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b. 2. Graph (a) the relationships between capital costs and leverage as measured by
D/V, and (b) the relationship between value and D.
Answer: Figure 1 plots capital costs against leverage as measured by the debt/value ratio.
Note that, under the MM no-tax assumption, rd is a constant 8 percent, but rs increases
c. Using the data given in part B, but now assuming that firms L and U are both
subject to a 40 percent corporate tax rate, repeat the analysis called for in B(1)
and B(2) under the MM with-tax model.
Answer: With corporate taxes added, the MM propositions become:
Proposition I: VL = VU + TD.
There are two very important differences between these propositions and the zero-tax
propositions: (1) when corporate taxes are added, VL does not equal VU; rather, VL
increases as debt is added to the capital structure, and the greater the debt usage, the
higher the value of the firm. (2) rsL increases less rapidly when corporate taxes are
4
3
2
1
0 0.5 1.0 1.5 2.0 2.5
Firm Value ($3.6 Million)
Value of Firm, V
(Millions of $)
VUVL
($)
Debt ($)
Figure 2
website, in whole or in part.
This represents a 40% decline in value, and it is logical, because the 40% tax rate
takes away 40% of the income and hence 40% of the firm’s value.
Looking at VL, we see that:
D + SL = VL
$1,000,000 + SL = $2,542,857
SL = $1,542,857.
now,
= ($1,000/$2,543)(8%)(0.6) + (1,543/$2,543)(16.33%)
= 1.89% + 9.91% = 11.8%.
The WACC is lower for the levered firm than for the unlevered firm when corporate
taxes are considered.
Figure 3 below plots capital costs at different D/V ratios under the MM model
downward-sloping WACC curve.
Figure 4 shows that, when corporate taxes are considered, the firm’s value
increases continuously as more and more debt is used.
Mini Case: 21 - 12
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website, in whole or in part.
Figure 3
With Taxes
0%
5%
35%
40%
45%
50%
0% 20% 40% 60% 80% 100%
Debt/Value Ratio
rs
WACC
rd x (1-T)
Mini Case: 21 - 13
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d. Suppose that Firms U and L are growing at a constant rate of 7% and that the
investment in net operating assets required to support this growth is 10% of
EBIT. Use the compressed adjusted present value (APV) model to estimate the
value of U and L. Also estimate the levered cost of equity and the weighted
average cost of capital.
Answer: If a firm is growing, the assumptions that MM made are violated. The extension to
shield.
First, calculate expected free cash flow:
Next, note that WACC = unlevered cost of equity if there is no debt so
WACC = rsU = 14%
Therefore, the value of the firm = $3,571,429 + $457,143 = $4,028,571.
The value of the equity is the value of the firm less the value of the
This is less than the increase in the non-growing firm’s value as calculated using the
And the new levered WACC:
WACCL = (D/V)rd(1 - T) + (S/V)rs
= (1,000,000/4,028,571)8%(1-.40)
+ ($3,028,571/4,028,571)15.98%
= 13.2%.
e. Suppose the expected free cash flow for Year 1 is $250,000 but it is expected to
grow unevenly over the next 3 years: FCF2 = $290,000 and FCF3 = $320,000,
after which it will grow at a constant rate of 7%. The expected interest expense
at Year 1 is $80,000, but it is expected to grow over the next couple of years
before the capital structure becomes constant: Interest expense at Year 2 will be
$95,000, at Year 3 it will be $120,000 and it will grow at 7% thereafter. What is
the estimated horizon unlevered value of operations (i.e., the value at Year 3
immediately after the FCF at Year 3)? What is the current unlevered value of
operations? What is the horizon value of the tax shield at Year 3? What is the
current value of the tax shield? What is the current total value? The tax rate and
The unlevered value of operations is the present value of the free cash flows and the
horizon value. In Excel, the formula is:
=NPV(Rate,value1,value2….) = NPV(0.14,250,290,320+4891.43) = $3,960.01.
website, in whole or in part.
The horizon value of the tax shield can be found by applying the constant growth
=NPV(Rate,value1,value2….) = NPV(0.14,32,38,48+733.71) = $584.94.
The value of operations is the sum of the unlevered value of operations and the value
of the tax shield:
Vop = VU + VTS = $3,960.01 + $584.94 = $4,544.95.
