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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            Total withdrawn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            Total withdrawn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            Total withdrawn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

 

 

 

Text Box:
 

 

 


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