Class 12 | Unit 4 BASIC OF BUSINESS MATHEMATICS | Banking | CBSE |

 Calculation of Simple Interest and Compound Interest

As the major activities of a bank are accepting deposits and giving loans, Depositors require monetary

compensation for partingwith their moneybywayof deposits with the bank. ‘Interest’ is the incentive

or reward paid to the depositors for postponing their other expenditures and keeping the money with

the bank.

Banks use the deposits to lend to the needy borrowers. Borrowers borrow money because they do

not have money which they need it. So if somebody gives them the needed money, the borrower

should be prepared to compensate the lender. This is called ‘interest’.

So it is clear that whenever funds are lent or borrowed, the question of receiving or paying ‘interest’

arises;

Factors affecting Market interest rates: 

Opportunity cost: Opportunity cost refers to any other use to which the money could be put, for

example lending to others or investing elsewhere. If one can get a higher return elsewhere, the interest

rate on loans will also rise.

Inflation: Simplyput, inflation is the movement in prices in % per annum terms. Usuallyprices move

only upwards. For example, now if you can buy some articles for Rs. 100 and after one year you

require Rs. 111 to purchase similar articles, then the inflation is 11%. Since the lender is postponing his

consumption now to use it later, he will require as a bare minimum compensation to recover enough tomake up for the inflation and plussomethingmore as anincentive. Because futureinflation is unknown,

the lender will always add some ‘premium’to the expected inflation rate and demand that as his interest

rate.

Demand and supply: Demand for and supplies of money are the crucial factors in determining the

interest rates.

Borrower Default: There is always the risk that the borrower will become bankrupt, abscond or

otherwise default in repaying the loan. In order to limit the bad consequences of such situations, the

lender usually adds some ‘risk premium’ to the interest rate already decided and quotes such rates to

the borrower.

Length of time: Shorter terms are less riskyfrom the point of view of default and exposure to inflation

because the near future is easier to predict. In these circumstances, short term interest rates are lower

than longer term interest rates.

Government intervention: Government’s and RBI’s actions may also influence short-term interest

rates.

Banks keenly follow the above factors and decide interest rates. Such rates are known as Market

Interest Rates.

Default or Penal interest: Default or penal interest is the rate of interest thataborrowermustadditionally

payif he does not fulfil anyof the conditions set bythe lender at the time of lending.

Interest is always expressed in terms of percentage (%) per annum, as a normal practice, which means

that the interest is calculated for period of 1 year. However, in real life situations, interest is calculated

for different periods for different products, and also as per the usual practice of the bankers (as lenders

while giving loans or as borrowers while accepting deposits). So appropriate adjustments have to be

made in the calculations, whenever, the intervals between interest calculations differ from a year.

Calculation of Simple Interest and Compound Interest:

 Risk Free Interest Rate:Risk-free interest rate is the theoretical rate of return of an investment with

no risk of financial loss. One view is that the risk-free rate represents the interest that an investor would

expect from an absolutelyrisk-free investment over a given period of time. Since the risk free rate can

be obtained with no risk, it is implied that any additional risk taken byan investor should be rewarded

with an interest rate higher than the risk-free rate.

Time Value of Money: The time value of money implies that a certain amount of money todayhas a

different buying power (value) (usuallymore) than the same currencyamount of moneyin the future.

While talking about interest, there are two types of calculating it: they are

 Simple interest and

 Compound interest.

“Simple” interest is easy to understand. If you deposit an amount of Rs. 10,000 in a bank as a deposit

for a year carrying interest @8.50% (Remember the convention that the % is always % per annum and

is not repeated every time). Here to calculate the interest element, you require three elements to

calculate the interest (denoted bythe letter “i”).

Theyare:

 PrincipalAmount:The amount for which you arecalculatingthe interest iscalled the “Principal”

usuallynoted bythe letter ‘p’ in the formula;

 Time:The period for which you are calculatingthe interest,called “Time” denoted bythe letter ‘t’

or usuallyin number of years denoted by the letter ‘n’ and

 Rate:The % rate at which interest is calculated, denoted by the letter ‘r’

Now look at the formula:

i =ptr/100 or pnr/100

Depending upon whether you use t or n for denoting the period.

So using the formula in the given example, the interest ‘i’ is given by:

i = p*n*r/100 = Rs.10,000 * 1 * 8.50 /100 = Rs. 850.

