Principles of Finance with Excel (Основы финансов c Excel)
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PFE Chapter 19, Capital structure and valuation page 43
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ABC DEF
T
C
, corporate tax rate
40%
T
E
, personal tax rate on equity
10%
T
D
, personal tax rate on debt
30%
Machine cost, year 0 100,000
Free cash flow (FCF) calculation
Additional sales, annually 40,000
Additional annual cost of sales 15,000
Annual depreciation 10,000 <-- =B6/10
Annual FCF, years 1-10 19,000 <-- =(1-B2)*(B9-B10-B11)+B11
Discount rate for machine FCFs 15% Loan to buy machine 50,000
r
D
, loan interest rate
8%
Net annual advantage of debt
financing, (1-T
D
)-(1-T
E
)*(1-T
C
)
16% <-- =(1-B4)-(1-B3)*(1-B2)
Year
Machine
FCF
Tax
advantage
of interest
0 -100,000 <-- =-B6
1 19,000 <-- =$B$12 640 <-- =$E$16*$E$15*$E$14
2 19,000 640 <-- =$E$16*$E$15*$E$14
3 19,000 640
4 19,000 640
5 19,000 640
6 19,000 640
7 19,000 640
8 19,000 640
9 19,000 640
10 19,000 640
Machine NPV -4,643 <-- =B19+NPV(B14,B20:B29) Loan NPV 4,801 <-- =E19+NPV(E15*(1-B4),E20:E29)
NPV: Machine + Loan 158 <-- =B31+E31
THE SONDERTURF STURFING MACHINE
In Upper Fantasia debt is not always valuable!
The Lower Fantasia tax system—which has only a corporate tax T
C
but no other taxes on
personal income—always makes it more valuable to finance with debt. You can see this from
the following formula drawn from Figure 20.2, which holds in Lower Fantasia:
()
()
1
*
1
Lower Fantasia
Ct
L
UUU
t
t
D
T Interest
V V PV interest taxshields V V
r
∞
=
=+ =+ >
+
∑
.
The same formula in Upper Fantasia—with its more complicated (but more realistic) tax system
which combines a corporate income tax T
C
with a personal tax on equity income T
E
and a
personal tax on ordinary income T
D
—is given by:
()
(
)
(
)
(
)
()
()
1
11*1*
11
N
D
EC t
Upper Fantasia
LU U
t
t
DD
T T T Interest
V V PV interest taxshields V
Tr
=
⎡⎤
−−− −
⎣⎦
=+ =+
+−
∑
The last expression need not always be positive. For example:
PFE Chapter 19, Capital structure and valuation page 44
(
)
(
)
(
)
()
()
()()()
()()()
()
()
()()()
()()()
()
()
()()()
1
1
1
11*1*
01 1*1 0
11
11*1*
01 1*1 0
11
11*1*
01 1*10
11
N
DEC t
DEC
t
t
DD
N
DEC t
DEC
t
t
DD
N
DEC t
DEC
t
t
DD
TTTInterest
if T T T
Tr
TTTInterest
if T T T
Tr
T T T Interest
if T T T
Tr
=
=
=
⎡⎤
−−− −
⎣⎦
>−−−−>
+−
⎡⎤
−−− −
⎣⎦
=
−−− −=
+−
⎡⎤
−−− −
⎣⎦
<
−−− −<
+−
∑
∑
∑
The conclusion is that in Upper Fantasia, financing with debt need not make a project
more valuable. Suppose, for example, that T
C
= 40%, T
E
= 3%, and T
D
= 50%. Then the
spreadsheet below shows that financing the sturfing machine with debt decreases the NPV:
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ABC DEF
T
C
, corporate tax rate
40%
T
E
, personal tax rate on equity
3%
T
D
, personal tax rate on debt
50%
Machine cost, year 0 100,000
Free cash flow (FCF) calculation
Additional sales, annually 40,000
Additional annual cost of sales 15,000
Annual depreciation 10,000 <-- =B6/10
Annual FCF, years 1-10 19,000 <-- =(1-B2)*(B9-B10-B11)+B11
Discount rate for machine FCFs 15% Loan to buy machine 50,000
r
D
, loan interest rate
8%
Net annual advantage of debt
financing, (1-T
D
)-(1-T
E
)*(1-T
C
)
-8% <-- =(1-B4)-(1-B3)*(1-B2)
Year
Machine
FCF
Tax
advantage
of interest
0 -100,000 <-- =-B6
1 19,000 <-- =$B$12 -328 <-- =$E$16*$E$15*$E$14
2 19,000 -328 <-- =$E$16*$E$15*$E$14
3 19,000 -328
4 19,000 -328
5 19,000 -328
6 19,000 -328
7 19,000 -328
8 19,000 -328
9 19,000 -328
10 19,000 -328
Machine NPV -4,643 <-- =B19+NPV(B14,B20:B29) Loan NPV -2,660 <-- =E19+NPV(E15*(1-B4),E20:E29)
NPV: Machine + Loan -7,304 <-- =B31+E31
THE SONDERTURF STURFING MACHINE
different taxes make debt disadvantageous!
