10.2.1 Steady-State Business under Inflationary Conditions

To analyze the issues linked to the use of capitalization formulas, it is necessary to clarify all the different critical aspects.

As previously seen, the assumption that the company needs to be in equilibrium (i.e., in a steady state) is fundamental for the use of synthetic valuation formulas. In particular, in a neutral inflation scenario, the most important assumptions implied by the model are:

The steady-state scenario without real terms growth implies:

• Constant revenues and costs
• New investment equal to yearly depreciation and absence of working capital changes
• A constant debt ratio

When all the aforementioned assumptions hold, cash flows are perpetual and net income is equal to the cash flow available for the shareholders (FCFE).

The neutral inflation scenario requires:

• Uniform price increases of products and services.
• Discount rates reflecting expected price changes.
• No increases in working capital needs; this is consistent with the assumption that current assets are balanced by current liabilities.
• No “paper profit”. This implies that tax-deductible depreciation adjusts to match net investments and that inventory is valued with the LIFO method.

By combining the steady-state assumption with that of neutral inflation, we can define the following equilibrium state for a company:

• Revenues and costs increase linearly every year.
• Net operating cash flow (FCFO) increases proportionally to the inflation rate every year.
• Net financial debt increases every year proportionally to the inflation rate. As a matter of fact, if it did not, the debt ratio would decrease assuming that operating capital increases in nominal terms proportionally to the inflation rate. This would conflict with the steady-state assumption and it would require a different modeling of and WACC*.
• Due to the previous assumption, the FCFE is higher than net income by an amount equal to , which is equal to the increase in debt necessary to maintain the target leverage.

10.2.2 An Illustration of the Steady-State/Neutral Inflation Assumption

Exhibit 10.1 shows the cash flow dynamics of a company when the steady-state assumption is combined with a neutral inflation scenario. In particular, we assume that at time zero prevailing expectation is a 10 percent annual inflation rate. Notice that this inflation rate is particularly high to emphasize its effects.

Inflation rate
	t−1	t0	t1	t2
Operating margin	3,000	3,300	3,630	3,993
Depreciation	(500)	(550)	(605)	(665.5)
Operating result	2,500	2,750	3,025	3,327.5
Interests	(200)	(620)	(682)	(750.2)
Taxes	(828)	(766.8)	(843.5)	(927.8)
Net income	1,472	1,363.2	1,499.5	1,649.5
Leverage at the end of period	4,000	4,400	4,840	5,324
FCFO	1,600	1,760	1,936	2,129.6
FCFE	1,472	1,763	1,939.5	2,133.5
Wequity	17,440	22,064	24,270	26,697
D/E	22.9%	19.9%	19.9%	19.9%

Column 2 in the table reports the basic data in a company's income statement in a no-inflation scenario . On the other hand, the other three columns show the results and cash flows for a neutral inflation scenario. The comparison between column 2 and the other three shows the effects of an inflation jump (from 0 to 10 percent).

The analysis of the last three columns allows us to understand the results and cash flow dynamics in a scenario with inflation (in other words, the analysis of the last three columns allows us to understand the dynamics of the results when inflation is stable at the 10 percent level).

Cash Flow Dynamic in an Inflation Scenario

The last three columns of Exhibit 10.1 show that the FCFO and the FCFE increase in nominal terms linearly to the inflation rate under a constant inflation rate. It is important to highlight that even interests increase linearly to the inflation rate. As a matter of fact, after the inflation jump, net debt increases annually in a linear fashion with respect to . This adjustment is necessary to maintain the debt ratio constant.

Let us now verify how the company value can be determined. Let us assume the following parameters:

• Real : 8%
• Nominal : 18.8%
• Real : 5%
• Nominal : 15.5%
• : 36%

Notice that the valuation takes place at the end of .

Let us begin with the business valuation procedure carried out using the adjusted discount rate approach. Remember that:

Levered enterprise value	=	Unlevered enterprise value	+	Tax shield
	=		+

Estimating the Unlevered Value

The unlevered value of the company can be determined as the discount of a perpetuity that grows at the inflation rate :

10.1

Using the Real Rate

The aforementioned formulas can be presented in another form. In particular, we could split K_eu into two elements: the real rate and the expected price change, that is:

We can rewrite [10.1] as:

from which:

which can be simplified as:

10.2

The new expression shows a well-known concept. As a matter of fact, in a neutral inflation scenario, the value of an asset generating cash flows and changing proportionally to the inflation rate can be determined by discounting the real cash flow by a real discount rate.