In the same wayinterest on the same deposit for 5 years assuming that the interest rate is 9.25%, would

be:

i = Rs. 10,000 * 5 * 9.25/100 = Rs. 4,625.

The formula given above is the standard in calculations. In real practice, banks in India pay interest

every quarter.

So how do we adjust the formula?

Recall that in the formula, ‘n’ stands for the number of years.

So we put n =1/2 (0.5) if the interest is to be calculated for 6 months i.e. ½ a year.

On the same lines, we put n = ¼, if the interest is to be calculated for 3 months which is a ¼ or 0.25 of

a year.

Similarlyinterest for one month is obtained byputting n = 1/12.

Practice Question: Calculate the interest on a 5 year deposit for Rs.25,000 if the interest rate is

9.50% - on a quarterly basis and on a half yearly basis.

Answer: Quarterly interest: Working: Rs. 25,000 * ¼ * 9.50/100 = Rs.593.75.

In this case, the depositor will get Rs. 593.75 at the end of every quarter from the bank.

So at the end of 5 years from the date of deposit, the bank will paythe depositor the interest for the last

quarter Rs. 593.75 and also the original Principal amount of Rs. 25,000.

Half-yearly interest

Working: Rs 25,000 * ½ * 9.50/100 = Rs 1.187.50

In this case, the depositor will get Rs. 1,187.50 at the end of every half year from the bank.

So at the end of 5 years from the date of deposit, the bank will paythe depositor the interest for the last

half year Rs1,187.50 and also the original Principal amount of Rs 25,000.

What we have discussed is the case of a deposit in a bank – where the bank is the borrower (debtor)

and the depositor is the lender (creditor).

In the case of a loan from the bank to a customer, the roles are reversed. The bank becomes the lender

(creditor) and the customer becomes the borrower (debtor). So the borrower goes on paying interest

at periodic intervals to the bank.

Let us see this with an example: Calculate the interest on a loan of Rs. 1,00,000 for a period of 7

years @12.75%.at monthly, quarterlyand half yearlyintervals.

Answer: Monthlyintervals (12 months *7 years =84 months):

Rs 1,062.50 every month and also Rs 1,00,000 in the last month i.e. at the end of 7 years.

Quarterly intervals (4 quarters in a year * 7 years = 28 times)

Rs 3,187.50 at the end of every quarter and also Rs 1,00,000in the last quarter end i.e. at the end of

7 years.

Half-yearly intervals (2 half-years in a year * 7 years = 14times):

Rs 6,375 at the end of every half year and also Rs 1,00,000 in the last half- year i.e. at the end of 7

years.

Generallyloansarerepayableinmonthlyorquarterlyorhalfyearlyinstallmentsandrarelyyearlyinstallment

or entire amount at the end.

Since interest is calculated on the actual amount due, as and when the borrower repays the installments,

the principal will be going on reducing and hence the interest amount also will be going on reducing as

more and more installments are paid bythe borrower. We will come back to this situation later. Before

that let us try to understand what ‘compound interest’ is.

Compound Interest: Let us go back to the case of deposit in a bank.

You deposited Rs 1,00,000 in a bank for 5 years which pays you interest @ 10% .The bank has got a

Fixed Deposit scheme in which the bank will not payyou interest everyquarter, but will payyou all the

interest and the principal in one lump sum at the end of 5 years. Since you are not in need of moneyfor

next 5 years, you agree to this scheme.

The interest calculations will change as follows:

I Quarter: The interest amount at the end of the first quarter will be, as you already know:

Rs 1,00,000 * ¼ *10/100 =Rs 2,500.

Since you are not withdrawing the interest amount of Rs 2,500, the bank is adding this Rs 2,500 to the

Principal of Rs 1,00,000 and will calculate the interest for the 2nd quarter on Rs (1,00,000 + 2,500).

II Quarter: So interest for the 2nd quarter

Rs 1,02,500 *1/4 * 10/100 = Rs 2,562.50.

Again, since, you are not withdrawing the interest amount of Rs. 2,562.50.

The bank is adding this Rs2,562.50 to the Principal of Rs. 1,00,000 + the first quarter interest of Rs.

2,500.

