s little benefit to

waiting (other than planning for your future).

Annuity: is a lump sum of cash invested to

produce a monthly stream of income for a fixed

period or for life. The income can start now

(immediate annuity) or in the future (deferred

annuity). Funds are not protected or insured by

the issuers. The size of the future monthly check

isn’t always a given – it depends if the annuity is

fixed or variable.



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SIMPLE INTEREST

Simple interest is paid by the borrower at the end of each year at a fixed rate (called rate of interest). In other words no interest is paid on the amount of interest. Another word, 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 (not to be confused with nominal as opposed to real interest rates). The simple interest can be calculated as: Interest = Principal x Rate x Time i.e. I=PRT (where P is principal, R is rate of interest and T is time)

Illustration A lends Rs.30000 to B at 10% interest rate. The annual interest would be Rs.30000

By Formula, I=PRT 300000 x (10/100) x 1 = 300000 x .10 x 1 = 30000 Hence total amount payable by the borrower to the lender = Principal + interest = (300000+30000) = 330000



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AMOUNT OF INSTALMENTS

Repayment of the loan can be made on a Yearly, Half-yearly, Quarterly, Monthly or even Weekly periodicity. Hence the total amount repayable can be divided by the units of time period in a year. Illustration

1. For example in the above case, the total loan repayable is if repayment is Yearly

Lender = Principal + interest = (300000+30000) = 330000

2. If repayment is half-yearly (year/2), the amount of installment would be Lender = Principal + interest/2 = (300000+30000/2) = 315000

3. If repayment is quarterly (year/4), the amount of installment would be Lender = Principal + interest/4 = (300000+30000/4) = 307500

COMPOUND INTEREST

Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words,

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.



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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 also known as the nominal interest rate

(not to be confused with nominal as opposed to real interest rates)

CALCULATION OF INTEREST

The total accumulated value, including the principal sum P plus compounded interest I, is given by the formula:

Fv=Pv(r/n)nt

Where: P is the original principal sum P' is the new principal sum r is the nominal annual interest rate n is the compounding frequency t is the overall length of time the interest is applied (usually expressed in years). The total compound interest generated is:

P′ = P + I

I = P (1 + r/n) n t

− P



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1. Illustration

Suppose a principal amount of Rs.1500 is deposited in a bank paying an annual interest rate of 4.3%,

compounded quarterly.

Then the balance after 6 years is found by using the formula above, with

P = 1,500, r = 4.3% = 0.043, n = 4 and t = 6:

So the new principal P′ after 6 years is approximately Rs.1, 938.84.

Subtracting the original principal from this amount gives the amount of interest received:

2. Illustration Suppose a principal amount of Rs.1500 is deposited in a bank paying an annual interest rate of 4.3%,

compounded half yearly

Then the balance after 6 years is found by using the formula above, with P = 1,500, r = 4.3% = 0.043, n = 4 and t = 6:



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So the new principal P′ after 6 years is approximately Rs.1, 921.24.

The amount of interest received can be calculated by subtracting the principal from this amount. Rs. 1 ,

Rs. (1,921.24 − 1,500) = Rs. 421.

(The interest is less compared with the previous case, as a result of the lower compounding frequency.)

CONTINUOUS COMPOUNDING

As n, the number of compounding periods per year, increases without limit, we have the case known as

continuous compounding, in which case the effective annual rate approaches an upper limit of er − 1,

where e is a mathematical constant that is the base of the natural logarithm.

Continuous compounding can be thought of as making the compounding period infinitesimally small,

achieved by taking the limit as n goes to infinity.

See definitions of the exponential function for the mathematical proof of this limit. The amount after t

periods of continuous compounding can be expressed in terms of the initial amount

P0 as, P (t) = P0ert



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RULE OF 72

In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's

doubling time. The rule number (e.g., 72) is divided by the interest percentage per period to obtain the

approximate number of periods (usually years) required for doubling. Although scientific calculators and

spread sheet programs have functions to find the accurate doubling time, the rules are useful for mental

calculations and when only a basic calculator is available.

