Text cash for annuity payments

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```					Chapter 06 - Time Value of Money Concepts

Chapter 6    th
Time Value of Money Concepts
(Spiceland 6 ed)

QUESTIONS FOR REVIEW OF KEY TOPICS
Question 6-1
Interest is the amount of money paid or received in excess of the amount borrowed or lent.

Question 6-2
Compound interest includes interest not only on the original invested amount but also on the
accumulated interest from previous periods.

Question 6-3
If interest is compounded more frequently than once a year, the effective rate or yield will be
higher than the annual stated rate.

Question 6-4
The three items of information necessary to compute the future value of a single amount are the
original invested amount, the interest rate (i) and the number of compounding periods (n).

Question 6-5
The present value of a single amount is the amount of money today that is equivalent to a given
amount to be received or paid in the future.

Question 6-6
Monetary assets and monetary liabilities represent cash or fixed claims/commitments to
receive/pay cash in the future and are valued at the present value of these fixed cash flows. All other
assets and liabilities are nonmonetary.

Question 6-7
An annuity is a series of equal-sized cash flows occurring over equal intervals of time.

Question 6-8
An ordinary annuity exists when the cash flows occur at the end of each period. In an annuity
due the cash flows occur at the beginning of each period.

Question 6-9
Table 2 lists the present value of \$1 factors for various time periods and interest rates. The
factors in Table 4 are simply the summation of the individual PV of \$1 factors from Table 2.

6-1
Chapter 06 - Time Value of Money Concepts

Question 6-10
Present
Value
?
0         Year 1      Year 2      Year 3     Year 4

___________________________________________
\$200        \$200        \$200           \$200
n = 4, i = 10%

Question 6-11
Present
Value
?
0         Year 1      Year 2      Year 3     Year 4

___________________________________________
\$200       \$200         \$200        \$200
n = 4, i = 10%

Question 6-12
A deferred annuity exists when the first cash flow occurs more than one period after the date
the agreement begins.

Question 6-13
The formula for computing present value of an ordinary annuity incorporating the ordinary
annuity factors from Table 4 is:
PVA = Annuity amount x Ordinary annuity factor
Solving for the annuity amount,
PVA
Annuity amount = Ordinary annuity factor
The annuity factor can be obtained from Table 4 at the intersection of the 8% column and 5
period row.

Question 6-14
\$500
Annuity amount =          3.99271
Annuity amount =          \$125.23

6-2
Chapter 06 - Time Value of Money Concepts

Question 6-15
Companies frequently acquire the use of assets by leasing rather than purchasing them. Leases
usually require the payment of fixed amounts at regular intervals over the life of the lease. Certain
long-term, noncancelable leases are treated in a manner similar to an installment sale by the lessor
and an installment purchase by the lessee. In other words, the lessor records a receivable and the
lessee records a liability for the several installment payments. For the lessee, this requires that the
leased asset and corresponding lease liability be valued at the present value of the lease payments.

6-3
Chapter 06 - Time Value of Money Concepts

BRIEF EXERCISES
Brief Exercise 6-1
Fran should choose the second investment opportunity. More rapid compounding
has the effect of increasing the actual rate, which is called the effective rate, at which
money grows per year. For the second opportunity, there are four, three-month
periods paying interest at 2% (one-quarter of the annual rate). \$10,000 invested will
grow to \$10,824 (\$10,000 x 1.0824*). The effective annual interest rate, often
referred to as the annual yield, is 8.24% (\$824 ÷ \$10,000), compared to just 8% for
the first opportunity.

* Future value of \$1: n=4, i=2% (from Table 1)

Brief Exercise 6-2
Bill will not have enough accumulated to take the trip. The future value of his
investment of \$23,153 is \$347 short of \$23,500.

FV = \$20,000 (1.15763* ) = \$23,153
* Future value of \$1: n=3, i=5% (from Table 1)

Brief Exercise 6-3

FV factor =         \$26,600 = 1.33*
\$20,000
* Future value of \$1: n=3, i=? (from Table 1, i = approximately 10%)

Brief Exercise 6-4
John would be willing to invest no more than \$12,673 in this opportunity.

PV = \$16,000 (.79209* ) = \$12,673
* Present value of \$1: n=4, i=6% (from Table 2)

Brief Exercise 6-5

PV factor       = \$13,200 = .825*
\$16,000
* Present value of \$1: n=4, i=? (from Table 2, i = approximately 5%)

6-4
Chapter 06 - Time Value of Money Concepts

Brief Exercise 6-6
Interest is paid for 12 periods at 1% (one-quarter of the annual rate).

FVA           = \$500 (12.6825* )         = \$6,341
* Future value of an ordinary annuity of \$1: n=12, i=1% (from Table 3)

Brief Exercise 6-7
Interest is paid for 12 periods at 1% (one-quarter of the annual rate).

