(A) Annual Compounding: $10,000 X (FV1, 8%, 5) (Table 6A-1, 1.46933) = $14,693

Exercise 6-3

Situation 1

(a) Annual compounding: $10,000 x (FV1, 8%, 5) (Table 6A-1, 1.46933) = $14,693.

(b) Semiannual compounding: $10,000 x (FV1, 4%, 10) (Table 6A-1, 1.48024) = $14,802.

The future value amounts increase because the compounding is more frequent; therefore there is more interest on interest.

Situation 2

(a) Annual discounting: $40,000 x (PV1, 8%, 6) (Table 6A-2, .63017) = $25,207.

(b) Semiannual discounting: $40,000 x (PV1, 4%, 12) (Table 6A-2, .62460) = $24,984.

The present value amounts decrease because the discounting is more frequent; therefore there is more discounting of interest on interest.

b. Present value of 1:
$108,100 ÷ $200,000 = .54050, Table 6A-2 value for n = 8, i = ?%.
Reference to Table 6A-2, line for n = 8, shows .54027 under the 8% column; therefore, the discount interest rate applied by the creditor is about 8%.

Exercise 6-12

a)Table value .72198 is a present value of 1 because it is less than 1. Future values of one always are more than 1.

b)Future value of 1: (FV1, 3%, 22) (Table 6A-1) 1.91610.

c)Present value of 1: (PV1, 2%, 24) (Table 6A-2) 0.62172.

d)$48,822 ÷ $6,000 = 8.13706 Table 6A-1 value for n = 15 and i = ?%. Reference to Table 6A-1, line for 15 years (n) shows 8.13706 under 15%; therefore, the approximate interest rate on the fund is 15%.

e)$40,000 x (PV1, 7%, 3) (Table 6A-2, .81630) = $32,652.

Exercise 6-15

Value Based
on 1
Symbol / Formula / (n = 9; i = 18%) / Source
1. FV of 1 / FV1 / (1 + i)n / 4.43545 / Table 6-1
2. PV of 1 / PV1 / / .22546 / Table 6-2
3. FV of ordinary annuity of n
payments of 1 each / FVA / / 19.08585 / Table 6-3
4. PV of ordinary annuity of n
payments of 1 each / PVA / / 4.30302 / Table 6-4
5. FV of annuity due of n
payments of 1 each / FVAD / FVA x (1 + i) / 22.52131 / Table 6-5
6. PV of annuity due of n
payments of 1 each / PVAD / PVA x (1 + i) / 5.07757 / Table 6-6

Problem 6-17

(a) Future value of 1:

1. Annual compounding:
$30,000 x (FV1, 16%, 5) (Table 6A-1, 2.10034) = $63,010.

2. Semiannual compounding:
$30,000 x (FV1, 8%, 10) (Table 6A-1, 2.15892) = $64,768.

3. Quarterly compounding:
$30,000 x (FV1, 4%, 20) (Table 6A-1, 2.19112) = $65,734.

(b) Present value of 1:

1. Annual discounting:
$20,000 x (PV1, 12%, 5) (Table 6A-2, .56743) = $11,349.

2. Semiannual discounting:
$20,000 x (PV1, 6%, 10) (Table 6A-2, .55839) = $11,168.

3. Quarterly discounting:
$20,000 x (PV1, 3%, 20) (Table 6A-2, .55368) = $11,074.

$12,798 ÷ $6,000 = 2.133, table value for future value of 1, n = 13; i = ?%.
Reference to Table 6A-1, line n = 13, shows 2.13293 under 6%; therefore, the approximate compound interest rate is about 6%.

(d) Present value of 1:

$5,864 ÷ $15,000 = .39093, table value for present value of 1 for i = 11%; n = ?
Reference to Table 6A-2, column i = 11%, shows .39092 on line n = 9; therefore, the number of periods is 9.

(e) Future value of an ordinary annuity:

$6,000 x (FVA, 9%, 10) (Table 6A-3, 15.19293) = $91,158.

(f) Future value of an annuity due:

$9,000 x (FVAD, 7%, 10) (Table 6A-5, 14.78360) = $133,052

(g) Present value of an ordinary annuity:

$40,000 x (PVA, 10%, 5) (Table 6A-4, 3.79079) = $151,632.

(h) Present value of an annuity due:

$8,000 x (PVAD, 14%, 5) (Table 6A-6, 3.91371) = $31,310.

(i) Future value of ordinary annuity:

$552,026 $60,000 = 9.20043, table value for future value of an ordinary annuity,
n = 7% = ?%.
Reference to Table 6A-3, line for n = 7, shows 9.20043 under the 9% column; therefore, the implicit interest rate is 9%.

