Question

How do I calculate compound interest annually \[?\]

Answer 1

Hint: We can calculate compound interest annually by directly using formula of amount after applying compound interest and then subtract Principal from that amount, by simply knowing the terms which are Principal, Rate of interest and the total time for which the interest is to be calculated.

Complete step by step answer:
First we must know what Compound interest is:
 Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from the simple interest where interest is not added to the principal while calculating the interest during the next period. Compound interest finds its usage in most of the transactions in the banking and finance sectors and also in other areas as well.
Although there is a formula for calculating Amount on which compound interest is applied,
And that is:
\[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]
Where,
\[A=\] Amount
\[P=\] Principal
\[R=\] Rate of interest
\[n=\] Number of times interest is compounded per year
Above formula is the general formula for the number of times the principal is compounded in a year.
But here interest is compounded annually so above formula reduced to,
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}\]
Now after finding the amount after applying compound interest for the given time period we can calculate the compound interest by simply taking the difference of the amount and principal.
Therefore,
\[\text{Compound Interest=Amount-Principal}\]
So this is the one way we can find the compound interest.
Although the above formula one need not be remembered as it simply comes from or derived by simple interest formula for every number of annual times and add it to the principal the total number of times the amount is to be calculated.
So, here we can see the derivation of above formula also,
Let us assume,
\[P=\]Principal
\[n=\]Time (in years)
\[R=\]Rate of interest.
Simple Interest for the first year can be calculated as:
\[\Rightarrow S{{I}_{1}}=\dfrac{P\times R\times T}{100}\]
Amount after first year:
\[=P+S{{I}_{1}}\]
\[=P+\dfrac{P\times R\times T}{100}\]
\[=P\left( 1+\dfrac{R}{100} \right)\]
\[={{P}_{2}}\]
Simple Interest for second year can be calculated the amount of first year will become principle for second year:
\[\Rightarrow S{{I}_{2}}=\dfrac{{{P}_{2}}\times R\times T}{100}\]
Amount after second year:
 \[\begin{align}
  & ={{P}_{2}}+S{{I}_{2}} \\
 & ={{P}_{2}}+\dfrac{{{P}_{2}}\times R\times T}{100} \\
 & ={{P}_{2}}\left( 1+\dfrac{R}{100} \right) \\
 & =P\left( 1+\dfrac{R}{100} \right)\left( 1+\dfrac{R}{100} \right) \\
 & =P{{\left( 1+\dfrac{R}{100} \right)}^{2}} \\
\end{align}\]
Similarly if we proceed further to \[n\] years, we can deduce the expression as:
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]
Compound interest can be calculated as:-
\[\text{Compound Interest=Amount-Principal}\]
\[\Rightarrow CI=A-P\]
\[\therefore CI=P\left[ {{\left( 1+\dfrac{r}{100} \right)}^{n}}-1 \right]\]
This is the second way to calculate compound interest.

Note: Compound interest opens doors to sources of profits for a company. For example, businesses can please investors by earning them higher profits than expected. Financial managers are expected to give dividends to investors, and there are many more applications of compound interest.