FVAD          = \$500 (12.8093* )         = \$6,405
* Future value of an annuity due of \$1: n=12, i=1% (from Table 5)

Brief Exercise 6-8

PVA           = \$10,000 (4.10020* )                = \$41,002
* Present value of an ordinary annuity of \$1: n=5, i=7% (from Table 4)

Brief Exercise 6-9
PVAD          = \$10,000 (4.38721*)                 = \$43,872
* Present value of an annuity due of \$1: n=5, i=7% (from Table 6)

Brief Exercise 6-10
PVA = \$10,000                x         4.10020*           =         \$41,002
* Present value of an ordinary annuity of \$1: n=5, i=7% (from Table 4)

PV      = \$41,002           x         .87344*             =         \$35,813
* Present value of \$1: n=2, i=7% (from Table 2)

Or alternatively:
From Table 4,
PVA factor, n=7, i=7%                                     =       5.38929
– PVA factor, n=2, i=7%                                =       1.80802
= PV factor for deferred annuity                        =       3.58127
PV = \$10,000 x 3.58127 = \$35,813 (rounded)

6-5
Chapter 06 - Time Value of Money Concepts

Brief Exercise 6-11
Annuity       = \$100,000          = \$14,903 = Payment
6.71008*
* Present value of an ordinary annuity of \$1: n=10, i=8% (from Table 4)

Brief Exercise 6-12

PV = \$6,000,0001 (12.40904* ) + 100,000,000 (.13137** )

PV = \$74,454,240 + 13,137,000 = \$87,591,240 = price of the bonds
1
\$100,000,000 x 6% = \$6,000,000
* Present value of an ordinary annuity of \$1: n=30, i=7% (from Table 4)
** Present value of \$1: n=30, i=7% (from Table 2)

Brief Exercise 6-13

PVAD = \$55,000 (7.24689* ) = \$398,579 = Liability
* Present value of an annuity due of \$1: n=10, i=8% (from Table 6)

6-6
Chapter 06 - Time Value of Money Concepts

EXERCISES
Exercise 6-1
1. FV = \$15,000 (2.01220* ) = \$30,183
* Future value of \$1: n=12, i=6% (from Table 1)

2. FV = \$20,000 (2.15892* ) = \$43,178
* Future value of \$1: n=10, i=8% (from Table 1)

3. FV = \$30,000 (9.64629* ) = \$289,389
* Future value of \$1: n=20, i=12% (from Table 1)

4. FV = \$50,000 (1.60103* ) = \$80,052
* Future value of \$1: n=12, i=4% (from Table 1)

Exercise 6-2
1. FV = \$10,000 (2.65330* ) = \$26,533
* Future value of \$1: n=20, i=5% (from Table 1)

2. FV = \$10,000 (1.80611* ) = \$18,061
* Future value of \$1: n=20, i=3% (from Table 1)

3. FV = \$10,000 (1.81136* ) = \$18,114
* Future value of \$1: n=30, i=2% (from Table 1)

Exercise 6-3
1. PV = \$20,000 (.50835* ) = \$10,167
* Present value of \$1: n=10, i=7% (from Table 2)

2. PV = \$14,000 (.39711* ) = \$5,560
* Present value of \$1: n=12, i=8% (from Table 2)

3. PV = \$25,000 (.10367* ) = \$2,592
* Present value of \$1: n=20, i=12% (from Table 2)

4. PV = \$40,000 (.46651* ) = \$18,660
* Present value of \$1: n=8, i=10% (from Table 2)

6-7
Chapter 06 - Time Value of Money Concepts

Exercise 6-4
PV of \$1
Payment           i=8%           PV           n
First payment:             \$5,000     x    .92593    =   \$ 4,630        1
Second payment              6,000     x    .85734    =     5,144        2
Third payment               8,000     x    .73503    =     5,880        4
Fourth payment              9,000     x    .63017    =     5,672        6
Total                        \$21,326

Exercise 6-5
PV = \$85,000 (.82645* ) = \$70,248 = Note/revenue
* Present value of \$1: n=2, i=10% (from Table 2)

Exercise 6-6
1.      PV = \$40,000 (.62092* ) = \$24,837
* Present value of \$1: n=5, i=10% (from Table 2)

2.      \$36,289       =      .55829*
\$65,000
* Present value of \$1: n=10, i=? (from Table 2, i = approximately 6%)

3.      \$15,884       =      .3971*
\$40,000
* Present value of \$1: n=?, i=8% (from Table 2, n = approximately 12 years)

4.       \$46,651 =           .46651*
\$100,000
* Present value of \$1: n=8, i=? (from Table 2, i = approximately 10%)

5.      FV = \$15,376 (3.86968* ) = \$59,500
* Future value of \$1: n=20, i=7% (from Table 1)

6-8
Chapter 06 - Time Value of Money Concepts

Exercise 6-7
1.    FVA        = \$2,000 (4.7793* ) = \$9,559
* Future value of an ordinary annuity of \$1: n=4, i=12% (from Table 3)

2.    FVAD = \$2,000 (5.3528* ) = \$10,706
* Future value of an annuity due of \$1: n=4, i=12% (from Table 5)

3.                                        FV of \$1
Deposit            i=3%            FV             n
First deposit:           \$2,000       x    1.60471   =   \$ 3,209         16
Second deposit            2,000       x    1.42576   =     2,852         12
Third deposit             2,000       x    1.26677   =     2,534          8
Fourth deposit            2,000       x    1.12551   =     2,251          4
Total                         \$10,846

