How to Apply the Quarterly Compound Interest Formula in Exams
Compound interest is interest that builds up over a set length of time on both principal and interest. The principal is also used to account for the interest that has accrued on a principal over time. It is the idea that we employ the most frequently on a daily basis. It is the new way of calculating interest now utilized in all international financial and commercial operations. Some of its major applications are:
Population Growth or Decline.
Growth of Bacteria
Increase or Decrease in an Item's Value.
Terminology of Compound Interest
Principal (P): The amount of money lent for a specific amount of time at a specific interest rate.
Time (t): It is the length of time the principal is lent, commonly expressed in years.
Interest (I): It is the return on an investment made by lending a principal for a specific amount of time.
Rate(r): It is the percentage of interest received for a loan of a certain amount.
Amount (A): The total sum of money remaining at the end is the amount. It is the total of the initial principal and all compound interest that has been earned.
Compound Interest Equation
Following the computation of the total amount over a period of time using the initial principal and the interest rate, the compound interest is determined. The formula for calculating the amount is given below for an initial principal of P, an annual interest rate of r, a time period of t in years, and a frequency of n times the interest is compounded annually.
Compound Interest: \[CI\, = \,P\,{\left( {1 + \frac{R}{{100}}} \right)^t} - P\]
Amount: \[A = P{(1 + \dfrac{r}{n})^{nt}}\]
What does Compounded Quarterly Mean?
Different formulas can be used to calculate compound interest for a given principal over various time periods. If the interest calculation period is quarterly, the sum is compounded four times a year and the interest is calculated every three months. The money left over after the first three months will be used to compute interest for the subsequent three months (second quarter). Additionally, interest will be computed for the third quarter on the amount left over after the first six months and for the final quarter on the amount left over after the first nine months. Thus, the following is the quarterly compound interest formula:
Compounded quarterly equation: \[C.I\, = \,P\,{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}} - \,P\]
How to Calculate Quarterly Compound Interest?
When the amount compounds every three months, it indicates that it does so four times per year. i.e., n = 4.We will learn how to calculate compound interest quarterly by solving the following examples:
1.Find the compound interest when Rs 100000 is invested for 9 months at 6% per annum, compounded quarterly.
Explanation: Here principal (P) = Rs 100000
Rate of interest (r) = 6%
Time (n)=\[\frac{9}{{12}} = \frac{3}{4}\] year.
Therefore, the amount of money accumulated for n years=
\[A\, = \,P\,{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4n}}\]
\[ = \,100000\,{\left( {1 + \frac{{\frac{8}{4}}}{{100}}} \right)^{4\, \times \frac{3}{4}}}\]
\[ = 100000{\left( {1 + \frac{2}{{100}}} \right)^3}\]
\[ = 100000{\left( {\frac{{51}}{{50}}} \right)^3}\]
\[ = 100000 \times \frac{{51}}{{50}} \times \frac{{51}}{{50}} \times \frac{{51}}{{50}}\]
\[ = 106120.8\]
Therefore, Compound Interest = Total Amount - Principal
= 106120.8 – 100000
= Rs 6120.8
Summary
The profit made from lending money is known as interest. It is always calculated using a specific interest rate for a specific amount of time. In compound interest, the principal (amount on which interest is calculated) is renewed each year, and compound interest is calculated in the same way as simple interest. Adding interest to the current principal sum is referred to as "compounding." We can calculate compound interest weekly, monthly, quarterly, half-yearly or yearly.
Solved Questions
1. Find the amount and the compound interest on Rs 1, 20,000 compounded quarterly for 9 months at the rate of 4% per annum.
Ans: Here Principal (p) = Rs 1, 20,000
Rate of interest = 4%
Time = 9 months which will be \[\frac{9}{{12}} = \frac{3}{4}\] years
Hence, Amount will be \[A\, = \,P\,{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4n}}\]
\[ = \,120000\,{\left( {1 + \frac{{\frac{4}{4}}}{{100}}} \right)^{4\, \times \frac{3}{4}}}\]
\[ = 120000{\left( {1 + \frac{1}{{100}}} \right)^3}\]
\[ = 120000{\left( {\frac{{101}}{{100}}} \right)^3}\]
\[ = 123636.12\]
Therefore, Compound Interest = Total Amount – Principal
= 1, 23,636.12 – 1, 20,000
= Rs 3636.12
2. If Rs 1200 is invested at a compound interest rate 8% per annum compounded quarterly for 12 months, find the compound interest.
