Rate Compounded Annually or Half Yearly: Definition, Formula, Examples

Rate Compounded Annually or Half Yearly: We know that interest is that extra or additional money taken from the borrower over the original amount initially given to the borrower. When we borrow money from a bank or other finance companies, we have to pay interest while returning the sum, i.e., extra money for keeping that money for a specific period.

There are two different ways of calculating interest, i.e., simple interest and compound interest. Interest is the additional money besides the actual money paid by the borrower to the moneylender instead of the money used by him. The principal amount of a loan or deposit is used to calculate simple interest. Compound interest, on the other hand, is calculated using the principal amount and the interest that accumulates on it over time.

What is Interest?

Interest is the additional money paid by the borrower to the moneylender like a bank, financial agency or individual, along with the actual sum of money borrowed in place of the money used by him.

Let us get through with the basic terms that we will use throughout this article.

1. Principal \(\left( P \right):\) The money borrowed or the money lent is called the principal. The amount to be paid back and the interest calculated are all depended on this principle.

2. Amount \(\left( A \right):\) The sum of the principal and the interest is called the amount.

3. Rate of interest \(\left( R \right):\) It is the measure on which the interest for a particular principal is calculated over a period. It is considered as the interest paid on \(₹100\) for a specific period.
(a) The total interest on the principal borrowed or lent depends on the principal sum, the rate of interest, the interest calculating frequency, and the length of time over which it is borrowed or deposited.

4. Time \(\left( T \right):\) It is the time for which money is borrowed.

5. Simple Interest \(\left( {SI} \right):\) It is the interest calculated on the original money or the principal for any given time and rate. The formula to find simple interest is
\(SI = \frac{{{\rm{Principal}} \times {\rm{Rate}} \times {\rm{Time}}}}{{100}}\) or in short, \(SI = \frac{{P \times R \times T}}{{100}}\)

Compound Interest

Compound interest is thought of as interest on interest. It makes a sum of money grow faster than simple interest. In compound interest, the interest calculated for a specific term is reinvested and added to the principal for the next term.

At the end of the first year, if the interest accrued is not paid to the moneylender but is added to the principal, this amount becomes the principal for the next year. This process is repeated until the amount for the whole time is found.

Thus, the difference between the final amount and the original amount is called compound interest or CI.

For example, \(₹2000\) borrowed for \(2\) years at \(8\% \) per annum and compounded annually means the interest at the end of the first year \( = ₹\frac{{2000 \times 1 \times 8}}{{100}} = ₹160.\)
The amount \(\left( {₹2000 + ₹160} \right) = ₹2160\) will be the principal for the next year.
Therefore, interest at the end of the second year \( = ₹\frac{{2160 \times 1 \times 8}}{{100}} = ₹172.80.\)
Amount at the end of the second year \( = ₹2160 + ₹172.80 =₹ 2332.80.\)
Here, total interest gained \(₹160 +₹ 172.80 = ₹332.80.\)

The period for which interest is calculated is called its conversion period. Interest can be compounded yearly, i.e., compounded annually, or half-yearly basis, i.e., twice in a year, or quarterly basis, i.e., four times in a year etc.

In simple interest, the principal remains constant for the whole time, but in compound interest, the principal keeps on changing every year (or any other fixed period).  If the interest is compounded annually, the principal changes after every year and if the interest is compounded half-yearly (or any other fixed period), the principal changes after every six months (or any other fixed period).

The formula to calculate compound interest annually is shown as follows;

Formula for Rate Compounded Annually

The compound interest may be compounded more than once a year. The period and rate of interest are converted accordingly.

The amount after \(T\) years is calculated as
\(A = P{\left( {1 + \frac{R}{{100}}} \right)^T}\)
And the compound interest for this period can be calculated by the formula:
\(CI = A – P = P{\left( {1 + \frac{R}{{100}}} \right)^T} – P\)
Where, \(A = \)Amount at the end of a term
\(P = \) Initial principal
\(R = \) Annual interest rate per cent per annum
\(T = \) Number of years for which the interest is to be calculated

What is Compounded Half-yearly?