In the example shown in Exhibit 10.1, we obtain:

Tax Shield Valuation

Assuming the steady-state/neutral inflation scenario, we know that the value of debt will have to be adjusted for inflation every year. The tax shield value will therefore be equal to:

[10.3]³

It is important to highlight that measures the level of debt at the beginning of the period used in estimations. The reason for this choice comes from the assumption that the first interest costs, and therefore the first tax shield, is determined based on the current debt level at the beginning of the period. Interest should, however, be calculated based on the average level of debt for the period under consideration. This is dealt with using acceptable simplification.

In our example, assuming , we find:

Determining the Levered Value

We can now determine the enterprise levered value as:

Note that both the unlevered value and increase in value every year due to inflation. As a matter of fact, also the amount of nominal debt increases linearly to .

This confirms our conclusion that the unlevered and levered company values increase systematically proportionally to the inflation rate in a steady-state/neutral inflation scenario. Even more importantly, this means that both levered and unlevered companies keep their real value unchanged.

Once is known, we can then determine the equity value:

The amount of debt at the end of is subtracted from to determine the value of equity. is then the level of debt at the time of the valuation.

10.2.3 Discount Rate Adjustment Approach

We can now determine the value of the company using adjusted discount rate formulas. To do so, it is necessary to use the formulas presented in the previous chapters for a growth scenario. In particular, the proper formulation is the one consistent with the discount of tax benefits at the rate .

Equity-Side Approach

Let us begin with the equity-side approach:

can be determined through the following formula introduced in previous chapters

In our case, we find that:

Interestingly, is paradoxically lower than , even if by a small amount. This seems illogical since the risk taken by shareholders is higher when the company is levered. However, we have to go back to the previous results to understand how this result is in fact correct. As we discussed then, the opportunity cost of capital will increase less when tax benefits are present than when they are not. This emphasizes the incremental flow for shareholders coming from tax shields. In our example, tax benefits are so high that () becomes negative. The tax shield value to shareholders is then greater than the loss of value caused by the worsened risk profile caused by higher leverage.

Asset Side Approach

We can now consider the asset side approach:

Based on our example, we can determine that:

And the value is:

10.2.4 Summary of Valuation Formulas Consistent with a Steady-State/Neutral Inflation Scenario

Exhibit 10.2 presents a summary of valuation formulas consistent with the steady-state/neutral inflation hypothesis:

ASSETS SIDE	APV APPROACH
	STANDARD APPROACH
EQUITY SIDE	STANDARD APPROACH

To simplify, let us quickly review the meaning of the above symbols:

• : value of the operating capital (asset value)
• FCFO: operating cash flow
• WACC*: weighted average cost of capital when tax shields are present as determined by using [d.1]
• _: equity value
• FCFE: cash flow available to shareholders
• : cost of equity, when tax shields are present as determined by using [d]
• : cost of equity of an unlevered company
• : expected inflation rate

At this point, it is important to highlight some important considerations for the application of the above formulas:

• The APV approach has two advantages: transparency and flexibility. Those who perform the valuation appreciate the value component given by and the assumptions behind this scenario—that is, the linear relationship between debt and the inflation rate and the presence of tax shields based on the tax rate .
• However, these assumptions are not always credible. For instance, if the company generates cash, it is more realistic to think that the debt amount will decrease. Under these conditions, the APV approach would work better as it allows us to adjust the tax shield value () to a debt profile consistent with the business plan, the long-term cash flow, and the leverage policy that the management intends to adopt. These specific aspects will be discussed in detail in the following paragraph when analyzing the “Hydroelectric Co.” example.
• On the other hand, adjusted discount rate formulas face another well-known problem. As a matter of fact, as we previously discussed, the debt ratio used to determine and the WACC* should be computed on the basis of the economic value of the business, which, in this case, should include the tax benefits linked to the increase in debt.

Therefore, we can conclude that the process of adjusting discount rates is often used in an approximate way and without a thorough analysis of the underlying assumptions. These assumptions, however, are the key to a correct application of the formulas.

10.2.5 Valuing Hydroelectric Co

The business in this example operates in the electric generation industry.

