Compound InterestAn ExampleSlide 3Slide 4Slide 5Slide 6Slide 7Compound Interest FormulaYieldSlide 10Slide 11Finding the YieldMore on YieldSlide 14Continuous CompoundingSlide 16Slide 17Present ValuesRatiosExample RatiosSlide 21The ProjectSlide 23Preliminary ReportsCompound InterestAn ExampleSuppose that you were going to invest $5000 in an IRA earning interest at an annual rate of 5.5%How would you determine the amount of interest you’ve made on your investment after one year? 275$055.050001iAn ExampleHow much money would you have in your IRA account?How much interest would you get after two years? 5275$055.015000)055.0(50005000500011 iF 13.290055.052752iAn ExampleHow much money would you have in your IRA account after two years?What about 10 years? 055.01055.015000055.015275055.052755275527522 iF 72.8540$055.0150001010FCompound InterestNotice that the interest in our account was paid at regular intervals, in this case every year, while our money remained in the account. This is called compounding annually or one time per year.Compound InterestSuppose that instead of collecting interest at the end of each year, we decided to collect interest at the end of each quarter, so our interest is paid four times each year. What would happen to our investment?Since our account has an interest rate of 5.5% annually, we need to adjust this rate so that we get interest on a quarterly basis. The quarterly rate is:%375.14/5.5 Compound InterestSo for our IRA account of $5000 at the end of a year looks like:After 10 years, we have:72.5280$4055.015000141F85.8633$4055.01500010410FCompound Interest FormulaP dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years.Think of P as the present value, and F as the future value of the deposit.tnnrPF 1YieldOne may compare investments with different interest rates and different frequencies of compounding by looking at the values of P dollars at the end of one year, and then computing the annual rates that would produce these amounts without compounding.YieldSuch a rate is called the effective annual yield, annual percentage yield, or simply yield.In our previous example when we compounded quarterly, after one year we had:72.5280$4055.015000141FYieldTo find the effective annual yield, y, notice that we gained $280.72 on interest after a year compounded quarterly. That interest represents a gain of 5.61% on $5000:0561.0500072.280yFinding the YieldTo find the effective annual yield, find the difference between our money after one year and our initial investment and divided by the initial investment.Therefore, interest at an annual rate r, compounded n times per year has yield y:111nnnrPPnrPyMore on YieldThere may be times when we need to find the annual rate that would produce a given yield at a specified frequency of compounding. In other words, we need to solve for r : rynnrynrynrynrynnnnn1111111111/1/1/1Compound Interest FormulaNotice that when we collected our interest more times during each year, i.e. we compounded more frequently, the amount of money in our account was actually greater than if only collected interest one time a year.What would happen to our money if we compounded a really large number of times?Continuous CompoundingAs n increases, approaches a constant value in the Frequency Worksheet. Here’s why:As n gets really large, m also becomes really large, and:rtmmrtntmPmPnrP11111n=mremm 59057182818284.211Continuous CompoundingThe value of P dollars after t years, when compounded continuously at an annual rate r , isOn the calculator use the button (on TI-83: 2nd + LN )In Excel, use the function EXP(x )rtePF YieldThe effective annual yield, y, for compounding continuously at an annual interest rate of r is:To find the annual interest rate r if we know the yield, y, we would have to solve for r in the above equation. To do this you would use logarithms:1rrePPePy 1ln yrPresent ValuesIf we are given the future value, F, the annual interest rate r, the number of times compounded per year n, and the length of time invested t, we may solve the present value P :trtneFPnrFP 1RatiosSometimes we are not interested in the percentage that an investment increases by. Rather, we would like to know by what factor the investment increased or decreased. Such factors are computed by find the ratio of the future value to the present value. This ratio, R, for continuous compounding is:This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period.rtrtePPePFR Example RatiosSuppose that in our IRA example, the annual interest rate of 5.5% is compounded continuously.If we wanted to know the weekly rate our investment would increase, we would simply have 0.055/52 or 0.00105 or 0.105%. This would mean that the ratio of the future value to the present value between consecutive weeks compounded continuously would be e0.055/52 or 1.00105Example RatiosMultiplying by this weekly ratio 52 times yields a yearly ratio of (e0.055/52)52 = e(0.055/52)52 = e0.055. As we would expect, this corresponds to the annual rate of 0.055.The Project•How can compound interest help us price a stock option?•Our annual risk-free rate of 4%, compounded continuously, gives a weekly risk-free rate of rrf = 0.04/52 0.0007692. The weekly ratio corresponding to this weekly rate is e0.04/52.•We call Rrf = e0.04/52 1.0007695 the risk-free weekly ratio for the Walt Disney option.The ProjectCompound interest can help us with option pricing in a second way. Suppose that we know a future value F for our 20 week option at the end of the 20 weeks. We suppose that money will earn at the risk-free annual interest rate or 4% compounded
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