These rules apply to exponential growth and are therefore used for compound interest as opposed to

simple interest calculations. They can also be used for decay to obtain a halving time. The choice of

number is mostly a matter of preference: 69 is more accurate for continuous compounding, while 72

works well in common interest situations and is more easily divisible. There are a number of variations to

the rules that improve accuracy. For periodic compounding, the exact doubling time for an interest rate of

r per period is:

Where T is the number of periods required. The formula above can be used for more than calculating the

doubling time. If one wants to know the tripling time, for example, simply replace the constant 2 in the

numerator with 3. As another example if one wants to know the number of periods it takes for the initial

value to rise by 50%, replace the constant 2 with 1.5.



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FLOATING INTEREST RATE

A floating interest rate, also known as a variable or adjustable rate, refers to any type of debt instrument,

such as a loan, bond, mortgage, or credit that does not have a fixed rate of interest over the life of the

instrument.

Floating interest rates typically change based on a reference rate (a benchmark of any financial factor,

such as the Consumer Price Index). One of the most common reference rates to use as the basis for

applying floating interest rates is the London Inter-bank Offered Rate, or LIBOR (the rates at which large

banks lend to each other).

The rate for such debt will usually be referred to as a spread or margin over the base rate: for example, a

five-year loan may be priced at the six-month LIBOR + 2.50%. At the end of each six-month period, the

rate for the following period will be based on the LIBOR at that point (the reset date), plus the spread.

The basis will be agreed between the borrower and lender, but 1, 3, 6 or 12 month money market rates

are commonly used for commercial loans.

Typically, floating rate loans will cost less than fixed rate loans, depending in part on the yield curve. In

return for paying a lower loan rate, the borrower takes the interest rate risk: the risk that rates will go up

in future. In cases where the yield curve is inverted, the cost of borrowing at floating rates may actually

be higher; in most cases, however, lenders require higher rates for longer-term fixed-rate loans, because

they are bearing the interest rate risk (risking that the rate will go up, and they will get lower interest

income than they would otherwise have had).



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FIXED INTEREST RATE

A fixed interest rate loan is a loan where the interest rate doesn't fluctuate during the fixed rate period of

the loan. This allows the borrower to accurately predict their future payments. A fixed interest rate is

based on the lender's assumptions about the average discount rate over the fixed rate period.

For example, when the discount rate is historically low, fixed rates are normally higher than variable

rates because interest rates are more likely to rise during the fixed rate period. Conversely, when interest

rates are historically high, lenders normally offer a discount to borrowers to fix their interest rate over

time, as rates are more likely to fall during the fixed rate period.

FRONT-END AND BACK-END INTEREST RATE

In case of front-end interest where deduction of interest is done from principal amount and after

deducting the interest from the principal amount, the net amount is disbursed to borrower.

For example when a customer approaches for a Gold Loan to a bank, and if that bank disburses the

amount on gold loan after deducting interest for agreed period, say (monthly, quarterly, yearly etc.)

months, and net amount after deduction of interest on principal is disbursed to customer such interest is

called front-end interest

However in banking practice other than gold loan, normal practice for interest calculation is back-end

interest, in which full loan amount is disbursed to borrower and interest is calculated on the basis of

agreed monthly, quarterly, yearly basis.



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CALCULATION OF BANK INTEREST ON DAILY BASIS

From April 1, 2010, interest on all savings bank account deposits is being calculated on a daily basis,

thereby earning account holders higher interest income. This is due to the fact that the Reserve Bank of

India has instructed banks to change the mechanism of interest income calculation.

The calculation is done on the 'daily balance method'

Earlier, banks would pay interest on the lowest available balance in the account between the tenth and

the last day of the month.

Any deposit in the account between the tenth and the end of the month, would not earn the account

holder any interest as it is not part of the interest rate calculation. Any withdrawal between the same

periods would result in lower interest income as the lowest balance would be taken into account for the

calculation.

Illustration Example: Manisha had an account balance of Rs 85,000 on April 10. He received a payment of Rs

300,000 on April 17 from the sale of some mutual fund units.

