D. Grab Bag
For each of the following scenarios, determine whether or not the payments describe an annuity. If they do, determine whether
they are an annuity due or an ordinary annuity. Also, determine whether the amount indicated is a present value or a future value.
- For the past 20 years, Steve has put $100 in a special savings account at the beginning of each month. He now has
$31,575.16 in that account. - Ever since their daughter was born, Jim and Lara have been setting aside money to pay for her college. At the end
of each month, they have always deposited at least $200 into a college savings account for her. They presently have
$18,735.92 in the account. - Delsimar Devices Corp. sets aside $275,000 at the end of each quarter. The money goes into a special fund the
company plans to use for a long-term expansion plan. The company expects that in 5 years it will have accumulated
$5,800,000 in this account. - Two years ago, I took out a $25,000 business loan. The payments are $500 per month.
- Sonja paid an insurance company $400,000 for a policy that will pay her $3,000 a month in income for the rest of her life.
146 Chapter 4 Annuities
4.2 Future Values of Annuities
In the previous section, we saw several examples of present and future values of annuities,
but did not discuss how to actually find the present or future value of any given annuity.
This will require more sophisticated mathematical tools than the ones we have been using
so far. In every calculation we have done so far, there has always been a single starting
amount (the principal, proceeds, or present value) and a single ending amount (the maturity
value or future value), with no payments made along the way. Yet with annuities, there are
lots of payments being made along the way.
In this section, we’ll begin to develop the mathematical tools we’ll need to work with
annuities. At first, we will focus just on finding future values, though later on in the chapter,
we will also address present values. At first we will concern ourselves only with ordinary
annuities. Later in this section we will see how to handle annuities due as well.
Looking ahead, you will no doubt notice that this is a very long section. It takes quite a
bit of space to develop the formulas and techniques used to handle annuity future values.
Depending on the level and objectives of the course that you are taking, your instructor may
or may not be covering all of the material in this chapter. If you have any doubts about which
material is being covered in your class, make sure to check with your instructor.The Future Value of an Ordinary Annuity
Suppose that you deposit $1,200 at the end of each year into an investment account that
earns 7.2% compounded annually. Assuming you keep the payments up, how much would
your account be worth in 5 years? Your annual payments constitute an ordinary annuity,
and since we are asking about their accumulated value with interest at the end, we are look-
ing for the future value.
It turns out that we actually can find this future value by using the tools we already have.
We will consider this example problem from several different lines of attack, in the hope of
developing approaches that we can use with annuities in general.