Download 1 A $1000 bond has a coupon of 6% and matures after

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#1 A $1,000 bond has a coupon of 6% and matures after 10 years; a)What would be
th ebond's price if comparable debts yield 8% b)What would be the price if
comparable debt yields 8% and the bond matures after 5 years c)Why are the prices
different in a and b d)What are the current yields and the yields to maturity in a
and b
a. Price of the bond:
PB = $30
(1 + .04)
+....+
$30
+ $1,000
(1 + .04)20 (1 + .04)20
= $30(13.590) + $1,000(0.456)
= $408 + $456
= $864
(PV = ?; PMT = 30; FV = 1000; N = 20, and I = 4.
PV = -864.)
Point out that the coupon is halved [$60/2], and the number of time periods is doubled
[10 x 2].
b. Price of the bond:
PB = $30
(1 + .04)
+....+
$30 + $1,000
(1 + .04)10 (1 + .04)10
= $30(8.111) + $1,000(0.676)
= $919
(PV = ?; PMT = 30; FV = 1000; N = 10, and I = 4.
PV = -919.)
c. In both cases the bond's price is less than par because the current rate of interest is
greater than the rate paid on these bonds (i.e., 8% versus 6%). However, the amount of
price decline is affected by the term of the bond, and the bond with the longer term
experiences the larger price decline because the investors will collect the smaller interest
payment for a longer period of time.
d. The current yields are
$60 = 6.94% and $60 = 6.53%
$864
$919
The yields to maturity are 8% in both cases.
#2 a)A $1,000 bond has a 7.5% coupon and matures after 10 years. If current
interest rates are 10%, what should be the price of the bond b)If after 6 years
interest rates are still 10%, what should be the price of the bond c)Even though
interest rates did not change in a and b, why did the price of the bond change
d)Change the interest rate in a and b to 6% and rework your answer. Even though
the interest rate is 6% ib both calculations, why are the bond prices different
a. Price of the bond:
PB = $37.50 +....+ $37.50 + $1,000
(1 + .05)
(1 + .05)20 (1 + .05)20
= $37.50(12.462) + $1,000(.377)
= $844
(PV = ?; PMT = 37.50; FV = 1000; N = 20, and I = 5.
PV = -844.)
b. Price of the bond:
PB = $37.50 +....+ $37.50 + $1,000
(1 + .05)
(1 + .05)8 (1 + .09)8
= $37.50(6.463) + $1,000(0.677)
= $919
(PV = ?; PMT = 37.50; FV = 1000; N = 8, and I = 5.
PV = -919.)
c. The term to maturity has diminished which increases the value of the bond (i.e., the
investor gets the principal back in only four instead of ten years).
d. Price of the bond (ten years to maturity):
PB = $37.50 +....+ $37.50 + $1,000
(1 + .03)
(1 + .03)
20
20
(1 + .03)
= $37.50(14.877) + $1,000(0.554) = $1,112
(PV = ?; PMT = 37.50; FV = 1000; N = 20, and I = 3.
PV = -1112.)
Price of the bond (four years to maturity):
PB = $37.50 +....+ $37.50 + $1,000
(1 + .03)
(1 + .03)6 (1 + .03)6
= $37.50(7.020) + $1,000(0.789) = $1,052
(PV = ?; PMT = 37.50; FV = 1000; N = 8, and I = 3.
PV = -1053.)
The term to maturity is less, but in this case the value of the bond declines. In the
previous example, the bond sold for a discount, now it sells for a premium, which
declines as the bond approaches maturity. The investor earns the higher coupon interest
for a shorter time period, which decreases the attractiveness of the bond.
#3 (actually #4) Blackstone, Inc. has a 5 year bond outstanding that pays $60
annually. The face value of each bond is $1,000, an d the bond sells for $890. a)What
is the bond's coupon rate b)What is the current yield c)What is the yield to maturity
#4 (actually #9) A bond has the following features •Coupon rate of interest: 8%
•Principle: $1.000 •Term of maturity: 10 years a)What will the holder receive when
the bond matures b)If the current rate of interest on comparable debt is 12%, what
should be the price of this bond? What would you expect the firm to call this bond?
Why? c)If the bond has a sinking fund that requires the firm to set aside annually
with a trustee sufficient funds to retire the entire issue at maturity, how much must
the firm remit each year for 10 years if the funds earn 9% annually and there is $10
million outstanding?
a. The coupon rate: 6% ($60/$1,000)
b. The current yield: $60/$890 = 6.7%
c. Determination of the yield to maturity (r):
$890 = $60 +....+ $60 + $1,000
(1 + r)
(1 + r)5 (1 + r)5
Select an interest rate (e.g., 8%) and substitute
into the equation:
$60(3.993) + $1,000(.681) = $921
Repeat the process (e.g., 9%):
$60(3.890) + $1,000(.650) = $883
The yield to maturity is between 8% and 9%.
