1. Capital Structures

2. Topics to consider: MM models, with and without corporate taxes
Compressed adjusted present value model
MM proofs

3. Determinants of Intrinsic Value: The Capital Structure Choice
Net operating
profit after taxes
Required investments
in operating capital −
Free cash flow
(FCF) =
FCF1
FCF2
FCF∞
Value = ··· +
(1 + WACC)1
(1 + WACC)2
(1 + WACC)∞
Firm’s
debt/equity
mix
Weighted average
cost of capital
(WACC)
Market interest rates
Cost of debt
Cost of equity
Market risk aversion
Firm’s business risk

4. Who are Modigliani and Miller (MM)?
They published theoretical papers that changed the way people thought about financial leverage.
They won Nobel prizes in economics because of their work.
MM’s papers were published in 1958 and Miller had a separate paper in The papers differed in their assumptions about taxes.

5. What assumptions underlie the MM and Miller Models?
Firms can be grouped into homogeneous classes based on business risk.
Investors have identical expectations about firms’ future earnings.
There are no transactions costs.
No agency or financial distress costs.
(More...)

6. What assumptions underlie the MM and Miller Models?
All debt is riskless, and both individuals and corporations can borrow unlimited amounts of money at the risk-free rate.
All cash flows are perpetuities. This implies perpetual debt is issued, firms have zero growth, and expected EBIT is constant over time.

7. MM Detailed Assumptions and Notation
Company U is Unlevered. It has no debt, only stock S.
Company L is Levered. It has debt amount D, with interest rate rd. It has stock SL.
Both companies are identical except for debt. Same EBIT and operating risk.
Zero growth, g = 0. No net additions to capital.
The required return on U’s equity is rsU.
The required return on L’s equity is rsL.
Debt is riskless and investors can borrow as well as lend at this rate. The cost of debt is rd.
Tax savings due to deductibility of interest expense have the same risk as debt and should be discounted at the cost of debt.

8. MM with Zero Taxes (1958) Proposition I: VL = VU. Proposition II:
rsL = rsU + (rsU – rd)(wd/ws)
The levered beta: b = bU + bU(D/S)
Notes: The ratios wd/ws and D/S are have the same value, so they can be used interchangeably. For a proof of the propositions, right click here and open link to page in this same file: Proof of MM with Zero Taxes (1958)

9. Numerical Illustration of the MM No-Tax Propositions
Firms U and L are in same risk class.
EBITU = EBITL = $500,000.
Firm U has no debt; rsU = 14%.
Firm L has $1,000,000 debt at rd = 8%.
The basic MM assumptions hold.
There are no corporate or personal taxes.

10. 1. Find VU and VL. VU = = = $3,571,429. VL = VU = $3,571,429. EBIT rsU
$500,000
0.14

11. 2. Find the market value of Firm L’s debt and equity.
VL = D + S = $3,571,429
$3,571,429 = $1,000,000 + SL
SL = $2,571,429.

12. 3. Find rsL. rsL = rsU + (rsU - rd)(D/SL) = 14.0% + (14.0% - 8.0%)( )
= 14.0% + (14.0% - 8.0%)( )
= 14.0% % = 16.33%.
$1,000,000
$2,571,429

13. 4. Proposition I implies WACC = rsU. Verify for L using WACC formula.
WACC = wdrd + wsrs = (D/V)rd + (S/V)rs
= ( )(8.0%)
+( )(16.33%)
= 2.24% % = 14.00%.
$1,000,000
$3,571,429
$2,571,429

14. MM Relationships Between Capital Costs and Leverage (D/V)
Without taxes
Cost of Capital (%) 26 20 14 8
Debt/Value Ratio (%) rs
WACC rd

15. MM No-Tax Conclusions:
The more debt the firm adds to its capital structure, the riskier the equity becomes and thus the higher its cost.
Although rd remains constant, rs increases with leverage. The increase in rs is exactly sufficient to keep the WACC constant.
Because the WACC and FCF’s don’t change, the firm’s value doesn’t change.

16. Graph value versus leverage.
Value of Firm, V (%) 4 3 2 1
Debt (millions of $) VL VU
Firm value ($3.6 million)
With zero taxes, MM argue that value is unaffected by leverage.

