Mortgages Analyzed

Continuous Compounding

# Understanding Continuous Compounding

Compound interest is a process where the interest earned in any period is added to the principal and then in subsequent periods the interest is calculated on both the principal and previously earned interest. In short, you earn interest on interest. Continuous compounding is an extreme case of compounding where the compounding period is infinitely small. This causes the interest to be compounded infinitely in any given period. In other words, the time interval between which interest is compounded is infinitely small causing the compounding periods per year (c) to become infinite, which has the effect that interest is compounded continuously.

Source :www.MortgagesAnalyzed.com
Source :www.MortgagesAnalyzed.com

Continuous compounding allows the lender to earn maximum interest amount for a given interest rate. However, it is not generally used for mortgage loans.

# Continuous Compounding Formula

The formula for calculating interest in case of continuous compounding is:

I = P × (ert - 1)

The formula for calculating the amount (future value) based on continuous compounding is:

A = Pert

Source :www.MortgagesAnalyzed.com

Where,

 P = principal amount (Present Value/original loan/investment amount) I = interest amount r = annual interest rate expressed as % per year. Also known as nominal or coupon rate t = time in years (loan term/investment period) e = Exponential constant A = amount (future value)

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# Examples

## Example 1: Single Year

Assume that Amit took a \$100,000 loan on Jan 1, 2000 for one year at an interest rate of 6% per annum compounded continuously. Principal and interest is payable at the end of the loan term.

Where,

 P = \$100,000 r = 6% t = 1

Interest (I) = 100,000 × (e0.06 × 1 - 1) = \$6,183.65

Amount (A) = 100,000 × e0.06 × 1 = \$106,183.65

On Dec 31, 2000, Amit will have to pay \$106,183.65 which includes an interest amount of \$6,183.65

Source :www.MortgagesAnalyzed.com

## Example 2: Multi Year

Assume that Amit took a \$100,000 loan on Jan 1, 2000 for three years at an interest rate of 6% per annum compounded continuously. Principal and interest is payable at the end of the loan term.

Where,

 P = \$100,000 r = 6% t = 3

Interest (I) = 100,000 × (e0.06 × 3 - 1) = \$19,721.74

Amount (A) = 100,000 × e0.06 × 3 = \$119,721.74

On Dec 31, 2000, Amit will have to pay \$119,721.74 which includes an interest amount of \$19,721.74

Source :www.MortgagesAnalyzed.com

Updated: Mar 01, 2013

Continuous compounding is an extreme case of compounding where the compounding period is infinitely small
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