 CHAPTER 6 SOLUTIONS

EXERCISE 6-4 (5-10 minutes)

 (a) Simple interest of \$1,600 per year X 8 \$12,800 Principal 20,000 Total withdrawn \$32,800 (b) Interest compounded annually—Future value of        1 @ 8% for 8 periods 1.85093 X   \$20,000 Total withdrawn \$37,018.60

 (c) Interest compounded semiannually—Future        value of 1 @ 4% for 16 periods 1.87298 X   \$20,000 Total withdrawn \$37,459.60

EXERCISE 6-6 (15-20 minutes)

 (a) Future value of an ordinary    annuity of \$4,000 a period    for 20 periods at 8% \$183,047.8 (\$4,000 X 45.76196) Factor (1 + .08) X         1.08 Future value of an annuity    due of \$4,000 a period at 8% \$197,691.66

 (b) Present value of an ordinary    annuity of \$2,500 for 30    periods at 10% \$23,567.28 (\$2,500 X 9.42691) Factor (1 + .10) X         1.10 Present value of annuity    due of \$2,500 for 30 periods    at 10% \$25,924.00 (Or see Table 6-5 which  gives \$25,924.03) (c) Future value of an ordinary    annuity of \$2,000 a period    for 15 periods at 10% \$63,544.96 (\$2,000 X 31.77248) Factor (1 + 10) X         1.10 Future value of an annuity    due of \$2,000 a period    for 15 periods at 10% \$69,899.46 (d) Present value of an ordinary    annuity of \$1,000 for 6    periods at 9% \$4,485.92 (\$1,000 X 4.48592) Factor (1 + .09) X       1.09 Present value of an annuity    date of \$1,000 for 6 periods    at 9% \$4,889.65 (Or see Table 6-5)

EXERCISE 6-8

 (a) Future value of \$12,000 @ 10% for 10 years (\$12,000 X 2.59374) = \$31,124.88 (b) Future value of an ordinary annuity of \$600,000 at 10% for 15 years (\$600,000 X 31.77248) \$19,063,488.00 Deficiency (\$20,000,000 – \$19,063,488) \$936,512.00 (c) \$70,000 discounted at 8% for 10 years: \$70,000 X .46319 = \$32,423.30 Accept the bonus of \$40,000 now. (Also, consider whether the 8% is an appropriate discount rate since the president can probably earn compound interest at a higher rate without too much additional risk.)

EXERCISE 6-9

 (a) \$50,000 X .31524 = \$15,762.00 + \$5,000 X 8.55948 = 42,797.40 \$58,559.40 (b) \$50,000 X .23939 = \$11,969.50 + \$5,000 X 7.60608 = 38,030.40 \$49,999.90

The answer should be \$50,000; the above computation is off by 10¢ due to rounding.

 (c) \$50,000 X .18270 = \$  9,135.00 + \$5,000 X 6.81086 = 34,054,30 \$43,189.30

EXERCISE 6-10

 (a) Present value of an ordinary annuity of 1 for 4 periods @ 8% 3.31213 Annual withdrawal X  \$20,000 Required fund balance on June 30, 2010 \$66,242.60

 (b) Fund balance at June 30, 2010 \$66,242.60 = \$14,700.62 Future amount of ordinary annuity at 8% 4.50611 for 4 years Amount of each of four contributions is \$14,700.62 PROBLEM 6-9

(a)               Time diagram (alternative one):

i = ?

PV–OA =

\$572,000        R =

\$80,000     \$80,000                             \$80,000    \$80,000       \$80,000  0            1           2                                 10         11          12

n = 12

Formulas:        PV–OA = R (PVF–OAn, i)

\$572,000 = \$80,000 (PVF–OA12, i)

PVF–OA12, i = \$572,000 ¸ \$80,000

PVF–OA12, i = 7.15

7.15 is present value of an annuity of \$1 for 12 years discounted at approximately 9%.

Time diagram (alternative two):

i = ?

PV = \$572,000                                                                             FV = \$1,900,000  n = 12

 Future value approach Present value approach FV = PV (FVFn, i) PV = FV (PVFn, i) or \$1,900,000 = \$572,000 (FVF12, i) \$572,000 = \$1,900,000 (PVF12, i) FVF12, i = \$1,900,000 ¸ \$572,000 PVF12, i = \$572,000 ¸ \$1,900,000 FVF12, i = 3.32168 PVF12, i = .30105 3.32 is the future value of \$1   invested at between 10% and   11% for 12 years. .301 is the present value of \$1   discounted at between 10%   and 11% for 12 years.