If the interest rate is annual and interest is compounded half-yearly, then the annual interest rate is halved, i.e., the rate becomes \(\frac{R}{2}\) and the number of years is doubled, i.e. \(2.\)
When the interest is compounded twice in a year, i.e., half-yearly, then \(T = 2n.\)
Therefore, the formula becomes,
\(A = P{\left( {1 + \frac{{\frac{R}{2}}}{{100}}} \right)^{2n}}\)
The rate is divided by two, and the number of years is multiplied by two as a whole year has two half years.
And the compound interest can be calculated as:
\(CI = P{\left( {1 + \frac{{\frac{R}{2}}}{{100}}} \right)^{2n}} – P\)

If the interest is calculated for any other fixed period (like \(3\) months), then the principal keeps on changing every term of the fixed period (like \(3\) months). The time from one specified interest period to the next period is called a conversion period.

If this specified period is \(1\) year (i.e., the interest is compounded annually), then there is \(1\) conversion period in a year. If this period is \(6\) months (i.e., the interest is compounded semi-annually or half-yearly), then there are \(2\) conversion periods in a year. If this period is \(3\) months (i.e., the interest is compounded quarterly), then there are \(4\) conversion periods in a year. In view of this discussion, we can restate the formula as:
\(A = P{\left( {1 + \frac{r}{{100}}} \right)^n}\)
Where \(A\) is the final amount, \(P\) is the principal, \(r\) is the rate of interest per conversion period, and n is the number of conversion periods.

Learn All the Concepts on Compound Interest

Applications of Compound Interest

There are some situations in our day to day life where we can use the formula of compound interest. Some of those instances of a situation are as follows:
1. Increase or decrease in population, production of an item.
2. Depreciation in values of machines at a given rate.
3. The growth of bacteria when the rate of growth is known. So, the concept of compound interest is applied to growth and depreciation.

Solved Examples

Q.1. Monika borrowed a certain sum at the rate of \(15\% \) per annum. If she paid at the end of two years \(₹2580\) as interest compounded annually, find the sum she borrowed?
Ans: Rate of interest \( = 15\% ,\) Time period \( = 2\) years and Compound interest \(\left( {C.I.} \right) = ₹2580\)
\(CI = P{\left( {1 + \frac{R}{{100}}} \right)^T} – P\)
\( \Rightarrow ₹2580 = P{\left( {1 + \frac{{15}}{{100}}} \right)^2} – P\)
\( \Rightarrow ₹2580 = P\left( {0.3225} \right)\)
\( \Rightarrow P = \frac{{₹2580}}{{0.3225}}\)
\( \Rightarrow P = ₹8000\)
Therefore, the sum borrowed by Monika is \(₹8000.\)

Q.2. Find the compound interest on \(₹10000\) for \(1\) year at \(8\% \) per annum, compounded half-yearly.
Ans: Principal amount \( = ₹10000\)
Rate of interest \( = 8\% \)
Time \( = 1\) year \( = 2\) half-years
We know that compound interest for half-yearly is \(P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2n}}\)
\( = 10000{\left( {1 + \frac{8}{{200}}} \right)^2}\)
\( = ₹10816\)
Hence, the compound interest \(CI = A – P = ₹10816 – ₹10000 = ₹816\)

Q.3. Find the amount and the compound interest on \(₹16000\) for \(1\frac{1}{2}\) years at \(10\% \) per annum, the interest is compounded half-yearly.
Ans: Since the rate of interest is \(10\% \) per annum, the rate of interest half-yearly \( = \frac{1}{2}\) of \(10\% = 5\% \)
Principal for the first half-year \(=₹16000\)
Interest for the first half-year \( =₹ \frac{{16000 \times 5 \times 1}}{{100}} = ₹800\)
Amount after the first half-year \( = ₹16000 + ₹800 = ₹16800\)
Principal for the second half-year \( = ₹\frac{{16800 \times 5 \times 1}}{{100}} = ₹840\)
Amount after one year \( = ₹16800 + ₹840 = ₹17640\)
Principal for the third half-year \( = ₹\frac{{17600 \times 5 \times 1}}{{100}} = ₹882\)
Amount after \(1\frac{1}{2}\) years \( = ₹17640 + ₹882 = ₹18522\)
Therefore, compound interest for \(1\frac{1}{2}\) years \(=\) Final amount \(-\) Original principal
\( = ₹18522 – ₹16000 = ₹2522\)