4.     \$2,000 x 4 = \$8,000

6-9
Chapter 06 - Time Value of Money Concepts

Exercise 6-8
1.    PVA        = \$5,000 (3.60478* )               = \$18,024
* Present value of an ordinary annuity of \$1: n=5, i=12% (from Table 4)

2.    PVAD = \$5,000 (4.03735* )                     = \$20,187
* Present value of an annuity due of \$1: n=5, i=12% (from Table 6)

3.                                        PV of \$1
Payment            i = 3%           PV             n
First payment:           \$5,000       x    .88849     =   \$ 4,442          4
Second payment            5,000       x    .78941     =     3,947          8
Third payment             5,000       x    .70138     =     3,507         12
Fourth payment            5,000       x    .62317     =     3,116         16
Fifth payment             5,000       x    .55368     =     2,768         20
Total                          \$17,780

6-10
Chapter 06 - Time Value of Money Concepts

Exercise 6-9
1.      PVA = \$3,000 (3.99271* ) = \$11,978
* Present value of an ordinary annuity of \$1: n=5, i=8% (from Table 4)

2.      \$242,980 =           3.23973*
\$75,000
* Present value of an ordinary annuity of \$1: n=4, i=? (from Table 4, i =
approximately 9%)

3.      \$161,214 =           8.0607*
\$20,000
* Present value of an ordinary annuity of \$1: n=?, i= 9% (from Table 4, n =
approximately 15 years)

4.      \$500,000 =           6.20979*
\$80,518
* Present value of an ordinary annuity of \$1: n=8, i=? (from Table 4, i =
approximately 6%)

5.      \$250,000 =           \$78,868
3.16987*
* Present value of an ordinary annuity of \$1: n=4, i=10% (from Table 4)

6-11
Chapter 06 - Time Value of Money Concepts

Exercise 6-10
Requirement 1
PV = \$100,000 (.68058* ) = \$68,058
* Present value of \$1: n=5, i=8% (from Table 2)

Requirement 2
Annuity amount = \$100,000
5.8666*
* Future value of an ordinary annuity of \$1: n=5, i=8% (from Table 3)

Annuity amount           = \$17,046
Requirement 3
Annuity amount = \$100,000
6.3359*
* Future value of an annuity due of \$1: n=5, i=8% (from Table 5)

Annuity amount = \$15,783

6-12
Chapter 06 - Time Value of Money Concepts

Exercise 6-11
1. Choose the option with the highest present value.

(1) PV = \$64,000

(2) PV = \$20,000 + \$8,000 (4.91732* )
* Present value of an ordinary annuity of \$1: n=6, i=6% (from Table 4)

PV = \$20,000 + \$39,339 = \$59,339

(3) PV = \$13,000 (4.91732* ) = \$63,925

Alex should choose option (1).

2. FVA = \$100,000 (13.8164* ) = \$1,381,640
* Future value of an ordinary annuity of \$1: n=10, i=7% (from Table 3)

Exercise 6-12
PVA = \$5,000                x         4.35526*       =       \$21,776
* Present value of an ordinary annuity of \$1: n=6, i=10% (from Table 4)

PV      = \$21,776           x         .82645*        =       \$17,997
* Present value of \$1: n=2, i=10% (from Table 2)

Or alternatively:
From Table 4,
PVA factor, n=8, i=10%                               =     5.33493
– PVA factor, n=2, i=10%                          =     1.73554
= PV factor for deferred annuity                   =     3.59939

PV = \$5,000 x 3.59939 = \$17,997

6-13
Chapter 06 - Time Value of Money Concepts

Exercise 6-13
Annuity = \$20,000 – 5,000 = \$670 = Payment
22.39646*
* Present value of an ordinary annuity of \$1: n=30, i=2% (from Table 4)

Exercise 6-14
PVA factor = \$100,000 = 7.46938*
\$13,388
* Present value of an ordinary annuity of \$1: n=20, i=? (from Table 4, i =
approximately 12%)

Exercise 6-15
Annuity =            \$12,000        = \$734 = Payment
16.35143*
* Present value of an ordinary annuity of \$1: n=20, i=2% (from Table 4)
5 years x 4 quarters = 20 periods
8% ÷ 4 quarters = 2%

6-14
Chapter 06 - Time Value of Money Concepts

Exercise 6-16
PV      =        ?           x        .90573*          =   1,200

PV      =     \$1,200 =           \$1,325
.90573*
* Present value of \$1: n=5, i=2% (from Table 2)

PVA =             ?          x        14.99203*            =       \$1,325
annuity amount

PVA =          \$1,325   =             \$88   =      Payment
14.99203*
* Present value of an ordinary annuity of \$1: n=18, i=2% (from Table 4)

Exercise 6-17
To determine the price of the bonds, we calculate the present value of the 40-
period annuity (40 semiannual interest payments of \$12 million) and the lump-sum
payment of \$300 million paid at maturity using the semiannual market rate of interest
of 5%. In equation form,

PV = \$12,000,0001 (17.15909* ) + 300,000,000 (.14205** )
PV = \$205,909,080 + 42,615,000 = \$248,524,080 = price of the bonds
1
\$300,000,000 x 4 % = \$12,000,000
* Present value of an ordinary annuity of \$1: n=40, i=5% (from Table 4)
** Present value of \$1: n=40, i=5% (from Table 2)