17. MM with Corporate Taxes (1963)
Proposition I:
VL = VU + VTax shield = VU + TD
Proposition II:
rsL = rsU + (rsU – rd )(1 – T)(D/S)
The levered beta:
b = bU + bU (1 – T)(D/S)
b = bU [1+(1 – T)(wd/ws)]
Note: For a proof of the propositions, right click here and open link to page in this same file: Proof: MM with Corporate Taxes (1963)

18. V, S, rs, and WACC for Firms U and L (40% Corporate Tax Rate)
With corporate taxes added, the MM propositions become:
Proposition I:
VL = VU + TD.
Proposition II:
rsL = rsU + (rsU - rd)(1 - T)(D/S).

19. Notes About the New Propositions
1. When corporate taxes are added, VL ≠ VU. 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 with leverage at a slower rate when corporate taxes are considered.

20. 1. Find VU and VL. EBIT(1 - T) rsU $500,000(0.6) 0.14
Note: Represents a 40% decline from the no taxes situation.
VL = VU + TD = $2,142, ($1,000,000)
= $2,142,857 + $400,000
= $2,542,857.

21. 2. Find market value of Firm L’s debt and equity.
VL = D + S = $2,542,857
$2,542,857 = $1,000,000 + S
S = $1,542,857.

22. 3. Find rsL. rsL = rsU + (rsU - rd)(1 - T)(D/S)
= 14.0% + (14.0% - 8.0%)(0.6)( )
= 14.0% % = 16.33%.
$1,000,000
$1,542,857

23. 4. Find Firm L’s WACC. WACCL = (D/V)rd(1 - T) + (S/V)rs
= ( )(8.0%)(0.6)
+( )(16.33%)
= 1.89% % = 11.80%.
When corporate taxes are considered, the WACC is lower for L than for U.
$1,000,000
$2,542,857
$1,542,857

24. MM: Capital Costs vs. Leverage with Corporate Taxes
Cost of Capital (%) 26 20 14 8
Debt/Value Ratio (%) rs
WACC
rd(1 - T)

25. MM: Value vs. Debt with Corporate Taxes
Value of Firm, V (%) 4 3 2 1
Debt
(Millions of $) VL VU TD
Under MM with corporate taxes, the firm’s value increases continuously as more and more debt is used.

26. What does capital structure theory prescribe for corporate managers?
MM, No Taxes: Capital structure is irrelevant--no impact on value or WACC.
MM, Corporate Taxes: Value increases, so firms should use (almost) 100% debt financing.

27. Do firms follow the recommendations of capital structure theory?
Firms don’t follow MM/Miller to 100% debt. Debt ratios average about 40%.
However, debt ratios did increase after MM. Many think debt ratios were too low, and MM led to changes in financial policies.

28. How is analysis different if firms U and L are growing?
Under MM (with taxes and no growth)
VL = VU + TD
This assumes the tax shield is discounted at the cost of debt.
Assume the growth rate is 7%
The debt tax shield will be larger than TD because it is growing.

29. 7% growth, TS discount rate of rTS
Value of (growing) tax shield =
VTS = rdTD/(rTS – g)
So value of levered firm =
VL = VU + rdTD/(rTS – g)

30. The Compressed Adjusted Present Value (APV)Model
The smaller is rTS, the larger the value of the tax shield. If rTS < rsU, then with rapid growth the tax shield becomes unrealistically large—rTS must be equal to rU to give reasonable results when there is growth.
The APV model assumes rTS = rsU.

31. Levered cost of equity in the APV model
In this case, the levered cost of equity is rsL = rsU + (rsU – rd)(D/S)
This looks just like MM without taxes even though we allow taxes and allow for growth. The reason is if rTS = rsU, then larger values of the tax shield don't change the risk of the equity.

32. Levered Beta in the APV Model
If there is growth and rTS = rsU then the equation that is equivalent to the Hamada equation is
bL = bU + (bU - bD)(D/S)
Notice: This looks like Hamada without taxes. Again, this is because in this case the tax shield doesn't change the risk of the equity.

