CHAPTER 3
THE TIME VALUE OF MONEY
Teaching Guides for Questions and Problems in the Text
QUESTIONS
Perhaps no concept causes more difficulty for students of finance than the time value of
money. Chapter 3 presents and illustrates this concept (i.e., future value and present value).
The problems range from easy to moderately hard and should help build students’
confidence in the use of interest tables, financial calculators, or spreadsheets. Background
in and retention of time value concepts varies considerably among students. In my
investments class, I assume prior knowledge and do not teach the material. Sufficient
problems are assigned to consume one class, and three or four problems are used on the
first test as a means of keeping the students honest.
3-1. A lump sum payment is a single payment while an annuity is a series of equal, annual
payments. IF the payments are not equal, the series is not an annuity. The equal payments
may occur at the beginning of the time period (an annuity due) or may occur at the end of
each time period (an ordinary annuity). You may also point out that a series of unequal
3-2. Compounding (determination of future value) is taking an amount from the present
3-3. a. and b. As time increases, more interest is being earned over the longer time period
c. As time increases, the amount received is further into the future and worth less now.
d. As time increases, the present value of each additional payment is less than the
3-4. a. and b. As the interest rate increases, more interest is being earned in each time
c. and d. The higher the current rate, the more desirable to have the funds now, hence
3-5. Assets generate cash flows in the future. The present value depends on both the
PROBLEMS
Answers to problems may differ resulting from rounding errors from using the interest
tables or from using a financial calculator set to two decimal places. The following
explanations employ both interest tables and a financial calculator.
3-1. This problem illustrates the difference between compounding and simple interest.
a. $1,000(1 + .04)20 = x
b. If the interest is withdrawn each year, the investor receives $40 annually and $800 over
the lifetime of the investment.
c. The difference in the amounts of interest ($1,191 – $800 = $391) is the result of
compounding.
3-2. a. x = $1,500(40.955) = $61,493
b. x = $61,493(1.403) = $86,275
c. x = $1,500(63.249) = $94,874
d. Leaving the funds in the account and continuing the annual contribution for only five
3-3. a. This problem also illustrates the future value of a sum of an annuity except the
amount of the annual contribution is the unknown and the future sum is known.
14.487 is the interest factor for the future sum of an annuity of $1 at 8 percent for ten
years. (PV = 0; N = 10; I = 8; FV = 100,000, and PMT = ? = -6,902.95.)
b. This problem illustrates the impact of a lower rate of interest. For the ordinary annuity:
12.578 is the interest factor for the future sum of an annuity of $1 at 5 percent for ten
3-4. This problem is an introduction to valuation. It asks what is the present value of a
series of future payments.
3-5. This problem illustrates the potential impact of inflation.
If inflation continues at 2 percent annually for twenty years, salaries must rise to $66,870
to maintain the purchasing power of $45,000. At a 4 percent annual rate of inflation,
salaries must rise to $98,595 to maintain the purchasing power of $45,000:
3-6. If a student uses the interest tables, this problem is one of the hardest problems of the
set because it involves two separate questions. First, the present value of the $30,000 must
be determined:
9.818 is the interest factor for the present value of an annuity of $1 for twenty years at 8
percent. The individual may spend $19,714 annually for the next twenty years if the
available funds are $193,550 and 8 percent is the rate earned on the declining balance.
3-7. The first and second parts of this problem illustrate the future value of an annuity due.
a. x = $2,000(40.995)(1.07) = $87,729
b. x = $2,000(57.275)(1.1) = $126,005
c. The funds are being withdrawn at the beginning, so the problem illustrates the present
value of an annuity due.
3-8. This question illustrates the impact of different rates.
b. $150(6.610) = $991.50 > $900
3-9. a. Annual compounding: $100(1 + .12)1 = $112
b. Annual compounding:
3-10. This problem is designed to illustrate the impact of com- pounding an annuity due
vis-à-vis an ordinary annuity. The problem concerns a retirement account. The two
individuals make the same annual contribution for the same number of years and earn the
same rate of interest. The only difference is the timing of the payment.
3-11. If you consider life insurance as an investment, than the question becomes how long
will it take for the $35,000 to grow into $100,000. That is
Find the interest factor 2.857 for 9 percent in the future value of $1 table. That gives the
answer to be approximately 12 percent annually. (PV = -35000; I = 9; PMT = 0, FV =
100000, and N = ? = 12.18)
Find the interest factor 2.875 for ten years in the future value of $1 table. That gives the
answer to be approximately 11 percent annually (PV = -35000; N = 10; PMT = 0, FV =
100000, and I = ? = 11.07)
3-12. This problem asks for the present value of an annuity that commences five years into
the future. The first step is to determine the present value of the annuity and then to
determine the present value of that lump sum. That is
3-14. The easiest means to approach this problem (when using interest tables) is to treat
the payments as two $1,000 annuities, one annuity for twenty years and one for ten years.
Determine the terminal value of each annuity and then add the amounts:
3-15. This problem may also be treated in more than one way. The easiest is probably to
consider the present value of one $5,000 annuity for twelve years plus one $5,000 annuity
for six years. The present value then is