Valuing levered projects
6.3 Project values with different debt ratios
1 − τ rd 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
Vrdτ 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)).
6.3 Project values with different debt ratios
...
The variety of formulas we have developed so far may look confusing at first sight, but their application is fairly straightforward. In this section we demonstrate how they are used by going through the different methods step by step.
6.3.1 Outline
Recall that our problem is to find the value of a project with the same business risk but a different debt ratio, compared with the existing operations. We find that value either by adjusting the base case present value or by adjusting the WACC. The WACC can be adjusted stepwise, by calculating it from new values for re and rd, or directly with a formula. The choice of the method and formulas to be used depends on the characteristics of the project, mainly whether its debt is rebalanced or predetermined.
The formulas enable us to calculate a complete set of returns for the project from the returns of the existing operations plus one new rate. The new rate usually is the project’s cost of debt, given its level of leverage. In practice, this rate is easily obtained by asking the bank for an offer. In addition, we use the opportunity cost of capital, which can be calculated from the existing operations. The OCC and the project’s cost of debt are used in the formulas that give the project’s cost of equity or WACC. We then have all the information to calculate the project’s value, either with the WACC or APV. Figure 6.3 places the methods in a decision tree that shows when the methods are appropriate.
Business risk:
ւ ց
Different Same
1. New OCC ↓
2. New WACC Financial
risk:
ւ ց
Different Same
↓ ↓
Debt: Same WACC
ւ ց
Predetermined Rebalanced:
1. Unlever and relever ւ ց
2. Mod.–Miller Periodically Continuously
3. APV 1. M.–Ezzell 1. Unlever and relever
2. APV 2. APV
Figure 6.3 Decision tree for calculation methods
From the discussion so far it should be clear that the main distinction is between predetermined or rebalanced debt. As we have seen, the value of tax shields can be con-siderably lower if they vary with the project’s value. The distinction between continuous and periodical rebalancing is seldom made, and the formulas are used interchangeably.
Similarly, formulas for perpetuities are often used for shorter-lived projects. The errors introduced by this are probably small compared with the estimation errors of the future cash flows.
6.3.2 Debt rebalanced
Our starting position is that we know the returns (reand rd) and relative sizes (Ve/Vand Vd/V) of debt and equity in the existing operations. We use bold symbols to denote data of the existing operations. We also know the firm’s financial policy (rebalanced debt, in this case) and we have obtained the interest rate against which the new project can borrow its chosen amount of debt. We can then calculate the project value by adjusting the WACC (stepwise or with a formula) or by using APV. We will go through the different methods step by step.
First way: stepwise adjust the WACC
This method requires continuous rebalancing and involves the following three steps:
1. Unlever: calculate the opportunity cost of capital from the existing operations, i.e.
using the returns and capital structure of the existing operations:
r = rd
D V+ re
E V
2. Use this OCC plus the project’s cost of debt and debt–equity ratio to calculate the project’s cost of equity:
re = r + (r − rd)D E
175 6.3 Project values with different debt ratios
These steps can also be done in terms of βs, after which the CAPM can be used to calculate the returns.
3. Relever: calculate the after-tax WACC using the project’s costs and weights:
WACC= re
E
V + rd(1 − τ )D V
Second way: adjust the WACC using Miles–Ezzell’s formula This method requires periodical rebalancing and is done in two steps.
1. Unlever: we use Miles–Ezzell and the returns and capital structure of the existing operations to calculate the opportunity cost of capital by solving:
r′ = r − τ rd
D V
(1 + r) (1 + rd)
for r, i.e. we use Miles–Ezzell ‘in reverse’.
2. Relever: we use Miles–Ezzell again, this time with OCC and the project’s cost of debt and debt-to-value ratio to calculate the project’s WACC:
r′= r − τ rd
D V
(1 + r) (1 + rd)
Third way: use APV
To calculate the APV, we first have to calculate the OCC using either the first or the second way above, depending on whether debt is continuously or periodically rebalanced.
We can then discount the project’s (unlevered) cash flows to calculate its base case NPV.
The value of the tax shields is found by discounting them at opportunity cost of capital and, if debt is rebalanced periodically, by multiplying the result with (1 + r)/(1 + rd).
The sum of the base-case NPV and the value of the tax shields is the APV.
6.3.3 Debt amounts predetermined
If debt is predetermined, we can use the same three ways to calculate project values, we just have to use different formulas. Our starting position is also the same, i.e. we know the returns (re and rd) and relative sizes (Ve/Vand Vd/V) of debt and equity in the existing operations. We also know the firm’s financial policy (predetermined debt, in this case) and we have obtained the interest rate against which the new project can borrow its chosen amount of debt. We can then, as before, calculate the project value by adjusting the WACC (stepwise or with a formula) or by using APV.
First way: stepwise adjust the WACC
1. Unlever: calculate the opportunity cost of capital r = rafrom the existing operations, i.e. using the returns and capital structure of the existing operations:
ra = re
As we have seen, this is not a very practical way because we have to calculate the present value of the tax shields first. But under the Modigliani–Miller assumption that all cash flows are perpetuities, this method is more practical:
r = ra= rd(1 − τ ) D
V− τ D+ re
E V− τ D
2. Use the project’s cost of debt and the project’s debt–equity ratio to calculate the project’s cost of equity using:
3. Relever: calculate the after-tax WACC using the project’s costs and weights:
WACC= reE
V + rd(1 − τ )D V
Second way: adjust the WACC using Modigliani–Miller’s formula
This, of course, requires the Modigliani–Miller assumptions and is done in two steps:
1. Unlever: use the Modigliani–Miller formula and the WACC and capital structure of the existing operations to calculate the opportunity cost of capital by solving:
r′= ra(1 − τ L) for ra (run MM in reverse).
2. Relever: use Modigliani–Miller again, with the OCC (ra) and the project’s debt-to-value ratio to calculate the project’s WACC:
r′= ra(1 − τ L)
Although the Modigliani–Miller formula assumes that debt is predetermined and permanent, it gives a good approximation for projects with limited lives if debt is predetermined.
Third way: APV
Also in the case of predetermined debt we first have to calculate the OCC to calculate the base-case NPV. We can then add the value of the tax shields by using the predetermined schedule for interest payments and discount the tax shield at the cost of debt.
177 6.4 Some examples