Q.4. At what rate of interest per annum will \( ₹10000\) amount to \(₹12100 \) in two years if the interest is compounded annually?
Ans: Given, \(P = ₹10000,\,A = ₹12100\) and \(T = 2\) years.
Now, let \(R\) be the rate of interest.
\(A = P{\left( {1 + \frac{R}{{100}}} \right)^T}\)
\(₹12100 = ₹10000{\left( {1 + \frac{R}{{100}}} \right)^2}\)
\(\frac{{12100}}{{10000}} = {\left( {1 + \frac{R}{{100}}} \right)^2}\)
\(\frac{{121}}{{100}} = {\left( {1 + \frac{R}{{100}}} \right)^2}\)
\({\left( {\frac{{11}}{{10}}} \right)^2} = {\left( {1 + \frac{R}{{100}}} \right)^2}\)
\(\frac{{11}}{{10}} = 1 + \frac{R}{{100}}\)
\(\frac{{11}}{{10}} – 1 = \frac{R}{{100}}\)
\(\frac{{11 – 10}}{{10}} = \frac{R}{{100}}\)
\(\frac{1}{{10}} = \frac{R}{{100}}\)
\(R = 10\)
Therefore, the rate of interest is \(10\% \) per annum.

Q.5. Find the amount which Kiara would get on \(₹8912,\) if she invested it for 18 months at \(12\frac{1}{2}\% ,\) per annum, interest being compounded half-yearly.
Ans: Principal \( = ₹8912,\) rate \( = 12\frac{1}{2}\% \) per annum \( = \frac{{25}}{4}\% \) half-yearly and time \( = 18\) months or \(T = 3.\)
Now, using the formula \(A = P{\left( {1 + \frac{R}{{100}}} \right)^T}\)
We have, \(A = ₹8192{\left( {1 + \frac{{25}}{4} \times \frac{1}{{100}}} \right)^3}\)
\( = ₹8192 \times {\left( {1 + \frac{1}{{16}}} \right)^3} = 8192{\left( {\frac{{17}}{{16}}} \right)^3}\)
\( = ₹\frac{{8192 \times 17 \times 17 \times 17}}{{16 \times 16 \times 16}}\)
\( = ₹2 \times 17 \times 17 \times 17 = 9826\)
Thus, Kiara will get \(₹9826\) after \(18\) months.

Summary

In this article, we learnt the concept of finding compound interest with the help of its definition and then further made our concept stronger by taking an example. In addition to this, we learnt to find the compound interest when compounded annually and half-yearly. To master the concepts further, we solved some examples where we compounded the interest annually and half-yearly.

Learn Important Compound Interest Formula

FAQs

We have provided some frequently asked questions here:

Q.1.How many years is compounded annually?
Ans: Interest for any number of years can be compounded annually.

Q.2. What is the simple interest rate formula?
Ans: Simple interest is calculated on the original money or the principal for any given time and rate. The formula to find simple interest is
\(SI = \frac{{{\rm{Principal}} \times {\rm{Rate}} \times {\rm{Time}}}}{{100}}\) or in short, \(SI = \frac{{P \times R \times T}}{{100}}\)

Q.3. Define the concept of compound interest.
Ans: Compound interest is thought of as interest on interest. It makes a sum of money grow faster than simple interest. In compound interest, the interest calculated for a specific term is reinvested and added to the principal for the next term.
At the end of the first year, if the interest accrued is not paid to the moneylender but is added to the principal, this amount becomes the principal for the next year. This process is repeated until the amount for the whole time is found.
Thus, the difference between the final amount and the original amount is called compound interest or CI.

Q.4. What is the formula of compounded annually?
Ans: In, \(A = P{\left( {1 + \frac{R}{{100}}} \right)^T},\) where, \(A = \) Amount at the end of a term
\(P = \) Initial principal
\(R = \) Annual interest rate per cent per annum
\(T = \) Number of years for which the interest is to be calculated.

Q.5. How do you calculate compounded annually?
Ans: If the interest is compounded annually or yearly, the interest calculated for the first year is added to the principal and used as the principal for the next year. The difference between the final amount and the original principal gives the compound interest.

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