Problem 6-17 (concluded)

(j) Present value of ordinary annuity: $141,366 ÷ $30,000 = 4.7122, table value for present
value of ordinary annuity, i = 11%; n = ?

Reference to Table 6A-4, for i = 11%, shows 4.71220 on line for n = 7; therefore, the implicit number of year-end cash payments is 7.

(k) This plan can be diagrammed as follows:

Now /
$30,000 /
$30,000 /
$30,000 /
$30,000 /
$30,000
Years 1–10 / 11 / 12 / 13 / 14 / 15
(PVI, 10%, 10) / $ / $ / $ / $ / $
(PVA, 10%, 5)

$X $Y

Step 1—Fund balance needed on Jan. 1, Year 11:
$30,000 x (PVA, 10%, 5) (Table 6A-4, 3.79079) = $113,724 = Y.

Step 2—Single deposit required on Jan. 1, Year 1 to obtain fund balance on Jan. 1, Year 11:
$113,724 x (PVI, 10%, 10) (Table 6A-2, .38554) = $43,845 =X.

Problem 6-18

Item a:
$3,000,000 - $1,000,000 = $2,000,000

Assuming both payments have been made to date. Since the liability bears interest, no discounting is required.

Item b:
$800,000

The note bears interest and, therefore, does not need to be discounted.

Item c:
Let X = the note's present value on April 1, 1999:

X (FV1, 15%,1) / = / $100,000 / 100,000 (PV1, 15%, 1) / = / X
X (1.15) / = / $100,000 / or / 100,000 (.86957) / = / X
X / = / $86,957 / $86,957 / = / X

Value on December 31, 1999 = $86,957 + $86,957 (.15)(9/12) = $96,740

Item d:

100,000 (PVAD, 16%, 4) / =
$100,000(3.24589) / = / $324,590

Problem 6-19

a. Working capital to total assets: Periodic interest will reduce the working capital of both firms. The periodic cash interest payments will reduce current assets and therefore working capital for Brooks Fiber while the accrued interest expense will increase the liabilities of Station Casinos. The amounts are:

BFP: $425(.58804)(10 and 7/8 th % ÷ 2) and for
SC: $198 (.99362)(10 and 1/8th % ÷ 2).

The accrued interest for BFP is larger and thus will be larger for the 8 and ½ months of 1996. Hence the impact in dollar terms will be greater for BFP.
The issuance of the notes increases cash, hence working capital, and total assets in 1996 by:

BFP: $425(.58804) = $250 million
SC: $198(.99362) = $196.7 million.

This results in a larger increase in the total assets for BFP than for SC. Both effects reduce the ratio more for BFP.

b. Net cash flow to current liabilities: BFP pays no interest until 2006. The accrued interest increases the bond liability, not current liabilities. Hence there is no impact on BFP’s ratio. SC pays interest and this reduces cash flow. Again there is no effect on current liabilities except at the close of the reporting year at which time interest for the partial interest period is accrued (but not yet paid). The increase in accrued interest paid will be less than the interest paid. Thus this ratio declines for SC.

c. Debt to equity: Equity is assumed for each firm, at the start of the year. Issuing the notes does not affect equity for either firm. The debt issued for BFP exceeds the debt issued for SC:

$425(.58804) > $198(.99362)

Hence the ratio will be larger for BFP after the issue than for SC.

d. Times interest earned: Because the semiannual interest for BFP exceeds that for SC

$250(10 and 7/8ths % ÷ 2) > $196(10 and 1/8th % ÷ 2)

This ratio will be larger for SC.

e. Cash flow per share: Only SC will pay out cash up to 2006. These payments will reduce cash flow for SC. The denominator of the ratio is not affected. Therefore the ratio declines for SC. There is no effect on either the numerator or the denominator for BFP.

f. Return on total assets: In the calculation of this ratio, interest after tax is added back to the numerator. Thus there is no effect on the numerator. The cash obtained from the issue is larger for BFP and (assuming the same earning power for assets for the 2 firms), this means BFP’s ratio will be reduced more than SC’s ratio. Relatively SC’s ratio will increase compared to BFP’s.

CASES

Case 6-1

$114,000 ÷ $60,000 = 1.90000—(Table 6A-1 value for future value of 1 for n = 5; i = ?%). Refer to table 6A-1, line for n = 5, shows 1.92541 under 14%; therefore the implicit rate is very close to 14%.

Choice of the option should depend, in a pure economic sense, on whether you can earn more or less than 14% at the same level of risk. If a return of 14% or more can be earned (with no more risk), then the $60,000 should be accepted. It can be invested at the higher rate for 5 years, thus yielding more than $114,000. Of course, personal “needs" for immediate cash may be a controlling factor (such as the payment of current debt, or the purchase of a sail boat—for fun!).