# Compound Interest

Guide to Understanding the Compound Interest Concept

• What is the definition of compound interest?
• How are compound interest and simple interest different?
• What are the components of the compound interest formula?
• How does the number of compounding periods impact compound interest?

## How to Calculate Compound Interest

The notion of compound interest can be described as earning “interest on interest.”

Compound interest stems from growth in the principal amount from the accumulation of interest, resulting in more interest being received (i.e. “interest on interest”).

Compound interest is when interest is earned on two components:

1. Original Principal – Initial Amount Invested, Borrowed, or Lent
2. Accumulated Interest – Interest from Earlier Periods

The accumulated interest is added to the principal amount, which subsequently determines the interest amount in the next period in a continuous cycle until the end of the term.

Therefore, even with a low-interest rate, the effects of compounding can cause the principal to grow substantially over a long time horizon.

## Compounding Frequency Chart

Compounding is a central piece of the decision-making process by investors, borrowers, and lenders.

The rate at which compound interest accumulates is a function of the frequency of compounding periods.

The greater the number of compounding periods, the greater the effects of compound interest (i.e. the “snowball effect”).

## Compound Interest Formula

The formula for calculating the future value of an interest-earning financial instrument with the effects of compound interest is shown below:

###### Future Value (FV) Formula
• Future Value (FV) = PV [1 + (r ÷ n)] ^ (n × t)

Where:

• PV = Present Value
• r = Interest Rate (%)
• t = Term in Years
• n = Number of Compounding Periods

The number of compounding periods is equal to the term in years multiplied by the corresponding factor.

• Daily Compounding: 365x Per Year
• Monthly Compounding: 12x Per Year
• Quarterly Compounding: 4x Per Year
• Semi-Annual Compounding: 2x Per Year
• Annual Compounding: 1x Per Year

If we subtract the present value (PV) from the future value (FV), the impact of compound interest can be isolated.

## Compound Interest vs Simple Interest

Compound interest is based on the principal amount plus any accrued interest.

In each compounding period, the interest accrued in the previous period is rolled-forward into the current period and increases the principal amount.

By contrast, the accumulated interest is not added to the principal in simple interest calculations.

Instead, simple interest is calculated off of the original principal amount.

###### Simple Interest Formula
• Simple Interest = P × r × t

Where:

• P = Present Value (PV)
• r = Interest Rate (%)
• t = Term in Years

## Compound Interest Calculator – Excel Template

We’ll now move to a modeling exercise, which you can access by filling out the form below.

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## Compound Interest Example Calculation

For our illustrative example, let’s say that you deposited \$100,000 into a bank account.

If we assume the annual interest rate (r) is 5% and the deposit was left untouched for 10 years, how much the original \$100,000 is worth in the future is determined by the compounding frequency.

• Interest Rate (r) = 5%
• Present Value (PV) = \$100,000
• Term (t) = 10 Years

The “FV” Excel function can be used to calculate how much your \$100,000 deposit is now worth after 10 years.

###### Future Value (FV) Excel Function
• “= FV (rate, nper, pmt, pv)”

Where:

• rate = Interest Rate (%)
• nper = Term in Years x Number of Compounding Periods
• pmt = 0
• pv = – Present Value (Principal)

Since the \$100,000 was an outflow from your perspective (i.e. an investment), it should be entered as a negative figure.

Under each scenario, the future value (FV) of the \$100,000 deposit and the percentage change compared to the original value is shown below:

• Annual Compounding: \$162,899 (62.9%)
• Semi-Annual Compounding: \$163,862 (63.9%)
• Quarterly Compounding: \$164,362 (64.4%)
• Monthly Compounding: \$164,701 (64.7%)
• Daily Compounding: \$164,866 (64.9%)

The deposit earns the difference between the future value (FV) and present value (PV).

• Annual: \$162,899 – \$100,000 = \$62,899
• Semi-Annual: \$163,862 – \$100,000 = \$63,862
• Quarterly: \$164,362 – \$100,000 = \$64,362
• Monthly: \$164,701 – \$100,000 = \$64,701
• Daily: \$164,866 – \$100,000 = \$64,866

For instance, if the compounding frequency is monthly, your \$100,000 deposit has grown to \$164,701, netting a total of \$64,701 in interest after 10 years.

The more frequently interest is compounded, the more interest is earned.