III Quarter: So for calculation of interest for the 3rdquarter:

PrincipalAmount: Rs 1,00,000 + Rs2,500 + Rs2,562.50 = Rs 1,05,062.50.

So the interest for the third quarter will be

=Rs.1,05,062.50 *1/4 * 10/100

= Rs. 2626.56.

So proceeding like this, the interest for the last quarter of the 5th year, i.e. 20th quarter, will be =

Rs.1,59,865.02 *1/4 *10/100 = 3,996.62

Assignment: Calculate the interest for every quarter

Thus the total amount to be received by y

Rs1,59,865.02 + Rs3,996.62= Rs1,63,861.64.

So you can observe in this process that you have earned further interest on the interest amount you have

not withdrawn.

This process of calculating interest on interest is called ‘compounding’.

To put in other words:

Your deposit of Rs 100,000 has earned an interest amount of Rs 63,861.64 @ 10% compounded on

quarterlybasis.

Compare this with the position if you had withdrawn interest at every quarter end. You would have

earned Rs 2,500 * 20 times = Rs50,000 for the 5 year period towards interest and so you would have

got back a total amount of Rs1,50,000 from the bank.

It is clear that due to the effect of ‘compounding’ the interest, you got Rs 13,861.64 more compared to

the ‘simple’ interest calculation method.

Period along the X – axis (denoted by ‘Y’ for years) and

Principal along theY– axis (denoted by‘P’for Principal)

Please note that calculations are the same and so whether you are paying or receiving interest depends

upon whether you are a borrower or a depositor.

You have seen how compound interest is calculated period by period and going on with the same

process till you reach the end of the total term. This is reallymonotonous and time consuming.Take the

help of a Scientific Calculator ora computer. (Nowadays most of the Cell Phones have good calculators

inthem).

In the above example, use the formula:A= P* (1 +r)^n

,

Where:

 A is the total amount of the initial Principal and accrued interest;

 P is theinitialPrincipalamount;

 r is the interest rate in % per annum converted for the period

 of compounding; and

 n is the number of times you have to compound.

In the previous example:Ahas to be found out/calculated;

P = Rs 1,00,000;

r = 10%/4 i.e. 2.5% for quarterly period interest, we have to compound 4 times in a year and

n = 20 (since there are 20 quarters in 5 years)

So usingthe formula,

A = Rs.1,00,000 * (1 + 2.5/100)^20

A = Rs. 1,63,861.64

Home work: Calculate ‘A’if the compounding is to be done

 Every one year,

 Every Half year and

 Everymonth.

Answers:

 Everyone year: Rs.1,61,051

 EveryHalf year: Rs.1,62,889.46 and

 Everymonth Rs. 1,64,530.89

By looking at these answers, you can easily understand that as the frequency of compounding (n)

increases, ‘A’also increases for the same Principal (P), interest (i) and period (no. of years)

This followinggraph will help youunderstand better:

Rate of Return:In finance, return is a profit on an investment. It comprises any change in value, and

interest or dividendsfrom the investment. Conversely,a loss instead of a profit is described as a negative

return.

Rate of return is a profit on an investment over a period of time, expressed as a proportion of the

original investment.The time period is typicallya year, in which case the rate of return is referred to as

annual return. Return, in the second sense, and rate of return, are commonlypresented as a percentage.

4.2 Fixed and Floating Interest Rates:

Fixed Rate Interest: When the rate of interest applied to a loan or a deposit remains constant and

unchanged from the beginning till the maturityof the loan/deposit, it is called “Fixed Rate of Interest’.

Interest rate remains fixed irrespective of market conditions. The borrower or the depositor has peace

of mind as he need not worryabout the future cash-flows since theyare all fixed and known in advance.

The uncertaintyabout the quantum of cash-flows over the period is removed. Thus it brings a sense of

certaintyand security.

Floating Rate Interest: Floating interest rate as the name implies is the rate of interest which varies

with market conditions. The biggest benefit with floating rate home loans is that theyare cheaper than

fixed interest rates. So, if you are getting a floatinginterest rate of 11.5 per cent while the fixed rate loan

is being offered at 14 per cent, you still save money if the floating interest rate rises by up to 2.5

percentage points. Even if the floatingrate goes over the fixed rate, it will be for some period of the loan

and not for the entire tenure. The interest rates mayfall over a longperiod and, thus, the floating interest

rate brings a lot of savings.

