- INTRODUCTION
- SIMPLE INTEREST
- COMPOUND INTEREST
- COMPOUNDING MORE THAN ONCE A YEAR
- POPULATION FORMULA
- DEPRECIATION OF VALUE025
- GROWTH RATE
- OFFERING LOAN ON A DISCOUNT BASIS
Interest and Growth Rates
Money borrowed today is repaid with a higher amount tomorrow. This gives rise to the concept of interest.
A Creditor is a person who lends money to someone who wishes to borrow it, the latter termed as the Debtor.
The amount of money which the creditor lends initially is known as the Principal (P) or Capital and the time frame for which he lends the money is known as Time or Period (T or n).
The difference between the amount of money which the debtor borrows today (i.e. the principal) and the amount of money which he needs to repay at the end of the time period is called the Interest (I) over the Principal amount. Also, the total money which he repays at the end is termed as the Amount (A).
In other words,
Amount = Principal + Interest.
The Interest is calculated based on the Rate of Interest (R), which is specified in terms of percent per annum (p. c. p. a) unless specified otherwise.
There are two ways in which interests are calculated.
- Simple Interest (SI)
- Compound Interest (CI)
The interest calculated for the given time duration only on the original principal, is called Simple interest.
Let us understand this concept using an example. Suppose Himesh borrows Rs. 100 from Rakesh for a period of 2 years and agrees to pay simple interest at the rate of 10% per annum. Now, the amount of interest due to Rakesh at the end of year 1 is
So, Himesh owes Rakesh Rs. 110 at the end of year 1. But since he has agreed to pay Simple Interest, he pays interest only on the original Rs. 100 for the second year too. So, he pays an interest of Rs. 10 in the second year too. So cumulatively, he pays Rakesh Rs. 120 at the end of two years.
This can also be calculated together for two years using
Amount = P + I = 100 + 20 = Rs. 120
Thus, when we calculate simple interest, the accumulated interest is not taken into account for the purpose of calculating interest for the later years. The interest remains constant every year.
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Find the simple interest on a principal of Rs. 3,500 at the rate of 4% per annum for a period of 8 years.
Solution:
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Solution:
Amount = P + SI = 75000 + 29375 = Rs. 1,04,375 |
to be paid after 2½ years.
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In 4 years, Rs. 6000 amounts to Rs. 8000. In what time at the same rate will Rs. 525 amount to Rs. 700? [SNAP 2009]
(1) 2 years(2) 3 years (3) 4 years(4) 5 years
Solution:
Hence, option 3. |
The time that 525 requires to grow to 700 will be the same as the time
that 6000 requires to grow to 8000.
Now consider the example of Himesh and
Rakesh again. We saw that Himesh owes Rakesh Rs. 110 at the end of year 1. If
Himesh pays Rakesh an interest of 10% on Rs. 110 for the second year, we say
that he pays compound interest. So in this case, interest for the second year is
10/100 
110 = Rs. 11
Thus, in all, Himesh pays an interest of 10 + 11 = Rs. 21 to Rakesh at the end of 2 years. Note that interest to be paid is more when it is compounded.
Here, we considered the frequency of compounding to be yearly.
This frequency of compounding is specified in questions and can be quarterly, semi-annually, 2 yearly etc.
When money is lent at compound interest, at the end of a fixed period, the interest for that fixed period is added to the principal, and this amount is considered to be the principal for the next year or period. This is repeated until the amount for the last period has been calculated. The difference between the final amount and the original principal is the Compound Interest (CI). This amount can be calculated using the following formula:
In the case of Rakesh and Himesh,
Amount = 100
(1.1)2 = Rs. 121
Hence,
Compound Interest = Amount – Principal = 121 – 100 = Rs. 21
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If Rs. 12,000 has been lent out at 10% rate of interest, the interest being compounded annually, then what is the interest for the third year?
Solution: The amount at the end of the second year will be the principal for the third year.
= 12000
The simple interest on this sum will be the interest for the third year.