PFE Chapter 19, Capital structure and valuation page 45
20.9. Relevering Smotfooler Inc., an Upper Fantasia company
In Section 20.4 we offered a question from a Finance 101 exam at Eastern Lower
Fantasia State University. This section offers a similar question from an exam at Upper Fantasia
University (their football team is called the Ufus).
Here’s the question: Smotfooler, Inc. is a well-known Upper Fantasia company. Here
are some facts about the company:
• Smotfooler expects to have an annual free cash flow of $2 million at the end of years 1, 2,
3, … forever. Recall that the free cash flow is the after-tax amount of cash that the
company generates from its business activities.
• Smotfooler currently has 100,000 shares outstanding on the Upper Fantasia stock
exchange. The Smotfooler share price is $100 per share.
• Smotfooler currently has no debt. However, a financial analyst has suggested that the
company issue $3,000,000 of perpetual debt and use the proceeds to repurchase shares.
The current interest rate on debt in Upper Fantasia is 8%, and the interest payments on
the debt will be made annually.
• Tax rates in Upper Fantasia are: T
C
= 40%, T
D
= 30%, T
E
= 10%.
Students on the finance exam were asked to answer the following questions:
Question 1: What is the current market value of Smotfooler?
Answer: Smotfooler currently has 100,000 shares outstanding, each of which is worth
$100. Thus the company’s equity value is currently $10,000,000 = $100*100,000. Since the
company has no debt, this is also its market value. In short: $10,000,000
U
V
=
.
PFE Chapter 19, Capital structure and valuation page 46
Question 2: After Smotfooler issues $3,000,000 of debt, what will be its market
value?
Answer: Since Upper Fantasia has only a corporate income tax, the relation
LU
VVTD=+ holds, where:
()()
(
)
()
()
(
)
(
)
()
1 1 * 1 1 30% 1 40% 1 10%
22.86%
1 1 30%
DCE
D
TTT
T
T
−−− − − −− −
== =
−−
.
(See also cell B7 on the spreadsheet below).
This means that after the company issues its debt, its market value will be
10,000,000 22.86%*3,000,000 10,685,714
LU
VVTD=+ = + = .
Question 3: After Smotfooler issues debt of $3,000,000 and uses the proceeds to
repurchase shares, what will be the company’s total equity value, E?
Answer: After Potfooler issues the debt and repurchases the shares, the total value of its
equity, E, plus the total value of its debt, D, have to sum to the company’s total market value V
L
.
In short:
10,685,714
But $3,000,000, and therefore:
L
VED
D
E =10,685,714-3,000,000=7,685,714
==+
=
Question 4: At what price will Smotfooler repurchase its shares?
Answer: By issuing $3 million of debt, Smotfooler has raised its total market value by
$685,714 (from $10 million to $10,685,714). This increase in value belongs to all the
shareholders. Since there are 100,000 shares outstanding before the share repurchase, this means
PFE Chapter 19, Capital structure and valuation page 47
that each share’s price increases by
$685,714
$6.86
100,000
= . Thus the answer to this question is that
the share price for repurchase is $106.86. Of this amount $100 is the share price before the
repurchase, and $6.86 is the increase in the share price as a result of the debt issue.