Using a financial calculator:
PV = -890; PMT = 60; FV = 1000; N = 5, and I = ?.
I = 8.81%
(Since 8.81 percent is more accurate, you may use this problem
to illustrate the usefulness of a financial calculator. This problem explicitly uses annual
interest payments to facilitate illustrating the concept of the yield to maturity.)
Chapter 19 Problem Set Do problems 2, 3, and 4 on pages 357-358 of the textbook.
#2 The management of a firm wants to introduce a new product. The product will
sell for $4 a unit, and can be produced by either of two scales of operation. In the
first, total cost are: TC = $3,000 + $2.8Q In the second scale of operations, total cost
are: TC = $5,000 + $2.4Q a)What is the break-even level of output for each scale of
operation b)What will be the firm's profits for each scale of operation if sales reach
5,000 units c)Ne-half of the fixed costs are noncash (depreciation). All other
expenses are for cash. If sales are 2,000 units, will cash receipts cover cash expenses
for each scale of operation d)The anticipated levels of sales are: YearUnit Sales
14,000 25,000 36,000 47,000 If management selects the scale of production with
higher fixed cost, what can it expect in years 1 and 2? On what grounds can
management justify selecting this scale of operation ? Is sales reach 5,000 a year,
was the correct scale of operation chosen?
a. The break-even levels of output:
$3,000/($4 - 2.80) = 2,500
and $5,000/($4 - 2.40) = 3,125
b. Earnings = total revenues - total costs
Total revenue = $4(5,000) = $20,000
Earnings under the two alternatives:
1. $20,000 - $3,000 - $2.80(5,000) = $3,000
2. $20,000 - $5,000 - $2.40(5,000) = $3,000
Sales of 5,000 units equates earnings. If output is less than 5,000 units, then the cost
function with the lower fixed costs and higher variable costs produces the higher profits
(or smaller losses). If output exceeds 5,000 units, then the scale of operation with higher
fixed costs and lower per unit costs will generate the higher earnings.
c. At sales of 2,000 and scale of operation with
TC = $3,000 + $2.80Q, then earnings are
$4(2,000) - $3,000 - $2.80(2,000) = ($600).
However, $1,500 of the expenses is non-cash depreciation, so the cash flow generated by
operations is
earnings
($600)
depreciation 1,500
$900
(Be certain to point out that a firm may operate at an accounting loss but still generate
positive cash flow.)
If the second scale of operations is used and TC = $5,000 + $2.4Q, the earnings would be
earnings = $4(2,000) - $5,000 - $2.4(2,000) = ($1,800).
Cash flow would be
earnings ($1,800)
depreciation 2,500
$ 700
Either scale of operation generates positive cash flow.
d. If the firm selects the scale with more fixed costs (i.e., higher operating leverage), its
earnings will be lower in year 1. This could be justified on the grounds that the lower
earnings are only temporary.
If the level of sales only reaches 5,000 units, earnings are the same for either scale of
operation. The student should not conclude that the scale of operation is unimportant in
this case. The scale with the higher fixed costs is riskier. The use of that scale increases
risk. Since earnings are the same under both alternatives, the scale with the less risk (i.e.,
lower operating leverage) is to be preferred.
#3 A firm has the following total revenue and total cost schedules: TR = $2Q TC =
$4,000 + $1.5Q a)What is the break-even level of output? What is the level of profits
at sales of 9,000 units? b)As a result of a major technological breakthrough, the total
cost schedule is changed to: TC = $6,000 + $0.5Q What is the break-even level of
output? What is the level of profits at sales of 9,000 units?
a. Break-even level of output:
$4,000/($2 - 1.50) = 8,000 units
Earnings: $2(9,000) - $4,000 - $1.50(9,000) = $500
b. Break-even level of output:
$6,000/($2 - .50) = 4,000 units
Earnings: $2(9,000) - $6,000 - $.50(9,000) = $7,500
Generally a substitution of fixed for variable costs increases the level of output necessary
to break even. However, that need not necessarily always be the case, as this problem
illustrates. The very large decrease in per unit variable costs more than offsets the
increase in fixed costs with the result that the break-even level of output declines.
