Compounding

With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding.

compounding

Compounding refers to the process where the interest earned on an investment or the interest charged on a loan is added to the principal amount, so that interest can be earned or charged on a larger amount over time.

Suppose that we deposit [latex]$1000[/latex] in a bank account offering [latex]3\%[/latex] interest, compounded monthly. How will our money grow?

The [latex]3\%[/latex] interest is an annual percentage rate (APR) – the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\frac{3\%}{12}= 0.25\%[/latex] per month.

In the first month,

• [latex]P_0 = $1000[/latex]
• [latex]r = 0.0025 (0.25\%)[/latex]
• [latex]I= $1000 (0.0025) = $2.50[/latex]
• [latex]A = $1000 + $2.50 = $1002.50[/latex]

In the first month, we will earn [latex]$2.50[/latex] in interest, raising our account balance to [latex]$1002.50[/latex].

In the second month,

• [latex]P_0 = $1002.50[/latex]
• [latex]I= $1002.50 (0.0025) = $2.51[/latex] (rounded)
• [latex]A = $1002.50 + $2.51 = $1005.01[/latex]

Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original [latex]$1000[/latex] we deposited, but we also earned interest on the [latex]$2.50[/latex] of interest we earned the first month. This is the key advantage that compounding interest gives us.

Calculating out a few more months gives the following:

Month	Starting balance	Interest earned	Ending Balance
[latex]1[/latex]	[latex]1000.00[/latex]	[latex]2.50[/latex]	[latex]1002.50[/latex]
[latex]2[/latex]	[latex]1002.50[/latex]	[latex]2.51[/latex]	[latex]1005.01[/latex]
[latex]3[/latex]	[latex]1005.01[/latex]	[latex]2.51[/latex]	[latex]1007.52[/latex]
[latex]4[/latex]	[latex]1007.52[/latex]	[latex]2.52[/latex]	[latex]1010.04[/latex]
[latex]5[/latex]	[latex]1010.04[/latex]	[latex]2.53[/latex]	[latex]1012.57[/latex]
[latex]6[/latex]	[latex]1012.57[/latex]	[latex]2.53[/latex]	[latex]1015.10[/latex]
[latex]7[/latex]	[latex]1015.10[/latex]	[latex]2.54[/latex]	[latex]1017.64[/latex]
[latex]8[/latex]	[latex]1017.64[/latex]	[latex]2.54[/latex]	[latex]1020.18[/latex]
[latex]9[/latex]	[latex]1020.18[/latex]	[latex]2.55[/latex]	[latex]1022.73[/latex]
[latex]10[/latex]	[latex]1022.73[/latex]	[latex]2.56[/latex]	[latex]1025.29[/latex]
[latex]11[/latex]	[latex]1025.29[/latex]	[latex]2.56[/latex]	[latex]1027.85[/latex]
[latex]12[/latex]	[latex]1027.85[/latex]	[latex]2.57[/latex]	[latex]1030.42[/latex]