Question 5: How many shares will Smotfooler repurchase?
Answer: According to the previous question, Smotfooler will repurchase its shares at
$106.86 per share. Since the company has issued $3 million in debt to repurchase the shares, this
means that it will repurchase
$3,000,000
28,074.87
$106.86
=
.
Question 6: What was Smotfooler’s cost of equity before the repurchase of shares?
Answer: Smotfooler has an annual free cash flow (FCF) of $2,000,000. Thus its
unlevered cost of equity,
()
2,000,000
20%
10,000,000
EU
U
FCF
rU r
V
== = = .
Question 7: What is Smotfooler’s cost of equity after the repurchase of the shares
on the open market?
Answer: Smotfooler issues $3 million in 8% debt in order to repurchase shares. Thus its
annual interest bill is 8%*8,000,000 = $240,000. Since interest is an expense for tax purposes,
the company’s shareholders will have an annual expected cash flow of:
(
)
()
,1
2,000,000 1 40% *240,000
1,856,000
C
Annual equity cash flow after debt issuance FCF T *interest=−−
=−−
=
PFE Chapter 19, Capital structure and valuation page 48
The value of the equity after the share repurchase is $7,685,714, so that the cost of equity of the
levered company is
()
1,856,000
24.15%
7,685,714
E
rL==.
Note from Figure 20.4. that there’s another way to do this calculation:
() ( ) ( )
()()
11
3,000,000
20% 20% 1 22.86% 8% 1 40% 24.15%
7,685,714
EUU DC
D
rL r r T r T
E
=+ −− − =⎡⎤
⎣⎦
=+ − − − =
⎡⎤
⎣⎦
Question 8: What is Smotfooler’s weighted average cost of capital (WACC) before
the repurchase of the shares?
Answer: Recall the definition of the WACC:
() ()
**1*
EDC
ED
WACC r L r T
ED ED
=+−
++
.
The answer to question 8 is easy: Since Smotfooler, before the share repurchase, has only
equity, its WACC = r
U
= 20%.
Question 9: What is Smotfooler’s weighted average cost of capital (WACC) after
the repurchase of the shares?
Answer:
() ()
()
**1*
7,685,714 3,000,000
24.15%* 8%* 1 40% 18.72%
7,685,714 3,000,000 7,685,714 3,000,000
EDC
ED
WACC r L r T
ED ED
=+−
++
=+−=
++
PFE Chapter 19, Capital structure and valuation page 49
Question 10: Why is
()
EU
rL r> ?
Answer: Before Smotfooler issued its bonds, the only risk borne by shareholders was the
business risk inherent in the company’s free cash flow. After the company issues its bonds,
shareholders have to bear two kinds of risk: business risk and financial risk. Thus r
E
(L)
represents a discount rate for cash flows which are riskier than the discount rate for the FCFs, r
U
.
Since riskier cash flows have higher discount rates, it follows that
(
)
EU
rL r>
.
Question 11: Why does the market value of Smotfooler increase after the issuance
of the debt and repurchase of the equity?
Answer: By issuing the debt, Smotfooler increases the amount of cash it produces by
()()()
11*1*
DCE
T T T Interest payment−−− −⎡⎤
⎣⎦
for every year which it has debt. This additional
cash flow is riskless. Since the holders of riskless cash flows in Upper Fantasia use a discount
rate of
()
1*
D
D
Tr−
to value the cash flows, it follows that:
(
)
(
)
(
)
()
()
()()()
()
()()()
()
()
()
()
()
N
1
11*1
1
11*1*
-
11
11*1
1
11*1
**
1
DCE
D
DCE
t
t
DD
DCE
DD
DCE
D
DD
T
TTT
T
T T T Interest payment
Value of additional debt related cash flows
Tr
TTT
Interest payment
Tr
TTT
rD T D
Tr
∞
=
↑
=
−−− −
−
−−− −⎡⎤
⎣⎦
=
+−
−−− −
=
−
−−− −
==
−
∑
The present value of the tax shield accounts for the increase in Smotfooler’s market value:
N
N
The PV of
Smotfooler's value
additional
before the debt
debt-related
issuance
cash flows
LU
VV TD
↑
↑
=+.
