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Annuities Section 5.3

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Introduction Let’s say you want to save money to go on a vacation, or you want to save money now for your baby’s college education. A strategy for saving a little bit of money in the present and having a big payoff in the future is called an annuity. An annuity is an account in which equal regular payments are made. There are two basic questions with annuities: Determine how much money will accumulate over time given that equal payments are made. Determine what periodic payments will be necessary to obtain a specific amount in a given time period.

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**Calculating short-term annuities**

Claire wants to take a nice vacation trip, so she begins setting aside $250 per month. If she deposits this money on the first of each month in a savings account that pays 6% interest compounded monthly, how much will she have at the end of 10 months? Claire’s first payment will earn 10 months interest. So F = 250( /12)12(10/12). Note that the time t is 10/12. Therefore F = 250(1.005)10 = $ Claire’s second payment will earn 9 months interest. Thus F = 250(1.005)9 = $

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**Table of future values Payment Future Value 1st 250(1.005)10 = $262.79**

2nd 250(1.005)9 = $261.48 3rd 250(1.005)8 = $260.18 4th 250(1.005)7 = $258.88 5th 250(1.005)6 = $257.59 6th 250(1.005)5 = $256.31 7th 250(1.005)4 = $255.04 8th 250(1.005)3 = $253.77 9th 250(1.005)2 = $252.51 10th 250(1.005)1 = $251.25 Totaling up the future value column, we see that Claire has $ to use for her vacation. She earned $69.80 in interest.

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**Ordinary Annuity and Annuity Due**

There are two types of annuity formulas. One formula is based on the payments being made at the end of the payment period. This called ordinary annuity. The annuity due is when payments are made at the beginning of the payment period. We will derive the ordinary annuity formula first.

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**Calculating Long Term Annuities**

The previous example reflects what actually happens to an annuity. The problem is what if the annuity is for 30 years. Future Value of the 1st payment for an ordinary annuity is F1 = PMT(1+r/n)m-1 The future value of the next to last payment is Fm-1 = PMT(1+r/n) The future value of the last payment is Fm = PMT. The total future value F = F1 + F2 + F3 + … + Fm-1 + Fm

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**Continuing the calculation of a long term annuity**

The future value is Eq1 Now multiply the equation above by (1+r/n) Eq2 Take Eq2 – Eq1 Note that m = nt. Simplifying gives the ordinary annuity future value formula

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**Formulas ORDINARY ANNUITY**

ANNUITY DUE – receives one more period of compounding than the ordinary annuity so the formula is

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Example Find the future value of an ordinary annuity with a term of 25 years, payment period is monthly with payment size of $50. Annual interest is 6%. F = $34,649.70 Note: We only put in $15,000. This means that interest earned was $19,649.70!

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Sinking Funds A sinking fund is when we know the future value of the annuity and we wish to compute the monthly payment. For an ordinary unity this formula is For an annuity due the formula is

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Sinking Fund Example Suppose you decide to use a sinking fund to save $10,000 for a car. If you plan to make 60 monthly payments (5 years) and you receive 12% annual interest, what is the required payment for an ordinary annuity?

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Real – Life Example In 18 years you would like to have $50,000 saved for your child’s college education. At 6% annual interest, compounded monthly, what monthly deposit must be made to accomplish this goal? The question does not specify when the payments will be made so we use both formulas for comparison. For the ordinary annuity For the annuity due

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