162 Chapter 4 Annuities
- Suppose that Ron deposits $125 per month into an account paying 8%. His brother Don deposits $250 per month into
an account paying 4%. How much will each brother have in his account after 40 years? - Suppose that Holly deposits $125 per month into an account paying 8%. Her sister Molly deposits $250 per month into
an account paying 4%. How much will each sister have in her account after 16 years? - The members of a community church, which presently has no endowment fund, have pledged to donate a total of
$18,250 each year above their usual offerings in order to help the church build an endowment. If the money is invested
at a 5.39% rate, how much will they endowment have grown to in 10 years? - Jack’s fi nancial advisor has encouraged him to start putting money into a retirement account. Suppose that Jack
deposits $750 at the end of each year into an account earning 8¾% for 25 years. How much will he end up with? How
much would he end up with if he instead made his deposits at the start of each year?
H. Additional Exercises
- A group of ambitious developers has begun planning a new community. They project that each year a net gain of
850 new residents will move into the community. They also project that, aside from new residents, the community’s
population will grow at a rate of 3% per year (due to normal population changes resulting from births and deaths). If
these projections are correct, what will the community’s population be in 15 years? - a. Find the future value of $1,200 per year at 9% for 5 years, fi rst as an ordinary annuity and then as an annuity due.
Compare the two results.
b. Find the future value of $100 per month at 9% for 5 years, fi rst as an ordinary annuity and then as an annuity due.
Compare the two results.
c. In both (a) and (b) the total payments per year were the same, the interest rate was the same, and the terms
were the same. Why was the difference between the ordinary annuity and the annuity due smaller for the monthly
annuity than for the annual one? - Suppose that Tommy has decided that he can save $3,000 each year in his retirement account. He has not decided yet
whether to make the deposit all at once each year, or to split it up into semiannual deposits (of $1,500 each), quarterly
deposits (of $750 each), monthly, weekly, or even daily. Suppose that, however the deposits are made, his account
earns 7.3%. Find his future value after 10 years for each of these deposit frequencies. What can you conclude? - (Optional.) As discussed in this chapter, we normally assume that interest compounds with the same frequency as the
annuity’s payments. So, one of the reasons Tommy wound up with more money with daily deposits than with, say,
monthly deposits, was that daily compounding results in a higher effective rate than monthly compounding.
Realistically speaking, the interest rate of his account probably would compound at the same frequency regardless of
how often Tommy makes his deposits. Rework Problem 43, this time assuming that, regardless of how often he makes
his deposits, his account will pay 7.3% compounded daily.