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Present and Future Value of Annuities

    Annuitites differ from ordinary simple and compound interest problems in that payments are made on a regular basis. For eaxmple, monthly, quarterly, semiannual or yearly payments. This topic illustrates future and present value of annuities using several examples. Also discussed are sinking funds: if a borrower makes periodic deposits that will produce a specified amount on a later specified date, then this borrower has established a sinking fund.

Definition (Future Value of an Ordinary Annuity) If dollars is invested at the end of each period for periods in an annuity that earns interest at a rate of per period, the future value of the ordinary annuity will be

.

Example (Future Value of an Ordinary Annuity) Someone qualifies to invest $5000 in an IRA each June 30 for the next 20 years. If they make these investments, and if the certificates pay 12%, compounded semiannually, how much will they have at the end of 20 years?
    We use the formula with and and so we have



If they did not invest their money they would only have $200,000 instead of the $773,810.             

Example (Payment for an Ordinary Annuity) What size payments must be put into an account at the end of each month to establish an ordinary annuity that has future value of $20,000 in 7 years, if the investment pays 7.3%, compounded monthly?
    We use the formula with and and so we have









If the payments are not invested then is obtained which is not as good as the investment which obtains $20,000.

Definition (Sinking Fund) If a borrower makes periodic deposits that will produce a specified amount on a later specified date, then this borrower has established a sinking fund.

Example (Sinking Fund) A small company establishes a sinking fund to discharge a debt of $30,000 due in 10 years by making semiannual payments, the first due in 6 months. If the deposits are placed into an account that pays 6%, compounded semiannual, what is the size of the deposits?
    We use the formula with and and so we have









Therefore, payments of $1,116.47 will discharge a debt of $30,000 even though So the point is to invest the money rather than paying the debt at once.

Definition (Present Value of an Ordinary Annuity) If a payment of dollars is to be made at the end of each period for periods from an account that earns interest at a rate of per period, then the account is an ordinary annuity, and the present value is

Example (Present Value of an Ordinary Annuity) Find the present value of an annuity that pays $500 at the end of each month for 3 years, if the interest rate is 6%, compounded monthly.
    We use the forumula with and and so we have
    


If the 36 payments of $500 were not invested it would take

Example (Payments from an Ordinary Annuity) (a) If $1,000,000 is invested in an annuity that earns 5.8% compounded monthly, what size of  payments will it provide at the end of each month for the next 30 years?
    We use the formula with and and so we have









Now the payments of for 360 payments leads to    

Example (Using Present and Future Values) Is it more economical to buy an automobile for $29,000 cash or to pay $8000 down and $3000 at the end of each quarter for 2 years, if money if worth 8% compounded quarterly?
    The automobile can be bought now for $29,000  or can be bought for $8000 plus the present value of the investment. The present value is given by the formula where and and so we have Thus the automobile can be bought for $29,000 or for Thus, it is cheaper to pay cash.

Exercises (Present and Future Values)

(1) $10000 is deposited for 10 years in an account paying 8% compounded quarterly. At the end of the 10 year period, I want to make 20 quarterly withdrawals. What is the size of each withdrawal?

    We can find the future value of the first investment, using the formula where and so we have
    


To find the quarterly withdrawals, we use the formula with and and so we have









Thus, is the size of each withdrawal.

(2) $500 is deposited each six months for 5 years into an account paying 6% compounded semiannually. No more deposits are made but the account still earns the interest. How much is in the account 10 years after the last deposit?

    To find the future value after 5 years, we use the formula with and and so we have



We can find the future value of the second investment, using the formula where and so we have
    


Thus, is in the account 10 years after the last deposit?

(3) $2000 is deposited each year for 20 years into an IRA account paying 6% compounded annually. Then 20 annual withdrawals are made from the account. (a) How  much is in the account just after the 20th deposit? (b) How much was deposited? (c) What is the size of each withdrawal? (d) How much is withdrawn?

    For part (a) we find the future value of the annuity, we use the formula with and and so we have



Thus, is in the account 20 years after the last deposit?
    For part (b), we want to know how much was deposited. Since we made 20 deposites of 2000, we have
    For part (c), to find the annual withdrawals, we use the formula with and and so we have









Thus, is the size of each withdrawal.
    For part (d) the amount that is withdrawn is for a total of 20 withdrawals and so the total amount that is withdrawn is

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