The drawback with floating interest rates is the uneven nature of monthly instalments. This makes it

difficult to budget with floating interest rate home loans.As seen in recent times, due to the hike in

floating home loan interest rates, the borrowers had to shell out thousands per month extra as their

EMIs, throwing their entire budget out of order.

In the case of floating interest rate, the total amount of interest is not determined for the entire period of

loan, at the time of borrowing,but is dependent on someunderlyingindex, which goeson changing, i.e.

‘floating’infinancialterms.

For example, a person may borrow Rs 1,000,000 at an interest rate equal to the Bank’s Base Rate

(BR) + 1% per annum. Here the bank’base rate is the floating index.ABank’s ‘Base Rate’is normally

the minimum rate of interest that the bank will charge from anyborrower.

Every Bank in India has to decide its BR based on some factors as prescribed by RBI and BR goes on

changing periodicallydepending upon the changes in the parameters prescribed byRBI.

However, during the period when the bank’s BR does not change, the total interest that has to be paid

bythe borrower also does not change. Further in the above example, where the interest rate fixed is BR

+ 1%, 1% is called the ‘margin’ over the index. But once the margin over the BR is fixed (like 1% as

mentioned above) for a particular loan, will remain constant till the loan is fullyrepaid. So, thetotal rate

of interest will change onlywhen the BR will change (either upwards or downwards).

So a ‘Floating Rate of interest’ borrower mayfind that his total interest rate maycome down or maygo

updependinguponthemovementsintheBR.BR isgenerallyusedastheFloatingIndexforallborrowers

like Retail customers.

For well informed corporate borrowers more, sophisticated indices like MIBOR is used. MIBOR

stands for Mumbai Inter Bank Offered Rate This is the rate at which banks in Mumbai offer loans to

one another.

But there is a huge difference between BR and MIBOR. BR does not change dailybut changes as and

when economic situations warrant. MIBOR is decided on a daily basis based on the demand for and

supplyof funds among banks.

The procedure is to take the MIBOR prevalent on the date of arranging the loan. Bymutual agreement,

the chosen MIBOR will be re-set at the end of 3 months or 6 months as agreed between the lender and

the borrower. Depending on the period of reset, the lender also borrows from the market for the same

period and on-lends to the borrower with the mark up of the margin.

Let us taken an example for easyunderstanding:

MIBOR on the date of agreement for loan between a bank and a corporate say 20th March 2014:

7.5%; Margin agreed over the MIBOR: 3%, agreed reset is every3 months. What it implies is that the

bank will charge interest @ (7.5%+ 3%) i.e. 10.5% to the corporate for the three month period from

20/03/2014 to 19/06/2014. Even though MIBOR changes daily, neither the bank nor the Corporate

is affected bysuch changes since both of them are committed.

On the reset date viz. 20/06/2014, suppose MIBOR is 7.65%, then from that date onwards the bank

will charge interest at 10.65% to the corporate up to 19/09/2014. This process goes on till the maturity

of the loan.

4.3 Calculation of EMIs:

What is EMI?

An Equated Monthly Instalment (EMI) is defined as “Payment of a fixed payment amount made by a

borrower to a lender at a specified date each calendar month. Equated monthlyinstalments are used to

payoff both interest and principal each month, so that over a specified number of years, the loan is fully

paid off along with interest.”

The benefit of an EMIfor borrowers is that theyknow preciselyhow much moneytheywill need to pay

toward the repayment of loan each month, thus making the personal budgeting process easier.

For example, a borrower has taken a loan from a bank for Rs 60,000 and has agreed to pay Interest

@ 10% per annum on monthlyinterval basis and also payan installment of Rs 10,000. So the loan has

to be cleared in 6 months. See how the interest is calculated for the first month: see the cell no.B33.

This cell calculates the interest on the previous month’s balance which is in the cell D32. Interest rate of

10% per annum is shown as 0.1 and period of a month is shown as 1/12, since interest is calculated on

monthlyintervals.

Column B shows the interest payable by the borrower in the respective months. The same amount is

repeated in Column C to show that the borrower pays the interest and also pays the installment.

Column Dis arrived at bytakingthe previous month’s balance, adding the interest calculated in column

B and then deducting the repayments bythe borrower as shown in the column C. This process goes on

till the loanis fullypaid.