On April 29, he made a down payment of Rs 320,000 to a builder for a property. This resulted in her account balance reducing to Rs 65,000. For the interest income calculation for the month of April, the bank would take Rs 65,000 as the base and pay her interest on that amount. So interest due to Manisha would be on Rs 65,000 for 30 days @ 3.5% p.a. which would be Rs 187. In spite of having a high account balance for most period of the month, Manisha lost interest income for the month.



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Under this method of interest rate calculation, the best thing Manisha could do is ensure that all transactions are done between the first and ninth of any month so that he would get benefit of interest. This required proper planning. New method of interest rate calculation Interest will be paid @3.5% p.a. on the daily balance in the account at the end of the day. Here, the account holder will get interest on the actual day end balance. Under this method, Manisha’s interest income calculation would be: For the first 14 days of April, interest to be paid would be calculated on Rs 85,000; For the next 14 days of April, interest to be paid would be calculated on Rs 385,000 and; For the balance 2 days, interest to be paid would be calculated on Rs 65,000.

So the total interest due to Manisha would be Rs 643. Under this method, Manisha's interest income is higher by Rs 456! Besides, she did not have to plan her withdrawals and deposits as he would receive interest on the actual account balance. As a savings bank account holder, you should be pleased with the latest change. Who would not like to see higher balance on account of higher interest income?

PS: Interest rate calculation Formula

Daily interest = Amount (Daily balance) * Interest (3.5/100) / days in the year



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ANNUITIES

An annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits

to a savings account, monthly home mortgage payments, monthly insurance payments and pension

payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may

be made weekly, monthly, quarterly, yearly, or at any other regular interval of time.

TYPES OF ANNUITIES

1. ANNUITY-IMMEDIATE OR ORDINARY ANNUITIES

If the payments are made at the end of the time periods, so that interest is accumulated before the

payment, the annuity is called an annuity-immediate or ordinary annuity. Mortgage payments are

annuity-immediate, interest is earned before being paid.

2. ANNUITIES DUE

An annuity-due is an annuity whose payments are made at the beginning of each period. Deposits

in savings, rent or lease payments, and insurance premiums are examples of annuities due.

3. CONTINGENT ANNUITIES

Annuities that provide payments that will be paid over a period known in advance are annuities

certain or guaranteed annuities. Annuities paid only under certain circumstances are contingent

annuities. A common example is a life annuity, which is paid over the remaining lifetime of the



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annuitant. Certain and life annuities are guaranteed to be paid for a number of years and then

become contingent on the annuitant being alive.

4. FIXED ANNUITIES

These are annuities with fixed payments. If provided by an insurance company, the company

guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the

Securities and Exchange Commission.

5. VARIABLE ANNUITIES

Registered products that are regulated by The Insurance Regulatory and Development Authority

(IRDA) They allow direct investment into various funds that are specially created for Variable

annuities. Typically, the insurance company guarantees a certain death benefit or lifetime

withdrawal benefits.

6. DEFERRED ANNUITIES

An annuity which begins payments only after a period is a deferred annuity. An annuity which

begins payments without a deferral period is an immediate annuity.

7. ANNUITY CERTAIN OR GUARANTEED ANNUITY

If the number of payments is known in advance, the annuity is an annuity certain or guaranteed annuity.



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CALCULATION OF ANNUITIES

1. FUTURE VALUE OF AN ORDINARY ANNUITY

The future value of an annuity formula is used to calculate what the value at a future date would be

for a series of periodic payments.

The future value of an annuity formula assumes that

1. The rate does not change

2. The first payment is one period away

3. The periodic payment does not change

If the rate or periodic payment does change, then the sum of the future value of each individual

cash flow would need to be calculated to determine the future value of the annuity. If the first cash

flow, or payment, is made immediately,

The future value of annuity due formula would be used.



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Illustration

Ayush decides to save by depositing Rs.1000 into an account per year for 5 years. The first

deposit would occur at the end of the first year. If a deposit was made immediately, then the future

value of annuity due formula would be used.

The effective annual rate on the account is 2%. If he would like to determine the balance after 5

years, he would apply the future value of an annuity formula to get the following equation.

