Principles of Finance with Excel (Основы финансов c Excel)
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PFE Chapter 20, Capital structure and valuation page 33
()
(
)
(
)
After-tax After-tax After-tax
ordinary income equity income corporate income
(including interest)
Net after-tax personal
income from pre-tax corporate
cash flows
111
DEC
TTT
↑↑↑
↑
−−− −
If this term is positive, as in the previous example (see cell B32), then XYZ corporation should
borrow; if it’s negative—as in the next example (in which the corporate tax rate is T
C
= 20%),
then Arthur should borrow and not the firm:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
ABC D
Computing the family income
T
C
, corporate tax rate
20%
T
E
, personal equity tax rate
10%
T
D
, personal debt tax rate on ordinary income
30%
r
D
, interest rate
8%
D, Debt 3,000
FCF, free cash flow (already after corporate taxes) 1,000
Company
borrows
A
rthur
borrows
FCF, after personal tax 1,000.00 1,000.00
Corporate debt 3,000.00 0.00
Corporate pre-tax interest payment 240.00 0.00
Corporate after-tax interest payment 192.00 0.00 <-- =C13*(1-$B$3)
Payout to equity owners 808.00 1,000.00 <-- =C11-C14
Arthur's income
Pre-tax equity income from XYZ 808.00 1,000.00 <-- =C15
Post-tax equity income from XYZ 727.20 900.00 <-- =C18*(1-$B$4)
Arthur's debt 0.00 3,000.00
Arthur's pre-tax interest payment 0.00 240.00 <-- =$B$6*C20
Arthur's after-tax interest payment 0.00 168.00
Arthur's post-tax income 727.20 732.00 <-- =C19-C22
Mom's pre-tax income 240.00 240.00 <-- =C20*B6
Mom's post-tax income 168.00 168.00 <-- =C25*(1-$B$5)
Total family income 895.20 900.00 <-- =C23+C26
Who should borrow--Arthur or company? Arthur <-- =IF(B27>C27,"Company",IF(B27<C27,"Arthur","Indifferent"))
Net advantage of corporate debt
(1-T
D
)-(1-T
E
)*(1-T
C
)
-0.02
FINANCING ARTHUR'S PURCHASE OF XYZ
Upper Fantasia tax code: Corporate income tax, T
C
= 20%
(instead of 40% in previous example)
Personal taxes: Tax on equity income, T
E
= 10%,
Tax on all other income, T
D
= 30%
PFE Chapter 20, Capital structure and valuation page 34
Some Finance History (2)
The Modigliani-Miller model dates from two articles published in 1958 and 1963. In
1977 Merton Miller (half of the MM team), reconsidered the problem of capital structure. He
still focused on taxation, but this time considered the case where both corporate and personal
income was taxed.
Miller’s reasoning, incorporated in our example of XYZ Corp, was that the corporate tax
rate T
C
gives an advantage to corporations wishing to finance with debt. On the other hand, for
individuals equity income is generally taxed at a lower rate T
E
than the tax rate T
D
on debt
income. The primary reason for this is that the major part of income from equity is received by
shareholders as capital gains; these are not only taxed at a lower tax rate, but the taxes on capital
gains are also postponable (as a shareholder, you can decide when to sell your shares and realize
your capital gains). This postponability lowers the T
E
below the statutory rate (see some
discussion in Chapter 21). Thus, Miller reasoned, there is a tradeoff:
•
On the corporate level, the deductibility of interest means that corporations produce
higher before-personal-tax payouts to stakeholders (bondholders and shareholders) when
they have more debt financing.
•
On the personal level, giving stakeholders (bond and shareholders) more interest income
instead of equity income means taxing them at higher personal rates.
This tradeoff is summarized in the expression
(
)
(
)
(
)
111
D
CE
TTT−−− −:
()
(
)
(
)
Payments to debtholders are Equity income is taxed twice:
only taxed at the personal tax of once at the firm
the debtholder, since the firm makes these
payments out of pre-tax income
11*1
DCE
TTT
↑↑
−−−−
level (since
payments to shareholders are paid
out of after-tax earnings), and then again at the
personal level
On the other hand, , so that there is a tradeoff ...
ED
TT
↑
<
PFE Chapter 20, Capital structure and valuation page 35
XYZ CORP--ARTHUR AND MOM'S INCOME
XYZ Corp
A
fte
r
-tax cost of debt: (1-T
C
)r
D
Each $ of before tax interest paid:
1) Decreases shareholder income by (1-T
C
)
2) Increases Arthur's Mom's interest payments by $1.
