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# Simple and Compound Interest Fundamentals

Interest is the additional money paid to the lender by the borrower for using the borrowed money for a specific period of time. When money is deposited in a bank, the bank pays interest on the amount deposited.

### Simple Interest

Simple interest is calculated only on the principal amount, or on that portion of the principal amount which remains unpaid.

Simple interest = Pnr / 100 where,
P - Principal or the original sum borrowed.
n -Time for which money is borrowed. It is expressed in number of periods which is normally 1 year.
r - Rate of interest. It’s the rate at which interest is calculated on the original sum.

Amount is the sum of principal and interest and is given by,
A = P + Pnr/100 = P (1 + nr/100)

### Compound Interest

When the interest is added to the principal at the end of each period to arrive at the new principal for the next period it is termed as compound interest.
Under compound interest, the amount at the end of the first year will become principal for the second year; the amount at the end of the second year becomes the principal for the third year and so on.

Amount after ‘n’ years = P(1 + r/100)n

Given amount and principal to find interest, I = A - P = P[(1 + r/100)- 1]

The frequency with which the interest is compounded can be different. If the interest is added to the principal every six months, then it is said to be compounded half yearly or twice a year. Similarly, if the interest is calculated and added four times in a year, then it is said to be compounded quarterly.

For example, if Rs. 8000 is lent at the rate of 12% per annum and compounded every three months, then the amount at the end of the year is calculated as follows:
The interest for 12 months is 12%.
Therefore for 3 months, the effective rate of interest is 12/4 = 3.
Also, the compounding is done 4 times instead of 1 time.
Therefore the amount = 8000 [1 + (12/4)/100]4

If compounding is done k times a year (i.e.) once every 12/k months at the rate of r% p.a.
then in n years the principal of P will amount to P[1 + r / (k  x 100)]kn
a. When interest is compounded annually amount =  P[1 + r / 100)]n
b. When interest is compounded half - yearly amount = P[1 + r / 2.100)]2n
c. When interest is compounded quarterly amount = P[1 + r / 4.100)]4n
d. When interest is compounded annually but time is in fraction, say 3(2/5) years
Amount = P(1 + r/100). [1 + (2/5)r / 100]

Compound interest may be computed in different ways. The most common methods are the effective rate method and the capitalized simple interest. Capitalized simple interest is an easy method of computing interest (often used in spreadsheets or with pocket calculators) but is not always fair for the lender and borrower. With this method, the same amount of interest is computed daily throughout the period and at the point of capitalization (compounding) the total interest generated is added to the principal. This balance becomes the new principal upon which interest is computed until the next capitalization.

In contrast, the effective rate method of calculating interest computes interest on a day-by-day basis (as opposed to a lump sum of interest at the capitalization date) using the effective interest rate. Based on an exponential formula, interest earned daily at the start of a period is less than that earned daily later on in the loan.

In loans, with the capitalized simple interest method, the borrower pays more interest in the early months than he would pay using the effective rate method. Interest compounding annually with either method will yield, at the loan anniversary date (which is the capitalisation date with the capitalised simple interest method), exactly the same total amount of interest.

### Value of Money

Present Value: It is the current worth of a future sum of money given a specified rate of return. Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows.
For example, if a lender is due to receive Rs. 121 at 20% compound interest after two years, the actual amount lent (Rs. 100) can be calculated using the given data. This actual amount left is called the present value.
If P is the present value of a sum and A is the amount at the end of n years, then P can be calculated as follows:

A = P(1 + r/100)

Therefore, P = A / (1 + r/100)

When a loan is repaid in equal annual installments, the value of the installment amount can be found by equating the sum of the present values of the installments made every year to the actual loan amount. If P is the amount loaned at r% and A is the equal installment paid every year, then

P = A / (1 + r/100) + A / (1 + r/100)1 + A / (1 + r/100)2 + ... + A / (1 + r/100)n

This forms a G.P and hence its sum can be calculated from which the value of the equal installment can be computed.

### Important Points to Note

1. In case of simple interest the principal remains the same every year. The interest for any year is the same as that for any other year.
2. In case of compound interest the amount at the end of a year is the principal for the next year. The interest for different years is not the same.
3. If the number of times of compounding in a year is increased to infinity we say compounding is done every moment and amount is given by P.enr/100
4. When rates are different for different years, say r1%, r2% and r3% for 1st, 2nd and 3rd year respectively, Amount = P(1 + r1/100)(1 + r2/100)(1 + r3/100)
5. The difference between the compound interest and simple interest on a certain sum for 2 years is equal to the interest calculated for one year on one year’s simple interest.
6. The difference between the compound interest for k years and the compound interest for (k + 1) years is the interest for 1 year on the amount at the end of kth year.
7. The difference between the compound interest for the kth year and compound interest for (k + 1)th  year is equal to the interest for one year on the compound interest for the kth year.
8. In Simple Interest loans, the repayments are adjusted against the principal, until the principal becomes zero. After the principal becomes zero, the payments received are adjusted against the interest, once the principal becomes zero; there is no further accumulation of interest.
9. In Compound Interest loans, the repayment money is adjusted against the principal of the compounding period during which repayment is made, until the principal becomes zero. After the principal becomes zero, the remaining money is adjusted against the interest due.
10. The specified rate of interest is called the nominal rate of interest. The effective rate of interest is that rate of simple interest which will result in an interest equal to that of the compound interest over the same period of time.

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