Here,

P = 1000

r = 2%

n = 5

As per formula,

=

The balance after the 5th year would be Rs.5204.04.



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2. PRESENT VALUE OF AN ORDINARY ANNUITY

The present value of annuity formula determines the value of a series of future periodic payments at a given time. The present value of annuity formula relies on the concept of time value of money, in that one rupee present day is worth more than that same rupee at a future date.

Illustration

Mr. Reddy wants to determine today's value of a future payment series with cash flow schedule as

follow, receiving Rs. 1000 into an account per year for 5 years with the effective annual rate on the

account is

Since Mr. Reddy wants to determine today’s value of future payment series, we will use formula

that calculates the present value of an ordinary annuity. This is the formula we would use as part



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of a bond pricing calculation. The PV of an ordinary annuity calculates the present value of the

coupon payments that we will receive in the future.

P = 1000 (Cash flow per period)

r = 5%

n = 5

Assumption

The formula shown has assumptions, in that it must be an ordinary annuity. These assumptions

are that

1) The periodic payment does not change

2) The rate does not change

3) The first payment is one period away



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3. FUTURE VALUE OF ANNUITY DUE

The future value of annuity due formula is used to calculate the ending value of a series of

payments or cash flows where the first payment is received immediately. The first cash flow

received immediately is what distinguishes an annuity due from an ordinary annuity. An annuity

due is sometimes referred to as an immediate annuity.

The future value of annuity due formula calculates the value at a future date. The use of the future

value of annuity due formula in real situations is different than that of the present value for an

annuity due. For example, suppose that an individual or company wants to buy an annuity from

someone and the first payment is received today. To calculate the price to pay for this particular

situation would require use of the present value of annuity due formula. However, if an individual is

wanting to calculate what their balance would be after saving for 5 years in an interest bearing

account and they choose to put the first cash flow into the account today, the future value of

annuity due would be used.



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Illustration

Ayush would like to calculate their future balance after 5 years with today being the first deposit.

The amount deposited per year is Rs.1,000 and the account has an effective rate of 3% per year.

It is important to note that the last cash flow is received one year prior to the end of the 5th year.

For this example, we would use the future value of annuity due formula to come to the following

equation:

After solving, the balance after 5 years would be Rs.5468.41.

4. PRESENT VALUE OF ANNUITY DUE

For the present value of an annuity due formula, we need to discount the formula one period

forward as the payments are held for a lesser amount of time. When calculating the present value,

we assume that the first payment was made today.

We could use this formula for calculating the present value of your future rent payments as

specified in a lease you sign with your landlord. Let's say for Example 4 that you make your first



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rent payment at the beginning of the month and are evaluating the present value of your five-

month lease on that same day. Your present value calculation would work as follows:

Present value of an annuity due formula

PV

If cash flow schedule are same as above example,

SINKING FUND

A fund created, by gradual periodic deposits, with the objective of getting a targeted amount to pay

off future debts, is called a sinking fund. The sinking funds can be created for a no. of purposes

such as repayment of debt in lump sum, redemption of bonds, replacement of worn out equipment,

buying of a new equipment etc.

This can be done by knowing the future value of an annuity, by using the following formula



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Illustration

In 10 years, a Rs. 40,000 machine will have a salvage value of Rs. 4,000. A new machine at that

time is expected to sell for Rs. 52,000. In order to provide funds for the difference between the

replacement

cost and the salvage value, a sinking fund is set up into which equal payments are placed at the

end of each year. If the fund earns 7 per cent compounded annually, how much should each

payment be?

r = 0.07 , n = 10

F = (52,000-4,000) = 48,000

48,000 = A[(1+07)10

-1/0.07]

48,000 = Ax13.82

A = 48,000/13.82 = 3,474.12



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BUSINESS MATHEMATICS AND FINANCE - MyOnlinePrep · PDF fileBUSINESS MATHEMATICS AND FINANCE MODULE-A ACCOUNTING & FINANCE FOR BANKERS - JAIIB . ... mock test and for JAIIB & CAIIB