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Compound interest (adapted from: HYPERLINK " )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:Here are the calculations for a 5 Year Loan at 10%:YearLoan at StartInterestLoan at End0 (Now)?1,000.00(?1,000.00 × 10% = ) ?100.00?1,100.001?1,100.00(?1,100.00 × 10% = ) ?110.00?1,210.002?1,210.00(?1,210.00 × 10% = ) ?121.00?1,331.003?1,331.00(?1,331.00 × 10% = ) ?133.10?1,464.104?1,464.10(?1,464.10 × 10% = ) ?146.41?1,610.515?1,610.51As you can see, it is simple to calculate if you take 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 yearA simple job, with lots of calculations. But there are quicker ways, using some clever mathematics.Make A FormulaLet us make a formula for the above ... just looking at the first year to begin with:?1,000.00 + (?1,000.00 × 10%) = ?1,100.00We can rearrange it like this:So, adding 10% interest is the same as multiplying by 1.10Note: the Interest Rate was turned into a decimal by dividing by 100: 10% = 10/100 = 0.10, read HYPERLINK " to learn more, but in practice all you need to do is move the decimal point 2 places, so for example 12% ? 1.2 ? 0.12So, now we have just one step:Multiply the "Loan at Start" by (1 + Interest Rate) to get "Loan at End"(But remember the Interest Rate should be a decimal value first! 0.10, not 10%)A simple calculation shows you they are the same:?1,000 + (?1,000 x 10%) = ?1,000 + ?100 = ?1,100??is the same as:?1,000 × 1.10 = ?1,100Now, 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 = ?1,610.51But it is easier to write down a series of multiplies using HYPERLINK " (or Powers) like this:The FormulaWe 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)nwhere FV = Future ValuePV = Present Valuer = annual interest raten = number of periodsThis is the basic formula for Compound Interest. Remember it, because it is very useful.ExamplesHow 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 the6% 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 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)nAnd now we can calculate the answer:PV = ?2,000 / (1+0.10)5 = ?2,000 / 1.61051 = ?1,241.84In 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 = ?4,631.93So, ?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 backAnd it is also possible to have yearly interest but with several compoundings within the year, which is called HYPERLINK " 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 backThis 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!APRBecause 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).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: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 RateWorking 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 - 1Note: the little "1/n" is a HYPERLINK " Exponent, first calculate 1/n, then use that as the exponent on your calculator.For example 20.2 would be 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 = 0.1487And 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 = 0.0838And 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)The "ln" function should be on a good calculator.You could also use log, just don't mix the two.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 = 7.27Magic! 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 = 47.1947 Years! But we are talking about a 10-fold increase, at only 5% interest.CalculatorI made a HYPERLINK " Interest Calculator that uses these formulas, if you are interested.SummaryThe basic formula for Compound Interest is:FV = PV (1+r)nTo find the Future Value, where:FV = Future Value,PV = Present Value,r = Interest Rate (as a decimal value), andn = Number of PeriodsAnd by rearranging that formula (see HYPERLINK " Interest Formula Derivation) we can find any value when we know the other three:??PV = FV / (1+r)nFind the Present Value when you know a Future Value, the Interest Rate and number of Periods.??r = ( FV / PV )1/n - 1Find 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

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