Mortgages Analyzed
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Compound Interest

What is Compound Interest?

Compound Interest

Compound Interest is a method of interest calculation where the periodic interest is calculated as a percentage of the principal amount and accrued interest. Interest earned to date is added to the principal amount to calculate future interest. With compound interest the interest is earned on interest and the returns are magnified for the lender.

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Formula for Calculating Compound Interest

Formulae

Compound Interest is calculated using the formula below.

I = PV × [(1+i)^n - 1 ]

The formula for calculating the Future Value of an amount payable in single installment is below.

FV = PV × (1+i)^n
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Where,

P  =  principal amount (original loan/investment amount). When using compound interest, it is usually denoted as PV
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)
n  =  number of periods. It equals time in years × compounding per year. (t × c)
c  =  compounding periods per year
i  =  periodic interest rate (r/c)
PV  =  Present Value
FV  =  Future Value or balance at the end of the period

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Relation with Simple Interest

Compound interest formula is derived from simple interest. In fact, compound interest is nothing but combination of series of simple interest calculations where the principal at the beginning of each period is the sum of original principal and accrued interest.

Let us take an example, assume that you took a $100,000 loan on Jan 1, 2001 for three years at an interest rate of 6% per annum compounded annually.

Where,

P  =  $100,000
r  =  10%
t  =  3 years
t  =  time in years (loan term/investment period)
i  =  10%
n  =  3

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At the end of period 1 (Dec 31, 2001):

Interest (I) = p × r × t = 100,000 × 0.10 × 1 = 10,000

Total Amount = Principal + Interest = 100,000 + 10,000 = 110,000

Or

Total Amount = P × (1 + i) = 100,000 × (1 + 0.10) = 110,000

At the end of period 2 (Dec 31, 2002):

Interest (I) = p × r × t = 110,000 × 0.10 × 1 = 11,000

Total Amount = Principal + Interest = 100,000 + 11,000 = 121,000

Or

Total Amount = P × (1 + i) × (1 + i) = 100,000 × (1 + 0.10) × (1 + 0.10) = 121,000

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At the end of period 3 (Dec 31, 2003):

Interest (I) = p × r × t = 121,000 × 0.10 × 1 = 12,100

Total Amount = Principal + Interest = 121,000 + 12,100 = 133,100

Or

Total Amount = P × (1 + i) × (1 + i) × (1 + i) = P × (1 + i)^n = 133,100

Similarly, we can calculate the future value for any amount using the formula PV × (1 + i)^n

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 annually. Principal and interest is payable at the end of the loan term.

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Where,

P  =  $100,000
r  =  6%
t  =  1
i  =  6%
n  =  1
c  =  1

Interest (I) = 10,000 × [(1 + 0.06)^1 - 1] = $6,000

Future Value (FV) = 100,000 × [(1 + 0.06)^1] = $106,000

On Dec 31, 2000, Amit will have to pay $106,000 which includes an interest amount of $6,000.

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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 annually. Principal and interest is payable at the end of the loan term.

Where,

P  =  $100,000
r  =  6%
t  =  3
i  =  6%
n  =  3
c  =  1

Interest (I) = 100,000 × [(1 + 0.06)^3 - 1] = $19,101.60

Future Value (FV) = 100,000 × [(1 + 0.06)^3] = $119,101.60

On Dec 31, 2002, he will have to pay $119,101.60 which includes an interest amount of $19,101.60.

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Example 3: Monthly Compounding

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

Where,

P  =  $100,000
r  =  6%
t  =  1
n  =  12
c  =  12
i  =  6%/12 = 0.5%

Interest (I) = 100,000 × [(1 + 0.005)^12 - 1] = $6,167.78

Future Value (FV) = 100,000 × [(1 + 0.005)^12] = $106,167.78

On Dec 31, 2000, Amit will have to pay $106,167.78 which includes an interest amount of $6,167.78.

In this example, if we change the loan term to 3 years then the interest $19,668.05 will be and the future value $119,668.05, calculated below.

Interest (I) = 100,000 × [(1 + 0.005)^36 - 1] = $19,668.05

Future Value (FV) = 100,000 × [(1 + 0.005)^36] = $119,668.05

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Example 4: Daily Compounding

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

Where,

P  =  $100,000
r  =  6%
t  =  1
n  =  365
c  =  365
i  =  6%/365 = 0.016438356%

Interest (I) = 100,000 × [(1 + 0.00016438356)^365 - 1] = $6,183.13

Future Value (FV) = 100,000 × [(1 + 0.005)^365] = $106,183.13

On Dec 31, 2000, Amit will have to pay $106,183.13 which includes an interest amount of $6,183.13.

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Example 5: Monthly Compounding With Monthly Interest Payment

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

Where,

P  =  $100,000
r  =  6%
t  =  1
n  =  12
c  =  12
i  =  6%/12 = 0.5%

Periodic Interest (I) = 100,000 × [(1 + 0.005)^1 - 1] = $500

Total Interest = 500 × 12 = $6000

Future Value (FV) at the end of each month = $100,000 (100,000 + 500 – 500)

Amit will have to pay $500 at the end of each month and on Dec 31, 2000 he would repay the principal amount of $100,000.

In this example the periodic interest amount would be same if the interest rate was stated as simple interest. This is because interest is not recapitalized. It is being paid out and no compounding takes place.

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Example 6: Loan Term Less Than Compounding Period

Assume that Amit took a $100,000 loan on Jan 1, 2000 for three years at an interest rate of 6% per annum compounded monthly. Principal and interest is payable at the end of the loan term. However, Amit decides to repay the loan in 15 days.

Where,

P  =  $100,000
r  =  6%
t  =  3
n  =  12 × 3 = 36
c  =  12
i  =  6%/12 = 0.5%

In this example the periodic interest rate will be prorated by the number of days the loan was held.

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Periodic Interest (I) = 100,000 × [(1 + (0.005 × (15/30)))^1 - 1] = $250

Future Value (FV) = 100,000 × [(1 + (0.005 × (15/30)))^1] = $100,250

On Jan 15, 2000, Amit will have to pay $100,250 which includes an interest amount of $250.

A much simpler way to solve this problem is to recognize that all that needs to be done is to calculate simple interest during the period of the loan. This is because the loan was repaid before the interest gets compounded and during the interim period the interest is calculated based on simple interest. Therefore, on Jan 15, 2000 Amit will have to repay $100,250 comprising of the principal amount of $100,000 and interest portion of $250.

Interest (I) = 100,000 × 6%/12 × 15/30 = $250 (or simply $100,000 × 6% × 15/360)

Amount (A) = 100,000 + 250 = $100,250

We are not using 15/365 in this problem because interest is compounded based on monthly compounding which assumes that each month is equal (30 days) and the year has 360 days.

This example also shows that we can solve any problem using the compound interest formula. Simple interest calculations are nothing but simplified versions of compound interest calculations.

Source :www.MortgagesAnalyzed.com

Updated: Mar 01, 2013

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With compound interest the interest is earned on interest and the returns are magnified for the lender
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