#4The manufacturer of a product that has a variable cost of $2.50 per unit and total
fixed cost of $125,000 wants to determine the level of output necessary to avoid
losses. a)What level of sales is necessary to break even if the product is sold for $4.25?
What will be the manufacturer ‘s profit or loss on the sales of 100,000 units? b)If
fixed cost ris to $175,000, what is the new level of sales necessary to break even? c)If
variable cost decline to $2.25 per unit, what is the new level of sales necessary to
break-even d)If fixed cost were to increase to $175,000, while variable cost declined
to $2.25 per unit, what is the new break-even level of sales? e)If a major proportion
of fixed costs were noncash (depreciation), would failure to achieve the break-even
level of sales imply that the firm cannot pay its current obligations as they come due?
Suppose $100,000 of the above fixed costs of $125,000 were depreciation expense.
What level of sales would be the cash break-even level of sales?
a. Break-even level of output:
$125,000/($4.25 - $2.50) = 71,429 units
Earnings at sales of 100,000 units:
$4.25(100,000) - $125,000 - $2.50(100,000)
= $50,000
b. At fixed costs of $175,000, the break-even level of output is $175,000/($4.25 - 2.50)
= 100,000 units.
c. With variable costs of $2.25 per unit, the break-even level of output is
$175,000/($4.25 - 2.25) = 62,500 units.
d. If both variable and fixed costs change, the break-even level of output is
$175,000/($4.25 - 2.25) = 87,500 units.
e. If a substantial proportion of the costs is non-cash depreciation expense, the firm
may still generate sufficient cash to meet its current obligations as they come due.
If $100,000 of the $125,000 fixed costs is depreciation, then only $25,000 are fixed cash
outlays. The firm's cash break-even level of output is $25,000/($4.25 - 2.50) = 14,286
units.
Chapter 21 Problem Set Do problems 1, 2, and 3 on pages 396-397 of the textbook.
HBM, Inc. has the following capital structure : Assets $400,000Debt$140,000
Preferred Stock20,000 Common Stock 240,000 The common stock is currently
selling for $15 a share, pays a cash dividend of $0.75 per share, and is growing
annually at 6%. The preferred stock pays a $9 cash dividend and currently sells for
$91 a share. The debt pays interest of 8.5% annually, and the firm is in 30%
marginal tax bracket. a)What is the after –tax cost of debt b)What is the cost of
preferred stock c)What is the cost of common stock d)What is the firm's weightedaverage cost of capital
a. After-tax cost of debt:
8.5%(1 - tax rate) = 8.5(1 - .30) = 5.95%
b. Cost of preferred stock:
Dividend/Price of the stock = $9/$91 = 9.9%
c. Cost of the common stock:
Dividend(1 + g)/Price of the stock + growth rate =
$0.75(1 + .06)/$15 + 6% = 11.3%
(This answer assumes the current $0.75 dividend grows by 6 percent during the year.)
d. The cost of capital (k) is a weighted average:
k = (weight)(cost of debt) + (weight)(cost of preferred) +
weight(cost of common stock)
= (.35)(5.95%) + (.05)(9.9%) + (.60)(11.3%) = 9.3575%
#2 Sun Instruments expects to issue new stock at $34 a share with estimated
flotation costs of 7% of the market price. The company currently pays a $2.10 cash
dividend and has 6% growth rate. Where are the costs of retained earnings and new
common stock?
Cost of retained earnings: $2.10(1 + .06)/$34 + 6% = 12.55%
Cost of new stock:
Dividend(1 + g)/(stock price minus any flotation costs)
+ growth rate
$2.10(1 + .06)/($34 - $2.38) + .06 = 13.04%
#10 A firm's current balance sheet is as follows: Assets= $100, Debt= $10, Equity=
$90. a) What is the firm's weighted-average cost of capital at various combinations
of debt and equity, given the following information? </p><p>Debt/Assets, After-Tax
Cost of Debt, Cost of Equity ,Cost of Capital </p><p>0% 8% 12% ? </p><p>10812 ?
</p><p>20 8 12?</p><p>30 813 ? </p><p>40 9 14? </p><p>50 1015 ? </p><p>6012
16?</p><p>b) Construct a pro forma balance sheet that indicates the firm's optimal
capital structure. Compare this balance sheet with the firm's current balance sheet.
What course of action should the firm take? Assets =$100, Debt $?, Equity $? c) As
a firm initially substitutes debt for equity financing, what happens to the cost of
capital, and why? d) If a firm uses too much debt financing, why does the cost of
capital rise?