Interest for the third year = 14520 |
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If the compound interest on a certain sum for 3 years at 4% is Rs. 1,500, then what would be the simple interest on the same sum at the same rate and the same time period?
Solution: Let the sum be Rs. P.
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Let us illustrate the difference between SI and CI with the following example. Consider the principal is Rs. 1,000; rate of interest is 10% per annum for four years.
As seen from the table, in the case of SI, the principal remains the same every year and interest for all the years is equal. In case of CI, the amount at the end of one year becomes the principal for the next year. The interest gets compounded every year. This difference can be shown graphically as follows:
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Mungeri Lal has two investment plans- A and B, to choose from. Plan A offers interest of 10% compounded annually while plan B offers simple interest of 12% per annum. Till how many years is plan B a better investment? [XAT 2009]
(1) 3(2) 4(3) 5 (4) 6(5) 7
Solution: Let Mungeri Lal have Rs. 100 to be invested in both the plans. From plan A, we can calculate total amount along with the compound interest of 10% for every year. From plan B, we can calculate total amount along with the simple interest of 12% for every year. From the calculations, we can show that till 4 years, plan B gives more amount than plan A with the same initial investment. Hence, option 2. |
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Mr. Jeevan wanted to give some amount of money to his two children, so that although today they may not be using it, in the future the money would be of use to them. He divides a sum of Rs.18,750/- between his two sons of age 10 years and 13 years respectively in such a way that each would receive the same amount at 3% p.a. compound interest when he attains the age of 30 years. What would be the original share of the younger son? [IIFT 2007]
(1) 8959.80(2) 8559.80 (3) 8969.80(4) 8995.80
Solution: Let P1 be the amount (i.e. principal) given to the10-year old child. Then, (18750 – P1) will be the amount given to the 13-year old child. The 10-year old will attain the age of 30 after a period of 20 years, while the 13-year old will become 30 after a period of 17 years. Since the amounts received by both of them when they attain the age of 30 are equal, we have,
Thus, Rs. 8959.60 is the original share of the younger son. Option 1 is the closest among the given ones. Hence, option 1. |
As mentioned earlier, the frequency of compounding can vary. It can be done half yearly (semi-annually), quarterly, monthly etc. When compounding is done more than once a year, the rate of interest for that time period will be less than the effective rate of interest for the entire year. For example, if the annual rate of interest is 10%, then the rate of interest when the amount is compounded half-yearly will be 5%. Thus, as the time period of compounding doubles, the corresponding rate of interest is halved.
REMEMBER:
- The difference between the simple interest and the compound
interest (calculated on the same principal and with the same rate of interest)
for the second year is equal to the interest calculated for one year on one
year’s simple interest. In mathematical terms, the difference between the
compound interest and simple interest for the second year will be equal to
P
(r/100)2
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The compound interest on a certain sum for 2 years is Rs. 360 and the simple interest on the same sum for 2 years is Rs. 300. Find the principal and the rate percent.
Solution: Since SI remains same for all years, SI for the 1st year = 300/2 = Rs. 150
The SI and CI remain same for the first year. So CI for the 1st year = Rs. 150
Now the compound interest is more because it has an additional component of interest on the SI for the first year. The additional component is equal to Rs. 60.
Principal = 150/0.4 = Rs. 375 |
If the original population of a town is P and the annual increase is r%, then the population in n years (P’) is
For example, if the rate of growth of population of rabbits in a warren is 100% per year or if the population doubles every year, 2 rabbits will become 16 rabbits in a matter of 3 years.
If the annual decrease is r%, then the population in n years is given by a change of sign in the formula:
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The population of a city currently is 30 million. The number has been increasing at a steady rate for the past 10 years. If it is observed that the rate of increase is 15% every year, then what will be the population of the city 3 years from now?
