Compound Interest

Explanation

Compound interest is paid both on the original principal and on the accumulated past interest.

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Formula symbols

 S Final value of investment P Initial value of investment i Interest rate per period n Number of interest periods

At start

S   =   P

Final value equals initial value.

After 1 period

S   =   P + Pi   =   P(1 + i)

Amount at the start of this period

Amount of interest earned during this period on the starting amount of this period

[Algebra: Factor P out of two terms]

Final value at end of this period

After 2 periods

S   =   P(1 + i) + P(1 + i)i   =   P(1 + i)(1 + i)   =   P(1 + i)2

Amount at the start of this period, that is, the final value at the end of the last period

Amount of interest earned during this period on the starting amount of this period

[Algebra: Factor P(1 + i) out of two terms]

Final value at end of this period

After 3 periods

S   =   P(1 + i)2 + P(1 + i)2i   =   P(1 + i)2(1 + i)   =   P(1 + i)3

Amount at the start of this period, that is, the final value at the end of the last period

Amount of interest earned during this period on the starting amount of this period

[Algebra: Factor P(1 + i)2 out of two terms]

Final value at end of this period

After n periods

S   =   P(1 + i)n

Often the interest per period, i, is expressed in terms of the annual percentage rate (APR), r, and the number of interest periods per year, k.

Under these conditions the interest per period is equal to the annual percentage rate divided by the number of interest periods per year, as in:

i = r / k

This would make the above formula for the final value of an investment after n interest periods look like this:

S = P(1 + r/k)n

Notice that the output, S, is an exponential function of n. That is, if we consider the final value of the investment as a function of the length of time for the investment, then n, the length of time for the investment, is in the exponent position, and this makes S an exponential function of n.

Example calculation

If \$4000 is invested at an annual rate of 6.0% compounded monthly, what will be the final value of the investment after 10 years? So, P = 4000 and r = 0.06.

Since the interest is compounded monthly, there are 12 periods per year, so, k = 12.

Since the investment is for 10 years, or 120 months, there are 120 investment periods, so, n = 120.

S = P(1 + r/k)n

S = 4000(1 + 0.06/12)120

S = 4000(1.005)120

S = 4000(1.819396...)

S = \$7277.59

Calculator for compound interest

S = P(1 + r/k)n

 S Final value of investment P Initial value of investment r Annual percentage rate (APR) k Interest periods per year i = r/k Percentage rate per interest period n Number of interest periods

Enter values below for the above formula. When this page loads, the calculator is set up for a \$100 initial investment earning 5.0% (0.05) APR compounded monthly for two years.

(Example: For r enter 5.0% as 0.05, etc.)

P: r: k: n:

After entering values into the above input areas, click the following 'Calculate' button to get S, the final value of the investment.

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