The cost of capital (k) is a weighted average:
k = (weight)(cost of debt) + weight(cost of equity)
Debt/
Assets
0%
10
20
30
40
50
60
Weight x + Weight x = Cost of
Cost
Cost
Capital
of Debt
of Equity
(.0)(.08) + (1.0)(.12) = .120
(.1)(.08) + (.9)(.12) = .116
(.2)(.08) + (.8)(.12) = .112
(.3)(.08) + (.7)(.13) = .115
(.4)(.09) + (.6)(.14) = .120
(.5)(.10) + (.5)(.15) = .125
(.6)(.12) + (.4)(.16) = .136
b. The optimal capital structure is that combination, which minimizes the firm's cost of
capital. In this case that occurs where debt is 20% of capital and the cost of capital is
11.2%. The balance sheet is
Assets
$100 Liabilities $20
Equity
80
Since the firm is currently using only 10% debt financing, it is not at its optimal capital
structure and should substitute some debt for equity.
c. The cost of capital initially declines because the effective cost of debt is less than the
cost of equity.
d. As the firm continues to substitute debt for equity, the firm becomes more
financially leveraged and riskier. This causes the interest rate to rise and the cost of
equity to increase. These increases in the cost of debt and equity cause the cost of capital
(i.e., the weighted average) to increase.
#11 You purchased machinery for $23,958 that generates cash flow of $6,000 for five
years. What is the internal rate of return on the investment?
Determination of the internal rate of return:
$23,958 = $600 +
(1 + r)
$600 + $600 + $600 + $600
2
(1 + r)
(1 + r)3 (1 + r)4 (1 + r)5
$23,958 = $6000(PVAIF for 5 years at ? percent)
IF = 3.993
Locate 3.993 in the interest table for 5 years and determine the internal rate of return to
be 8%.
(PV = -23958; N = 5; I = ?; PMT = 600, and FV = 0. I = 8.)
#12 The cost of capital for a firm is 10%. The firm has two possible investments
with the following cash inflows:</p><p>A B </p><p>Year 1$300 $200 </p><p> 2
200200 </p><p> 3 100 200 </p><p>a) Each investment cost $480. What investment(s)
should the firm make according to net oresent value? b) What is the internal rate of
return for the two investments? Which investment(s) should the firm make? Is this
the same answer you obtained in part a? c) If the cost of capital rises to 14%, which
investment(s) should the firm make?
a. Determination of net present value:
NPVA = $300 +
$200 + $100 - $480
(1 + .1) (1 + .1)2 (1 + .1)3
= $300(.909) + 200(.826) + 100(.751) - 480
= $513 - 480 = $33
NPVB = $200 +
$200 + $200 - $480
(1 + .1) (1 + .1)2 (1 + .1)3
= $200(.909) + 200(.826) + 200(.751) - 480
= $497 - 480 = $17
The firm should make both investments because their net present values are positive.
b. Determination of the internal rate of return for A:
$480 = $300 +
(1 + r)
$200 + $100
(1 + r)2 (1 + r)3
Since this is not an annuity, select an interest rate and attempt to equate both sides of the
equation. For example, use 14%:
$300(.877) + 200(.770) + 100(.675) = $484.60 which is
approximately equal to $480.
The internal rate of return on investment A is approximately 14% (14.68% using a
financial calculator that accepts uneven cash flows).
Determination of the internal rate of return for B:
$480 = $200 + $200 + $200
(1 + r) (1 + r)2 (1 + r)3
$480 = $200(PVAIF for 3 years at ? percent)
IF = 2.000
Locate 2.000 in the interest table for 3 years and
determine the internal rate of return to be 12%.
(PV = -480; N = 3; I = ?; PMT = 200, and FV = 0.
I = 12.04.)
Since the internal rates of return exceed the cost of capital (10%), the firm should make
both investments. (This is the same answer determined in part a.)
c. If the cost of capital were to increase to 14%, the
net present values would be
A: $484.60 - 480 = $4.60
B: $464.20 - 480 = ($15.80)
The net present values decline in both cases, but the net present value of A is still positive,
so the firm should make that investment. (You should point out that the increase in the
cost of capital does not change the investments' internal rates of return. However, since
investment B's internal rate of return is now less than the cost of capital, that investment
should not be made. This is the same conclusion derived from using the net present value.)
#13 A firm has the following investment alternatives: Cash Inflows </p><p>Year A
B C</p><p>1 $1,100 $3,600 ____ </p><p>2 1,100_____ ____ </p><p>3 1,100 _____
$4,562 </p><p>Each investment costs $3,000; investments B and C are mutually
exclusive, and the firm's cost of capital is 8%. a) What is the net present value of
each investment? b) According to the net present values, which investment(s) should
the firm make? Why? c) What is the internal rate of return on each investment? d)
According to the internal rates of return, which investment(s) should the firm make?