Solution: P = 30 million, r = 15%, n = 3 years
Hence, using the formula, the population after 3
years = 30 million
So the population becomes 45.6 million after 3 years. |
45.6 million
The value of any asset decreases with time due to any of a number of factors including wear and tear, outdated technology, usage etc. This decrease is called its depreciation. If P is the original value and r is the rate of depreciation per year, then the final value (F) after n number of years is given by the formula,
For example, a vehicle bought for Rs. 2,50,000 is sold at Rs. 1,28,000 at a depreciation of 20% after 3 years.
Growth rates, namely Average Annual Growth Rate (AAGR) and Compound Annual Growth Rate (CAGR), are generally used to state the growth of investments over a period of years.
The absolute growth is the difference between the final value and the initial value.
The Simple Annual Growth rate, also known as the Average Annual Growth Rate, is the arithmetic mean of the growth rates over a number of years.
For example, if an investment grows 10% in the first year, 15% in the second year and 14% in the third year, then,
Compound Annual Growth Rate is the consistent growth rate that an investment follows year after year. CAGR is rarely real. It is a figure that gives the rate at which an investment would grow if it were to grow at a consistent rate.
For example, in the example stated for AAGR, an amount of Rs. 10,000 would grow to Rs. 14,421 in 3 years.
This is the same value as you would have got had you used Geometric Mean to find compound interest, as follows:
The value of CAGR is always less than AAGR.
List of Formulae Concerning Growth Rates
Absolute Growth = Final Value – Initial Value
Growth Rate for a year 
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Bennett distribution company, a subsidiary of a major cosmetics manufacturer Bavlon, is forecasting the zonal sales for the next year. Zone I with current yearly sales of Rs.193.8 lakh is expected to achieve a sales growth of 7.25%; Zone II with current sales of Rs.79.3 lakh is expected to grow by 8.2%; and Zone III with sales of Rs.57.5 lakh is expected to increase sales by 7.15%. What is the Bennett’s expected sales growth for the next year? [IIFT 2009]
(1) 7.46%(2) 7.53% (3) 7.88%(4) 7.41%
Solution: Total sales this year = 193.8 + 79.3 + 57.5 = Rs. 330.6 lakhs Expected sales next year = 193.08
= 7.46% Hence, option 1. |
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Refer to the graph below and answer the questions that follow.
Turnover of ABC Ltd. from 2000 to 2005
Question 1: By how much percent has the turnover increased from 2000 to 2005?
Solution: Turnover in 2000 = Rs. 8 crores and the turnover in 2005 = Rs. 18 crores.
Question 2: What is the Simple Annual Growth Rate of turnover in the given period 2000 to 2005?
Solution:
Question 3: What is the compounded annual rate of growth of
turnover in the period 2000
Solution:
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If a bank or a company offers loan on a
discount basis at a% then if the original value of the loan is Rs. 100
then the bank or the company loans out Rs. [100 
(a% of 100)] or Rs. (100
a) [In this case] and get Rs. 100 back at time of maturity.
In general, if d is the discount rate, then the
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ICICI bank offers a 1-year loan to a company at an interest rate of 20 percent payable at maturity, while Citibank offers on a discount basis at a 19% interest rate for the same period. How much should the ICICI Bank decrease/increase the interest rate to match up the effective interest rate of Citibank? [FMS 2009]
(1) Increase by 3.5% (2) Decrease by 1.8% (3) Increase by 1% (4) Decrease by 1.4%
Solution: Assume that ICICI Bank gives a loan of Rs. 100 at the interest of 20% payable at maturity. This means that ICICI will receive a Rs. 20 as interest at the end of the period. Thus the total amount received by ICICI at the end of the period will be Rs. 120. To calculate ICICI’s effective rate such that it is competitive with Citibank’s, we need to find out how much Citibank earns if it lends out Rs. 100. Citibank lends at 19% on a discount basis. Discount basis means that if the original value of the loan is Rs. 100, the client gets only Rs. 81 right now and pays Rs. 100 at the end of the period.
Thus, for the interest rate of ICICI Bank to match up the effective interest rate of Citibank, it has to be increased by (23.45 – 20) = 3.45%. The closest answer option is option 1, i.e. increase by 3.5%.Hence, option 1. |
