Continuous interest is when interest is continuously reinvested.

The formula for calculating the accumulated value via continuous interest is as follows:

A = Pe^rt

where A is the accumulated value, P is Principal, r is the annual interest rate, t is time in years.

Example 1  Calculate the interest earned from a $4,500 loan compounded continuously at 5% for 6 years.

Here, P = $4,500, r = 5% = .05, and m = 4 (quarterly).  Over 6-years, there are n = 4*6 = 24 quarters.  The accumulated value is:

A = (4500)[1 + .05/4]²⁴ = 6063.08

I = A - P = 6063.08 - 4500 = 1563.08

Example 2  A savings account pays 4% interest and is compounded daily (365 days).  When will the accumulated value be twice the original principal?

We are solving for n:

2P = P( 1 + .04/365 )n

log 2 = n log (1 + .04/365)

n = log 2 / log (1 + .04/365) = 6325.32

n is the number of periods which for this example, is in days.  It will take 6325 days to double the principal, which is roughly 6325/365 = 17.32 years.

The online calculator below calculates simple interest.

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