33. Relevant information for valuation
EBIT = $500,000
T = 40%
rU = 14% = rTS
rd = 8%
Required reinvestment in net operating assets = 10% of EBIT = $50,000.
Debt = $1,000,000

34. Calculating VU NOPAT = EBIT(1-T) = $500,000 (.60) = $300,000
Investment in net op. assets
= EBIT (0.10) = $50,000
FCF = NOPAT – Inv. in net op. assets
= $300,000 - $50,000
= $250,000 (this is expected FCF next year)

35. Value of unlevered firm, VU
VU = FCF/(rsU – g)
= $250,000/(0.14 – 0.07)
= $3,571,429

36. Value of tax shield, VTS and VL
VTS = rdTD/(rsU – g)
= 0.08(0.40)$1,000,000/( )
= $457,143
VL = VU + VTS
= $3,571,429 + $457,143
= $4,028,571

37. Cost of Equity and WACC in the APV Model
Just like with MM with taxes, the cost of equity increases with D/V, and the WACC declines.
But since rsL doesn't have the (1-T) factor in it, for a given D/V, rsL is greater than MM would predict, and WACC is greater than MM would predict.

38. Cost of Capital for MM and APV

39. Dynamic Capital Structures and the APV Model
The capital structure will change for several years before becomes constant.
It is expected to stay at its target
Can’t use MM or free cash flow valuation model.
Use the compressed APV

40. Example Data Tax rate = 40% Unlevered cost of equity = rsU = 14%
Long term growth rate = gL = 7%
Forecast of nonconstant period ($ thousands):
 Year 1 2 3
Free Cash Flow
$250
$290
$320
Interest expense
$80
$96
$120

41. APV Approach with Nonconstant Cash Flows
Calculate the unlevered value of operations, VU
Find terminal value, TVU,3
Find PV of FCFs and terminal value
Calculate the value of the tax shield, VTS
Find horizon value of tax shield, HVTS,3
Find PV of tax shields and terminal value
Sum VU and VTS to get Vop

42. ($3,960 if no rounding in intermediate steps)
Estimating Current Unlevered Value of Operations (Nonconstant g in FCF until after Year 3; gL = 7%; rsU = 14%)
($ thousands)
gL = 7%
Year 1 2 3 4
FCF
$250
$290
$320 ↓
$219
← $250/(1+rsU)1
PVs of FCF
$223
← $290/(1+rsU)2
$216
← $320/(1+rsU)3
PV of HVU,3
$3,301
$4,891/(1+rsU)3 ←
HVU,3= $4,891
 VU =
$3,959
($3,960 if no rounding in intermediate steps) 

43. ($584.94 if no rounding in intermediate steps)
Estimating Current Value of the Tax Shield(Nonconstant g in TS until after Year 3; gL = 7%; rsU = 14%)
($ thousands)
gL = 7%
Year 1 2 3 4
Interest Exp.
$80
$96
$120
Int(T) ↓ TS
$32
$38
$48
$51.36
$28.1
← $32/(1+rsU)1
PVs of TS
$29.2
← $38/(1+rsU)2
$32.4
← $48/(1+rsU)3
PV of HVTS,3
$494.8
$4,891/(1+rsU)3 ←
HVTS,3= $733
 VTS =
$584.5
($ if no rounding in intermediate steps) 

44. Total Current Value of Operations
The value of operations is the sum of the unlevered value and the value of the tax shield:
Vop = VU + VTS
Vop = $3,959 + $584.5 = $4,543.5
Vop = $4,543.5
Note: If no rounding in intermediate steps, Vop = $4,445).

45. What if L's debt is risky?
If L's debt is risky then, by definition, management might default on it. The decision to make a payment on the debt or to default looks very much like the decision whether to exercise a call option. So the equity looks like an option.

46. Equity as an option
Suppose the firm has $2 million face value of 1-year zero coupon debt, and the current value of the firm (debt plus equity) is $4 million.
If the firm pays off the debt when it matures, the equity holders get to keep the firm. If not, they get nothing because the debtholders foreclose.