Why? e) According to both the net present values and internal rates of return,
which investments should the firm make? f) If the firm could reinvest the $3,600
earned in year one from investment B at 10%, what effect would that information
have on your answer to part e? Would the answer be different if the rate were 14%?
g) If the firm's cost of capital had been 10%, what would be investment A's internal
rate of return? h) The payback method of capital budgeting selects which
investment? Why?
a. Determination of the net present values:
Net present value of investment A:
$1,100(PVAIF 8I, 3N) - $3,000 = $1,100(2.577) - $3,000
= ($165)
Net present value of investment B:
$3,600/(1 + .08) - $3,000 = $3,600(.926) - $3,000 = $333
Net present value of investment C:
$4,562/(1 + .08)3 - $3,000 = $4,562(.794) - $3,000 = $621
b. Investment A is not acceptable because its net present value is negative. Since B and
C are mutually exclusive, the firm selects C; it has the higher net present value.
c. Determination of the internal rates of return:
Investment A:
$1,100/(1 + rA)t = $3,000
interest factor = $3,000/$1,100 = 2.727
rA = approximately 5%
(PV = -3000; N = 3; I = ?; PMT = 1100, and FV = 0.
I = 4.92.)
Investment B:
$3,600/(1 + rB) = $3,000
interest factor = $3,000/$3,600 = .833
rB = 20%
(PV = -3000; N = 1; I = ?; PMT = 0, and FV = 3600.
I = 20.)
Investment C:
$4,562/(1 + rC)3 = $3,000
interest factor = $3,000/$4,562 = .6575
rC = 15%
(PV = -3000; N = 3; I = ?; PMT = 0, and FV = 4562.
I = 14.99.)
Investment A is not acceptable because its internal rate of return is less than the cost of
capital. Since B and C are mutually exclusive, the firm selects B since it has the higher
internal rate of return.
d. Investment A is not acceptable because the internal rate of return is less than the
firm's cost of capital. Since B and C are mutually exclusive, the firm selects B because it
has the higher internal rate of return.
e. Net present value selects C, but internal rate of return selects B. Since the
investments are mutually exclusive, the firm must resolve the conflict.
f. At 10% reinvestment rate:
$3,600(1 + .1)2 = X
$3,600(1.210) = $4,356
The terminal value of investment B is less than the terminal value of C ($4,356 versus
$4,562). The conflict is resolved in favor of C.
At 14% reinvestment rate:
$3,600(1 + .14)2 = X
$3,600(1.297) = $4,669
The terminal value of B is now greater than the terminal value of C. The conflict is
resolved in favor of B.
Question f illustrates the importance of the reinvestment rate when the financial manager
must choose among competing investments that meet the acceptance criteria specified by
the net present value and internal rate of return methods for capital budgeting.)
g. The internal rate of return is not affected by the firm's cost of capital. The internal
rate of return of investment A remains approximately 5%.
h. The payback method will select that investment which returns the cost of the
investment the fastest. In this example, the payback method would select B, which as part
g above indicates, may not be the best choice.
#14 The chief financial officer has asked you to calculate the net present values and
internal rates of return of two $50,000 mutually exclusive investments with the
following cash flows: </p><p>YearProject ACash Flow Project B Cash
Flow</p><p>1$10,000 $ 0 </p><p>2 25,000 22,000 </p><p>3 30,000 48,000
</p><p>If the firm's cost of capital is 9%, which investment(s) would you
recommend? Would your answer be different if the cost of capital were 14%?
Using a 9 percent cost of capital:
Present value of A:
$10,000 x .917 = $ 9,170
25,000 x .842 = 21,050
30,000 x .772 = 23,160
$53,380
Net present value: $53,380 - $50,000 = $3,380
Present value of B:
$ 0 x .917 = $ 0
22,000 x .842 = 18,524
48,000 x .772 = 37,056
$55,580
Net present value: $55,580 - $50,000 = $5,580
Since B has the higher net present value, select B.
Internal rate of return A:
at 12%: $10,000 x .893 = $ 8,930
25,000 x .797 = 19,925
30,000 x .712 = 21,360
$50,215
at 14%: $10,000 x .877 = $ 8,770
25,000 x .769 = 19,225
30,000 x .675 = 20,250
$48,245
The internal rate of return is between 12 and 14% (12.2%).