Mark Grace, Inc. should choose alternative two since it provides a higher rate of return.

(b)               Time diagram:

i = ?

(\$824,150 – \$200,000)

PV–OA =      R =

\$624,150      \$76,952                             \$76,952   \$76,952   \$76,952  0            1                                 8            9          10

n = 10 six-month periods

Formulas:        PV–OA = R (PVF–OAn, i)

\$624,150 = \$76,952 (PVF–OA10, i)

PV–OA10, i = \$624,150 ¸ \$76,952

PV–OA10, i = 8.11090

8.11090 is the present value of a 10-period annuity of \$1 discounted at 4%.  The interest rate is 4% semiannually, or 8% annually.

(c)               Time diagram:

i = 5% per six months

PV = ?

PV–OA =        R =

?         \$24,000     \$24,000                             \$24,000    \$24,000   \$24,000 (\$600,000 X 8% X 6/12)   0            1           2                                 8            9          10

n = 10 six-month periods [(7 – 2) X 2]

Formulas:

PV–OA = R (PVF–OAn, i)                                PV = FV (PVFn, i)

PV–OA = \$24,000 (PVF–OA10, 5%)                 PV = \$600,000 (PVF10, 5%)

PV–OA = \$24,000 (7.72173)                          PV = \$600,000 (.61391)

PV–OA = \$185,321.52                                                PV = \$368,346

Combined present value (amount received on sale of note):

\$185,321.52 + \$368,346 = \$553,667.52

(d)               Time diagram (future value of \$300,000 deposit)

i = 21/2% per quarter

PV =

\$300,000                                                                                                                                       FV = ?   12/31/03                                          12/31/04                12/31/12                                         12/31/13

n = 40 quarters

Formula:          FV = PV (FVFn, i)

FV = \$300,000 (FVF40, 2 1/2%)

FV = \$300,000 (2.68506)

FV = \$805,518

Amount to which quarterly deposits must grow:

\$1,300,000 – \$805,518 = \$494,482.

Time diagram (future value of quarterly deposits)

i = 21/2% per quarter

R             R           R             R                            R            R            R            R             R

R = ?          ?            ?             ?                            ?             ?            ?             ?             ?   12/31/03                                          12/31/04                 12/31/12                                         12/31/13

n = 40 quarters

Formulas:           FV–OA = R (FVF–OAn, i)

\$494,482 = R (FVF–OA40, 2 1/2%i)

\$494,482 = R (67.40255)

R = \$494,482 ¸ 67.40255

R = \$7,336.25

 PROBLEM 6-11

(a)        Time diagram for the first ten payments:

i = 10%

PV–AD = ?

R =

\$800,000 \$800,000 \$800,000 \$800,000                \$800,000 \$800,000 \$800,000 \$800,000   0           1              2             3                             7             8             9           10

n = 10

Formula for the first ten payments:

PV–AD = R (PVF–ADn, i)

PV–AD = \$800,000 (PVF–AD10, 10%)

PV–AD = \$800,000 (6.75902)

PV–OA = \$5,407,216

Time diagram for the last ten payments:

i = 10%

R =

PV–OA = ?                                                \$300,000  \$300,000                       \$300,000   \$300,000     0           1           2                       10         11                       18         19        20

n = 9                                                                 n = 10

Formula for the last ten payments:

PV–OA = R (PVF–OAn, i)

PV–OA = \$300,000 (PVF–OA19 – 9, 10%)

PV–OA = \$300,000 (8.36492 – 5.75902)

PV–OA = \$300,000 (2.6059)

PV–OA = \$781,770

Note: The present value of an ordinary annuity is used here, not the present value of an annuity due.

OR

Time diagram for the last ten payments:

i = 10%

PV = ?                                               R =      \$300,000                    \$300,000   \$300,000 \$300,000     0           1          2                       9         10                     17         18        19

FVF (PVFn, i)                                                  R (PVF–OAn, i)

Formulas for the last ten payments:

(i)                  Present value of the last ten payments:

PV–OA = R (PVF–OAn, i)

PV–OA = \$300,000 (PVF–OA10, 10%)

PV–OA = \$300,000 (6.14457)

PV–OA = \$1,843,371

(ii)        Present value of the last ten payments at the beginning of current year:

PV = FV (PVFn, i)

PV = \$1,843,371 (PVF9, 10%)

PV = \$1,843,371 (.42410)

PV = \$781,774*

*\$4 difference due to rounding.

Cost for leasing the facilities   \$5,407,216 + \$781,774 = \$6,188,990

Since the present value of the cost for leasing the facilities, \$6,188,990, is less than the cost for purchasing the facilities, \$7,200,000, Starship Enterprises should lease the facilities.

