Daily Compound Interest Formula
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.
What is Daily Compound Interest Formula?
The daily compound interest formula is where the interest is calculated 365 times in a year, hence the value of n is 365. By the given explanation, the daily compound interest formula 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.
When the amount compounds daily, it means that the amount compounds 365 times in a year. i.e., n = 365.
Where
 A is the total amount (A = Principal + Interest).

Example 1: You have invested $1000 in a bank where your amount gets compounded daily at an interest rate of 5%. Then what is the amount you get after 10 years? Calculate this by using the daily compound interest formula.
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 daily compound interest formula is:
A = P (1 + r / 365)^{365 t}
A = 1000 ( 1+ 0.05/365)^{365×10}
A =$1648.66
Answer: The amount after 10 years = $1648.66.

Example 2: How long does it take for $15000 to double if the amount is compounded daily at 10% annual interest? Calculate this by using the daily compound interest formula and round your answer to the nearest integer.
Solution:
To find: The time taken for $15000 to double.
The principal amount, 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 daily compound interest formula is:
A = P (1 + r / 365)^{365 t}
30000 = 15000 (1 + 0.1 / 365)^{365 t}
Dividing both sides by 15000,
2 =(1.0002739)^{365 t}
Taking ln on both sides
ln 2 = 365 × t × ln 1.0002739
t = ln 2/(365 ln 1.0002739)
t= 7Answer: It takes 7 years for $15000 to become double.
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