Shareholder Arthur
Equity income taxed at T
E
Each $ of before tax interest paid by XYZ
decreases Arthur's income from the
company by (1-T
C
)*(1-T
E
)
Bondholder Mom
Interest income taxed at T
D
Each $ of interest paid by the company
increases Mom's income by (1-T
D
)
Family income: Arthur + Mom
Each $ of before-tax interest paid by XYZ increases Mom's
income by (1-T
D
) and decreases Arthur's by (1-T
C
)*(1-T
E
).
Net effect: (1-T
D
) - (1-T
E
)*(1-T
C
).
If this is positive, it's good for the family and the firm should
increase its borrowing; if its negative, it's bad for the family and
the firm should decrease its borrowing.
Miller suggested that
in equilibrium
(1-T
D
) - (1-T
E
)*(1-T
C
) = 0
Figure 20.3. Cash flows of Arthur + Mom’s family income. In this flow diagram, Arthur is the
shareholder of XYZ Corp., and Mom is the bondholder of XYZ (meaning: she lends the
company money). Each $1 of interest income paid to Mom by XYZ Corp. changes the family
income by
()()()
11*1
D
EC
TTT−−− −. If this term is positive, then XYZ Corp.’s borrowing from
Mom adds to the family income; if it is negative, then XYZ Corp.’s borrowing detracts from the
family income.
PFE Chapter 20, Capital structure and valuation page 36
20.7. Valuing XYZ Corp.—taking account of leverage and taxes
We redo the calculations in section 20.2, but this time use all the taxes—the corporate tax
rate T
C
, the personal tax rate on equity income T
E
, and the personal tax rate on ordinary income
(including interest) T
D
. Without leverage XYZ Corp’s FCFs are worth $5,000:
()
()
1
,@
1,000 1,000
5,000
20%
1.20
U
t
t
U
V Unlevered value of XYZ
PV future FCFs discounted unlevered discount rate
Annual FCF
r
∞
=
=
=
== ==
∑
We use the additivity principle to value the levered version of XYZ Corp:
(
)
() ()
11
-
1,000 1,000
5,000
20%
11.20
The unlevered value of the firm is
the present value of its free cash flows
discounted an an appropriate (unlevered)
cost
LU
t
U
tt
tt
U
V V PV additional debt related CFs
FCF
V
r
∞∞
==
=+
====
+
=
∑∑
()()()
()
()
()
()
1
1
of capital
11*1*
11
8%*3,000*16% 38.4
685.71
5.6%
1 8%* 1 30%
The tax shields created
by the debt are discounted at the
con
U
DCE t
t
t
DD
t
t
r
T T T Interest
Interest tax
PV
shields
Tr
∞
=
∞
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
⎡− − − − ⎤
⎛⎞
⎣⎦
=
⎜⎟
⎝⎠
+−
===
+
+−
∑
∑
sumer's after-tax interest rate.
5,685.71
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
=
XYZ Corp is worth more as a levered firm than as an unlevered firm because it produces
more cash for its owners when it is levered. The additional cash produced—generated by the
PFE Chapter 20, Capital structure and valuation page 37
fact that the company has a cheaper cost of debt than Arthur—is worth $685.71, which is the
present value of the future tax shields on the interest:
(
)
(
)
(
)
()
()
()()()
()
()()()
()
()()()
()
1
11*1*
11
11*1
*
1
11*1
*
1
11*1
*
1
DCE
t
t
DD
DCE
DD
DCE
DD
DCE
D
T T T Interest
Interest tax
PV
shields
Tr
TTT
I
nterest
Tr
TTT
I
nterest
Tr
TTT
D
T
∞
=
⎡⎤
−−− −
⎛⎞
⎣⎦
=
⎜⎟
⎝⎠ +−
⎡⎤
−−− −
⎣⎦
=
−
⎡⎤
−−− −
⎣⎦
=
−
⎡⎤
−−− −
⎣⎦
=
−
∑
We use the letter T to denote the debt-valuation factor:
(
)
(
)( )
()
11*1
1
DCE
D
TTT
T
T
⎡
⎤
−−− −
⎣
⎦
=
−
. T is
the capitalized advantage of debt.
9
What about the cost of capital—r
E
and WACC with leverage?
The levered version of XYZ Corp. is worth $5,685.71, of which $3,000 is debt.
Subtracting the value of the debt from the total worth of the company, we see that the equity of
the company is worth $2,685.71. In order to calculate the firm’s cost of equity r
E
, we first
compute the after-tax cash flows accruing to the equity owners:
[
]
()
--
1,000 8%*3,000* 1 40% 856.00
annual after corporate tax equity cash flow FCF after -tax interest paid by XYZ=−
⎡⎤
=− −=
⎣⎦
The discounted value of this annual equity cash flow of $856.00 is the value of the equity; this
defines the cost of equity r
E
:
9
To relate this to the previous case with only corporate taxes, note that when T
E
= T
D
= 0, T = T
C
.
