Compound Interest Formula Derivations
Showing how the formulas are worked out, with Examples!
With Compound Interest, you work out the interest for the first period, add it to the total, and then calculate the interest for the next period, and so on ..., like this:
Make A Formula
Let's look at the first year to begin with:
$1,000.00 + ($1,000.00 × 10%) = $1,100.00
We can rearrange it like this:
So, adding 10% interest is the same as multiplying by 1.10
(Note: the Interest Rate was turned into a decimal by dividing by 100: 10% = 10/100 = 0.10, read Percentages to learn more.)
And that formula works for any year:
· We could do the next year like this: $1,100 × 1.10 = $1,210
· And then continue to the following year: $1,210 × 1.10 = $1,331
So it works like this:
In fact we could go straight from the start to Year 5 if we multiply 5 times:
$1,000 × 1.10 × 1.10 × 1.10 × 1.10 × 1.10 = $1,610.51
But it is easier to write down a series of multiplies using Exponents (or Powers) like this:
We have been using a real example, but let us make it more general by using letters instead of numbers, like this:
(Can you see it is the same? Just with PV = $1,000, r = 0.10, n = 5, and FV = $1,610.51)
How about some examples ...
... what if the loan went for 15 Years? ... just change the "n" value:
... and what if the loan was for 5 years, but the interest rate was only 6%? Here:
The Four Formulas
So, the basic formula for Compound Interest is:
|FV = PV (1+r)n||
To find the Future Value, where:
With that we can work out FV when we know PV, the Interest Rate and Number of Periods
But by rearranging that formula we can find FV, the Interest Rate or the Number of Periods when we know the other three, like this:
|PV = FV / (1+r)n||Find the Present Value when you know a Future Value, the Interest Rate and number of Periods.|
|r = ( FV / PV )1/n - 1||Find the Interest Rate when you know the Present Value, Future Value and number of Periods.|
|n = ln(FV / PV) / ln(1 + r)||Find the number of Periods when you know the Present Value, Future Value and Interest Rate|
How did we get those other three formulas? Read On!
Working Out the Present Value
Let's say you want to reach $2,000 in 5 Years at 10%. How much should you start with?
In other words, you know a Future Value, and want to know a Present Value.
We can just rearrange the formula to suit ... dividing both sides by (1+r)n to give us:
So now we can calculate the answer:
PV = $2,000 / (1+0.10)5 = $2,000 / 1.61051 = $1,241.84
It works like this:
Another Example: How much would you need to invest now, to get $10,000 in 10 years at 8% interest rate?
PV = $10,000 / (1+0.08)10 = $10,000 / 2.1589 = $4,631.93
So, $4,631.93 invested at 8% for 10 Years would grow to $10,000
Working Out The Interest Rate
If you have $1,000, and want it to grow to $2,000 in 5 Years, what interest rate do you need?
We need a rearrangement of the first formula to work it out:
(Note: to understand the step "take nth root" please read Fractional Exponents).
Now we have the formula, it is just a matter of "plugging in" the values to get the result:
r = ( $2,000 / $1,000 )1/5 - 1 = ( 2 )0.2 - 1 = 1.1487 - 1 = 0.1487
And 0.1487 as a percentage is 14.87%,
So you would need a 14.87% interest rate to turn $1,000 into $2,000 in 5 years.
Another Example: What interest rate would you need to turn $1,000 into $5,000 in 20 Years?
r = ( $5,000 / $1,000 )1/20 - 1 = ( 5 )0.05 - 1 = 1.0838 - 1 = 0.0838
And 0.0838 as a percentage is 8.38%. So 8.38% will turn $1,000 into $5,000 in 20 Years.
Working Out How Many Periods
If you want to know how many periods it will take to turn $1,000 into $2,000 at 10% interest, you can also rearrange the basic formula.
But we need to use the natural logarithm function ln() to do it.
(Note: to understand the step "use logarithms" please read Working with Exponents and Logarithms).
Now let's "plug in" the values:
n = ln( $2,000 / $1,000 ) / ln( 1 + 0.10 ) = ln(2)/ln(1.10) = 0.69315/0.09531 = 7.27
Magic! It will need 7.27 periods to turn $1,000 into $2,000 at 10% interest.
Another Example: How many years to turn $1,000 into $10,000 at 5% interest?
n = ln( $10,000 / $1,000 ) / ln( 1 + 0.05 ) = ln(10)/ln(1.05) = 2.3026/0.04879 = 47.19
47 Years! But we are talking about a 10-fold increase, at only 5% interest.
Now that you see how each formula was derived and how to use it, hopefully it will be easier for you to remember them, and to be able to use them in different situations.