47. Equity as an option
The equity holder's position looks like a call option with
P = underlying value of firm = $4 million
X = exercise price = $2 million
t = time to maturity = 1 year
Suppose rRF = 6%
 = volatility of debt + equity = 0.60

48. Use Black-Scholes to price this option
VC = P[N(d1)] - Xe -rRFt[N(d2)]
d1 =
 t 0.5
d2 = d1 -  t 0.5
ln(P/X) + [rRF + (2/2)]t

49. Black-Scholes Solution
V = $4[N(d1)] - $2e -(0.06)(1.0) [N(d2)].
ln($4/$2) + [( /2)](1.0)
d1 =
(0.60)(1.0) =
d2 =
d1 – (0.60)(1.0) = d1 – 0.60
= – =

50. Black-Scholes Solution (Continued)
N(d1) = N(1.5552) =
N(d2) = N(0.9552) =
Note: Values obtained from Excel using NORMSDIST function.
V = $4(0.9401) - $2e-0.06(0.8303)
= $ $2(0.9418)(0.8303)
= $2.196 Million = Value of Equity

51. Value of Debt The value of debt must be what is left over:
Value of debt = Total Value – Equity
= $4 million – million
= $1.804 million

52. This value of debt gives us a yield
Debt yield for 1-year zero coupon debt
= (face value / price) – 1
= ($2 million/ million) – 1
= 10.9%

53. How does  affect an option's value?
Higher volatility  means higher option value.

54. Managerial Incentives
When an investor buys a stock option, the riskiness of the stock () is already determined. But a manager can change a firm's  by changing the assets the firm invests in. That means changing  can change the value of the equity, even if it doesn't change the expected cash flows:

55. Managerial Incentives
So changing  can transfer wealth from bondholders to stockholders by making the option value of the stock worth more, which makes what is left, the debt value, worth less.

56. Value of Debt and Equity for Different Volatilities

57. Bait and Switch
Managers who know this might tell debtholders they are going to invest in one kind of asset, and, instead, invest in riskier assets. This is called bait and switch and bondholders will require higher interest rates for firms that do this, or refuse to do business with them.

58. If the debt is risky coupon debt
If the risky debt has coupons, then with each coupon payment management has an option on an option—if it makes the interest payment then it purchases the right to later make the principal payment and keep the firm. This is called a compound option.

59. Proof of MM with Zero Taxes (1958)
Proposition I: VL = VU.
Steps in proof:
Show that total investor cash flows are the same for both firms.
Show that if VL ≠VU, then investors can create arbitrage profits.
But this would lead to buying and selling activities that would drive VL and VU, to the same value.

60. Annual Cash Flow to U’s Investors (CFU)
Cash flow to shareholders:
No growth, so dividends equal net income (NI).
No interest payments or taxes, so NI =EBIT.
No debt, so no debtholders.
CFU = EBIT.

61. Annual Cash Flow to L’s Investors (CFL)
Debtholders receive interest payments, so their cash flow is: rd D
Zero taxes, so the cash flow to shareholders is:
EBIT − rd D
CFL = rd D + (EBIT − rd D) = EBIT

62. Cash Flows and Firm Values
Note that CFU = EBIT = CFL
U has no debtholders, so
VU = SU
L has debtholders, so
VL = SL + D
Proposition I: VL = SL + D = VU = SU.
The value of a levered firm is equal to the value of the firm if it had no debt.

63. Proof By Using an Arbitrage Argument
If VL ≠ VU, then an investor could:
Sell the expensive asset
Buy the cheaper asset
Have money left over
Have zero net future annual cash flow.
This would be arbitrage, which should not exist in well-functioning markets.