PFE Chapter 20, Capital structure and valuation page 38
()
()
1
1
1
856.00 856.00
2,685.71
1
856.00
31.87%
2685.71
t
t
t
E
t
t
E
E
E
equity cash flow
Equity value
r
r
r
r
∞
=
∞
=
=
+
==
+
⇒= =
∑
∑
With a little mathematical flimflammery, we can show that:
() ( )
N
()()
U
r is the discount
rate for the FCFs,
When XYZ borrows, it's shareholders
which represents
bear an additional
the firm's business
risk
*1 *1
3,000
20% 20% 1 22.86% 8% 1 40%
2,685.71
EU U D C
fi
D
rr r Tr T
E
↑
↑
⎡⎤
=+ −− −
⎣⎦
⎡⎤
=+−−−
⎣⎦
. The
term above represents the financial risk
premium for the equity holders
31.87%
nancialrisk
=
We can now compute the WACC:
()
()
1
2,685.71 3,000
31.87% 8% 1 40%
2,685.71 3,000 2,685.71 3,000
17.59%
EDC
ED
WACC r r T
ED ED
=+−
++
=+−
++
=
With a little more “flimflammery” we can show that discounting the FCFs at the WACC
gives the total value of the firm:
()()
11
1,000 1,000
5,685.71
17.59%
1 1 17.59%
t
tt
tt
FCF
WACC
∞∞
==
===
++
∑∑
Here’s all of this summarized in a spreadsheet:
PFE Chapter 20, Capital structure and valuation page 39
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
ABC
FCF, annual free cash flow (already after corporate taxes) 1,000
r
U
, unlevered cost of capital
20%
D, Debt 3,000
r
D
, interest rate
8%
T
C
, corporate tax rate
40%
T
E
, personal equity tax rate
10%
T
D
, personal debt tax rate on ordinary income
30%
Tax advantage of debt, (1-T
D
)-(1-T
C
)*(1-T
E
)
16.00% <-- =(1-B8)-(1-B6)*(1-B7)
T= [(1-T
D
)-(1-T
C
)*(1-T
E
)]/(1-T
D
) , tax factor
22.86% <-- =B10/(1-B8)
V
alue of firm
V
U
, unlevered value
5,000.00 <-- =B2/B3
Value of tax shield on interest 685.71 <-- =B10*B5*B4/((1-B8)*B5)
V
L
, levered value of firm
5,685.71 <-- =B15+B14
E, value of equity 2,685.71 <-- =B16-B4
Cash flow to equity 856.00 <-- =B2-(1-B6)*B5*B4
Return on equity, r
E
(L)
31.87% <-- =B20/B18
WACC 17.59% <-- =B21*B18/B16+(1-B6)*B5*B4/B16
Three checks
Return on equity, r
E
(L) = r
U
+ [r
U
*(1-T) - r
D
*(1-T
C
)]*D/E
31.87% <-- =B3+(B3*(1-B11)-B5*(1-B6))*B4/B18
Value of firm, V
L
= FCF/WACC
5,685.71 <-- =B2/B23
Value of firm, V
L
= V
U
+ T*D
5,685.71 <-- =B14+B11*B4
COMPUTING THE WACC IN THE MILLER MODEL
with corporate and personal taxes
Summarizing this section
We complete this section by restating its major conclusions. If corporate income is taxed,
and if the tax system differentiates between income derived from equity and ordinary income,
then leverage (borrowing) may increase or decrease the value of the firm, depending on the sign
of the tax factor
()()()
11*1
D
EC
TTT−−− −.
A summary table is given below:
PFE Chapter 20, Capital structure and valuation page 40
SUMMARY TABLE—CHANGING LEVERAGE WHEN CORPORATE AND PERSONAL INCOME ARE TAXED
Symbols: Corporate tax rate: T
C
, personal tax rate on equity income T
E
, personal tax rate on ordinary income T
D
Tax advantage of debt =
(
)
(
)
(
)
11*1
D
CE
TTT−−− −, Tax factor:
(
)
(
)
(
)
()
11*1
1
D
CE
D
TTT
T
T
−−− −
=
−
Item Formula Why
V
U
= Value of unlevered firm
()
1
1
t
U
t
t
U
FCF
V
r
∞
=
=
+
∑
The value of the unlevered firm is the PV of future
FCFs discounted at r
U
, the unlevered cost of
capital
(
)
()()()
()
()
1
111*
11
LU
N
D
EC t
U
t
t
DD
V V PV net interest taxshields
TTTInterest
V
rT
=
=+
⎡⎤
−−− −
⎣⎦
=+
+−
∑
Another way to write this is *
LU
VVTD
=
+ , where
The value of the levered firm is V
U
plus the present
value of future interest tax shields. When there are
both corporate and personal taxes, the PV of the tax
shields is given by:
(
)
(
)
(
)
()
()
1
111*
11
N
D
EC t
t
t
DD
TTTInterest
rT
=
⎡⎤
−−− −
⎣⎦
+−
∑
V
L
= Value of the levered firm
(
)
()()()
()
()
()()()
()
1
111*
11
*,
111
1
LU
DEC
U
t
t
DD
U
DEC
D
V V PV net interest taxshields
T T T Interest
V
rT
VTD
TTT
where T
T
∞
=
=+
⎡⎤
−−− −
⎣⎦
=+
+−
=+
−−− −
=
−
∑
The cell to the left contains the formula for the
value of the levered firm when the firm issues
perpetual debt. This formula is the same as the
parallel formula in Figure 20.2 for the case where
T
E
=T
D
=0.
