Quarterly Compound Interest Formula
Before learning the quarterly compound interest formula, let us recall a few things about the compound interest. In compound interest, the interest for every period is calculated on the amount for the previous period. This means the amount for the previous time period becomes the principal for the current time period. "Compounding" means adding interest to the current principal amount. The amount can be compounded either daily, weekly, monthly, quarterly, half-yearly, or yearly. In compound interest, the formula for the final amount is:
A = P (1 + r / n)n t
Here,
- P = the principal amount
- r = rate of interest
- t = time in years
- n = number of times the amount is compounding.
What Is Quarterly Compound Interest Formula?
When the amount compounds quarterly, it means that the amount compounds 4 times in a year. i.e., n = 4. We use this fact to derive the quarterly compound interest formula. Thus, the quarterly compound interest formula is:
A = P (1 + r/4)4t
Here, A is the total amount (Principal + Interest). Let us see the applications of the quarterly compound interest formula in the following section.
Solved Examples Using Quarterly Compound Interest Formula
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Example 1: You have invested $1000 in a bank where your amount gets compounded quarterly at 5% annual interest. Then what is the amount you get after 10 years?
Solution:
To find: The amount after 10 years.
The principal amount is, P = $1000.
The rate of interest is, r = 5% =5/100 = 0.05.
The time in years is, t = 10.
Using the quarterly compound interest formula:
A = P (1 + r / 4)4 t
\[ \begin{align} A &= 1000 \left( 1+ \dfrac{0.05}{4}\right)^{4 \times 10}\\[0.2cm] A &=\$ 1643.62 \end{align}\]
Answer: The amount after 10 years = $1643.62.
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Example 2: How long does it take for $15000 to double if the amount is compounded quarterly at 10% annual interest? Round your answer to the nearest integer.
Solution:
To find: The time taken for $15000 to double.
The principal amount is, P = $15000.
The rate of interest is, r = 10% =10/100 = 0.1.
The final amount is, A = 15000 x 2 = $30000
Let us assume that the required time in years is t.
Using the quarterly compound interest formula:
A = P (1 + r / 4)4 t
\[ \begin{align} 30000& = 15000 \left( 1+ \dfrac{0.1}{4}\right)^{4t}\\[0.2cm]
\text{Dividing } &\text{ both sides by 15000,}\\[0.2cm]
2 &=(1.025)^{4t}\\[0.2cm]
\text{Taking } \ln &\text{ on both sides}\\[0.2cm]
\ln 2 &= 4t \ln 1.025\\[0.2cm]
t &= \dfrac{\ln 2}{4\ln 1.025}\\[0.2cm]
t&= 7
\end{align}\]Answer: It takes 7 years for $15000 to become double.
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