6-15
Chapter 06 - Time Value of Money Concepts

Exercise 6-18
Requirement 1
To determine the price of the bonds, we calculate the present value of the 30-
period annuity (30 semiannual interest payments of \$6 million) and the lump-sum
payment of \$200 million paid at maturity using the semiannual market rate of interest
of 2.5%. In equation form,

PV = \$6,000,0001 (20.93029* ) + 200,000,000 (.47674)
PV = \$125,581,740 + 95,348,000 = \$220,929,740 = price of the bonds
1
\$200,000,000 x 3 % = \$6,000,000
* Present value of an ordinary annuity of \$1: n=30, i=2.5% (from Table 4)
** Present value of \$1: n=30, i=2.5% (from Table 2)

Requirement 2
\$220,929,740 x 2.5% = \$5,523,244

Because the bonds were outstanding only for six months of the year, Singleton
reports only ½ year’s interest in 2011.

Exercise 6-19
Requirement 1
PVA = \$400,000 (10.59401* ) = \$4,237,604 = Liability
* Present value of an ordinary annuity of \$1: n=20, i=7% (from Table 4)

Requirement 2
PVAD = \$400,000 (11.33560* ) = \$4,534,240 = Liability
* Present value of an annuity due of \$1: n=20, i=7% (from Table 6)

Exercise 6-20
PVA factor = \$2,293,984 = 11.46992*
\$200,000
* Present value of an ordinary annuity of \$1: n=20, i=? (from Table 4, i = 6%)

6-16
Chapter 06 - Time Value of Money Concepts

Exercise 6-21
List A                         List B

e     1. Interest                    a. First cash flow occurs one period after
agreement begins.
m 2.       Monetary asset            b. The rate at which money will actually grow
during a year.
j     3.   Compound interest         c. First cash flow occurs on the first day of the
agreement.
i     4.   Simple interest           d. The amount of money that a dollar will grow
to.
k    5.    Annuity                   e. Amount of money paid/received in excess of
amount borrowed/lent.
l     6.   Present value of a single f. Obligation to pay a sum of cash, the amount
of amount which is fixed.
c    7.    Annuity due               g. Money can be invested today and grow to a
larger amount.
d    8.    Future value of a single h. No fixed dollar amount attached.
amount
a    9.    Ordinary annuity          i. Computed by multiplying an invested amount
by the interest rate.
b 10.      Effective rate or yield   j. Interest calculated on invested amount plus
accumulated interest.
h 11.      Nonmonetary asset         k. A series of equal-sized cash flows.
g 12.      Time value of money       l. Amount of money required today that is
equivalent to a given future amount.
f 13.      Monetary liability       m. Claim to receive a fixed amount of money.

6-17
Chapter 06 - Time Value of Money Concepts

CPA / CMA REVIEW QUESTIONS
CPA Exam Questions

1. b. PV = FV x PV factor,
PV=\$25,458 x 0.3075 = \$7,828
2. d. The sales price is equal to the present value of the note payments:

Present value of first payment         \$ 60,000
Present value of last six payments:
\$60,000 x 4.36                        261,600
Sales price                            \$321,600
3. a. PVA = \$100 x 4.96764 = \$497
4. b. First solve for present value of a four-year ordinary annuity:
PVA = \$100 x 3.03735 = \$304
Then discount back two years:
PV = \$304 x 0.79719 = \$242
5. d. PVAD = \$100,000 x 9.24424 = \$924,424

6. a. PVA = \$100 x 5.65022 = \$565 (present value of the interest payments)
PV = \$1,000 x 0.32197 = \$322 (present value of the face amount)
Total present value = \$887 = current market value of the bond
7. a. PVA = PMT x PVA factor
\$15,000 = PMT x 44.955
PMT = \$334

6-18
Chapter 06 - Time Value of Money Concepts

CMA Exam Questions

1. d. Both future value tables will be used because the \$75,000 already in the
account will be multiplied times the future value factor of 1.26 to determine
the amount 3 years hence, or \$94,500. The three payments of \$4,000
represent an ordinary annuity. Multiplying the three-period annuity factor
(3.25) by the payment amount (\$4,000) results in a future value of the
annuity of \$13,000. Adding the two elements together produces a total
account balance of \$107,500.

2. a. An annuity is a series of cash flows or other economic benefits occurring at
fixed intervals, ordinarily as a result of an investment. Present value is the
value at a specified time of an amount or amounts to be paid or received
later, discounted at some interest rate. In an annuity due, the payments
occur at the beginning, rather than at the end, of the periods. Thus, the
present value of an annuity due includes the initial payment at its
undiscounted amount. This lease should be evaluated using the present
value of an annuity due.

6-19
Chapter 06 - Time Value of Money Concepts

PROBLEMS
Problem 6-1
Choose the option with the lowest present value of cash outflows, net of the
present value of any cash inflows (Cash outflows are shown as negative amounts;
cash inflows as positive amounts).