PFE Chapter 19, Capital structure and valuation page 50
Question 12: Does debt always increase corporate value in Upper Fantasia?
Answer: No. It depends on the sizes of the three tax rates T
C
, T
D
, and T
E
. In the example
below, there is a net tax disadvantage to debt—by issuing debt, Smotfooler lowers its market
value and raises its WACC:
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2
3
4
5
6
7
8
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10
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12
13
14
15
16
17
18
19
20
21
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35
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37
38
39
40
41
42
43
44
ABC
Upper Fantasia tax system
T
C
,
U
pper
F
an
t
as
i
a corpora
t
e
t
ax ra
t
e
40
%
T
E
,
U
pper
F
an
t
as
i
a persona
l
t
ax ra
t
e on equ
it
y
i
ncome
10
%
T
D
,
U
pper
F
an
t
as
i
a persona
l
t
ax ra
t
e on or
di
nary
i
ncome
30
%
A
nnua
l
d
e
bt
a
d
van
t
age:
(
1
-
T
D
)
-
(
1
-
T
E
)*(
1
-
T
C
)
16
%
<-- =
(
1-B5
)
-
(
1-B4
)*(
1-B3
)
PV
o
f
d
e
bt
a
d
van
t
age:
T
=
[(
1
-
T
D
)
-
(
1
-
T
E
)*(
1
-
T
C
)]/(
1
-
T
D
)
22.86
%
<-- =B6
/(
1-B5
)
Unlevered company
Annual free cash flow (FCF)
$2,000,000
Number of shares 100,000
Price per share $100
Total equity value $10,000,000 <-- =B12*B11
Question 1: V
U
, unlevered value of Smotfooler
$10,000,000 <-- =B13
Levered company
Debt issued $3,000,000
Interest rate on debt 8%
Question 2: V
L
, levered value of Smotfooler, V
L
= V
U
+ T*D
$10,685,714 <-- =B15+B7*B18
Question 3: Equity value after share repurchase, E = V
L
- D
$7,685,714
Incremental firm value from exchanging
equity by debt = V
L
- V
U
= T*D
$685,714 <-- =B20-B15
Incremental firm value on a per-share basis $7 <-- =B22/B11
Question 4: New share value, after repurchase $106.86 <-- =B12+B23
Question 5: Number of shares repurchased =
[debt used for repurchase]/[new share value]
28,074.87 <-- =B18/B24
Number of shares remaining after
repurchase = original number of shares
minus number of shares repurchased
71,925.13 <-- =B11-B26
Check: Market value of remaining shares =
number of remaining shares * new share value
$7,685,714 <-- =B27*B24
Question 6: Smotfooler's cost of equity when unlevered,
r
U
=FCF/V
U
20.00%
Annual interest costs, before taxes $240,000 <-- =B18*B19
Annual equity cash flow, after interest = FCF - (1-T
C
)*interest
$1,856,000 <-- =B10-(1-B3)*B32
Question 7: Smotfooler's cost of equity when levered,
r
E
(L)=[FCF-(1-T
C
)*interest]/[value of equity, E]
24.15% <-- =B33/B28
Note: See formula in row 44 below for another
way to compute the levered cost of equity
Question 8: Smotfooler's WACC before the debt issuance = rU 20.00%
Question 9: Smotfooler's WACC after the debt issuance
= r
E
(L)*E/(E+D)+r
D
*(1-TC)*D/(E+D)
Percentage of equity in Smotfooler = E/(E+D) 71.93% <-- =B28/B20
Percentage of debt in Smotfooler = D/(E+D) 28.07% <-- =B18/B20
WACC = r
E
(L)*E/(E+D)+r
D
*(1-T
C
)*D/(E+D)
