Copyright © 2008, The McGraw-Hill Companies, Inc.ixEXERCISES THAT BUILD BOTH
SKILLS AND CONFIDENCE
Each section of every chapter includes
a set of exercises that gives you the
opportunity to practice and master
the skills presented in the section.
These exercises are organized in three
groupings, designed to build your
skills and your confi dence so that you
can master the material.BUILDING FOUNDATIONS
In each exercise set, there are several
initial groupings of exercises under a
header that identifi es the type of prob-
lems that will follow and gives a good
hint of what type of problem it is.BUILDING CONFIDENCE
In each set there is also a grouping of
exercises labeled “Grab Bag.” These
sections contain a mix of problems
covering the various topics of the sec-
tion, in an intentionally jumbled order.
These exercises add an additional and
very important layer of problem solv-
ing: identifying the type of problem
and selecting an appropriate solution
technique.EXPANDING THE CONCEPTS
Each section’s exercise set has one
last grouping, labeled “Additional
Exercises.” These are problems that
go beyond a standard problem for the
section in question. This might mean
that some additional concepts are
introduced, certain technicalities are
dealt with in greater depth, or that the
problem calls for using a higher level
of algebra than would otherwise be
expected in the course.144 Chapter 4 Annuities
EXERCISES 4.
A. The Defi nition of an Annuity
Determine whether or not each of the following situations describes an annuity. If the situation is not an annuity, explain why it is not.- A car lease requires monthly payments of $235.94 for 5 years.
- Your cell phone bill.
- The money Adam pays for groceries each week.
- Ashok bought a guitar from his brother for $350. Since he didn’t have the money to pay for it up front, his brother agreed that he could pay him $25 a week until his payments add up to $350.
- Caries’ Candy Counter pays $1,400 a month in rent for its retail store.
- The rent for the Tastee Lard Donut Shoppe is $850 a month plus 2% of the monthly sales.
- Cheryl pays for her son’s day care at the beginning of every month. Her provider charges $55 for each day her son is scheduled to be there during the month.
- Every single morning, rain or shine, Cieran walks to his favorite coffee shop and buys a double redeye latte.
- According to their divorce decree, Terry is required to pay his ex-wife $590 a month in child support until their daughter turns 21.
- In response to her church’s annual stewardship campaign, Peggy pledged to make an offering of $20 each week.
B. Present and Future Values
Each of the following problems describes an annuity. Determine whether the amount indicated is the annuity’s present value or future value. - Artie bought a policy from an insurance company that will pay him $950 a month guaranteed for the next 20 years. Is the amount he paid a present value or future value?
- The Belcoda Municipal Electric Company expects that in 5 years’ time it will need to make signifi cant upgrades to its equipment. In order to set aside enough money to pay these expenses, the utility has begun depositing $98,000 each
quarter into an investment account each quarter. Is the amount they are trying to accumulate a present or future value?
Copyright © 2008, The McGraw-Hill Companies, Inc.- Find the future value of an annuity due of $502.37 per year for 18 years at 5.2%.
- Suppose that you deposit $3,250 into a retirement account today, and vow to do the same on this date every year. Suppose that your account earns 7.45%. How much will your deposits have grown to in 30 years?
- a. Lisa put $84.03 each month into an account that earned 10.47% for 29 years. How much did the account end up being worth?
b. If Lisa had made her deposits at the beginning of each month instead of the end of the month, how much more would she have wound up with?
F. Differing Payment and Compounding Frequencies (Optional)- Find the future value of an ordinary annuity of $375 per month for 20 years assuming an interest rate of 7.11% compounded daily.
- Find the future value of an ordinary annuity of $777.25 per quarter for 20 years, assuming an interest rate of 9% compounded annually, and assuming interest is paid on payments made between compoundings.
- Repeat Problem 29, assuming instead that no interest is paid on between-compounding payments.
G. Grab Bag - Anders put $103.45 each month in a long-term investment account that earned 8.39% for 32 years. How much total interest did he earn?
- J.J. deposits $125 at the start of each month into an investment account paying 7¼%. Assuming he keeps this up, how much will he have at the end of 30 years?
- A fisaved on a regular basis over time can grow into surprisingly large amounts. She is thinking of using the following example: nancial planner is making a presentation to a community group. She wants to make the point that small amounts
Suppose you spend $3.25 every morning on a cup of gourmet coffee, but instead decide to put that $3.25 into an investment account that earns 9%, which falls well within the average long-term growth rate of the investments my fi rm
offers. How much do you think that account could grow to in 40 years?Calculate the answer to her question. - Find the future value of a 25-year annuity due if the payments are $500 semiannually and the interest rate is 3.78%.
- How much interest will I earn if I deposit $45.95 each month into an account that pays 6.02% for 10 years? For 20 years? For 40 years?
- Find the future value annuity factor for an ordinary annuity with monthly payments for 22 years and an 8^5 ⁄ 8 % interest rate.
Exercises 4.2 161Walkthrough ixCopyright © 2008, The McGraw-Hill Companies, Inc.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 fideposits $750 at the end of each year into an account earning 8¾% for 25 years. How much will he end up with? How nancial advisor has encouraged him to start putting money into a retirement account. Suppose that Jack
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.
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