What are Interest Rates?
- Interest rates are very powerful and intriguing mathematical concepts. Our banking and finance sector revolves around these interest rates. One minor change in these rates could have tremendous and astonishing impacts over the economy. But why?
- Before determining the reason of this why? Let’s first know what is interest and these interest rates?
- Interest is the amount charged by the lender from the borrower on the principal loan sum. It is basically the cost of renting money. And, the rate at which interest is charged on the principal sum is known as the interest rate.
- The rate at which interest is charged depends on two factors:
(i) The value of money doesn’t remain same over time. It changes with time. The net worth of ₹ 100 today will not be same tomorrow i.e. If 5 pens could be bought presently with a INR 100 note then in future, maybe only 4 pens can be bought with the same ₹ 100 note. The reason behind this the inflation or price rise. So, the interest rate includes this factor of inflation
(ii) The credibility of the borrower, if there is more risk and chance of default on the borrower’s part then more interest will be charged. And, if there is less chance of payment failure on the part of borrower then the rate of interest would be lower. - The above two reason becomes the basis of why interest rates are so important and have a great effect on markets and economy.
- Since a minor rise in interest rates increases the cost of borrowing for the borrower and as a result, he has to pay more interest on his loan amount and thus, a decline in his money income that he could spend on other products which create a ripple effect of decreased spending throughout the economy and vice versa.
- Since change in interest rate has a chain effect in the market, it has a great deal of importance in the study of market, finance, and economy. And that’s why, forms an integral part of the curriculum in the MBA programs. But, a relatively simpler level of questions is asked in the CAT based on the concepts learned at the time of high school.
Types of Interest Rates
Interests are of two types:
- Simple Interest
- Compound Interest
1. Simple Interest
- Let’s first start and understand Simple Interest because as the name suggests it is simple and comparatively easy to comprehend.
- Simple interest is that type of interest which once credited does not earn interest on itself. It remains fixed over time.
- The formula to calculate Simple Interest is
SI = {(P x R x T)/ 100}
Where, P = Principal Sum (the original loan/ deposited amount)
R = rate of interest (at which the loan is charged)
T = time period (the duration for which money is borrowed/ deposited) - So, if P amount is borrowed at the rate of interest R for T years then the amount to be repaid to the lender will be A = P + SI
- Consider a basic example of SI to understand the application of above formula such as Find the simple interest on ₹68000 at 16 ^{2}/_{3 }% p.a. for 9 months.
Here, P = ₹68000
R = 16^{2}/_{3 }% = 50/3% p.a.
T = 9 months = 9/12 years = ¾ years
SI = (68000 x 50/3 x ¾ x 1/100) = ₹8500
Try yourself:Find the simple interest on Rs. 5200 for 2 years at 6% per annum.
Explanation
P = Principal Amount
T = Time period
R = Rate of Interest
I = P X T X R/100
I = 5200 X 2 X 6/100
I = 624
Table: Some useful results based on Simple Interest
RESULTS | APPLICATION | SOLUTION |
If rate to interest is r_{1}% for T_{1} years, r_{2}% for T_{2} years …. r_{n} for T_{n} years for an investment. And if the Simple Interest obtained is ₹a on the investment. Then the principal amount is given by a x 100/ (r_{1}T_{1} + r_{2}T_{2} + …+ r_{n}T_{n}) | Adam borrowed some money at the rate of 6% p.a. for the first two years, at the rate of 9% p.a. for the next three years, and at the rate of 14% p.a. for the period beyond 5 years. If he pays a total interest of ₹ 11400 at the end of nine years, how much money did he borrow? | In this case, r_{1} = 6%, T_{1} = 2 years r_{2} = 9%, T_{2} = 3 years r_{3} = 14%, T_{3} = 4 years (since, beyond 5 years rate is 14%) and Simple interest = ₹11400 Therefore, P = (11400 x 100)/ (6*2 +9*3 +14*4) = 1140000/ (12 + 27 + 56) = 1140000/ 95 = ₹12000 |
If a person deposits sum of ₹A at r_{1}% p.a. and sum of ₹B at r_{2}% p.a. then the rate of interest for whole sum is R = {(Ar_{1 }+ Br_{2})/ (A + B)} | A man invested 1/3 of his capital at 7%; ¼ at 8% and the remainder at 10%. If his annual income is ₹561, What is his capital? | Let x be his capital or principal. Therefore, R = (^{1}/_{3} x * 0.07 +¼ x *0.08 +^{5}/_{12} x*0.10)/x R = (^{1}/_{3} * 0.07 +¼ *0.08 +^{5}/_{12} *0.10) R = 0.08496 Total SI = ₹561 ₹561 = 0.08496x x = ₹6602 |
If a sum of money becomes “n” times in “T years” at Simple Interest, then the rate of interest p.a. is R = 100(n – 1) / T | The rate at which a sum becomes 4 times of itself in 15 years at S.I. will be? | It’s a very easy question you just need to use this formula and you will directly reach to an answer. Therefore, R = (100 x 3)/15 = 20% |
If a certain sum of money is lent out in n parts in such a manner that equal sum of money is obtained at simple interest on each part where interest rates are R1, R2, … , Rn respectively and time periods are T1, T2, … , Tn respectively, then the ratio in which the sum will be divided in n parts can be given by 1/R_{1}T_{1:} 1/R_{2}T_{2}:..:1/R_{n}T_{n} | A person invests money in three different schemes for 6 years, 10 years and 12 years at 10%, 12% and 15% Simple Interest respectively. At the completion of each scheme, he gets the same interest. What is the ratio of his investment? | Here, T_{1 }= 6, T_{2} = 10 and T_{3} = 12 years resp. And, R_{1} = 10%, R_{2} = 12%, and R_{3} = 15% resp. Hence, the ratio of his investment will be 100/60 : 100/120 : 100/180 1/6 : 1/12 : 1/18 1 : 1/2 : 1/3 6 : 3 : 2 |
Try yourself:How much time will it take for an amount of Rs.450 to yield Rs. 81 as interest at 4.5% per annum of simple interest?
