1. - CHAPTER 14 -
Financial Leverage (Revisited)

2. The “corporate tax shield”
In Chapter 8, we discussed how the tax-deductibility of the firm’s interest payments on its debt reduces the firm’s corporate tax liability
– the “corporate tax shield”.
For example, if the rate of corporate tax is equal to 30%, each $1 of interest repayment reduces the firm’s taxable income by $1, and, hence, reduces the firm’s tax burden by 30 cents.
If “less for the tax authorities” means “more for the shareholders”, it would appear that debt creates additional value for the firm’s shareholders in a direct relation to the tax savings due to debt.

3. The “corporate tax shield”(cont)
This is the position Modigliani and Miller (1963) argued. They said:
Suppose that the firm holds debt with value $DEBT in perpetuity with annual interest rate, iD.
In this case, the reduction in the firm’s corporate tax liabilities, or tax savings TS, on account of the debt (annually in perpetuity) is determined as:
TS = $DEBT iD Tc (14.1)
and the present value of the tax savings (PVTS) is determined as
𝑃𝑉𝑇𝑆= 𝑇𝑆 𝑘 𝑇𝑆 = 𝐷𝐸𝐵𝑇 𝑖 𝐷 𝑇 𝑐 𝑘 𝑇𝑆 (14.2)
for some discount factor, kTS.

4. The “corporate tax shield”(cont)
It follows that Modigliani and Miller’s proposition as Eq. 6.3:
Vfirm ≡ VE + VD = VU
must be replaced by
Vfirm ≡ VE + VD = VU + PVTS (14.3)
where PVTS denotes the present value of the tax savings, which is added to VU = Vunlevered (the value of the company in the case that it held no debt).
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5. The “corporate tax shield”(cont)
Modigliani and Miller considered that the appropriate discount rate (kTS) with which to discount the tax savings (TS) due to the tax deductibility of the firm’s interest payments in Eqn 14.2 should be the cost of the debt (iD).
They accordingly determined that the present value of the tax savings (PVTS) to the firm due to the tax savings TS should be calculated by discounting the TS by iD.
Which is to say, we have Eqn 14.2 as
𝑃𝑉𝑇𝑆= 𝑉 𝐷 𝑖 𝐷 𝑇 𝑐 𝑖 𝐷 = 𝑉 𝐷 𝑇 𝑐 (14.4)
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6. The “corporate tax shield”(cont)
In this case, Eq becomes
Vfirm ≡ VE + VD = VU + VDTc (14.5)
The equation appears in Modigliani and Miller’s 1963 paper, Corporate Income Taxes and the Cost of Capital: A Correction.
For example, if the firm’s rate of corporate tax liability (Tc) is 30%, a dollar of debt of held in perpetuity equates with a reduction of the firm’s future tax obligations which has a current value of VDTc = 30 cents.
Extraordinarily: each $1 of debt appears to add 30 cents of value to the firm.
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7. Adjustments to the cost of capital formulas
The question we now ask, is, How does the corporate tax deductibility of interest payments impact on the formulae of Chapter 8:
a project (firm’s) cost of equity (Eqns 8.5 and 8.6):
𝑘 𝐴𝑉 ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐸 + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐷 = 𝑘 𝑈
𝑘 𝐸 = 𝑘 𝑈 + 𝑉 𝐷 𝑉 𝐸 (𝑘 𝑈 − 𝑘 𝐷 )
and a project (firm’s) beta (Eqns 8.7 and 8.8):
𝛽 𝐴𝑉 ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 𝛽 𝐸 + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 𝛽 𝐷 = 𝛽 𝑈
𝛽 𝐸 = 𝛽 𝑈 + 𝑉 𝐷 𝑉 𝐸 𝛽 𝑈 − 𝛽 𝐷

8. Adjustments to cost of capital formulas (cont)
Recall that in Chapter 8 - with the assumption of zero corporate tax – we argued that the weighted average, kAV, of the firm’s costs of equity (kE) and debt (kD) as
𝑘 𝐴𝑉 ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐸 + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐷
should remain equal to the unlevered cost of equity (kU) (Eqn 8.5):
𝑘 𝐴𝑉 ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐸 + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐷 = 𝑘 𝑈

9. Adjustments to cost of capital formulas (cont)
Allowing corporate tax, we argue that kAV as above can be identified as the weighted average of
(i) what the firm’s cost of equity would be in the case that there was no corporate tax, which is to say, kU , and
(ii) the cost of capital (kTS) for the present value of the tax savings (PVTS) component of the firm’s value,
weighted, respectively, by
(i) the component value of the firm that does not depend on the corporate tax savings (which is the value of the firm if we assume no debt), which we can write as, Vfirm – PVTS, and
(ii) the component value of the tax savings, PVTS.

10. Adjustments to cost of capital formulas (cont): the “fundamental” equation
Thus, we write (by definition, Vfirm = VE + VD ):
kAV ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 kE + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 kD
= 𝑉 𝐸 + 𝑉 𝐷 −𝑃𝑉𝑇𝑆 𝑉 𝐸 + 𝑉 𝐷 kU + 𝑃𝑉𝑇𝑆 𝑉 𝐸 + 𝑉 𝐷 kTS (14.6)
Eqn 14.6 replaces Eqn 8.5.

11. Adjustments to cost of capital formulas (cont)
The previous Eqn 14.6 may not look too inspiring!
Nevertheless, it is the single
“fundamental” equation
that is required to ensure consistency in any calculations involving a change of leverage.

12. Adjustments to cost of capital formulas (cont)
If, following Modigliani and Miller, we allow kTS = kD, and thereby PVTS = VDTc, we derive from the “fundamental” Eqn 14.6 (with a little manipulation):
𝑘 𝐸 = 𝑘 𝑈 + 𝑉 𝐷 𝑉 𝐸 𝑘 𝑈 − 𝑘 𝐷 (1− 𝑇 𝑐 ) (14.9)
which relates the firm’s costs of equity (kE) to the unlevered cost of equity (kU) and the cost of debt (kD).
In their 1963 paper, Modigliani and Miller argued that the above equation replaces Eqn 8.6 :
𝑘 𝐸 = 𝑘 𝑈 + 𝑉 𝐷 𝑉 𝐸 (𝑘 𝑈 − 𝑘 𝐷 )
when we allow for corporate tax.

13. Adjustments to cost of capital formulas (cont)
By the same argument used to derive Eqn 14.6, we can argue that the betas are now related in the same manner. So that:
𝛽 𝐸 = 𝛽 𝑈 + 𝑉 𝐷 𝑉 𝐸 𝛽 𝑈 − 𝛽 𝐷 1− 𝑇 𝑐 (14.12)
replaces Eqn 8.8:
𝛽 𝐸 = 𝛽 𝑈 + 𝑉 𝐷 𝑉 𝐸 𝛽 𝑈 − 𝛽 𝐷

14. Break time

15. Alternative to Modigliani and Miller’s derivation of the corporate tax shield (PVTS)
In seeking to calculate the PVTS due to the tax deductibility of the firm’s debt, we are not obliged to accept Modigliani and Miller’s consideration that the appropriate rate (kTS) at which to discount the firm’s tax savings (TS) due to the debt (Eqn 14.1) is the firm’s cost of debt, kD (see over).

16. Alternative to Modigliani and Miller’s derivation of the corporate tax shield (PVTS) (cont)
It can be argued that the PVTS should be calculated by discounting the tax savings (TS)
not by kD,
but by the firm’s unlevered cost of equity, kU,
so that we have Eqn 14.2 as:
𝑃𝑉𝑇𝑆= 𝑇𝑆 𝑘 𝑇𝑆 = 𝐷𝐸𝐵𝑇 𝑖 𝐷 𝑇 𝑐 𝑘 𝑈 (14.13)
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17. Alternative to Modigliani and Miller’s derivation of the corporate tax shield (PVTS) (cont)
The argument for discounting the tax savings (TS) by the firm’s unlevered cost of equity, kU, is
that the tax-saving benefits have a forward- looking variability that accords with the firm’s ability to sustain debt, which, in turn, will depend on the firm’s unlevered cash flow.

18. Alternative to Modigliani and Miller’s derivation of the corporate tax shield (PVTS) (cont)
In this case (ie, kU = kTS), we can observe that the PVTS terms in our “fundamental” Eqn 14.6:
kAV = 𝑉 𝐸 + 𝑉 𝐷 −𝑃𝑉𝑇𝑆 𝑉 𝐸 + 𝑉 𝐷 kU 𝑃𝑉𝑇𝑆 𝑉 𝐸 + 𝑉 𝐷 kTS (14.6)
conveniently cancel with each other !!!
With the outcome that we recover Eqns 8.5 and 8.6 (for the costs of equity capital) and Eqns 8.7 and 8.8 (for their betas) which were derived assuming no corporate tax:
𝑘 𝐴𝑉 ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐸 + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 𝑘 𝐷 = 𝑘 𝑈 (8.5)
𝑘 𝐸 = 𝑘 𝑈 + 𝑉 𝐷 𝑉 𝐸 𝑘 𝑈 − 𝑘 𝐷 (8.6)
𝛽 𝐴𝑉 ≡ 𝑉 𝐸 𝑉 𝐸 + 𝑉 𝐷 𝛽 𝐸 + 𝑉 𝐷 𝑉 𝐸 + 𝑉 𝐷 𝛽 𝐷 = 𝛽 𝑈 (8.7)
𝛽 𝐸 = 𝛽 𝑈 + 𝑉 𝐷 𝑉 𝐸 𝛽 𝑈 − 𝛽 𝐷 (8.8)

19. The corporate tax shield challenged
The above equations relate the cost of capital with leverage (debt) to the unlevered cost (assuming no debt) without the need to worry about:
(i) how potential bankruptcy with debt might affect the calculations
(ii) how the calculations are affected if the firm’s interest payments on its debt exceeds the “fair” interest rate that accords with the CAPM.
We look briefly at each assumption.

