Chapter 6 Bid A should be accepted since its present value

PROBLEM 6-2

(a) Time diagram:

i = 8% FV – OA = $90,000

R R R R R R R R

R = ? ? ? ? ? ? ? ?

0 1 2 3 4 5 6 7 8

n = 8

(b) Time diagram:

i = 12%

FV – AD =

R R R R 500,000

R = ? ? ? ?

PROBLEM 6-2 (Continued)

1.

Future value of an ordinary annuity of 1 for

25 periods at 12% ……………………………………….

133.33387

Factor (1 + .12) ………………………………………………

(c) Time diagram:

i = 9%

PV = $20,000 FV = $47,347

0 1 2 3 n

Future value approach

Present value approach

FV = PV (FVFn, i)

PV = FV (PVFn, i)

$47,347 = $20,000 (FVFn, 9%)

$20,000 = $47,347 (PVFn, 9%)

PROBLEM 6-2 (Continued)

(d) Time diagram:

i = ?

PV = FV =

$19,553 $27,600

Future value approach

Present value approach

FV = PV (FVFn, i)

PV = FV (PVFn, i)

$27,600 = $19,553 (FVF4, i)

$19,553 = $27,600 (PVF4, i)

= $27,600 ÷ $19,553

= $19,553 ÷ $27,600

PROBLEM 6-3

Time diagram (Bid A):

i = 9%

$69,000

PV – OA = R =

? 3,000 3,000 3,000 3,000 69,000 3,000 3,000 3,000 3,000 0

0 1 2 3 4 5 6 7 8 9 10

n = 9

Present value of initial cost

12,000 X $5.75 = $69,000 (incurred today) ………………

Present value of maintenance cost (years 1–4)

12,000 X $.25 = $3,000

R (PVF – OA4, 9%) = $3,000 (3.23972) ……………………….

Present value of resurfacing

FV (PVF5, 9%) = $69,000 (.64993)………………………………

44,845

Present value of maintenance cost (years 6–9)

R (PVF – OA9–5, 9%) = $3,000 (5.99525 – 3.88965) ………

PROBLEM 6-3 (Continued)

Time diagram (Bid B):

i = 9%

$126,000

PV – OA = R =

? 1,080 1,080 1,080 1,080 1,080 1,080 1,080 1,080 1,080 0

0 1 2 3 4 5 6 7 8 9 10

n = 9

Present value of initial cost

12,000 X $10.50 = $126,000 (incurred today) ……….

Present value of maintenance cost

12,000 X $.09 = $1,080

R (PV – OA9, 9%) = $1,080 (5.99525) ……………………..

Bid A should be accepted since its present value is lower.

PROBLEM 6-4

Lump sum alternative: Present Value = $500,000 X (1 – .46) = $270,000.

Long should choose the annuity payout; its present value is $16,297 greater.

PROBLEM 6-5

(a) The present value of $55,000 cash paid today is $55,000.

(b) Time diagram:

i = 21/2% per quarter

PV – OA = R =

? $4,000 $4,000 $4,000 $4,000 $4,000

(c) Time diagram:

i = 21/2% per quarter

$18,000

PV – AD =

R = $1,800 $1,800 $1,800 $1,800 $1,800

PROBLEM 6-5 (Continued)

(d) Time diagram:

i = 21/2% per quarter

PV – OA = R =

? $1,500 $1,500 $1,500 $1,500

PV – OA = R =

? $4,000 $4,000 $4,000

n = 12 quarters n = 25 quarters

Formulas:

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

Present values:

(a) $55,000.

Option (c) is the best option, based upon present values alone.

Time diagram:

i = 12%

PV – OA = ? R =

($39,000) ($39,000) $18,000 $18,000 $68,000 $68,000 $68,000 $68,000 $38,000 $38,000 $38,000

Formulas:

PV – OA = R (PVF – OAn, i)

PV – OA =R (PVF – OAn, i)

PV – OA = ($39,000)(PVF – OA5, 12%)

PV – OA = $18,000 (PVF – OA10-5, 12%)

PV – OA = $68,000 (PVF – OA30–10, 12%)

PV – OA = $38,000 (PVF – OA40–30, 12%)

PV – OA = ($39,000)(3.60478)

PV – OA = $18,000 (5.65022 – 3.60478)

PV – OA = $68,000 (8.05518 – 5.65022)

PV – OA = $38,000 (8.24378 – 8.05518)

PV – OA = ($140,586)

PV – OA = $18,000 (2.04544)

PV – OA = $68,000 (2.40496)

PV – OA = $38,000 (.18860)

PV – OA = $36,818

PV – OA = $163,537

PV – OA = $7,167

Present value of future net cash inflows:

$(140,586)

36,818

163,537

7,167

$ 66,936

PROBLEM 6-7

(a) Time diagram (alternative one):

i = ?