Machine A:

PV = – \$48,000 – 1,000 (6.71008* ) + 5,000 (.46319** )
* Present value of an ordinary annuity of \$1: n=10, i=8% (from Table 4)
** Present value of \$1: n=10, i=8% (from Table 2)

PV = – \$48,000 – 6,710 + 2,316

PV = - \$52,394

Machine B:
PV = – \$40,000 – 4,000 (.79383) – 5,000 (.63017) – 6,000 (.54027)
PV of \$1: i=8%                        n=3               n=6               n=8
(from Table 2)

PV = - \$40,000 - 3,175 - 3,151 - 3,242
PV = - \$49,568
Esquire should purchase machine B.
Problem 6-2
1. PV = \$10,000 + 8,000 (3.79079* ) = \$40,326 = Equipment
* Present value of an ordinary annuity of \$1: n=5, i=10% (from Table 4)

2. \$400,000 = Annuity amount x 5.9753*
* Future value of an annuity due of \$1: n=5, i=6% (from Table 5)

Annuity amount = \$400,000
5.9753
Annuity amount = \$66,942 = Required annual deposit
3. PVAD = \$120,000 (9.36492* ) = \$1,123,790 = Lease liability
* Present value of an annuity due of \$1: n=20, i=10% (from Table 6)

6-20
Chapter 06 - Time Value of Money Concepts

Problem 6-3
Choose the option with the lowest present value of cash payments.

1. PV = \$1,000,000

2. PV = \$420,000 + 80,000 (6.71008* ) = \$956,806
* Present value of an ordinary annuity of \$1: n=10, i=8% (from Table 4)

3. PV = PVAD = \$135,000 (7.24689* ) = \$978,330
* Present value of an annuity due of \$1: n=10, i=8% (from Table 6)

4. PV = \$1,500,000 (.68058* ) = \$1,020,870
* Present value of \$1: n=5, i=8% (from Table 2)

Harding should choose option 2.

Problem 6-4
The restaurant should be purchased if the present value of the future cash
flows discounted at 10% rate is greater than \$800,000.

PV = \$80,000 (4.35526* ) + 70,000 (.51316** ) + 60,000 (.46651**)
n=7                     n=8

+ \$50,000 (.42410**) + 40,000 (.38554**) + 700,000 (.38554**)
n=9                     n=10                      n=10

* Present value of an ordinary annuity of \$1: n=6, i=10% (from Table 4)
** Present value of \$1:, i=10% (from Table 2)

PV = \$718,838 < \$800,000

Since the PV is less than \$800,000, the restaurant should not be purchased.

6-21
Chapter 06 - Time Value of Money Concepts

Problem 6-5
The maximum amount that should be paid for the store is the present value of the
estimated cash flows.

Years 1-5:
PVA = \$70,000                x        3.99271* =                  \$279,490
* Present value of an ordinary annuity of \$1: n=5, i=8% (from Table 4)

Years 6-10:

PVA = \$70,000                x        3.79079* =                  \$265,355
* Present value of an ordinary annuity of \$1: n=5, i=10% (from Table 4)

PV      = \$265,355           x        .68058*      =              \$180,595
* Present value of \$1: n=5, i=8% (from Table 2)

Years 11-20:

PVA = \$70,000                x        5.65022*     =              \$395,515
* Present value of an ordinary annuity of \$1: n=10, i=12% (from Table 4)

PV      = \$395,515           x        .62092*      =              \$245,583
* Present value of \$1: n=5, i=10% (from Table 2)

PV      = \$245,583           x        .68058*      =              \$167,139
* Present value of \$1: n=5, i=8% (from Table 2)

End of Year 20:

PV      = \$400,000           x    .32197* x .62092 x .68058 =       \$54,424
* Present value of \$1: n=10, i=12% (from Table 2)

Total PV = \$279,490 + 180,595 + 167,139 + 54,424 =                \$681,648

The maximum purchase price is \$681,648.

6-22
Chapter 06 - Time Value of Money Concepts

Problem 6-6
1.
PV of \$1 factor = \$30,000 = .5000*
\$60,000
* Present value of \$1: n=? , i=8% (from Table 2, n = approximately 9 years)

2.
PVA
Annuity factor = Annuity amount

Annuity factor = \$28,700 = 4.1000*
\$7,000
* Present value of an ordinary annuity of \$1: n= 5, i=? (from Table 4, i =
approximately 7%)

3.
PVA
Annuity amount = Annuity factor

Annuity amount = \$10,000 = \$1,558                   = Payment
6.41766*
* Present value of an ordinary annuity of \$1: n=10, i=9% (from Table 4)

6-23
Chapter 06 - Time Value of Money Concepts

Problem 6-7
Requirement 1
PVA
Annuity amount = Annuity factor

Annuity amount = \$250,000 = \$78,868 = Payment
3.16987*
* Present value of an ordinary annuity of \$1: n=4, i=10% (from Table 4)

Requirement 2
PVA
Annuity amount = Annuity factor

Annuity amount = \$250,000 = \$62,614 = Payment
3.99271*
* Present value of an ordinary annuity of \$1: n=5, i=8% (from Table 4)

