# Understanding Compound Interest

Learning the concept of compound interest is important to achieving financial goals. Retirement planning, as well as investment scenarios, often depends on calculations involving the compounding of interest. It's also important to become skilled at this concept while still young, because time plays a big factor in these calculations too.

In this article, we're going to cover the topic of compound interest. As part of that discussion, we'll first explain what it is, and why it's such a powerful investment concept. Next, we'll provide equations that demonstrate its calculation, including formulas that are available in spreadsheet tools such as Excel. Finally, we'll run through some examples demonstrating how it's used to solve everyday problems, as well as links to an online calculator.

## Compound Interest

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The compounding of interest occurs once interest is added to the principal of an investment. From that point forward, not only does the original principal earn interest, but the interest already received also earns interest. This is an important concept for investors to understand. Let's see how this works with a quick example.

Let's imagine someone was able to invest $2,000 and earn 10% each year on that investment. A 21 year-old that put this $2,000 away until age 65 would have over $132,000 to spend in retirement. A 50 year-old individual investing that same $2,000 until age 65 would only have around $8,300 to spend in retirement.

The above example demonstrates the two points mentioned earlier. Compound interest is an important investment concept to learn, and it's even more powerful when the investor has a lot of time on their side. That's because the dimension of time plays a very important role in this calculation.

### Compound Interest Formula

The most common calculation performed by financial planners when analyzing the impact of compound interest on an investment involves four factors: the future value of the investment, the present value of the investment, the rate of interest earned in each period, and the number of periods. By knowing these four components, the formula below can be used to perform daily, monthly, and annual compound interest calculations. The formula for compound interest is normally expressed as:

FV = PV x (1 +i)^{N}

where:

- FV = future value of the investment
- PV = present value of the investment
- i = rate of interest earned each period
- N = number of periods

### Excel Calculations

Using Microsoft Excel, it's also possible to perform compound interest calculations. The function used in Excel is FV (future value). This function is quite flexible, and the syntax used in Excel is as follows:

=FV(rate,nper,pmt,pv,type)

where:

- rate = the interest rate per period
- nper = the number of payment periods
- pmt = used if an additional payment is made each period (otherwise left blank)
- pv = the present value of the investment
- type (optional) = the number 0 or 1, and is used if additional payments are made whereby 0 = at the end of the period, and 1 = at the beginning of the period

Now that we've seen the two equations used to calculate compound interest, each equation will be put to use in a series of examples.

### Compound Interest Examples

The examples below will demonstrate three concepts. The first has to do with using the compound interest formula to solve a problem involving annual interest rates. The second problem will demonstrate the formula at work when performing monthly compounding. Finally, we'll show how to set up the FV function in Excel to solve a problem.

#### Example 1: Annual Interest Rates

We're going to revisit the quick example given earlier in this article. As a reminder, this involved a 21 year-old that invested $2,000 at 10% and determined the value of that investment at age 65. In this problem, the following factors were used to solve for the investment's future value:

FV = PV x (1 +i)^{N}

where:

- PV = $2,000
- i = 10% per year or 0.10
- N = 65 - 21 or 44 years

= $2,000 x (1 + 0.10)^{44} or = $2,000 x (1.10)^{44} or = $2,000 x 66.2640, or $132,528

#### Example 2: Monthly Interest Rates

The prior example will be expanded to include monthly compounding of interest, not annual. Once a monthly example is understood, the same approach is used for weekly or daily compounding. The concept is exactly the same, only the time period will vary. Once again, a 21 year-old invests $2,000 at 10% compounded monthly, determining the value of that investment at age 65. In this problem, the following factors were used to solve for the investment's future value:

FV = PV x (1 +i)^{N}

where:

- PV = $2,000
- i = 10% per year, compounded monthly
- N = 65 - 21 or 44 years x 12 months

= $2,000 x (1 + 0.10 / 12)^{44x12} or = $2,000 x (1.00833)^{528} or = $2,000 x 79.9793 = $159,959

When compounding monthly, the annual value was divided by 12 (months) to get a monthly interest rate of 0.833% or 0.00833. The number of periods was also adjusted to 44 years x 12 months = 528 months. The key to solving this problem is having a monthly interest rate and the number of periods also stated as months. This same method can be used to calculate daily or weekly compounding.

The monthly compounding result of $159,959 is higher than the annual result of $132,528. That's because a 10% rate of interest that compounds monthly is adding interest to the principal each month, and because we're earning interest on the interest during each compounding interval (month), the future value of the investment will be higher.

#### Example 3: Monthly Interest Rates, Excel Formula

This last example uses Example 2 above, and demonstrates how that problem would be solved in Excel. As a reminder, the syntax is as follows:

=FV(rate,nper,pmt,pv,type)

where:

- rate = 0.10 / 12
- nper = 44*12
- pmt = used if an additional payment is made each period, otherwise left blank
- pv = $2,000
- type (optional) = the number 0 or 1, and is used if additional payments are made whereby 0 = at the end of the period, and 1 = at the beginning of the period

=FV(0.10/12,44*12,,2000) = $159,959

Note that pmt = blank because this problem did not involve adding payments to our original investment of $2,000. It's also possible to omit "type" for that same reason. Of course, the answer found in Excel is exactly the same as Example 2.

### Online Compound Interest Calculator

As promised at the start of this article, we're going to provide a link to our online compound interest calculator. The calculator only needs three inputs: investment, interest rate, and duration. The results include the effective compound interest rates as well as the future values of the investment with compounding. This tool provides daily, weekly, monthly, quarterly, semi-annual as well as annual values for each of these measures.

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