CHAPTER 8
The Time Value of Money
THINKING BEYOND THE QUESTION
How much will it cost to borrow money?
A shorter borrowing period means that a creditor will receive repayment
of a loan sooner than if the borrowing period is longer. Consequently, the
money that is borrowed is at risk over a shorter period. The lender has
QUESTIONS
Q8-1 Future value and present value are based on the concept of interest. Both
recognize that, when invested at a rate of interest, an amount of money
will grow to a larger amount as time passes. Future value is the amount
202 Chapter 8
Q8-2 The concept of future value assumes that money has a time value. It as-
sumes, for example, that a specific amount of money invested today will
Q8-3 The concept of present value assumes that money has a time value. It as-
Q8-4 Any interest factor found on Table 1 reveals the amount that $1 will grow
Q8-5 Each year the interest earned will be larger than the interest earned in the
Q8-6 Table 1 reveals what happens to $1 when it is invested at varying periods
and interest rates. As one moves from the upper left corner of the table to
Q8-7 The problem can be solved in two steps using Table 1 only. First, find the
Q8-8 $572,066.55 FVA = Amount × Interest Factor
Q8-9 Any interest factor found on Table 2 reveals the amount that a $1 ordinary
Q8–10 Table 3 reveals what happens to $1 when it is discounted for varying pe-
riods and rates. In general, the farther (in time) that money is away from
The Time Value of Money 203
Q8–11 The problem can be solved in two steps using Table 3 only. First, dis-
Q8–12 Table 4 reveals what happens to an ordinary annuity of $1 when it is dis-
4. This is correct. But, as the length of the annuity increases, the effect of
the higher discount rate is offset by the effect of the additional annuity
Q8–13 If part of Jeraldo’s capital was returned to him at the end of each year
Q8–14 The rows in time value of money tables (whether in textbooks or pro-
grammed into calculators or computers) represent periods rather than
Q8–15 The size of the monthly payment is fixed at $288. When a payment is
made, the interest incurred since the previous payment is deducted first
and the remainder is subtracted from the balance of the loan. Therefore,
204 Chapter 8
EXERCISES
E8-2 Owed on
March 31, 2008
$26,750
FV = $25,000 × (1.07)1 = $26,750
March 31, 2009
$28,622.50
E8-3 a. The future value $10,000 deposited for 25 years at 6% is computed as
follows:
FVA = A × IF
E8-4 a. The future value of a seven-year, 5%, $2,000 annuity is computed as
follows:
E8-5 a. $1,829,252 Future value of a four-year, 9%, $400,000 annuity:
The Time Value of Money 205
c. The future value of an annuity grows from the contribution of addi-
E8-6 a. $226,566 =FV(0.08, 30, –2,000)
b. $237,991 =FV(0.04, 60, –1,000)
E8-7 $925.93 $1,000 ÷ 1.08 = $925.93 (or $1,000 × 0.92593)
E8-8 $740.74 $800 ÷ 1.08 = $740.74 (or $800 × 0.92593)
E8-9 $952.38 $1,000 ÷ 1.05 = $952.38 (or $1,000 × 0.95238)
E8–10 $600.00 The present value of $802.93 at 6% for five years is $802.93 ×
E8–12 $798.54 $200 × 3.99271 (from Table 4, 8%, 5 years) = $798.54
206 Chapter 8
E8–13 a. $408.15 $500 × 0.81630 (from Table 3, 7%, 3 years) = $408.15
E8–14
a. No. The present value of the net cash inflows to be received is less
than the present value of the investment made to obtain those cash
inflows.
Proof:
Expected additional cash inflow per year $50,500
Proof:
To earn exactly 8%, the present value of the future net cash
Current expectations
Expected new cash inflows $ 50,500.00
The Time Value of Money 207
E8–15
b. i. $2,464.51 FV = Amount of annuity × IF (Table 2)
iv. The total of column C is the amount of interest earned over the
six years.
A
C
D
E
Year
Int. Earned
(Col. B ×
Int. Rate)
Amount
Invested at
End of Year
Future Value at
End of Year
(Cols. B + C + D)
0.00
1,413.44
1,878.80
5,212.90
E8–16
a. Present value of Option 1:
= $79,025.80
b. Option 2 is worth more (has a higher present value) even though he
208 Chapter 8
E8–17 a.
Compounding
Frequency
Interest Factor
(Table 1)
Future
Value
Annual
1.25440
$1,254.40
c. The same effect should take place. An annuity is merely a series of
d.
Compounding
Frequency
Interest Factor
(Table 3)
Present
Value
Annual
0.79719
e. The more frequent the compounding period, the smaller is the pre-
f. More frequent compounding reduces the interest earned each peri-
od, but the annuity payments are received more frequently. The total
effect depends on frequency and interest rates.
E8–18 $410.02 (5 years) PVA = $100 × 4.10020 (from Table 4, 5 periods,
The Time Value of Money 209
E8–19 $1,070.24 $80 × 7.02358 (from Table 4, 7%, 10 years) + $1,000 ×
0.50835 (from Table 3, 7%, 10 years) = $561.89 + $508.35 =
pay. At any price above this amount, the company will earn
less than a 12% return on its investment. If the company is
able to pay less than this amount, its rate of return will be
higher than 12%.
b. Year 1 = $90,000 $1,000,000 × 9% = $90,000
Proof:
Period
Present Value
at Beginning
of Period
Interest
Expense
at 9%
Payment
Repayment
of Principal
Value
of Debt at
End of Period
Year 1
$1,000,000
$90,000
$395,055
$305,055
$694,945
210 Chapter 8
c.