64. Suppose VL > VU Get cash by selling 1% of VL
Use this cash to create a portfolio that reproduces L’s cash flows exactly, but is cheaper than VL:
Borrow 1% of D at an interest rate of rd.
Buy 1% of U

65. Summary of Transactions
Initial Cash Flow
Sell 1% of SL
Borrow 1% of D
Buy 1% of U
Total Net Initial Cash Flow
+0.01(SL)
+0.01(D)
−0.01(SU)
0.01(SL + D – SU)
= 0.01(VL − VU) > 0
Annual Cash Flows
Sell 1% of SL
Borrow 1% of D
Buy 1% of U
Total Net Annual Cash Flow
Annual dividends
−0.01(EBIT−rdD)
+0.01(EBIT)
+0.01(rdD)
Annual interest
−0.01(rdD)
Annual total

66. The arbitrage opportunity
Start off with no money!
Sell 1% of L’s stock, borrow an amount equal to 1% of D, and buy 1% of U.
The initial CF for this position is positive since we assumed VL > VU .
Net annual cash flows are zero.
This means after you enter the position, you have money in your pocket and no other net cash flows.

67. But arbitrage can’t last long!
Everyone would engage in the arbitrage transactions:
Selling pressure would cause VL to fall
Buying pressure would cause VU to rise
All of this would take place until VL = VU

68. Example… Suppose we have the following data: EBIT = $100 for U and L.
rsU = 10%.
rd = 6%.
L has D = 250. (Debt is zero for U.)

69. Firm Values in Example VU = SU = EBIT/ rsU VU = 100/0.10 = $1,000.
VL = SL + D (= VU because of Proposition I)
SL = VL − D
SL = $1,000 − $250 = $750.

70. Dividends in Example
U’s only cash flow is a dividend to shareholders. With no debt, no taxes, and no growth:
DividendU = EBIT= $100
L pays interest, so its dividend (which is also its net income) is:
DividendL = EBIT − rd D
DividendL = $100 − 0.06($250) = $85

71. Suppose SL = $800, instead of $750.
Initial Cash Flow
Sell 1% of SL
Borrow 1% of D
Buy 1% of U
Total Net Initial Cash Flow
+0.01($800) = $8
+0.01($250) = $2.5
−0.01($1,000)
= −$10
$8 + $2.5 −$10
= $0.50
Annual Cash Flows
Sell 1% of SL
Borrow 1% of D
Buy 1% of U
Total Net Annual Cash Flow
Annual dividends
−0.01($85)
= −$8.50
+0.01($100)
= $10
−$ $10
= $1.50
Annual interest
−0.01(0.06 x $250)
= −$1.50
−$1.50
Annual total
−$8.50
$10 $0
You have no net annual cash flows, but you have $0.50 in your pocket as a risk-free profit from the position.

72. How will investors use this?
Investors will bid up the price of U, and bid down the price of L until VL = VU .
What if there is no short selling?
Then investors who were considering purchasing L would, instead, purchase U and borrow on their own accounts. They get the same annual cash flows but the initial investment is less. So no one would purchase L at this price, and the price of L’s stock would have to drop until VL = VU.

73. In our example
In our example, investors will buy U’s stock and borrow rather than buy L’s stock. L is too expensive. This will bid down the price of L’s stock below $800. This will continue until SL = 1,000, or SL = $750.
Here, investors use homemade leverage to reproduce L’s cash flows, but cheaper.

74. What if VL < VU
Then reverse the position. Buy 1% of L, sell 1% of U and invest 0.01(D) in a bond paying interest of rd.
Annual net cash flows are still 0, just like before.
Because now VL < VU you have money left over, but no net annual liability.

75. How will investors use this?
Investors will bid up the price of L, and bid down the price of U until VL = VU .
What if there is no short selling?
Then investors who were considering purchasing U would, instead, purchase L and lend (invest in bonds) on their own accounts. They get the same annual cash flows as U but the initial investment is less. No one would purchase U at this price, and the price of U’s stock would have to drop until VL = VU.

76. Suppose SL = $700, instead of $750.
Initial Cash Flow
Sell 1% of SU
Lend 1% of D
Buy 1% of SL
Total Net Initial Cash Flow
+0.01($1000)
= $10
−0.01($250)
= −$2.5
−0.01($700)
= −$7
$10 − $2.5 −$7
= $0.50
Annual Cash Flows
Sell 1% of SU
Lend 1% of D
Buy 1% of SL
Total Net Annual Cash Flow
Annual dividends
−0.01($100)
= −$10
+0.01($85)
= $8.5
−$10 + $8.5
= $1.50
Annual interest
0.01(0.06 x $250)
$1.50
Annual total
−$10
$8.5 $0
You have no net annual cash flows, but you have $0.50 in your pocket as a risk-free profit from the position.
You have duplicated L’s position with homemade leverage again.

