# Compound Interest Formula Derivations

*Showing how the formulas are worked out, with Examples!*

With Compound Interest we 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** - etc...

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:

## The Formula

We have been using a real example, but let us make it more general by **using letters instead of numbers**, like this:

(Compare this to the calculation above it: PV = $1,000, r = 0.10, n = 5, and FV = $1,610.51)

- When the interest rate is annual, then
**n**is the number of years - When the interest rate is monthly, then
**n**is the number of months - and so on

### Examples

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:

(Note that it is **1.06**, not 1.6)

## The Four Formulas

So, the basic formula for Compound Interest is:

FV = PV (1+r)^{n}

- FV = Future Value,
- PV = Present Value,
- r = Interest Rate (as a decimal value), and
- n = Number of Periods

With that we can work out the Future Value **FV** when we know the Present Value **PV**, the Interest Rate **r** and Number of Periods **n**

And we can **rearrange** that formula to find FV, the Interest Rate or the Number of Periods when we know the other three.

Here are all four furmulas:

FV = PV (1+r)^{n} | Find the Future Value when we know a Present Value, the Interest Rate and number of Periods. | |

PV = FV / (1+r)^{n} | Find the Present Value when we know a Future Value, the Interest Rate and number of Periods. | |

r = ( FV / PV )^{1/n} - 1 | Find the Interest Rate when we know the Present Value, Future Value and number of Periods. | |

n = \frac{ln(FV / PV)}{ln(1 + r)} | Find the number of Periods when we know the Present Value, Future Value and Interest Rate |

How did we get those other three formulas? Read On!

## Working Out the Present Value

### Example: Sam wants to reach $2,000 in 5 Years at 10% annual interest. How much should Sam start with?

In other words, we 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:

^{n}

^{n}= FV

^{n}:PV = \frac{FV}{(1+r)^{n}}

So now we can calculate the answer:

### Example (continued):

**PV** = $2,000 / (1+0.10)^{5} = $2,000 / 1.61051 = **$1,241.84**

So Sam should start with **$1,241.84**

It works like this:

**Another Example:** How much do 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 grows to $10,000

## Working Out The Interest Rate

### Example: Sam has only $1,000, and wants it to grow to $2,000 in 5 Years, what interest rate should Sam be looking for?

We need a rearrangement of the first formula to work it out:

^{n}

^{n}= FV

^{n}= \frac{FV}{PV}

^{1/n}

^{1/n}− 1

*(Note: to understand the step "take nth root" please read Fractional Exponents)*

The result is:

r = ( FV / PV )^{1/n} − 1

Now we have the formula, it is just a matter of "plugging in" the values to get the result:

### Example (continued):

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 Sam needs **14.87%** to turn $1,000 into $2,000 in 5 years.

**Another Example:** What interest rate do 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

### Example: Sam can only get a 10% interest rate. How many years will it take Sam to get $2,000?

When we want to know how many periods it takes to turn $1,000 into $2,000 at 10% interest, we can rearrange the basic formula.

But we need to use the natural logarithm function * ln() *to do it.

^{n}

^{n}= FV

^{n}=

*FV / PV*

*FV / PV*)

*(Note: to understand the step "use logarithms" please read Working with Exponents and Logarithms).*

Now let's "plug in" the values:

### Example (continued):

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 years** to turn $1,000 into $2,000 at 10% interest.

Poor Sam will have to wait over 7 years.

**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.

## Conclusion

Knowing how the formulas are derived and used makes it easier for you to remember them, and to use them in different situations.