Financing rules and discount rates

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In this section we analyse the second way to express the effect of leverage, viz. by adjust-ing the discount rate. As we have seen, the rate needs to be adjusted downwards, from

the OCC to the WACC, to include the value of the interest tax shields. We have also seen that the two different financing policies, predetermined or rebalanced debt, have differ-ent implications for the riskiness of the tax shields. Hence, differdiffer-ent discount rates must be used to calculate their values. This may seem a bit complicated at first sight, but the analysis involves no more than rewriting the balance sheet identity with the two different discount rates for the tax shields.

6.2.1 Predetermined amounts of debt

Starting point of the analyses is the following balance sheet for the project:

Value assets Va Debt D

Value tax shields PV(TS) Equity E

Total value V Total value V

If the money amounts of debt are predetermined and independent of the future project value, the D/E ratio will go up and down over time with the project value and, hence, so will re and the WACC. The tax shields, meanwhile, follow from the predetermined amounts of debt and are therefore as risky as debt itself, so that they should be discounted at the cost of debt, rd. This means that we can write the balance sheet identity total assets = debt + equity in terms of the costs of capital as follows:

r_aV_a+ r_dPV(TS)= r_eE+ r_dD (6.2) Rearranging terms gives expressions for raand re:

r_a = re

E V_a + rd

D− PV(TS) V_a or ra

V_a V = re

E V + rd

D− PV(TS)

V (6.3)

and

r_e = ra+ (ra− rd)D− PV(TS)

E (6.4)

These are general expressions that can also be used for projects with limited lives. They are not, however, very useful since they call for the present value of the tax shields which is usually unknown before the project value is calculated. If we make the Modigliani–

Miller assumption that all cash flows are perpetuities, so that debt is permanent (as well as predetermined), then the present value of the tax shields is:

PV(TS)= τ (r_dD)

r_d = τ D (6.5)

Substituting (6.5) in (6.4) and (6.3) gives the Modigliani–Miller expressions for reand ra

that we derived in the previous chapter:

r_e = ra+ (ra− rd)D− τ D E r_e = ra+ (ra− rd)(1 − τ )D

E (6.6)

171 6.2 Financing rules and discount rates i.e. MM proposition 2 with taxes, and for ra:

r_aV_a

The WACC formula in (6.7) can be rewritten in two ways. The first gives an explicit relation between raand r^′(which is exact for predetermined, permanent values):

r_aV_a

Defining L = D/V , i.e. the debt–value ratio, we get the Modigliani–Miller formula:

WACC= r^′ = ra(1 − τ L) (6.8)

The MM formula can be used to ‘unlever’ and ‘relever’. Given the WACC, the formula can be used to unlever, i.e. to calculate ra, the required return of unlevered cash flows.

This is very useful because in most situations, re, r_d and τ are (at least in principle) observable, but r_a is not. The obtained value of r_a can then be used to calculate the WACC for a different debt ratio, i.e. to relever the return. This is elaborated in the next section. The Modigliani–Miller formula can also be derived by substituting (6.6) into the WACC formula (6.7).

The second way to rewrite the WACC formula (6.7) under the MM assumptions gives an alternative expression for ra,as we saw in the previous chapter:

r_a= r_d(1 − τ ) D

V − τ D + r_e E V − τ D

Finally, since βs are additive across investments, just as returns are, we can do the same analysis in terms of β:

Debt can be rebalanced continuously, so that the D/E ratio always has the same value, but also periodically, e.g. from year to year. In the latter case, we have to account for the periods in which debt is predetermined.

Continuous rebalancing

We start with the same project balance sheet as before:

Value assets VA Debt D

Value tax shields PV(TS) Equity E

Total value V Total value V

Continuous rebalancing means that the D/E ratio is constant over time. If we assume that the cost of debt, rd, is fixed, then reand the WACC will also be constant over time.

Meanwhile, the tax shields will go up and down with the value of the project. This makes them as risky as the assets, so that they should be discounted at the same rate as the assets, i.e. r_a= r. This means that both items on the left-hand side of the balance sheet, assets and tax shields, have the same return. Hence, their relative sizes (VA/V and PV(TS)/V) are irrelevant for the total return of the project. This, in turn, must then also apply to the right-hand side of the balance sheet: the total return required by debt and equity holders must equal ra= r, regardless of the relative sizes of VAand PV(TS).²The result is that taxes disappear from the equation and the opportunity cost of capital simply is the weighted average of the costs of debt and equity:

r_aV_a

This can also be rewritten in terms of reor β, which gives Modigliani–Miller proposition 2 without taxes:

If debt is rebalanced periodically, we have to use a combination of discounting at rd and r_a. More specifically,³the tax shield over the next period is predetermined and should be discounted at rd. That tax shield is τ rdDand its discounted value is (τ rdD)/(1 + rd). The tax shields further in future depend on the value of the project; they are more uncertain and should be discounted at ra.Their value is the total PV(TS) minus the first period’s value: PV(TS) − (τ rdD)/(1 + rd). If we include these two terms in the balance sheet identity in return terms the result is:

V_ar_a+ τ r_dD

2 In comparison: with predetermined debt the total return in (6.2) decreases with the relative size of the tax shields, since r_d< r_a.

3 The following elegant derivation is suggested by Inselbag and Kaufold (1997).

173 6.3 Project values with different debt ratios

This can be rewritten (using Va= E + D − PV(TS)) to give an expression for re: r_e = r_a+ (r_a− r_d)D

E 

1 − τ r_d 1 + rd

(6.9) The expression in (6.9) is the equivalent of Modigliani–Miller proposition 2, under the assumption that debt is periodically rebalanced. Substituting (6.9) into the formula for the WACC (6.1) gives (after extensive rewriting):

r^′= WACC = ra−D

Vr_dτ 1 + ra

1 + rd

(6.10) This formula is known as the Miles–Ezzell formula. It is the equivalent of the Modigliani–

Miller formula (6.8) under the assumption that debt is periodically rebalanced. The Miles–

Ezzell formula is used in the same way as the Modigliani–Miller formula for unlevering and relevering:

• For a given WACC, the formula gives ra,the opportunity cost of capital.

• When ra is known, the formula can be used to calculate the WACC for a different debt ratio (and different cost of debt).

The Miles–Ezzell formula can also be derived with a backward iteration procedure, that starts with the last period and works its way to the beginning (as was originally done by Miles and Ezzell (1980)).