ExamplesHow about some examples ... ... and what if the loan was for 5 years, but the interest rate was only 6%? Here:
... and what if the loan was for 20 years at 8%? ... you work it out! Going "Backwards" to Work Out the Present ValueLet'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: PV = FV / (1+r)^{n} 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 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 Compounding PeriodsCompound 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 would be 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
Here are some examples: Example 1: "1% per month" actually works out to be 12.683% APR (if no fees). And: 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 would be a good time to have a break before we look at two more topics:
Working Out The Interest RateYou 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? The formula is: r = ( FV / PV )^{1/n}  1
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 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 PeriodsYou can calculate the Interest Rate if you know a Present Value, a Future Value and how many Periods. 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): n = ln(FV / PV) / ln(1 + r)
Anyway, 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 years 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 10fold increase, at only 5% interest. CalculatorI made a Compound Interest Calculator that uses these formulas, if you are interested. SummaryThe basic formula for Compound Interest is:
AnnuitiesSo far we have talked about what happens to a value as time goes by ... but what if you have a series of values, like regular loan payments or yearly investments? That is covered in Annuities, coming soon.