Internal rate of return B:
at 12%: $ 0 x .893 = $ 0
22,000 x .797 = 17,534
48,000 x .712 = 34,176
$51,710
at 14%: $ 0 x .877 = $ 0
22,000 x .769 = 16,918
48,000 x .712 = 32,400
$49,318
Once again the internal rate of return is between 12 and 14% but closer to 14 since the
range of values has increased. Using a calculator that accepts uneven cash flows the IRR
is 13.4%.
Both techniques (the NPV and the IRR) accept B over A.
Using a 14 percent cost of capital:
Present value of A:
$10,000 x .877 = $ 8,770
25,000 x .769 = 19,225
30,000 x .675 = 20,250
$48,245
Net present value: $48,245 - $50,000 = ($1,755)
Present value of B:
$ 0 x .877 = $ 0
22,000 x .769 = 16,918
48,000 x .675 = 32,400
$49,318
Net present value: $49,318 - $50,000 = ($682)
Since both NPVs are negative, accept neither. Also the IRRs calculated above are less
than 14%, so accept neither.
#15 An investment with total cost of $10,000 will generate total revenues of $11,000
for one year. Management thinks that since the investment is profitable, it should be
made. Do you agree? What additional information would you want? If funds cost
12%, what would be your advice to management? Would your answer be different
if the cost of capital is 8%?
While the investment offers an accounting profit of $1,000, that is insufficient
information to make the investment. Alternative uses for the funds and their cost should
be considered.
PV = $11,000/(1 + .12) = $11,000(.893) = $9,823
NPV = $9,823 - $10,000 = ($177)
Since the present value of the $11,000 is less than the $10,000 cost of the investment, the
investment should not be made (i.e., net present value is a negative $177).
At 8% cost of capital, the NPV is
PV = $11,000/(1 + .08) = $11,000(.926) = $10,186
NPV = $10,186 - $10,000 = $186
Now the present value of the $11,000 exceeds the $10,000 cost of the investment, and the
investment should be made.
#16 The financial manager has determined the following schedules for the cost of
funds: </p><p>Cost of Debt ratio, Cost of Debt. Equity </p><p>0% 5% 13%
</p><p>10 5 13 </p><p>20 5 13 </p><p>30 5 13 </p><p>40 5 14 </p><p>50 6 15
</p><p>60816 </p><p>a) Determine the firm's optimal capital structure? b)
Construct a simple pro forma balance sheet that shows the firm's optimal
combination of debt and equity for its current level of assets. Asssts =$500, Debt
____ Equity ____ </p><p> $500</p><p>c) An investment cost $400 and offers
annual cash inflows of $133 for five years. Should the firm make the investment? d)
If the firm makes this additional investment, how should its balance sheet appear?
</p><p>Assets _____ Debt ____ Equity ____ </p><p>e) If the firm is operating with
its optimal capital structure and a $400 asset yields 20.0%, what return will the
stockholders earn on their investment in the asset?
a. The cost of capital (k) is a weighted average:
k = (weight)(cost of debt) + weight(cost of equity)
Debt/
Assets
0%
10
20
30
40
50
60
Weight x + Weight x = Cost of
Cost
Cost
Capital
of Debt
of Equity
(.0)(.05) + (1.0)(.13) = .130
(.1)(.05) + (.9)(.13) = .122
(.2)(.05) + (.8)(.13) = .114
(.3)(.05) + (.7)(.13) = .106
(.4)(.05) + (.6)(.14) = .104
(.5)(.06) + (.5)(.15) = .105
(.6)(.08) + (.4)(.16) = .112
b. The optimal capital structure is that combination, which minimizes the firm's cost of
capital. In this case that occurs where debt is 40% of capital and the cost of capital is
10.4%. The balance sheet is
Assets
$500 Liabilities $200
Equity
300
$500
c. The internal rate of return:
$133(PVAIF ?I, 5N) = $400
IF = 3.01
r = approximately 20%
(PV = -400; N = 5; I = ?; PMT = 133, and FV = 400.
I = 19.7)
Since the cost of capital is 10.4%, the investment should be made since the internal rate
of return exceeds the cost of capital. (The net present value approached cannot be used if
the student is limited to using the interest tables. If a financial calculator is used, the PV
is $499, and the NPV is $99, which indicates that the investment should be made.)
d. Assets
$900 Liabilities $360
Equity
540
$900
e. If a $400 asset yields 20 percent, that is $80. The $80 is divided between the debt
and equity sources of funds. Debt finances $160 (40 percent), and equity finances $240
(60 percent). Since the cost of debt is 5 percent, that requires only $8 which leaves $72
for the stockholders, which generates a $72/$240 = 30% return.