ASSETS
=
LIABILITIES
+
OWNERS’ EQUITY
Date
Accounts
Cash
Other
Assets
Contributed
Capital
Retained
Earnings
d.
ASSETS
=
LIABILITIES
+
OWNERS’ EQUITY
Date
Accounts
Cash
Other
Assets
Contributed
Capital
Retained
Earnings
E8–23 a. The size of the payment is solved by the following equation:
b. The loan would be entered into the accounting system as follows:
ASSETS
=
LIABILITIES
+
OWNERS’ EQUITY
The three payments would be entered into the accounting system as
follows:
ASSETS
=
LIABILITIES
+
OWNERS’ EQUITY
Date
Accounts
Cash
Other
Assets
Contributed
Capital
Retained
Earnings
Date
Accounts
Other
Assets
Contributed
Capital
Retained
Earnings
The Time Value of Money 211
ASSETS
=
LIABILITIES
+
OWNERS’ EQUITY
Other
Contributed
Retained
Cash
–18,173
Presentation of an amortization table was not part of the require-
ments. The following table, however, explains the entries to the ac-
counting system above.
Year
Beginning of
Year Balance
Interest
Expense
Cash
Payment
Reduction in
Loan Balance
End of Year
Balance
E8–24
a. PV = Amount × IF
i. $251.89 $300 × 0.83962 (Table 3)
Total $999.48
b. Implications: 1. The present value of an investment decreases as
the time until the investment is received increases.
E8–25
a. $3,851 PV(.11,18,500)
Interest Expense
Loan Payable
212 Chapter 8
PROBLEMS
P8-1
The difference relates to the length of time the money is invested.
Time is a powerful component of the value of money. Even though
an investor may be able to save only small amounts, it is important
to begin investing early rather than wait until later and have to invest
larger amounts or look for higher paying, but riskier investments.
P8-2 Future value of plan #1:
Employee contribution per year $3,000
Future value of plan #2:
Employee contribution per year $2,500
The Time Value of Money 213
Choosing between plan #1 and plan #2:
Effective arguments can be made in favor of either alternative. Plan #1
P8-3
A. Prudence = $16,000 $2,000 on each of 8 birthdays starting
C. $518,113.04 FV annuity = Amount × IF
= $2,000 × IF (40 periods @ 8%)
= $2,000 × 259.05652
= $518,113.04
E. Compound interest is a powerful financial tool when savings are
started early and continue over a long period of time. Here we see
two examples. First, Prudence invested only $16,000 yet ended up
214 Chapter 8
P8–4
A. FV = PV × IF = 44,400 × 2.25219 = 99,997. Almost. She will fall just
short of her goal.
C. 16 years To determine the approximate number of
equal annual deposits of $3,700 to equal
Investment B:
PV = $4,500 × 0.73503 (from Table 3, 8%, 4 years) = $3,308
Investment C:
The Time Value of Money 215
P8-6
Use the future value of the annuity formula in this problem. The
payments can be treated as 10 separate calculations of the future
value of a single amount. The relationship used is: FV = PV × IF (Ta-
ble 1)
Year
PV
Amount
Deposited
Years on
Deposit
IF
Interest
Factor
FV
Future Val-
ue
1
$ 3,000
10
2.36736
$ 7,102.08
2
3,000
9
2.17189
6,515.67
3
3,000
8
1.99256
5,977.68
4
3,000
7
1.82804
5,484.12
5
3,000
6
1.67710
5,031.30
6
3,000
5
1.53862
4,615.86
7
3,000
4
1.41158
4,234.74
8
3,000
3
1.29503
3,885.09
9
3,000
2
1.18810
3,564.30
1
1.09000
$49,680.84
C. The amounts are larger in part B because the amount of interest
earned is greater. The reason for the greater amount of interest is
that each $3,000 IRA contribution was on deposit one extra period.
216 Chapter 8
P8-7
(a)
(b)
(c)
Group
Number
of
employees
Annual
pension
benefit
Years
to be
paid
Cash
required
5
10,000
5
10,000
5
dates. The present value of each group’s benefits, if
payments started at year-end 2007, is obtained by
multiplying column (a) by column (b) by column (c).
do start.
(a)
(b)
(c)
(d)
(e)
(f)
benefit
Annual
pension
PV factor for
5 period
PV @
year-end
PV factor
for 4, 9 or
Year-end
2007 pension
C.
ASSETS
=
LIABILITIES
+
OWNERS’ EQUITY
Date
Accounts
Cash
Other
Assets
Contributed
Capital
Retained
Earnings
P8-8 A. and B.
Monthly rate
(annual ÷ 12)
Number
of periods
Amount
borrowed
Monthly
payment
The Time Value of Money 217
C. Responses will vary. Many students will automatically select the op-
D. Dealer financing (72 payments of $721.21) $51,927.12
E.
Total payments
Amount
borrowed
Interest
Dealer financing
$51,927.12
$35,000
$16,927.12
Bank financing
Credit union financing
P8-9
D. If the Taylors can afford the additional $305.31 per month, they will