77. MM no Taxes Proposition II
Proposition II: rsL = rsU + (rsU – rd)(D/SL) Proof: Just solve the following equation from Proposition I for rsL : (Algebra hint on next page)

78. To do the algebra…
To solve, note that by definition EBIT on the right hand side can be rewritten as EBIT = SLrsL + rdD Plug this into the equation on the previous slide. This makes the algebra a good deal easier!

79. In our example
rsU = 10%, D = $250, SL = $750, rd = 6%. rsL = 10% + (10% - 6%)(250/750) = % To check, the present value of L’s dividends at rsL is $85/ = $750. This must be L’s stock price if there is to be no arbitrage.

80. Proof: MM with Corporate Taxes (1963)
VL = VU + VTax shield = VU + TD The value of a levered company is equal to its value if it had no debt plus the value of the interest tax shield.

81. Proof of Proposition I
Begin by expressing U’s and L’s cash flows to investors in a way that can be compared to cash flows of known assets. As shown by MM’s arbitrage proof, assets with the same cash flows must have the same values. Start with U: U’s annual CF = EBIT(1 – T)

82. L’s Annual Cash Flows
The annual cash flows to L are the dividends plus the interest payments: L’s Annual CF = (EBIT – rdD)(1 – T) + rdD = EBIT(1 – T) + rdDT The first part, EBIT(1 – T), is the same as the cash flow to U. The second part, is the debt tax shield.

83. Proof of Proposition I with taxes cont..
The first term, EBIT(1 – T), is equal to U’s cash flows.
Therefore, the first term’s value is equal to the value of an unlevered firm, VU. Otherwise, arbitrage would be possible.

84. Proof of Proposition I with taxes cont..
The second term, the debt tax shield, is a perpetual stream of cash flows equal to rdDT.
Its value depends on the discount rate. Modigliani and Miller assumed the appropriate discount rate was the required return on debt, rd, so the tax shield value is:
Value of tax shield = rdTD/rd = TD.

85. Proof of Proposition I with taxes cont..
The total value of the levered firm is:
VL = VU+ TD
In other words, the value of a levered firm is equal to the value of an unlevered firm plus the value of side effects due to leverage.

86. Our example
Consider firms U and L, both with EBIT = $100 and T = 40%. rsU = 10%, D = $150 for L, rd = 6%. CF to U = Dividends = 100(1-0.40) = $60 per year VU = $60/0.10 = $600. This is different from before because now we have taxes.

87. Our example continued…
CF to L = Dividends + interest
= (100 – 150x0.06)(1-0.40) + 150x0.06
= = But this isn’t a useful way to look at it. Instead, rewrite it as:
= 100x x0.06x0.40 per year
= = 63.6
= CF to U + Debt tax shield
= EBIT(1-T) + rdTD

88. Our example: L’s cash flows
EBIT(1-T) Tax Shield Annual CF: $60 $3.60 Disc. rate: 10% 6% PV: 60/0.1 = $ /0.06 = $60 VL = VU + VTS VL = VU + VTS = $600 + $60 = $660.

89. Proposition II with taxes
rsL = rsU + (rsU – rd )(1 – T)(D/S) The required rate of return to a levered stock is the unlevered return plus a premium that depends on the tax rate, the rate on debt, and the amount of debt.

90. Proof of Proposition II with taxes
Proposition I says that VL = SL + D = VU + VTS SL + D = SU + VTS = EBIT(1 – T)/rsU + TD Algebra (with a hint on the next page) gives the result of Proposition II.

91. to do the algebra… Since SL = (EBIT – rd D)(1 – T)/rsL
you can solve for EBIT(1 – T):
EBIT(1 – T) = SLrsL + rdD(1 – T)
Plug this expression for EBIT(1 – T) into the right hand side of:
SL + D = SU + VTS = EBIT(1 – T)/rsU + TD
and solve for rsL to get the Proposition II result.