Requirement 3
PVA
Annuity factor = Annuity amount

Annuity factor = \$250,000 = 4.86845*
\$51,351
* Present value of an ordinary annuity of \$1: n=? , i= 10% (from Table 4, n = approximately 7
payments)

Requirement 4
PVA
Annuity factor = Annuity amount

Annuity factor = \$250,000 = 2.40184*
\$104,087
* Present value of an ordinary annuity of \$1: n= 3, i= ? (from Table 4, i = approximately
12%)

6-24
Chapter 06 - Time Value of Money Concepts

Problem 6-8
Requirement 1
Present value of payments 4-6:

PVA = \$40,000                x        2.48685*    =             \$99,474
* Present value of an ordinary annuity of \$1: n= 3, i= 10% (from Table 4)

PV      = \$99,474            x        .75131*     =             \$74,736
* Present value \$1: n= 3, i= 10% (from Table 2)

Present value of all payments:

\$ 62,171 (PV of payments 1-3: \$25,000                 x   2.48685* )

74,736 (PV of payments 4-6 calculated above)
\$136,907

The note payable and corresponding building should be recorded at \$136,907.

Or alternatively:

PV = \$25,000 (2.48685* ) + 40,000 (1.86841** ) = \$136,907
* Present value of an ordinary annuity of \$1: n=3, i=10% (from Table 4)

From Table 4,
PVA factor, n=6, i=10%                     = 4.35526
– PVA factor, n=3 i=10%                    = 2.48685
= PV factor for deferred annuity           = 1.86841**

Requirement 2

\$136,907 x 10% = \$13,691 = Interest in the year 2011

6-25
Chapter 06 - Time Value of Money Concepts

Problem 6-9
Choose the alternative with the highest present value.

Alternative 1:

PV = \$180,000

Alternative 2:

PV = PVAD = \$16,000 (11.33560* ) = \$181,370
* Present value of an annuity due of \$1: n=20, i=7% (from Table 6)

Alternative 3:

PVA = \$50,000                x        7.02358*       =            \$351,179
* Present value of an ordinary annuity of \$1: n=10, i=7% (from Table 4)

PV      = \$351,179           x         .54393*       =            \$191,017
* Present value of \$1: n=9, i=7% (from Table 2)

John should choose alternative 3.

Or, alternatively (for 3):

PV = \$50,000 (3.82037* )               = \$191,019
(difference due to rounding)

From Table 4,
PVA factor, n=19, i=7%                       = 10.33560
– PVA factor, n=9, i=7%                      = 6.51523
= PV factor for deferred annuity             = 3.82037*

or, From Table 6,

PVAD factor, n=20, i=7%                      = 11.33560
— PVAD factor, n=10, i=7%                    = 7.51523
= PV factor for deferred annuity             = 3.82037*

6-26
Chapter 06 - Time Value of Money Concepts

Problem 6-10
PV = \$20,000 (3.79079* ) + 100,000 (.62092** ) = \$137,908
* Present value of an ordinary annuity of \$1: n=5, i=10% (from Table 4)
** Present value of \$1: n=5, i=10% (from Table 2)

The note payable and corresponding merchandise should be recorded at \$137,908.

6-27
Chapter 06 - Time Value of Money Concepts

Problem 6-11
Requirement 1
PVAD = Annuity amount x Annuity factor

Annuity amount = Annuity factor

Annuity amount = \$800,000
7.24689*
* Present value of an annuity due of \$1: n=10, i=8% (from Table 6)

Annuity amount = \$110,392 = Lease payment
Requirement 2
Annuity amount = \$800,000
6.71008*
* Present value of an ordinary annuity of \$1: n=10, i=8% (from Table 4)

Annuity amount = \$119,224 = Lease payment
Requirement 3
PVAD = (Annuity amount x Annuity factor) + PV of residual

Annuity amount =            Annuity factor

PV of residual = \$50,000              x   .46319*   = \$23,160
* Present value of \$1: n=10, i=8% (from Table 2)

Annuity amount = \$800,000 – 23,160
7.24689*
* Present value of an annuity due of \$1: n=10, i=8% (from Table 6)

Annuity amount = \$107,196 = Lease payment

6-28
Chapter 06 - Time Value of Money Concepts

Problem 6-12
Requirement 1
PVA = Annuity amount x Annuity factor

PVA
Annuity amount = Annuity factor

Annuity amount = \$800,000
7.36009*
* Present value of an ordinary annuity of \$1: n=10, i=6% (from Table 4)

Annuity amount = \$108,694 = Lease payment
Requirement 2
Annuity amount = \$800,000
15.32380*
* Present value of an annuity due of \$1: n=20, i=3% (from Table 6)

Annuity amount = \$52,206 = Lease payment
Requirement 3
Annuity amount = \$800,000
44.9550*
* Present value of an ordinary annuity of \$1: n=60, i=1% (given)

Annuity amount = \$17,796 = Lease payment

6-29
Chapter 06 - Time Value of Money Concepts

Problem 6-13
Choose the option with the lowest present value of cash outflows, net of the
present value of any cash inflows. (Cash outflows are shown as negative amounts;
cash inflows as positive amounts)

PV = - \$160,000 - 5,000 (5.65022* ) + 10,000 (.32197** )
* Present value of an ordinary annuity of \$1: n=10, i=12% (from Table 4)
** Present value of \$1: n=10, i=12% (from Table 2)

PV = - \$160,000 - 28,251 + 3,220

PV = - \$185,031

2. Lease option:

PVAD = - \$25,000 (6.32825* ) = - \$158,206
* Present value of an annuity due of \$1: n=10, i=12% (from Table 6)

Kiddy Toy should lease the machine.