#17 Investment Quick and Slow cost $1,000 each, are mutually exclusive, and have
the following cash flows. The firm's cost of capital is 10%. </p><p>Year, Cash
Inflows Q, Cash Inflow S</p><p>1 $1,300 $386 </p><p>2 ____ 386 </p><p>3____
386 </p><p>4 ____ 386 </p><p>a) According to the net present value method of
capital budgeting, which investment(s) should the firm make? b) According to the
internal rate of return method of capital budgeting, which investment(s) should the
firm make? c) If Q is chosen, then $1,300 can be reinvested and earn 12%. Does this
information alter your conclusions concerning investing in Q and S? To answer,
assume that S's cash flows can be reinvested at its internal rate of return. Would
your answer be different if S's cash flow were reinvested at the cost of capital
(10%)? .
a. Net present value of Q:
NPVS = $1,300/(1 + .1) - $1,000
= $1,181.70 - $1,000 = $181.70
Net present value of S:
$386(PVAIF 10I, 4N) - $1,000 = $386(3.170) - $1,000
= $223.62
Since the investments are mutually exclusive, the firm can only make one and will select
S because its net present value is higher.
b. Internal rate of return for Q:
$1,000 = $1,300/(1 + rQ)
(1 + rQ) = $1,000/$1,300 = .769
rQ = 30%
Internal rate of return for S:
$386(PVAIF ?I, 4N) = $1,000
Interest factor = $1,000/386 = 2.591
rS = 20%
Since the investments are mutually exclusive, the firm can only make one and will select
Q because its internal rate of return is higher. (This answer contradicts the answer to part
a.)
c. Reconciliation depends on what the firm can do with the $1,300 earned in year 1 by
investment Q. In this part the funds can be reinvested at 12%, so the terminal value of
investment Q is
$1,300(1 + .12)t = $1,300(1.405) = $1,826.50
The terminal value of S also depends on the reinvestment rate, which is S's internal rate
of return of 20%.
$386(FVAIF 20I, 4N) = $386(5.386) = $2,072.05
Since the terminal value of S is larger, it is preferred to Q. However, if the reinvestment
rate of S had been the cost of capital (10%), then terminal value of S would have been
$386(4.641) = $1,791.42,
and investment Q would have been selected. This problem illustrates the importance of
the reinvestment rate.
#18 a) What is the EOQ for a firm that sells 5,000 units when the cost of placing an
order is $5 and the carrying cost are $3.50 per unit? b) How long will the EOQ last?
How many orders are placed annually? c) As a result of lower interest, the financial
manager determines the carrying cost are now $1.80 per unit. What are the new
EOQ and annual number of objects?
a. EOQ = [(2SO)/C].5 = [(2)(5,000)($5)/$3.50].5 = 119.5 units
b. Annual number of orders:
Sales per day: 5,000/365 = 13.7 units (i.e., 14 units)
Duration of the EOQ: 120/13.7 = 8.8 days (i.e., 9 days)
Annual number of orders: 365/8.8 = 41.4
(i.e., 42 orders a year)
Alternative calculation of annual number of
orders: 5,000/119.5 = 42 (orders per year)
c. EOQ = [(2SO)/C].5 = [(2)(5,000)($5)/$1.80].5 = 166.7 units
Annual number of orders: 5000/166.7 = 30
Alternative calculation:
Sales per day: 5,000/365 = 13.7 units (i.e., 14 units)
Duration of the EOQ: 166.7/13.7 = 12.2 days
(i.e., 12 days)
Annual number of orders: 365/12.2 = 29.9
(i.e., 30 orders a year)
#19 Given the following information: Annual sales in units =30,000, Cost of placing
an order =$60,00, Per-unit carrying costs =$ 1.50, Existing units of safety stock =300.
</p><p>a) What is the EOQ? b) What is the average inventory based on the EOQ
and the existing safety stock? c) What is the maximum level of inventory? d) How
many orders are placed each year?
a. EOQ = [2(30,000)($60)/$1.50].5 = 1,549 units
b. Average inventory: EOQ/2 + safety stock
= 1,549/2 + 300 = 1074.5 units
c. Maximum level of inventory: 1,549 + 300 = 1,849 units
d. Orders placed each year: 30,000/1,549 = 19 orders
#20 What is the effective, compound rate of interest you earn if you enter into a
repurchase agreement in which you buy a Treasury bill for $76,789 and agree to sell
it after a month (30 days) for $77,345? What is the compound rate of interest you
pay if you sell a Treasury bill for $76,789 and repurchase it after 30 days for $77,345?