In the general case where personal taxes are
perhaps not zero,
(
)
(
)
(
)
()
111
1
D
EC
D
TTT
T
T
−−− −
=
−
can
be positive, negative, or zero.
E = Value of equity
(
)
1
U
E
VTD=−−
The equity value of the levered firm =
(
)
1
LU
E
VDV TD=−=−−
D=Value of Debt D
()
E
rL
= Cost of equity of the
levered firm
() ( ) ( )
11
EUU DC
D
rL r r T r T
E
⎡
⎤
=+ −− −
⎣
⎦
WACC = weighted average
cost of capital
WACC=
L
FCF
V
Figure 20.4 . Corporate value and cost of capital when corporate income is taxed at rate T
C
, personal ordinary income is taxed at rate T
D
, and
personal equity income is taxed at rate T
E
.
PFE Chapter 19, Capital structure and valuation page 41
20.8. Buying a sturfing machine in Upper Fantasia
In this section and the next we return to the examples of sections 20.3 and 20.4. This
time we do these examples for a company in Upper Fantasia, where, as you will recall there are
three tax rates:
• In Upper Fantasia corporate income is taxed at the rate T
C
= 40%
• Personal income from equity (meaning: dividends and capital gains) is taxed at rate T
E
=
10%
• Personal income from all other sources is taxed at rate T
D
= 30%
Sonderturf considers buying a sturfing machine
Sonderturf Corp., a company is Upper Fantasia, is considering purchasing a new sturfing
machine. The sturfing machine costs $100,000; it has a ten-year life, during which it is straight-
line depreciated to zero salvage value. In each of the ten years of the machine’s life, it will
produce sales of $40,000. These sales will cost $15,000 to produce. The result is that the
machine has an annual free cash flow of $19,000 per year.
The Sonderturf financial wizards have determined that an appropriate risk-adjusted
discount rate for the sturfing machine’s free cash flows is r
U
= 15%. Discounting the machine’s
FCFs at this rate shows that it has a negative NPV of -$4,643. Thus the conclusion is that
Sonderturf should not acquire the sturfing machine. (For details of these calculations, refer to
section 20.3, page000.)
PFE Chapter 19, Capital structure and valuation page 42
Sonderturf gets a loan to buy the machine
Having heard the bad news from Sonderturf, the sturfing machine’s manufacturer offers
the company a loan of $50,000. The loan’s conditions are exactly the same as those of the loan
in section 20.??? which was offered to Sonderturf in Lower Fantasia: In years 1-9, Sonderturf
will pay only interest ($4,000), and in year 10 it will pay interest of $4,000 as well as repay the
loan principal.
It follows from Figure 20.4 that the value of the loan is
()
()()()
()
()
()()()
10
1
,
,,
111*
11
1 30% 1 10% 1 40%
CE
D
DEC t
t
t
DD
loan in Upper Fantasia where there are corporate income
PV taxes T taxes on equity income T
and taxes on ordinaryincome T
T T T Interest
PV net interest tax shields
rT
=
⎛⎞
⎜⎟
=
⎜⎟
⎜⎟
⎝⎠
−−− −
⎡⎤
⎣⎦
=
+−
−−− −
=
∑
()
()
10
1
*$4,000
$4,801
1 1 30%
t
t
D
r
=
⎡⎤
⎣⎦
=
+−
∑
The Sonderturf financial wizards conclude that the company should now purchase the
machine, taking the loan to finance part of the purchase. They calculate that:
()()
(
)
()( )
()()()
()
()
In Upper Fantasia the tax shield
takes account of corporate as well as
personal taxes:
10
DEC t
t
t=1
DD
1-T - 1-T 1-T *Interest
1+ 1-T * r
NPV machine loan NPV machine NPV loan
NPV machine PV loan interest tax shields
↑
⎡⎤
⎣⎦
+= +
=+
∑
$4,643 $4,801
$158
=− +
=
The calculations are shown in the following Excel spreadsheet: