Capital Structure With Taxes: The Modigliani-Miller Propositions
The story so far: Last time, we discussed how the value of a firm is changed in the presence of debt. We ignored taxes in determining the relationship between firm value and leverage. This relationship was given by the Modigliani-Miller propositions - three propositions which told us that (i) the value of a firm was unchanged when we added debt to the capital structure (ii) the rate of return required by shareholders went up in the presence of debt and (iii) the total weighted average cost of capital was unchanged in the presence of debt. Today, we find out what happens if we add taxes to our mix. Now read on ...
Capital Structure in the Presence of Corporate taxes
(but no other imperfections or inefficiencies) Assume the same definitions as before, but in addition, that corporations are taxed at the rate tc Modigliani-Miller proposition I The value of a firm increases when you add debt. VL = VU + Present value of the interest tax shields = VU + tc B Modigliani-Miller proposition II The rate of return demanded by shareholders, rs goes up but not as much as before. It is now given by the formula:
rs = ro +(ro −rb)(1−tc )
B S
Modigliani-Miller proposition III The weighted average cost of capital goes down and is now given by the formula:
MGMT 310: Financial Management Krannert School, Purdue University Raghavendra Rau Finance Department
B WACC = r0(1− tc ) V
Proof of Proposition I Intuition : The government takes away a part of the pizza. Interest is tax deductible, but payments to shareholders (“the cost of equity”) are not, so, it pays to borrow. Formal Proof of Proposition I Suppose you have two firms in front of you. One is levered, the other is not. Both have the same assets and cash flows from assets. Which firm is worth more? Consider the following two investments (as in the last class) 1. Buy a proportion k of the equity of firm U 2. Buy a proportion k of the equity and the debt of firm L. This debt is a perpetuity, paying rb × BL = R per year The first investment gives a payoff, next year, of k EBIT (1-tc) and costs k SU The second investment gives a payoff of k (EBIT - R)(1-tc) + k R = k EBIT (1-tc) + tc R k and costs k [SL + BL] tc R k is the interest tax shield. The second strategy is worth more than the first strategy. SU < SL + BL How much more? It is worth tc R k more. Therefore the cost of the second strategy today must be equal
MGMT 310: Financial Management Krannert School, Purdue University Raghavendra Rau Finance Department
to the cost of the first strategy plus the present value of the interest tax shield tc R k. What is the present value of the interest tax shield? If the debt is a perpetuity, the present value of the interest tax shield is equal to
PV (tc R) = tc
rbB = tcB rb
Therefore, the value of the levered firm is equal to the value of the unlevered firm plus the present value of the interest tax shield. VL = VU + tc B
Proof of Proposition II Start with an unlevered firm. Assume an all equity financed firm, with tax rate 45%. We have: EBIT Assets Debt Equity Number of Shares Stock Price ro = EBIT (1 - tc)/ Assets 20 100 0 100 100 1 11%
The above data is relevant for an all equity financed firm. Suppose the firm issued 50 Perpetual Debt paying 8% interest per year, and uses the money to buy back stock. Upon announcement, according to Modigliani-Miller proposition number 1, the value of the equity should increase with the present value of the (perpetual) tax subsidy tc × 50 = 22.5 So after the announcement the value of the firm's equity is equal to 122.5, or 1.225 per share. Hence, the company has to buy back 50/1.225 = 40.8 shares at 1.225 per share. After the Repurchase
MGMT 310: Financial Management Krannert School, Purdue University Raghavendra Rau Finance Department
EBIT Interest Assets Debt Equity Number of Shares Stock Price Debt/Equity EBIT - Interest Tax rs
20 4 100 50 72.5 59.2 1.225 68.97% 16 7.2 12.14%
The Modigliani-Miller formula predicts that this rate of return is equal to rS = 0.11 + (1 - 0.45)(0.11 - 0.08) × 0.6896 = 12.138% Q.E.D.
Therefore, the rate of return goes up from 11% to 12.14%. (Suppose the original rate of return demanded by shareholders in the absence of debt had been 11%. Then in the absence of taxes, if the company issued 50 perpetual debt, the rate of return would have gone to 14%.) Why does the rate of return demanded by shareholders go up but not by as much as before? Again, this is because of an increase of risk. Risk goes up but not by as much as before, because of the addition of a riskless tax shield with the increased debt. Again, we can relate the beta of the unlevered firm to the beta of a levered firm by the equation:
βSL = βSU 1+ (1− t ) S
c
B
Proof of Proposition III
MGMT 310: Financial Management Krannert School, Purdue University
Raghavendra Rau Finance Department
S B WACC = rs + rb(1− tc) V V
B S B = ro + (1− tc)(ro − rb) + rb(1− tc) S V V
B WACC = ro(1− tc ) V
Cost of capital falls if we borrow more!
Conclusion
I In a world without taxes, capital structure is irrelevant. II In a world with corporate taxes, it pays to borrow. III Borrowing increase expected return on equity, expected earnings per share, but also the risk.
Evidence : Belgian Tax Reform of 1982
The Belgian government assigned a cost of equity of 13% for all corporations - this was tax deductible at the corporate level for newly issued equity in 1982 - 1983. Consequences: 1. More equity issued in 1982 - 1983 than during the previous 13 years. 2. General increase in Belgian stock market by 40% in December 1981. 3. Stock prices increase around equity announcements in 82-83.
MGMT 310: Financial Management Krannert School, Purdue University
Raghavendra Rau Finance Department