Compound Interest

Compound Interest is a method of interest calculation where the periodic interest is a percentage of the principal amount and accrued interest. In case of compound interest, the interest is added to the principal amount to calculate future interest. Therefore, interest will earn interest and magnify the returns for the lender.

Formulae

The formula for calculating compound interest is:

 I = PV × [(1+i)^n - 1 ]

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

 FV = PV × (1+i)^n

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 multiplied by 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

Formula Derivation (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 Calculation, 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 Amit took a \$100,000 loan on Jan 1, 2000 for three years at an interest rate of 6% per annum compounded annually.

Where,

P = 100,000
r = 10%
t = 3 years
i = 10%
n = 3

At the end of period 1:

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:

Interest (I) = p × r × t = 110,000 × 0.10 × 1 = 11,000
Total Amount = Principal + Interest = 110,000 + 11,000 = 121,000
Or Total Amount = P × (1 + i) × (1 + i) = 121,000

At the end of period 3:

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.

Where,

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

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

Interest (I) = 100,000 × [(1 + 0.06)^1 - 1] = \$6,000
Future Value (FV) = \$100,000 × [(1 + 0.06)^1] = \$106,000

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,

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

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

Interest (I) = 100,000 × [(1 + 0.06)^3 - 1] = \$19,101.60
Future Value (FV) = \$100,000 × [(1 + 0.06)^3] = \$119,101.60

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,

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

On Dec 31, 2000, Amit will have to pay \$106,136 which includes an interest amount of \$6,136.

Interest (I) = 100,000 × [(1 + 0.005)^12 - 1] = \$6,167.7812
Future Value (FV) = \$100,000 × [(1 + 0.005)^12] = \$106,167.7812

In this case, if the loan term was 3 years then the interest and future value would be \$19,668.05 and \$119,668.05 respectively.

Interest (I) = 100,000 × [(1 + 0.005)^36 - 1] = \$19,668.0525
Future Value (FV) = \$100,000 × [(1 + 0.005)^36] = \$119,668.0525

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,

PV = \$100,000
r = 6%
t = 1
i = 6%/365 = 0.00016438356
n = 365
c = 365

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

Interest (I) = 100,000 × [(1 + 0.00016438356)^365 - 1] = \$6,183.1310
Future Value (FV) = \$100,000 × [(1 + 0.00016438356)^365] = \$106,183.1310

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,

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

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.

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)

In this case the periodic interest amount would be same if the interest rate was stated as simple interest. This is because the before interest is recapitalized, it is being paid out and no compounding takes place.

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 problem the periodic interest rate will be prorated by the number of days the loan was held. On Jan 15, 2000, Amit will have to pay \$100,250 which includes an interest amount of \$250

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

A much simpler way to solve this problem is to recognize that all that is need 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.