Calc

Compound Interest Formula - Explained

Article Category: Finance  |   


The compound interest formula incorporates your principal amount, interest rate, time factor and compound periods. Putting these figures into the formula, you can find out the amount of compounded interest you might gain (savings/investment) or owe (loan) over that time period.

The concept of compound interest is that interest is added back to the principal sum so that interest is gained on that already-accumulated interest during the next compounding period. In this article we'll take a look at the formula in more depth, go through an example and examine other versions of the formula that help you calculate interest rate, time factor and include additional regular contributions.

How do you calculate compound interest?

Compound interest can be calculated using the formula A = P (1 + r/n) (nt), entering into it the initial principal amount (P), annual interest rate (r as decimal), time factor (t) and the number of compound periods (n). Should you wish, you can then deduct the principal (P) from the result to get a figure for compound interest only.

Let's go through this process step by step:

The compound interest formula

The compound interest formula

The formula for compound interest, including principal sum, is:
A = P (1 + r/n) (nt)

Where:

  • A = the future value of the investment/loan, including interest
  • P = the principal investment amount (the initial deposit or loan amount)
  • r = the annual interest rate (decimal)
  • n = the number of times that interest is compounded per unit t
  • t = the time the money is invested or borrowed for

It's worth noting that this formula gives you the future value of an investment or loan, which is compound interest plus the principal. Should you wish to calculate the compound interest only, you need to deduct the principal from the result. So, your formula looks like this:

Compounded interest only = P (1 + r/n) (nt) - P


Let's look at an example

If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, the value of the investment after 10 years can be calculated as follows...

P = 5000.
r = 5/100 = 0.05 (decimal).
n = 12.
t = 10.

If we plug those figures into the formula, we get the following:

A = 5000 (1 + 0.05 / 12) (12 * 10) = 8235.05.

So, the investment balance after 10 years is $8,235.05.

Methodology

A few people have written to me asking me to explain step-by-step how we get the 8235.05. This all revolves around BODMAS / PEMDAS and the order of operations. Let's go through it:

A = 5000 (1 + 0.05 / 12) ^ (12(10))

(note that ^ means 'to the power of')

Using the order of operations we work out the totals in the brackets first. Within the first set of brackets, you need to do the division first and then the addition (division and multiplication should be carried out before addition and subtraction). We can also work out the 12(10). This gives us...

A = 5000 (1 + 0.00416) ^ 120

(note that the over-line in the calculation signifies a decimal that repeats to infinity. So, 0.00416666666...)

Then:

A = 5000 (1.00416) ^ 120

The exponent goes next. So, we calculate (1.00416) ^ 120.

This means we end up with:

5000 × 1.6470095042509848

= 8235.0475.

You may have seen some examples giving a formula of A = P ( 1+r ) t . This simplified formula assumes that interest is compounded once per period, rather than multiple times per period.

The benefit of compound interest

I think it's worth taking a moment to examine the benefit of compound interest using our example. The benefit hopefully becomes clear when I tell you that without compound interest, your investment balance in the above example would be only $7,500 ($250 per year for 10 years, plus the original $5000) by the end of the term. So, thanks to the wonder of compound interest, you stand to gain an additional $735.05.

To give a graphical example, the graph below shows the result of $1000 invested over 20 years at an interest rate of 10%. The principal figure is in green. The blue part of the graph shows the result of 10% interest without compounding. Finally, the purple part demonstrates the benefit of compound interest over those 20 years.

Diagram of compound interest benefits

You can read more about the benefits of compound interest in our article, what is compound interest?

Interactive compound interest formula

I have created the calculator below to show you the formula and resulting accrued investment/loan value (A) for the figures that you enter. Note that this calculator requires JavaScript to be enabled in your browser.




P ( 1 + r n ) ( n t ) = A

For a comprehensive set of tools for calculating compound interest on your savings, please use the compound interest calculators.

Formulae to find compound interest rate, time and principal

It may be that you want to manipulate the compound interest formula to work out the interest rate or principal investment/loan figure. Here are the formulae you need.

Formula for interest rate (r)

Should you wish to work out the average yearly interest rate you're getting on your savings, investment, personal loan or car loan, this formula can help. Note that you should multiply your result by 100 to get a percentage figure (%)

Formula for calculating interest rate

Formula for principal (P)

This formula is useful if you want to work backwards and find out how much you would need to start with in order to achieve a chosen future value.