If you observe column C, you can find that the borrower has paid Rs 10,500 in the first month and

paid Rs 10,083 in the last month. The variation in the installments paid is due to the variation in

the interest amounts.

The previous table is simple enough to construct if you know elementary Excel.

Now look at the same example re-worked with the assumption that the borrower pays a constant

amount of installment (EMI) uniformly from the beginning till full repayment of the loan. Here the

borrower has paid some amount which includes the actual interest amount + some varying amount

towards the Principal. Here the Principal amount paid per month has varied as against the previous

table where the interest amount has varied. However the loan is completely repaid in the same

period of 6 months as in the previous case.

But the question is how to calculate the EMI?

Use this formula: (You can use a calculator also)

EMI = [(P*r/1200)*(1+r/1200)^N] / [(1+r/1200) – 1]

P = Total loan amount, Rs.60,000;

N = Number of installments, 6

r = Rate of interest, 10%

What we have done is to add the Principal amount to the total accrued interest for the entire period of

the loan and then divided that total amount by the number of installments to arrive at that uniform

constant amount to be repaid by the borrower every month called EMI.

So applyingthe formula for the given example:

The EMI = [(60000*10/1200)*(1+10/1200)^6] / [(1+10/12000^6) – 1]

In this case this works out to Rs. 10,292.50, rounded off to Rs. 10,293.

See the actual working in the excel sheet as follows:

There is another easier way to work with Excel to solve this problem. Using this formula, whatever is

the rate of interest or period or amount, you can get the EMI.

 Open an Excel sheet

 Click ‘Formulas’ on the top line and then click the ‘fx inserts function’ option which you will find

on the top left.

 From the list which appears select ‘PMT’ and click it and you see this box.

Rate: input 10% / 12 (Since interest is calculated at monthly intervals, the rate in % is divided by 12)

Nper: Number of months in which the loan is to be paid = 6. (Example if the loan is for 7 years, then

Nper = 84)

PV: This is the total loan amount = 60000 (Enter the figure without commas)

Fv = 0 (At the end of the loan period, the outstanding amount in the loan should be zero)

Type=0 (At the end of the payment you are not making anypayment i.e. you are making zero payment)

Now you can get the result instantaneously as you see here below :( The amount used is US$ in the

Excel worksheet – Never mind it applies to any currency)

Do some exercises with Excel and cross-check with calculators.

Calculation of Front End and Back End Interest: Front end interest is interest calculated at the

beginning of the loan installments, in advance, whereas the back end interest is interest calculated at the

end of the installments when interest amounts fall due. To explain this let us go back to the previous

example and work it out both ways:

Loan amount Rs. 60,000 for six months @ 10% with EMI @Rs. 10293.

Back-end interest calculations are the same as what you have seen earlier.

Interest is calculated at end of each and every period and recovered.

If you observe the Front-end calculations you can find the difference: that is, the interest is calculated

up-front i.e. at the beginning of the month itself. So in the front end calculation, interest is calculated on

Rs. 60,000 for one month and added to Principal at the time of releasing the loan itself. In the above

screen shot, you can see interest appearing at the start itself. (Installment number 0 and interest Rs. 500

are marked in red color).

In the back end, interest is calculated at the end of the month and shown in red color against installment

number 1. If you compare the totals, you can immediatelyunderstand that in the Front-end calculation

method, the borrower is paying an additional amount of Rs 526. Front-end interest is not charged by

banks.

4.4 Calculation of Interest on Savings Accounts

In real life situations, banks charge interest on loans based on the actual balances in the accounts at the

end of each day, add them up till the end of the month and then calculate the interest. This applies to

payment of interest on Savings Bank accounts also.

Let us see some example as to how it works in an overdraft account.

Limit Rs. 100,000. Interest rate =12%.

(Balances are understood to be debit balances. If by chance there is credit balance in the account on

some days, then no interest is charged on those days and no interest is paid also for the credit balance.

(Limit Rs. 100000 means that the customer will not be allowed to draw more than Rs. 1 lakh at any

time.)

At the endof the month summations for all the columns are made (except the balance column).You can

find that the difference between the dr. column and Cr. Column tallies with the balance column and

hence the calculations are arithmeticallycorrect. The last column in the table shown above is called the

‘Product’ which is equal to the amount of loan outstanding in the account multiplied bythe number of

days such amount remained outstanding.