(b)               Time diagram:

i = 11%

PV–OA = ?

R =

\$12,000  \$12,000 \$12,000                    \$12,000 \$12,000 \$12,000 \$12,000   0           1              2             3                             6             7              8              9

n = 9

Formula:          PV–OA = R (PVF–OAn, i)

PV–OA = \$12,000 (PVF–OA9, 11%)

PV–OA = \$12,000 (5.53705)

PV–OA = \$66,444.60

The fair value of the note is \$66,444.60.

(c)               Time diagram:

Amount paid =

\$784,000

0               10                                 30

Amount paid =

\$800,000

Cash discount = \$800,000 (2%) = \$16,000

Net payment = \$800,000 – \$16,000 = \$784,000

If the company decides not to take the cash discount, then the company can use the \$784,000 for an additional 20 days.  The implied interest rate for postponing the payment can be calculated as follows:

(i)                  Implied interest for the period from the end of discount period to the due date:

Cash discount lost if not paid within the discount period

Net payment being postponed

= \$16,000/\$784,000

= 0.0204

(ii)                Convert the implied interest rate to annual basis:

Daily interest = 0.0204/20 = 0.00102

Annual interest = 0.00102 X 365 = 37.23%

Since Starship’s cost of funds, 10%, is less than the implied interest rate for cash discount, 37.23%, it should continue the policy of taking the cash discount.

 PROBLEM 6-12

1.                  Purchase.

Time diagrams:

Installments

i = 10%

PV–OA = ?

R =

\$300,000         \$300,000         \$300,000         \$300,000         \$300,000

0          1          2          3          4          5

n = 5

Property taxes and other costs

i = 10%

PV–OA = ?

R =

\$56,000     \$56,000             \$56,000    \$56,000   \$56,000   \$56,000   0          1             2                              9           10           11           12

n = 12

Insurance

i = 10%

PV–AD = ?

R =

\$27,000   \$27,000   \$27,000                \$27,000   \$27,000   \$27,000   0           1              2                             9            10          11           12

n = 12

Salvage Value

PV = ?                                                                                         FV = \$500,000   0           1              2                            9            10          11           12

n = 12

Formula for installments:

PV–OA = R (PVF–OAn, i)

PV–OA = \$300,000 (PVF–OA5, 10%)

PV–OA = \$300,000 (3.79079)

PV–OA = \$1,137,237

Formula for property taxes and other costs:

PV–OA = R (PVF–OAn, i)

PV–OA = \$56,000 (PVF–OA12, 10%)

PV–OA = \$56,000 (6.81369)

PV–OA = \$381,567

Formula for insurance:

PV–AD = R (PVF–ADn, i)

PV–AD = \$27,000 (PVF–AD12, 10%)

PV–AD = \$27,000 (7.49506)

PV–AD = \$202,367

Formula for salvage value:

PV = FV (PVFn, i)

PV = \$500,000 (PVF12, 10%)

PV = \$500,000 (0.31863)

PV = \$159,315

Present value of net purchase costs:

 Down payment \$   400,000 Installments 1,137,237 Property taxes and other costs 381,567 Insurance 202,367 Total costs \$2,121,171 Less: Salvage value 159,315 Net costs \$1,961,856

2.                  Lease.

Time diagrams:

Lease payments

i = 10%

PV–AD = ?

R =

\$240,000 \$240,000 \$240,000            \$240,000 \$240,000   0           1              2                          10           11           12

n = 12

Interest lost on the deposit

i = 10%

PV–OA = ?

R =

\$10,000 \$10,000                  \$10,000  \$10,000  \$10,000  0           1              2                          10           11           12

n = 12

Formula for lease payments:

PV–AD = R (PVF–ADn, i)

PV–AD = \$240,000 (PVF–AD12, 10%)

PV–AD = \$240,000 (7.49506)

PV–AD = \$1,798,814

Formula for interest lost on the deposit:

Interest lost on the deposit per year = \$100,000 (10%) = \$10,000

PV–OA = R (PVF–OAn, i)

PV–OA = \$10,000 (PVF–OA12, 10%)

PV–OA = \$10,000 (6.81369)

PV–OA = \$68,137*

Cost for leasing the facilities = \$1,798,814 + \$68,137 = \$1,866,951

Rijo Inc. should lease the facilities because the present value of the costs for leasing the facilities, \$1,866,951, is less than the present value of the costs for purchasing the facilities, \$1,961,856. *OR:  \$100,000 – (\$100,000 X .31863) = \$68,137