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Chapter 5 Section 5.3 Annuities

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The compound interest investment in which all of the money that is to be invested (i.e. the principal ( P )) is put in the investment all at once. This type of investment is called a lump sum investment. A second type of investment is where the money is invested in a series of regular fixed payments ( pymt ) at the end of each compounding period is called an ordinary annuity. This is the most common sort of annuity. If the money is invested in a series of regular fixed payments ( pymt ) at the beginning of each compounding period is called an annuity due. Holiday Club Investment In order to save money for Christmas presents a person will invest $200 at the end of September, October and November for Christmas shopping starting in December. The account pays 3.6% compounded monthly. How much will they have saved for Christmas shopping? This is a type of annuity investment. They want to know the future value ( FV ) of the annuity. We can look at it as 3 different lump sum investments, one for that is 2 periods ( n = 2), one that is 1 period ( n = 1) and 1 that is 0 periods ( n = 0).

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Annuity Formulas Annuities are often used for investment that run for many periods such as loans and retirement savings accounts. This may involve hundreds of payments and treating each one as a separate lump sum investment is not practical. Mathematicians have developed an ingenious method involving algebra to generate a formula for this. If we let pymt stand for the amount of each payment and n is the number of payments then the total future value of all payments is given by: Multiply both sides of the equation above (i.e. every term on the right side) by the value (1+ i ). That will cause each exponent to go up by 1. We then subtract the first expression from the second noticing what cancels. Some factoring, canceling and regrouping gives the formula!

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Annuity Formulas There are 2 types of annuities we mentioned, one where the payment is made at the end of each period call an ordinary annuity which its future value is given by FV (ord) and the other where each payment is made at the beginning of each period which we call and annuity due where the future value is given by FV (due). A person has $300 dollars taken out of their paycheck at the end of each biweekly period and put in a retirement account that pays 5.2% compounded biweekly. How much will they have in this account when they retire in 25 years? This is asking for the future value of an ordinary annuity. pymt = 300

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Example A person wants to have 250,000 in their retirement account when they retire in 20 years. The decide to invest at the end of each week into a Tax Deferred Account (TDA) that pays 7.8% compounded weekly. a) How much will their weekly payments need to be? We still use the FV (ord) formula but we want to know the value for pymt. We will use some algebra to solve for this value. FV (ord)=250,000 n = 20·52 = 1040 b) How much money does he make in interest over the life of this investment? He makes 1040 payments of $99.91. His total investment is 1040·99.91 = $103,906 He will have $250,000 in the TDA when he retires so the amount he has earned will be $250000-$103,906 = $146,094 in interest over the 20 years!

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c) We he retires he switches to an account that pays 6% compounded monthly and withdraws $1000 each month for living expenses. Fill in the table below. MonthBeginning Balance InterestWithdrawEnding Balance 1 2 3 250,000.00 1. He begins the first month with the 250,000 in the retirement account. 2. In month 1 he earns 250,000*(.06/12) = 1250 in interest. 3. After the 1000 dollar withdrawal he has a gain of $250. 4. The next month begins with what was at the end of the previous month. 5. In month 2 he earns 250,250*(.06/12) = 1,251.25 in interest. 6. After the 1000 dollar withdrawal he has a gain of $251.25 7. The next month begins with what was at the end of the previous month. 8. In month 3 he earns 250,501.25*(.06/12) = 1252.51 in interest. 9. After the 1000 dollar withdrawal he has a gain of $252.51 1,250.001,000.00250,250.00 1,000.00 1,251.25 1,252.51 250,501.25 250,753.76

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