Explanation
Time=(100 X 81/450 X 4.5) years
= 4 years
Try yourself:Mr. Thomas invested an amount of Rs. 13,900 divided in two different schemes A and B at the simple interest rate of 14% p.a. and 11% p.a. respectively. If the total amount of simple interest earned in 2 years be Rs. 3508, what was the amount invested in Scheme B?
2. Compound Interest
- This the most usual type of interest that is used in the banking system and economics. In this kind of interest along with one principal further earns interest on it after the completion of 1-time period.
- Suppose an amount P is deposited in an account or lent to the borrower that pays compound interest at the rate of R% p.a. Then after n years the deposit or loan will accumulate to:
P(1+R/100)^{n} - Consider this example, if an amount of 100 is deposited in saving bank account for 3 years at the interest rate of 6% p.a.
- Then, after one year the ₹100 will accumulate to ₹106. Since in compound interest, interest itself earns interest, therefore, after 1-year interest for the 2^{nd} will be calculated on ₹106 unlike to that of Simple interest where interest will be calculated on ₹100 only.
- Thus, after the end of the third year the total amount will become ₹100(1.06)^{3 }= ₹119.1016.
➢ Important Formulas
- When the interest is compounded Annually:
Amount = P (1 + R/100)^{ n} - When the interest is compounded Half-yearly:
Amount = P (1 + (R/2)/100)^{2n} - When the interest is compounded Quarterly:
Amount = P (1 + (R/4)/100)^{4n} - When the rates are different for different years, say R_{1}%, R_{2}% and R_{3}% for 1 year, 2 years and 3-year resp. Then,
Amount = P (1 + R_{1}/100) (1 + R_{2}/100) (1 + R_{3}/100) - Present worth of ₹ x due n years hence is given by:
Present worth = x/ (1 + R/100)^{n} - If a certain sum becomes “x” times in n years, then the rate of compound interest will be R = 100(x^{1/n} – 1)
- If a sum of money P amounts to A_{1} after T years at CI and the same sum of money amounts to A_{2} after (T + 1) years at CI, then
R = (A_{2} – A_{1})/ A_{1} x 100
➢ Solved Examples on Compound Interest
Question 1: A man invests ₹ 5000 for 3 years at 5% p.a. compounded interest reckoned yearly. Income tax at the rate of 20% on the interest earned is deducted at the end of each year. Find the amount at the end of the third year.
- Here, P = ₹5000, T = 3 years, r = 5%. Therefore, Interest at the end of 1^{st} year = 5000 (1 + 0.05) – 5000 = ₹250
- Now Income tax is 20% on the interest income so the leftover interest income after deducing income tax = (1 – 0.2) * 250 = ₹200
Total Amount at the end of 1^{st} year = ₹5000 + 200 = ₹5200 - Interest at the end of 2^{nd} year = 5200 (1 + 0.05) – 5200 = ₹260
Interest income after Income tax = 0.8 * ₹260 = ₹208
Total Amount at the end of 2^{nd} year = ₹5200 + 208 = ₹5408 - Interest at the end of 3^{rd} year = ₹5408 (1.05) – 5408 = ₹270.4
Interest income after Income tax = 0.8 * ₹270.4 = ₹216.32
Total Amount at the end of 2^{rd} year = ₹5408 + 216.32 = ₹5624.32
Question 2: A sum of ₹12000 deposited at compound interest becomes double after 5 years. After 20 years, it will become?
- The rate of interest at which ₹12000 doubles after 5 years is given by
R = 100(x^{1/n} – 1)
= 100(2^{1/5} – 1)
=100 x (1.1486 – 1)
= 100 x 0.1486 = 14.86% - Therefore, after 20 years it becomes:
A = ₹12000(1 + 14.86/100)^{20}
= ₹12000 (1.1486)^{20}
= ₹12000 x 15.97
= ₹ 191671.474
➢ Compound Interest Installments
Let a person takes a loan from bank at r% and agrees to pay loan in equal installments for n years. Then, the value of each installment is given by
P = X/ (1 + r/100)^{n}………X/ (1 + r/100)^{2} + X/ (1 + r/100)
Example: One can purchase a flat from a house building society for ₹ 55000 or on the terms that he should pay ₹ 4275 as cash down payment and the rest in three equal installments. The society charges interest at the rate 16% p.a. compounded half-yearly. If the flat is purchased under an installment plan, find the value of each installment.
- The cost of the flat is ₹ 55000. Now, if the person could either buy flat by paying ₹55000 or through installment plan. Since the flat was purchased through installment plan then the loan amount = ₹55000 – 4275 (down payment) = ₹50725.
- Here r = 16% compounded Half-yearly in 3 equal instalments. Let x be the amount of installment.
- Then, ₹50725 = x/ (1 + 16/200)^{3} + x/ (1 + 16/200)^{2} + x/ (1 + 16/200)
₹50725 = x (1/1.2591 + 1/1.1664 + 1/1.08)
₹50725 = x (0.79421 + 0.85722 + 0.9259)
₹50725 = x (2.577)
₹50725/2.5777 = x - x = ₹19683