Albert Einstein, one of the greatest physicists of all time, is said to have once described compound interest as “the eighth wonder of the world.” It is also claimed that he went on to add, “He who understands it, earns it; he who doesn't, pays it.” And he was right.

Typically, compound interest is based on your initial principal, all the accumulated interest from previous compounding periods and any contributions you add to your account regularly. It is sometimes referred to as interest on interest and is one of the most powerful tools to build wealth.

Many kinds of investments use compound interest with contributions. Examples include 401(k) and savings accounts. However, annuities lead in this kind of multiplication.

Compound interest may also apply to your stock investments if you reinvest all your dividends and continue to buy stocks regularly. Therefore, you need to know the compound interest with contributions formula to calculate how much you could make over time.

Generally, when an institution offers you a financial investment product at a specified annual percentage yield (APY), you will be dealing with the power of compounding.

## Compound Interest Formula

It would help if you first learned the standard compound interest formula before factoring in contributions. That way, you could do the math manually for each year, while adding the contributions at the beginning of each compounding period.

Below is the formula:

CA = P(1+(r/n)) ^{} ^{nt}

Where:

- CA = compound amount
- P = principal
- r = interest rate in decimal form
- n = number of times the principal is compounded annually
- t = the period the compounding takes place in years

It is worth noting though, the formula differs a little when the compounding takes place once per year instead of several times each year.

In such a case, the formula for compound interest would be:

**CI = P (1+ (r/100))**** ^{ t}**

**- P**

Where:

- r = Annual interest rate
- P = Principal

## Compound Interest With Contributions Formula

The standard compound interest formula does not factor in any contributions that you may add to the principal. When you add money regularly, your principal also increases, which means the interest rate works on a bigger amount.

Typically, contributions to an account earning compound interest are pretty common in annuities. So, you can use the formula for future values of an ordinary annuity to determine the compound interest with contributions.

Below is the compound interest with contributions formula:

P = (PMT [((1 + r) ^{n} - 1) / r]) (1 + r)

Where:

- P = The future value of the savings you expect to be paid in the future
- PMT = The amount of each contribution
- r = The interest rate
- n = The number of periods over which payments are to be made

## How to Calculate Compound Interest With Contributions

Below is an example that shows how to calculate compound interest with contributions.

### Example

Suppose you want to save money for 10 years at an annual interest rate of 8 percent compounding annually. Also suppose that for 10 years, you make annual contributions worth $3,000. Based on this information, you can calculate for the compound interest you would earn after 10 years if you left everything alone.

Using the annuity formula, the future value of your savings account would be:

P = (PMT [((1 + r) ^{n} - 1) / r]) (1 + r)

P = ($3,000[((1+0.08)^{10}-1)/0.08]) (1+0.08), which can be simplified as (3,000[((1.08^{10}-1)/0.08]) (1.08)

And that is:

$3,000(14.4866) *1.08, which equals $46,936.58

The interest alone would be $46,936.58-$30,000, which equals $16,936.58.

On the other hand, if you were to contribute $250 per month for 10 years at an annual interest of 8 percent compounded monthly, the values would be different.

In that case:

P = (250[((1.00667)^{120}-1)/0.00667]) (1+0.00667), which can be simplified as 250[((1.00667^{120}-1)/0.00667)(1.00667)

And that equals $46,051.85

In this case, the interest part alone is $16,051.85.

Matters can get more complicated if you invest an initial lump sum and make contributions regularly while compound interest works. In that case, it would be best to use an online compound interest calculator that factors in contributions.

References

- The Globe and Mail: Compound interest may not be Einstein’s eighth wonder, but it is a powerful tool for investors
- Forbes: Compound Interest and The Road to Retirement: How to Fund Your Future
- Forbes: What Is APY?
- AccountingTools: The formula for the future value of an annuity due
- Investor.Gov: Compound interest calculator

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I hold a BS in Computer Science and have been a freelance writer since 2011. When I am not writing, I enjoy reading, watching cooking and lifestyle shows, and fantasizing about world travels.