Annuities, Sinking Funds, and Amortization
Math Analysis and Discrete Math – Sections 5.3 and 5.4
I. Warm-Up Problem
Previously, we have computed the future value of an investment when a fixed amount of money is deposited in an account that pays interest compounded periodically. Often, however, people do not deposit money and then sit back and watch it grow. Rather, money is invested in small amounts at periodic intervals. Consider these problems: 1. Chrissy deposits $200 each year into a savings account that has an annual interest rate of 8% compounded annually. How much money will Chrissy have in her account after three years? Hint: Make up a table of how much she has in her account by ...view middle of the document...
We apply the geometric series formula and do some algebraic manipulation to come up with the formula below. For reference, the geometric series formula is
r n +1 " 1 r "1
So, in summary…
! Definition: An annuity is a sequence of equal periodic deposits. When the deposits are made at the same time the interest is credited, the annuity is termed ordinary. The amount of the annuity is the sum of all deposits made plus all interest accumulated.
Amount of an Annuity If P represents the deposit in dollars made at each payment period for an annuity at i percent interest per payment period, the amount A of the annuity after n payment periods is given by:
(1+ i )
Definition: A person with a debt may decide to accumulate sufficient funds to pay off the debt by agreeing to set aside enough money each month (or quarter or year) so that when the debt becomes payable, the money set aside each month plus ! the interest earned will equal the debt. The fund created by such a plan is called a sinking fund.
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III. Practice Problems for Annuities and Sinking Funds
For Problems #1-3, find the amount of each annuity. 1. The deposit is $1000 monthly for 1 year at 5% compounded monthly.
The deposit is $500 quarterly for 2 years at 4% compounded quarterly.
The deposit is $1000 annually for 20 years at 6% compounded annually.
For Problems #4-5, find the payment required for each sinking fund. 4. The amount required is $50,000 after 10 years at 7% compounded semiannually. What is the semiannual payment?
The amount required is $2500 after 2 years at 5% compounded quarterly. What is the quarterly payment?
Problem 1: Saving for a Car Emily wants to invest an amount every month so that she will have $5,000 in 3 years to make a down payment on a new car. Her account pays 8% compounded monthly. How much should she deposit each month?
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Problem 2: Saving for a House In 6 years, you would like to have $20,000 for a down payment on a beach house. How much should you deposit each quarter into a savings account paying 3% interest compounded quarterly?
Problem 3: Paying off Bonds The state has $5,000,000 worth of bonds that are due in 20 years. A sinking fund is established to pay off the debt. If the state can earn 10% annual interest compounded annually, what is the annual sinking fund deposit needed?
Problem 4: Managing a Condo The Crown Colony Association is required by law to set aside funds to replace its roof. It is estimated that the roof will need to be replaced in 20 years at a cost of $180,000. The condo can invest in treasuries yielding 6% paid semiannually. If the condo invests in the treasuries, what semiannual payment is required to have the funds to replace the roof in 20 years?
***Problem 5: Time Needed for a Million Dollars If Josh deposits $10,000 every year in an account paying 8% compounded annually, how long will it take him to accumulate $1,000,000?