18.72% <-- =B34*B40+B19*(1-B3)*B41
Additional formula: r
E
(L)=r
U
+[r
U
*(1-T)-r
D
*(1-T
C
))*D/E
24.15% <-- =B30+(B30*(1-B7)-B19*(1-B3))*B18/B21
SMOTFOOLER--DEBT ISSUED TO REPURCHASE SHARES
Smotfooler is located in Upper Fantasia
PFE Chapter 19, Capital structure and valuation page 51
20.10. Is there really an advantage to debt?
In this chapter we’ve laid out the theory of capital structure. We can answer the question
of the importance of capital structure in several ways:
Method 1: What are the relevant tax rates T
C
, T
D
, T
E
?
As you can see, the value of XYZ Corp. is critically dependent on 2 factors:
•
r
U
, the risk-adjusted rate of return for the free cash flows. This rate is unaffected by the
capital structure, since the free cash flows are operating cash flows and do not depend on
the financing of the firm.
•
()()()
111
D
CE
TTT−−− −--the relative after-tax costs of debt versus equity income.
Looking at this second parameter, we examine several cases. In the case below, the anticipated
dividend yield of 2% is taxed at 40% while the anticipated capital gains yield of 6% is taxed at
10%. The equity tax rate is 17.5%, and the net tax advantage of debt over equity is 8.02%:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
ABC
Corporate tax rate, T
C
37%
Anticipated equity tax
Tax rate
Dividend yield 2.00% 40%
Capital gains yield 6.00% 10%
Net after-tax yield 6.60% <-- =B5*(1-C5)+B6*(1-C6)
Before tax yield 8.00% <-- =B5+B6
Personal tax rate on equity income, T
E
17.50% <-- =1-B8/B9
Personal tax rate on ordinary income, T
D
40.00%
Tax advantage of debt
over equity: (1-T
D
)-(1-T
C
)*(1-T
E
)
8.02% <-- =(1-B12)-(1-B2)*(1-B11)
WHAT ARE THE RELATIVE TAX EFFECTS
With a somewhat different yield and tax configuration there is actually a net tax
disadvantage to debt:
PFE Chapter 19, Capital structure and valuation page 52
1
2
3
4
5
6
7
8
9
10
11
12
13
14
ABC
Corporate tax rate, T
C
37%
Anticipated equity tax
Tax rate
Dividend yield 0.00% 40%
Capital gains yield 6.00% 0%
Net after-tax yield 6.00% <-- =B5*(1-C5)+B6*(1-C6)
Before tax yield 6.00% <-- =B5+B6
Personal tax rate on equity income, T
E
0.00% <-- =1-B8/B9
Personal tax rate on ordinary income, T
D
40.00%
Tax advantage of debt
over equity: (1-T
D
)-(1-T
C
)*(1-T
E
)
-3.00% <-- =(1-B12)-(1-B2)*(1-B11)
WHAT ARE THE RELATIVE TAX EFFECTS
Below you will see a third case in which only corporate income is taxed. In this case
there is an overwhelming advantage to debt financing:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
ABC
Corporate tax rate, T
C
37%
Anticipated equity tax
Tax rate
Dividend yield 5.00% 0%
Capital gains yield 0.00% 0%
Net after-tax yield 5.00% <-- =B5*(1-C5)+B6*(1-C6)
Before tax yield 5.00% <-- =B5+B6
Personal tax rate on equity income, T
E
0.00% <-- =1-B8/B9
Personal tax rate on ordinary income, T
D
0.00%
Tax advantage of debt
over equity: (1-T
D
)-(1-T
C
)*(1-T
E
)
37.00% <-- =(1-B12)-(1-B2)*(1-B11)
WHAT ARE THE RELATIVE TAX EFFECTS
Method 2: What’s the evidence in firm behavior?
Instead of asking whether tax rates support a net tax advantage, we can also look at
different firms. We can ask whether in a particular industry there is a consistent behavior
towards debt. The answer is no, as you will see in Chapter 20. As you will see in Chapter 20,