For example, the opening balance in the account viz. Rs. 50000 was outstanding for two days i.e. on

01/02/2014 and 02/02/2014. At the end of 03/02/2014 the balance changes to 70000 which are

outstanding for one day…. and so on. The last transaction in the month was on 27/02/2014 which was

outstanding for two days viz. 27th and 28th February.

Now having found out the product, for the month, it is easyto calculate the interest chargeable on the

account to the borrower.

You know the formula: interest = p*n*r*/100.

In this example, you have to take p as the total product viz. 1810000;

n as the number of years for which you have to calculate interest – in this case n = 1/365 i.e. one day

since you have individuallyworked out the balances outstanding on each dayand added them up.And

r is given as 12%.

So the interest to be charged on the account for the month is:

= pnr / 100

= 1810000*1/365*12/100

= Rs. 595.07.

As per RBI’s rules, the amount is rounded off to the nearest rupee i.e. Rs. 595.

Savings Bank Account Interest Calculation: Instead of a loan account, if it is a Savings Bank

account, the bank will be giving interest to the customer in the same way as shown above at the

appropriate rate. Most of the banks in India pay interest on savings accounts at 4% p.a. So let us see

an example.

Interest payable to the customer at 4% = Rs. 2035780*4/100*1/365

= Rs. 223.10

= Rs. 223 on rounding off.

In banks nowadays, nobodydoes these calculations manuallybut the computers do them automatically

whentheinterestpaymentsfalldue.Whentheinterestisdueontheloanstheyareautomaticallycalculated

and debited to the accounts and credited to the income accounts.

Similarly when the interest is due on Savings Bank or other deposit accounts (usually at quarterly

intervals) theyare calculated and credited to the same accounts and debited to interest paid account.

4.5 Calculations of Date of Maturity of Bills of Exchange

In this section we will see how the date of maturityof the Bill of Exchange is calculated.

Maturity: The maturity of a promissory note or bill of exchange is the date at which it falls due for

payment.

Days of Grace: Every promissory note or bill of exchange which is not expressed to be payable on

demand, at sight or on presentment, is at maturityon the third dayafter the dayon which it is expressed

to be payable.

Calculating Maturity of Bill:Section 23: Calculating maturityof bill or note payable so manymonths

after date or sight:

In calculating the date at which a promissorynote or bill of exchange, made payable a stated number of

months after date or after sight, or after a certain event, is at maturity, the period stated shall be held to

terminate on the dayof the month which corresponds with the dayon which the instrument is dated, or

presented for acceptance or sight, or noted for non acceptance, or protested for non acceptance, or

the event happens, or, where the instrument is a bill of exchange made payable a stated number of

months after sight and has been accepted for honour, with the day on which it was so accepted.

If the month in which the period would terminate has no corresponding day, the period shall be held to

terminate on the last dayof such month.

 A negotiable instrument, dated 29th January 20XX, is made payable at one month after date.

The instrument is at maturity on the third day after the 28th February 20XX.

 Anegotiable instrument, dated 30thAugust 20XX, is made payable three months after date.

The instrument is at maturityon the 3rd December 20XX.

 Apromissory note or bill of exchange, dated 31stAugust 20XX, is made payable three months

after date.

The instrument is at maturity on the 3rd December, 20XX (after allotting 3 days of grace)

Section 24:Calculating maturityof bill or note payable so many days after Date or sight:

In calculating the date at which a promissorynote or bill of exchange made payable a certain number of

days after date or after sight or after a certain event is at maturity, the dayof the date, or of presentment

for acceptance or sight, or of protest for non-acceptance, or on which the event happens, shall be

excluded. Hence the a dayis excluded in the calculation of number of days

Section 25: When day of maturity is a holiday:

When the day on which a promissory note or bill of exchange is at maturity is a public holiday, the

instrument shall be deemed to be due on the next preceding business day. The expression “public

holiday” includesSundays and anyotherdaydeclared bytheCentral Government, bynotification in the

Official Gazette, to be a public holiday. Bills payable after 30 days of sight (drawn on 30August 20XX,

will mature on 28 Sept 20XX.If 28 Sept 20XX happens to be a Sunday, and then the maturitydate will

be 27 Sept 20XX.