Formula for calculating principal (starting amount)

Example: Let's say your goal is to end up with $10,000 in 5 years, and you can get an 8% interest rate on your savings, compounded monthly. Your calculation would be: P = 10000 / (1 + 0.08/12)(12×5) = $6712.10. So, you would need to start off with $6712.10 to achieve your goal.

Formula for time (t)

This variation of the formula works for calculating time (t), by using natural logarithms. You can see how this formula was worked out by reading this explanation on algebra.com.

t = ln(A/P) / n[ln(1 + r/n)]

Where:

  • A = the value of the accrued investment/loan
  • P = the principal amount
  • r = the annual interest rate (decimal)
  • n = the number of times that interest is compounded per unit t
  • t = the time the money is invested or borrowed for

Compound interest formula (with regular contributions)

A lot of people have asked me to include a single formula for compound interest with monthly additions. Believe me when I tell you that it isn't quite as simple as it sounds. In order to work out calculations involving monthly additions, you will need to use two formulae - our original one, listed above, plus the 'future value of a series' formula for the monthly additions.

At the request of readers, I've adapted the formula explanation to allow you to calculate periodic additions, not just monthly (added May 2016). These formulae assume that your frequency of compounding is the same as the periodic payment interval (monthly compounding, monthly contributions, etc). If you would like to try a version of the formula that allows you to have a different periodic payment interval to the compounding frequency, please see the ' periodic payments' section below.

If the additional deposits are made at the END of the period (end of month, year, etc), here are the two formulae you will need:

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × {[(1 + r/n)(nt) - 1] / (r/n)}

If the additional deposits are made at the BEGINNING of the period (beginning of year, etc), here are the two formulae you will need:

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × {[(1 + r/n)(nt) - 1] / (r/n)} × (1+r/n)

Where:

  • A = the future value of the investment/loan, including interest
  • P = the principal investment amount (the initial deposit or loan amount)
  • PMT = the monthly payment
  • r = the annual interest rate (decimal)
  • n = the number of times that interest is compounded per unit t
  • t = the time (months, years, etc) the money is invested or borrowed for

Let's put this into practice with another example

If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, with additional deposits of $100 per month (made at the end of each month). The value of the investment after 10 years can be calculated as follows...

P = 5000. PMT = 100. r = 5/100 = 0.05 (decimal). n = 12. t = 10.

If we plug those figures into the formulae, we get:

  • Total = [ Compound interest for principal ] + [ Future value of a series ]
  • Total = [ P(1+r/n)^(nt) ] + [ PMT × (((1 + r/n)^(nt) - 1) / (r/n)) ]
  • Total = [ 5000 (1 + 0.05 / 12) ^ (12 × 10) ] + [ 100 × (((1 + 0.00416)^(12 × 10) - 1) / (0.00416)) ]
  • Total = [ 5000 (1.00416) ^ (120) ] + [ 100 × (1.00416)^(120) - 1) / 0.00416) ]
  • Total = [ 8235.05 ] + [ 100 × (0.647009497690848 / 0.00416) ]
  • Total = [ 8235.05 ] + [ 15528.23 ]
  • Total = [ $23,763.28 ]

So, the investment balance after 10 years is $23,763.28.

You can learn more about this future value of a series formula, and use an interactive formula for it, in the future value formula article.

One thing you might notice is that this figure may differ slightly from the figure you get from the compound interest calculator. The reason for this is that the compound interest formula above assumes that the interest calculation occurs before the regular deposit is added on. The calculator, conversely, adds the deposit in first before calculating the interest. Both are legitimate ways of calculating.

Different periodic payments

A few people have commented that they would like a version of the above formula that takes into consideration the number of periodic payments (both formulae above assume your periodic payments match the frequency of compounding). For example, your money may be compounded quarterly but you're making contributions monthly. In this case, you may wish to try this version of the formula (suggested by Darinth Douglas):

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × p {[(1 + r/n)(nt) - 1] / (r/n)}

(With 'p' being the number of periodic payments in the compounding period)

I hope you have found this compound interest formula article helpful. If you have, I would be very grateful if you would consider sharing it on social media or on your website/blog. Thank you. Should you wish to leave a comment, please do so below.

Alastair Hazell




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Last update: 21 May 2019


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