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# Compound Interest

We can calculate it one step at a time:

- Calculate the Interest (= Loan at Start Interest Rate)
- Add the Interest to the Loan at Start to get the Loan at End of the year
- The Loan at End of the year is the Loan at Start of the
**next**year

A simple job, with lots of calculations.

**But there are quicker ways, using some clever mathematics.**

## Make A Formula

Let us make a formula for the above. just looking 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

$1,000 + ($1,000 x 10%) = $1,000 + $100 = **$1,100**

$1,000 1.10 = **$1,100**

Note: the Interest Rate was turned into a decimal by dividing by 100:

Read Percentages to learn more, but in practice just move the decimal point 2 places, like these examples:

The result is that we can do a year in one step:

**Multiply the Loan at Start by (1 + Interest Rate) to get Loan at End**

Now, here is the magic.

the same 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:

This does all the calculations in the top table in one go.

## The Formula

We have been using a real example, but let’s be 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)

Here is is written with FV first:

FV = PV (1+r) n

where **FV** = Future Value**PV** = Present Value**r** = annual interest rate**n** = number of periods

*This is the basic formula for Compound Interest.*

Remember it, because it is very useful.

### 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:

Did you see how we just put the

6% into its place like this:

and what if the loan was for 20 years at 8%. you work it out!

## Going Backwards to Work Out the Present Value

Let’s say your goal is to have $2,000 in 5 Years. You can get 10%, so **how much should you start with** ?

In other words, you know a Future Value, and **want to know a Present Value** .

We know that **multiplying** a Present Value (PV) by (1+r) n gives us the Future Value (FV), so we can go backwards by **dividing**. like this:

So the Formula is:

And now we can calculate the answer:

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

In other words, $1,241.84 will grow to $2,000 if you invest it at 10% for 5 years.

**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

## Compounding Periods

Compound Interest is not always calculated per year, it could be per month, per day, etc. **But if it is not per year it should say so!**

Example: you take out a $1,000 loan for 12 months and it says **1% per month** , how much do you pay back?

Just use the Future Value formula with n being the number of months:

FV = PV (1+r) n = $1,000 (1.01) 12 = $1,000 1.12683 = **$1,126.83** to pay back

And it is also possible to have yearly interest *but with several compoundings within the year*. which is called Periodic Compounding .

Example, 6% interest with **monthly compounding** does not mean 6% per month, it means 0.5% per month (6% divided by 12 months), and is worked out like this:

FV = PV (1+r/n) n = $1,000 (1 + 6%/12) 12 = $1,000 (1.005) 12 = $1,000 1.06168. = **$1,061.68** to pay back

This is equal to a **6.168%***($1,000 grew to $1,061.68)* for the whole year.

So be careful to understand what is meant!

## APR

Because it is easy for loan ads to be confusing (sometimes on purpose!), the **APR** is often used.

**APR** means * Annual Percentage Rate* . it shows how much you will actually be paying for the year (including compounding, fees, etc).

This ad looks like 6.25%, but **is really 6.335%**

Here are some examples:

Example 1: **1% per month** actually works out to be **12.683% APR** (if no fees)**.**

Example 2: **6% interest with monthly compounding** works out to be **6.168%****APR** (if no fees)**.**

If you are shopping around, ask for the APR.

## Break Time!

So far we have looked at using (1+r) n to go from a Present Value (PV) to a Future Value (FV) and back again, plus some of the tricky things that can happen to a loan.

Now is a good time to have a break before we look at two more topics:

- How to work out the
**Interest Rate**if you know PV, FV and the Number of Periods. - How to work out the
**Number of Periods**if you know PV, FV and the Interest Rate

## Working Out The Interest Rate

You can calculate the Interest Rate if you know a Present Value, a Future Value and how many Periods.

Example: you have $1,000, and want it to grow to $2,000 in 5 Years, what **interest rate** do you need?

r = ( FV / PV ) 1/n – 1

*Note: the little 1/n is a Fractional Exponent. first calculate 1/n, then use that as the exponent on your calculator.*

*For example 2 0.2 is entered as 2, x^y , 0. 2, =*

Now we just plug 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 need **14.87%** interest rate 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

You can calculate how many Periods if you know a Future Value, a Present Value and the Interest Rate.

Example: you want to know how many periods it will take to turn $1,000 into $2,000 at 10% interest.

This is the formula (note: it uses the natural logarithm function * ln* ):