20. (i) Financial distress and bankruptcy costs
The textbooks advocate that shareholders seek an optimal debt to equity ratio for the firm that occurs at the equilibrium point that the tax advantages of the tax deductibility of interest payments on debt and the benefits of mitigating the agency problem (Section 5.7) are balanced with the costs of bankruptcy and financial distress that too much debt might bring about.

21. (i) Financial distress and bankruptcy costs (cont): Miller’s objection to “bankruptcy” costs
Interestingly, as we observed in Chapter 8, Merton Miller himself chose to distance himself from the above developments that the Modigliani and Miller propositions precipitated.
Miller’s observation was that the theoretical tax savings contingent on debt financing are sufficiently significant that it cannot make sense to equate them with something as ill- defined as “financial distress” or even the costs of a restructure following bankruptcy – which for Miller, were not that significant.

22. (ii) Non-compliance with the CAPM
Miller pointed out that we cannot confidently assess the value of the tax deductibility of interest payments on debt without allowing for the complications of shareholder personal tax liabilities as they differ across bonds and equity.
He argued that because of individual tax preferences - some investors are taxed less at the individual level on their returns from equity than on returns from debt - firms would need at some point to raise their borrowing rate to induce more lending from those investors whose rates of tax liability on a return from bonds is greater than is the case for equity.
Thus, Miller allows that what the firm gains by the tax deductibility of the interest rates on debt, it is ultimately prepared
to give away with a higher interest rate at the margin of its borrowing.
In this case, the firm’s cost of debt is greater than the “fair value” determined by the CAPM.

23. (ii) Non-compliance with the CAPM (cont)
In addition to Miller’s argument based on tax consdieratoins, we can imagine how in the practical negotiation of the interest rate to be attached to a bond issue, institutional purchasers of the bond issues will be seeking to bid up the interest rate, in effect, seeking in practice to appropriate for the holders of the bond at least some of the value of the corporate tax deductibility of the firm’s interest payments.

24. (ii) Non-compliance with the CAPM (cont)
In this case, the additional interest rate over and above the rate determined by the beta of the debt in the CAPM represents an additional “cost” to the firm (project).
Specifically:
Cost to the firm when the debt interest rate (kD) is greater than
the CAPM-determined rate (kCAPM)
= $ 𝑉 𝐷 𝑘 𝐷 − 𝑘 𝐶𝐴𝑃𝑀 𝑘 𝑇𝑆 (14.13)

25. (ii) Non-compliance with the CAPM (cont)
Thus, the actual value added to the firm’s equity with debt, PVTSADJ, is determined as
𝑃𝑉𝑇𝑆 𝐴𝐷𝐽 = 𝑃𝑉𝑇𝑆 (𝑎𝑠 𝐸𝑞𝑛 14.2)− $ 𝑉 𝐷 𝑘 𝐷 − 𝑘 𝐶𝐴𝑃𝑀 𝑘 𝑇𝑆
= $ 𝑉 𝐷 𝑘 𝐷 𝑇 𝑐 𝑘 𝑇𝑆 − $ 𝑉 𝐷 𝑘 𝐷 − 𝑘 𝐶𝐴𝑃𝑀 𝑘 𝑇𝑆
= $ 𝑉 𝐷 𝑘 𝐶𝐴𝑃𝑀 − 𝑘 𝐷 (1− 𝑇 𝑐 ) 𝑘 𝑇𝑆 (14.14)
where we have either kTS = kCAPM (the cost of debt commensurate with the risk of the debt) or kTS = kU (the unlevered cost of equity).

26. (ii) Non-compliance with the CAPM (cont)
In either case, Eqn reveals that the tax benefits of debt are nullified at the point:
kCAPM = kD(1-Tc)
which is to say,
at the point that the cost of debt, kD, exceeds kCAPM /(1-Tc):
𝑘 𝐷 > 𝑘 𝐶𝐴𝑃𝑀 1− 𝑇 𝑐 (14.15)

27. (ii) Non-compliance with the CAPM (cont)
For example, suppose we refer back to Illustrative Example 14.1, and suppose that – for whatever reasons – the interest rate that Temptations negotiates for its bonds is 9.5% as opposed to the theoretical value that aligns with the CAPM (7.5%, part (f)).
See over. . .

28. (ii) Non-compliance with the CAPM (cont)
With Eqn 14.14, the value added by the debt is calculated as
𝑃𝑉𝑇𝑆 𝐴𝐷𝐽 = $ 𝑉 𝐷 𝑘 𝐶𝐴𝑃𝑀 − 𝑘 𝐷 1− 𝑇 𝑐 𝑘 𝑇𝑆
In Illustrative Example 14.1, we chose to discount adjustments due to debt at the market cost of debt (kD). Hence:
𝑃𝑉𝑇𝑆 𝐴𝐷𝐽 = $50,000,000(0.075− ) = $5,666,667,
Hence the new value of the firm, Vfirm = $100,000,000 + $5,666,666 = $105,666,667.

29. Review
The tax deductibility of the firm’s interest payments on its debt is of theoretical value to the firm’s shareholders.
Nevertheless, if the firm is paying at a higher interest rate on the debt than the “fair rate” for risk, the advantage of the tax-deductibility of the interest payments will tend to cancel with the higher interest rate that the firm is paying.