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	Total withdrawn	$32,800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	Total withdrawn	$37,018.60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	Total withdrawn	$37,459.60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–OA_{n, i})

$572,000 = $80,000 (PVF–OA_{12, i})

PVF–OA_{12, i} = $572,000 ¸ $80,000

PVF–OA_{12, 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–OA_{n, i})

$624,150 = $76,952 (PVF–OA_{10, i})

PV–OA_{10, i} = $624,150 ¸ $76,952

PV–OA_{10, 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–OA_{n, i}) PV = FV (PVF_{n, i})

PV–OA = $24,000 (PVF–OA_{10, 5%}) PV = $600,000 (PVF_{10, 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 (FVF_{n, i})

FV = $300,000 (FVF_{40, 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–OA_{n, i})

$494,482 = R (FVF–OA_{40, 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–AD_{n, i})

PV–AD = $800,000 (PVF–AD_{10, 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–OA_{n, i})

PV–OA = $300,000 (PVF–OA_{19 – 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 (PVF_{n, i}) R (PVF–OA_{n, i})

Formulas for the last ten payments:

(i) Present value of the last ten payments:

PV–OA = R (PVF–OA_{n, i})

PV–OA = $300,000 (PVF–OA_{10, 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 (PVF_{n, i})

PV = $1,843,371 (PVF_{9, 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–OA_{n, i})

PV–OA = $12,000 (PVF–OA_{9, 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–OA_{n, i})

PV–OA = $300,000 (PVF–OA_{5, 10%})

PV–OA = $300,000 (3.79079)

PV–OA = $1,137,237

Formula for property taxes and other costs:

PV–OA = R (PVF–OA_{n, i})

PV–OA = $56,000 (PVF–OA_{12, 10%})

PV–OA = $56,000 (6.81369)

PV–OA = $381,567

Formula for insurance:

PV–AD = R (PVF–AD_{n, i})

PV–AD = $27,000 (PVF–AD_{12, 10%})

PV–AD = $27,000 (7.49506)

PV–AD = $202,367

Formula for salvage value:

PV = FV (PVF_{n, i})

PV = $500,000 (PVF_{12, 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–AD_{n, i})

PV–AD = $240,000 (PVF–AD_{12, 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–OA_{n, i})

PV–OA = $10,000 (PVF–OA_{12, 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