$76,789(1 + i)30/365 = $77,345
76,789(1 + i).082192 = 77,345
(1 + i).082192 = 77,345/76,789 = 1.007241
i = 1.0917 - 1 = 9.17%
(PV = 76789; I = ?; N = .082192; PMT = 0, and FV = 77,345.
I = 9.17.)
The yield earned by the buyer and the cost of the loan to the seller are the same (9.17%).
#21 Tinker, Inc, finances its seasonal working capital need with short-term bank
loans, Management plans to borrow $65,000 for a year. The bank has offered the
company a 3.5% discounted loan with a 1.5% origination fee. What are the interest
payment and the origination fee required by the loan? What is the rate of interest
charges by the bank?
Interest paid: $65,000 x 0.1 x 120/360 = $2,167
Origination fee: $65,000 x 0.015 = $975
Funds the firm gets to use:
$65,000 - $2,167 - $975 = $61,858
Since the loan is discounted and the firm pays an origination fee, the firm gets the use of
$61,858 but must repay $65,000 when the loan is due for a total cost of $3,142.
The simple rate of interest:
i = $3,142 x 360 = 15.24%
$61,858 120
If the firm only borrows once during the year, this non-compounded rate is sufficient, but
if the loan process is repeated (as is often the case), an annual, compound rate should be
computed. See the next problem.
#22 An individual wishes to borrow $10,000 for a year and is offered the following
alternatives: a) A 10% loan discounted in advance? b) An 11% straight loan (ie.,
interest paid at maturity). Which loan is more expensive?
Cost of the discounted loan paper:
$1,000 x 360 = 11.1%
$9,000 360
Cost of the non-discounted loan with the higher
stated rate:
$1,100 x 360 = 11%
$10,000 360
The loan with the higher stated interest rate has the lower effective cost of borrowing.
#23 Which of the following terms of trade credit is the more expensive: a) A 3%
cash discount if paid on the 15th day with the bill due on the 45th day (3/15, net 45)?
b) A 2% cash discount if paid on the 10th day with the bill due on the 30th day (2/10,
net 30)?
a. The simple rate of interest rate:
i = .03 x 360 = 37.1%
1 - .03 45 - 15
The compound interest rate:
i = (1.0309)30/365 - 1 = .449 = 44.9%
b. The simple interest rate:
i = .02 x 360 = 36.7%
1 - .02 30 - 10
The compound interest rate:
i = (1.0204)18.25 - 1 = .446 = 44.6%
The interest rates are about the same.
#24 If $1 million face amount of commercial paper (270-day paper) is sold for
$982,500, what is the simple rate of interest being paid? What is the compound
annual rate?
The simple rate of interest:
i = $17,500 x 12 = 2.37%
$982,500 9
The compound rate of interest:
i = ($1,000,000/$982,500)365/270 - 1 = .0242 = 2.42%
#25 Bank A offers the following terms for a $10 million loan: </p><p>• Interest rate:
8% for one year on funds borrowed </p><p>• Fees: 0.5% of the unused balance for
the unused term of the loan </p><p>Bank B offers the following terms for a $10
million loan: </p><p>• Interest rate: 6.6% for one year on funds borrowed
</p><p>• Fees: 2% origination fee </p><p>a) Which terms are better if the firm
intends to borrow the $10 million for the entire year? b) If the firm plans to use the
funds for only three months, which terms are better?</
a. Cost of Bank A's loan:
interest payment: (.08)($10,000,000) = $800,000
fee: none if the loan is outstanding for the entire year
i = $800,000 x 12 = 8%
$10,000,000 12
Cost of Bank B's loan:
interest payment: (0.066)($10,000,000) = $660,000
origination fee: (0.02)($10,000,000) = $200,000
i =
$660,000
x 12 = 6.73%
$10,000,000 - 200,000 12
b. Cost of Bank A's loan:
interest payment: (.08)($10,000,000)(3/12) = $200,000
fee on the unused balance:
(0.05)($10,000,000)(.75) = $37,500
i = $237,500 x 12 = 9.5%
$10,000,000 3
Cost of Bank B's loan:
interest payment: (0.066)($10,000,000)(.25) = $165,000
commitment fee: (0.02)($10,000,000) = $200,000
i =
$165,000
x 12 = 6.73%
$10,000,000 - 200,000 3
Bank B's terms are better under both scenarios.