PV – OA =

$600,000 R =

$80,000 $80,000 $80,000 $80,000 $80,000

0 1 2 10 11 12

n = 12

7.50 is present value of an annuity of $1 for 12 years discounted at

approximately 8%.

Time diagram (alternative two):

i = ?

PV = $600,000 FV = $1,900,000

PROBLEM 6-7 (Continued)

Future value approach

Present value approach

FV = PV (FVFn, i)

PV = FV (PVFn, i)

$1,900,000 = $600,000 (FVF12, i)

$600,000 = $1,900,000 (PVF12, i)

3.16667 is the approximate future

value of $1 invested at 10%

for 12 years.

.31579 is the approximate present

value of $1 discounted at 10%

for 12 years.

Dubois should choose alternative two since it provides a higher rate

of return.

(b) Time diagram:

i = ?

PROBLEM 6-7 (Continued)

Formulas: PV – OA = R (PVF – OAn, i)

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:

PV = ?

PV – OA = R =

? $32,000 $32,000 $32,000 $32,000 $32,000 ($800,000 X 8% X 6/12)

0 1 2 8 9 10

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

Formulas:

PROBLEM 6-7 (Continued)

(d) Time diagram (future value of $200,000 deposit)

i = 21/2% per quarter

PV =

$200,000 FV = ?

12/31/14 12/31/15 12/31/23 12/31/22

n = 40 quarters

Time diagram (future value of quarterly deposits)

i = 21/2% per quarter

R R R R R R R R R

R = ? ? ? ? ? ? ? ? ?

PROBLEM 6-7 (Continued)

Formulas: FV – OA = R (FVF – OAn, i)

PROBLEM 6-8

Vendor A:

$ 18,000

payment

X 6.14457

(PV of ordinary annuity 10%, 10 periods)

$ 110,602

+ 55,000

down payment

+ 10,000

maintenance contract

$ 175,602

total cost from Vendor A

Vendor B:

$ 9,500

semiannual payment

X 18.01704

(PV of annuity due 5%, 40 periods)

$ 171,162

Vendor C:

$ 1,000

X 3.79079

(PV of ordinary annuity of 5 periods, 10%)

$ 3,791

PV of first 5 years of maintenance

$ 2,000

[PV of ordinary annuity 15 per., 10% (7.60608) –

X 3.81529

PV of ordinary annuity 5 per., 10% (3.79079)]

$ 7,631

PV of next 10 years of maintenance

$ 3,000

[(PV of ordinary annuity 20 per., 10% (8.51356) –

X .90748

PV of ordinary annuity 15 per., 10% (7.60608)]

$ 2,722

PV of last 5 years of maintenance

Total cost of press and maintenance Vendor C:

$ 150,000

cash purchase price

maintenance years 1–5

maintenance years 6–15

2,722

maintenance years 16–20

$ 164,144

The press should be purchased from Vendor C, since the present value of

the cash outflows for this option is the lowest of the three options.

PROBLEM 6-9

(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

0 1 2 3 7 8 9 10

n = 10

Formula for the first ten payments:

Formula for the last ten payments:

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

present value of an annuity due.

PROBLEM 6-9 (Continued)

OR

Time diagram for the last ten payments:

i = 10%

PV = ? R = $400,000 $400,000 $400,000 $400,000

Formulas for the last ten payments:

(i) Present value of the last ten payments:

PROBLEM 6-9 (Continued)

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

year:

Since the present value of the cost for leasing the facilities,

$6,449,581, is less than the cost for purchasing the facilities,

$7,200,000, McDowell Enterprises should lease the facilities.

(b) Time diagram:

i = 11%

PV – OA = ?

R =

PROBLEM 6-9 (Continued)

(c) Time diagram:

Amount paid =

$792,000

0 10 30

Amount paid =

$800,000

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

the due date:

PROBLEM 6-9 (Continued)

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