6-30
Chapter 06 - Time Value of Money Concepts

Problem 6-14
Requirement 1
Tinkers:

PVA = \$20,000                x        7.19087*     =          \$143,817
* Present value of an ordinary annuity of \$1: n=15, i=11% (from Table 4)

PV      = \$143,817           x        .81162*      =          \$116,725
* Present value of \$1: n=2, i=11% (from Table 2)

Evers:

PVA = \$25,000                x        7.19087*     =          \$179,772
* Present value of an ordinary annuity of \$1: n=15, i=11% (from Table 4)

PV      = \$179,772           x        .73119*      =          \$131,447
* Present value of \$1: n=3, i=11% (from Table 2)

Chance:

PVA = \$30,000                x        7.19087*     =          \$215,726
* Present value of an ordinary annuity of \$1: n=15, i=11% (from Table 4)

PV      = \$215,726           x        .65873*      =          \$142,105
* Present value of \$1: n=4, i=11% (from Table 2)

Or, alternatively:
Deferred annuity factors:

Deferred annuity
Employee       PVA factor, i=11%      - PVA factor, i=11%   =         factor
Tinkers         7.54879 (n=17)       -   1.71252 (n=2)     =        5.83627
Evers           7.70162 (n=18)       -   2.44371 (n=3)     =        5.25791
Chance          7.83929 (n=19)       -   3.10245 (n=4)     =        4.73684

6-31
Chapter 06 - Time Value of Money Concepts

Problem 6-14 (concluded)

Present value of pension obligations at 12/31/11:
Tinkers: \$20,000 x 5.83627 = \$116,725
Evers: \$25,000 x 5.25791 = \$131,448*
Chance: \$30,000 x 4.73684 = \$142,105

*rounding difference

Requirement 2
Present value of pension obligations as of December 31, 2014:

Employee        PV as of 12/31/11    x  FV of \$1 factor,      =    FV as of 12/31/14
n=3, i=11%
Tinkers            \$116,725        x       1.36763          =        \$159,637
Evers               131,448        x       1.36763          =         179,772
Chance              142,105        x       1.36763          =         194,347
Total present value,
12/31/14                   \$533,756

Amount of annual contribution:

FVAD = Annuity amount x Annuity factor

Annuity amount = Annuity factor

Annuity amount = \$533,756 =                \$143,881
3.7097*

* Future value of an annuity due of \$1: n=3, i=11% (from Table 5)

6-32
Chapter 06 - Time Value of Money Concepts

Problem 6-15
Bond liability:
PV = \$4,000,0001 (18.40158* ) + 100,000,000 (.17193** )
PV = \$73,606,320 + 17,193,000 = \$90,799,320 = initial bond liability
1
\$100,000,000 x 4 % = \$4,000,000
* Present value of an ordinary annuity of \$1: n=40, i=4.5% (from Table 4)
** Present value of \$1: n=40, i=4.5% (from Table 2)

Lease liability:
Lease A:
PVAD = \$200,000 (9.36492* ) = \$1,872,984 = Liability
* Present value of an annuity due of \$1: n=20, i=10% (from Table 6)

Lease B:
PVAD = \$220,000 x 8.82371* = \$1,941,216
* Present value of an annuity due of \$1: n=17, i=10% (from Table 6)

PV      = \$1,941,216 x .75131* = \$1,458,455
* Present value of \$1: n=3, i=10% (from Table 2)

Or, alternatively for Lease B:

PVA = \$220,000 x 8.02155* = \$1,764,741
* Present value of an ordinary annuity of \$1: n=17, i=10% (from Table 4)

PV       = \$1,764,741 x .82645** = \$1,458,470 (difference due to rounding)
**Present value of \$1: n=2, i=10% (from Table 2)

Or, alternatively for Lease B:

PV = \$220,000 (6.62938* )             = \$1,458,464 (difference due to rounding)
From Table 4,
PVA factor, n=19, i=10%                   =      8.36492
– PVA factor, n=2, i=10%                  =      1.73554
= PV factor for deferred annuity          =      6.62938*

The company’s balance sheet would include a liability for bonds of \$90,799,320 and
a liability for leases of \$3,331,439 (\$1,872,984 + 1,458,455).

6-33
Chapter 06 - Time Value of Money Concepts

CASES
Ethics Case 6-1
The ethical issue is that the 21% return implies an annual return of 21% on an
investment and misrepresents the fund’s performance to all current and future
stakeholders. Interest rates are usually assumed to represent an annual rate, unless
otherwise stated. Interested investors may assume that the return for \$100 would be
\$21 per year, not \$21 over two years. The Damon Investment Company ad should
explain that the 21% rate represented appreciation over two years.

6-34
Chapter 06 - Time Value of Money Concepts

Analysis Case 6-2
Sally should choose the alternative with the highest present value.