Ans: Here, Principal (p) = Rs 1200
Rate of interest = 8 %
Time = \[\dfrac{{12}}{{12}} = 1\] years
Hence, amount on the accumulated sum will be:
\[ = \,1200\,{\left( {1 + \dfrac{{\frac{8}{4}}}{{100}}} \right)^{4\, \times 1}}\]
\[ = 1200{\left( {1 + \frac{2}{{100}}} \right)^4}\]
\[ = 1200{\left( {\frac{{51}}{{50}}} \right)^4}\]
\[ = 1298.91\]
Therefore, Compound Interest = Total Amount – Principal
= 1298.91-1200
=Rs 98.91
3. What is the compound interest (CI) on Rs.6000 for 1 years at 12% per annum compounded annually?
Ans: Here Principal (P) = 6000
Rate of interest: 12 %
Time: 1 year
Hence, amount on the given sum will be:
\[ = \,6000\,{\left( {1 + \frac{{\frac{12}{4}}}{{100}}} \right)^{4\, \times \frac{1}{1}}}\]
\[ = 6000{\left( {1 + \frac{3}{{100}}} \right)^4}\]
\[ = 6000{\left( {\frac{{103}}{{100}}} \right)^4}\]
\[ = 6753.05\]
Hence, C.I = Amount – Principal
= 6753.05 – 6000
= Rs 735.05
FAQs on Compound Interest Quarterly Formula Explained
1. What is the specific formula used to calculate compound interest when it is compounded quarterly?
The formula to calculate the total amount (A) when interest is compounded quarterly is:
A = P (1 + (r/4)/100)4t or A = P (1 + R/4)4t, where R is the rate in decimal form (r/100).
Here are the components:
- P is the Principal amount (the initial sum of money).
- r is the annual interest rate in percent.
- t is the time period in years.
- The interest rate is divided by 4 because there are four quarters in a year.
- The time is multiplied by 4 to find the total number of compounding periods.
2. Can you provide a simple example of how to use the compound interest quarterly formula?
Certainly. Let's calculate the compound interest on Rs. 20,000 for 1 year at an annual interest rate of 8%, compounded quarterly.
Given:
- Principal (P) = Rs. 20,000
- Annual Rate (r) = 8% or 0.08
- Time (t) = 1 year
1. Adjust the rate for quarterly compounding: r/4 = 8%/4 = 2% per quarter.
2. Adjust the time for quarterly periods: 4t = 4 * 1 = 4 periods.
3. Use the formula: A = 20000 * (1 + 0.02)4
4. A = 20000 * (1.02)4 ≈ 20000 * 1.0824 = Rs. 21,648.64
5. Calculate Compound Interest: CI = A - P = 21,648.64 - 20,000 = Rs. 1,648.64.
3. If interest is 'compounded quarterly', does it mean it is calculated 3 times or 4 times a year?
This is a common point of confusion. 'Compounded quarterly' means the interest is calculated and added to the principal four times per year. A year has four quarters, and each quarter consists of three months. So, while a quarter is a 3-month period, the compounding action happens at the end of each of these four periods within a year.
4. How does the quarterly compound interest formula differ from the half-yearly formula?
The core difference between the quarterly and half-yearly formulas lies in how the annual interest rate (r) and the time period (t) are adjusted. This adjustment is based on the number of compounding periods in a year.
- For Quarterly Compounding, there are 4 periods per year. The rate is divided by 4 (r/4) and the time is multiplied by 4 (4t). The formula is A = P(1 + r/4)4t.
- For Half-Yearly Compounding, there are 2 periods per year. The rate is divided by 2 (r/2) and the time is multiplied by 2 (2t). The formula is A = P(1 + r/2)2t.
5. Why is it generally more beneficial for an investor to have their interest compounded quarterly rather than annually?
It is more beneficial because of the power of more frequent compounding. When interest is compounded quarterly, the interest earned is added back to the principal four times a year. This means that for each subsequent quarter, you earn 'interest on interest' on a slightly larger principal amount. Compounding annually only does this once a year. Over time, this frequent recalculation results in a higher total amount and more earnings for the investor compared to annual compounding with the same principal and annual rate.
6. How do you correctly set the time 't' in the quarterly formula if the duration is given in months, like 9 or 18 months?
The variable 't' in the formula must always be in years. If the time is given in months, you must first convert it to years by dividing the number of months by 12.
- For 9 months: t = 9 / 12 = 0.75 years. The exponent in the formula would be 4t = 4 * 0.75 = 3.
- For 18 months: t = 18 / 12 = 1.5 years. The exponent in the formula would be 4t = 4 * 1.5 = 6.
7. What are some real-world applications where understanding the quarterly compound interest formula is important?
Understanding the quarterly compound interest formula is crucial in many personal finance and economic scenarios. Key applications include:
- Bank Savings Accounts & Fixed Deposits (FDs): Many banks in India compound interest on savings accounts and FDs on a quarterly basis.
- Loans: Certain loans, especially in the private financial sector, may compound interest quarterly, affecting the total repayment amount.
- Investments: Evaluating the true return on investment products like bonds or certain mutual funds that may use quarterly compounding.
- Economic Growth Projections: Economists may use similar principles to model short-term economic growth or inflation rates.