Alternative 1:

PV = \$50,000

Alternative 2:

PV = PVAD = \$10,000 (5.21236* ) = \$52,124
* Present value of an annuity due of \$1: n=6, i=6% (from Table 6)

Alternative 3:
PVA = \$22,000                  x      2.67301*     =                \$58,806
* Present value of an ordinary annuity of \$1: n=3, i=6% (from Table 4)

PV         =    \$58,806        x      .89000*      =                \$52,337
* Present value of \$1: n=2, i=6% (from Table 2)

Sally should choose alternative 3.

Or, alternatively (for 3):

PV = \$22,000 (2.37897* ) = \$52,337

From Table 4,
PVA factor, n=5, i=6%                            = 4.21236
– PVA factor, n=2, i=6%                          = 1.83339
= PV factor for deferred annuity                 = 2.37897*
or, From Table 6,
PVAD factor, n=6, i=6%                           = 5.21236
– PVAD factor, n=3, i=6%                         = 2.83339
= PV factor for deferred annuity due             = 2.37897*

6-35
Chapter 06 - Time Value of Money Concepts

Communication Case 6-3
Content (65%)
_____ 25 Explanation of the method used (present value) to
compare the two contracts.
_____        30 Presentation of the calculations.
49ers PV = \$6,989,065
Cowboys PV = \$6,492,710
_____  10 Correct conclusion.
____
_____ 65 points

Writing (35%)
_____ 5 Proper letter format.

_____        6     Terminology and tone appropriate to the audience of
a player's agent.
_____        12 Organization permits ease of understanding.
____ Introduction that states purpose.
____ Paragraphs that separate main points.
_____        12 English
____ Sentences grammatically clear and well organized,
concise.
____ Word selection.
____ Spelling.
____ Grammar and punctuation.
___
_____        35 points

6-36
Chapter 06 - Time Value of Money Concepts

Analysis Case 6-4
The settlement was determined by calculating the present value of lost future
income (\$200,000 per year)1 discounted at a rate which is expected to approximate
the time value of money. In this case, the discount rate, i, apparently is 7% and the
number of periods, n, is 25 (the number of years to John’s retirement). John’s
settlement was calculated as follows:

\$200,000                   x            11.65358*                    =         \$2,330,716
annuity
amount

* Present value of an ordinary annuity of \$1: n=25, i=7% (from Table 4)

Note: In the actual case, John’s present salary was increased by 3% per year to reflect future salary
increases.

1In the actual case, John’s present salary was increased by 3% per year to reflect future salary increases.

6-37
Chapter 06 - Time Value of Money Concepts

Judgment Case 6-5
Purchase price of new machine                    \$150,000
Sales price of old machine                       (100,000)
Incremental cash outflow required                \$ 50,000

The new machine should be purchased if the present value of the savings in
operating costs of \$8,000 (\$18,000 - 10,000) plus the present value of the salvage
value of the new machine exceeds \$50,000.

PV = (\$8,000 x 3.99271* ) + (\$25,000 x .68058** )

PV = \$31,942 + 17,015

PV = \$48,957

* Present value of an ordinary annuity of \$1: n=5, i=8% (from Table 4)
** Present value of \$1: n=5, i=8% (from Table 2)

The new machine should not be purchased.

6-38
Chapter 06 - Time Value of Money Concepts

Real World Case 6-6
Requirement 1
The effective interest rate can be determined by solving for the unknown present
value of \$1 factor for 22 semiannual periods (2010-2020):

PV of \$1 factor = \$ 189 = .693578*
\$272.5
* Present value of \$1: n= 22, i= ? (from Table 2, i = approximately 1.5%)

There is no row 22 in Table 2. The 24-period factor in the 1.5% column is
.69954. So, 1.5% is the approximate effective semiannual interest rate. A financial
calculator or Excel will produce the same rate.

Requirement 2
Using a 1.5% effective semiannual rate and 40 periods:

PV = \$1,000 (.55126* ) = \$551.26
* Present value of \$1: n=40, i=1.5% (from Table 2)

The issue price of one, \$1,000 maturity-value bond was \$551.26.

6-39
Chapter 06 - Time Value of Money Concepts

Real World Case 6-7
Requirement 1
The effective interest rate can be determined by solving for the unknown present
value of an ordinary annuity of \$1 factor for 3 periods:

PV of an ordinary annuity of \$1 factor = \$39 = 2.7857*
\$14
* Present value of an ordinary annuity \$1: n= 3, i= ? (from Table 4, i = approximately 4%)

In row 3 of Table 4, the value of 2.77509 is in the 4% column. So, 4% is the
approximate effective interest rate. A financial calculator or Excel will produce the
same result.

Requirement 2
The effective interest rate can be determined by solving for the unknown present
value of an annuity due \$1 factor for 4 periods:

PV of an annuity due of \$1 factor =              \$39 = 2.7857*
\$14
* Present value of an annuity due \$1: n= 3, i= ? (from Table 6, i = approximately 8%)

In row 3 of Table 6, the value of 2.78326 is in the 8% column. So, 8% is the
approximate effective interest rate. A financial calculator or Excel will produce the
same result.

6-40

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