Assumptions for the valuation:

• The company manages a hydroelectric plant. No expansion of the plant is possible.
• Depreciation is equal, on average, to the investment necessary for the maintenance of the current capacity.
• Management thinks that the current leverage can be maintained in the long term.
• It is assumed that tariffs adjust to costs.
• Current assets and current liabilities tend to balance.

Exhibit 10.3 shows the balance sheet, income statement, FCFO, and FCFE of Hydroelectric Co. at the time of the estimate.

^* Determined as 2% of the financial debt outstanding at the beginning of the period .

The other important assumptions for the valuation are:

Hydroelectric Co. Valuation: Steady-State/Neutral Inflation Scenario

In order to verify the different results obtained following numerous assumptions, we will use the APV approach to value Hydroelectric Co. It implies, as we know, that:

In this specific case, can be determined as follows:

can be calculated with the following expressions:

From which:

With the available data, we can finally determine the equity value by subtracting the value of net financial debt from (Exhibit 10.3):

Alternative Debt Profiles

It is now possible to check the sensitivity of the value just computed to various debt profiles.

For instance, by observing that the business model of Hydroelectric Co. constantly generates cash flow, it is logical to believe that the debt level will tend to decrease or to at least stay constant.

Under the assumption of decreasing debt and assuming a 10-year debt repayment plan, the value of tax shields will change as shown in Exhibit 10.4.

The discounted value of the tax shield series using a discount rate K_d= 6.5% is 10.4 million. This amount is 80 percent lower than the result given in an increasing debt scenario (53.4 million).

If we consider the assumption that debt stays constant, the tax shields would become:

Hydroelectric Co. Valuation: The Adjusted Discount Rate Process

It can be interesting to verify our results of the valuation of Hydroelectric Co. with the quick-and-dirty adjusted discount rate process, which is similar to the approaches frequently used in practice.

We calculate the WACC* on the basis of the book value debt ratio:

Applying the synthetic approach, we find:

It is interesting to notice that, in a perpetual nominal growth scenario, the value computed under this approach is 25 million greater than the value found through the APV method. Such a result is explained by the use of the book value of debt in the formula, which is higher than the effective one. In fact, economic—and not book—values are supposed to be considered in these formulas. In this specific case, the economic value of equity is significantly higher than the book one, leading to this impressive difference in results.⁵

10.3.1 Perpetual Growth Model for Unlevered Companies

The perpetual growth model applied to unlevered businesses, that is, to companies with no debt, is known as the Gordon model, named after Myron Gordon, the person who spread its use in the United States of America during the 1950s and 1960s.⁶

The formula to compute the value of an unlevered company can be presented as follows (notice that when there is no debt, the FCFO and the FCFE coincide):

10.5

Let us use an example to better discuss its meaning. Assume that the income statement of a company is:

The cash flow dynamic is shown in Exhibit 10.5. The dashed line parallel to the horizontal axis shows the net cash flow of the business available to shareholders (FCFE) in a steady-state scenario. The flow is equal to 1,575 from to infinity since there are no new investments except renewals, which are equivalent to depreciation in a steady-state scenario.

What happens when the company invests 20 percent of its cash flow in new investments for each period? The rate of return for these new investments net of taxes is expected to be 12 percent.

The FCFE at time is equal to 1,260 (20 percent less than the flow in the steady-state scenario). Furthermore, notice that from onward, the cash flow is no longer equal to 1,575, but to 1,612.8 since the reinvestment of 315 in increases the future flow by 37.8 from to infinity. It is important to mention that the 12 percent rate of return is determined by dividing the reinvested equity amount by the incremental amount net of depreciation. We also assume that depreciation contributes to the systematic renewal of the reinvestment in .⁷

In , the business reinvests 20 percent of its cash flow in the same period (1,612.8). Consequently, the gross cash flow is equal to 1,651.5 (i.e., the sum of the flow in and the incremental flow of the new investment) from to infinity. The growth process is limited to a few periods in the table. Nevertheless, we can imagine a scenario in which this recurs for an unlimited period of time.

10.3.2 Constant Growth Model for Levered Companies (APV Approach)

We have shown that the value of the company increases proportionally to the rate when cash flows grow, conditionally to the new investment rate being equal to or greater than the opportunity cost of capital. Assuming that the level of leverage remains constant, the growth process implies that net financial debt will grow year-by-year based on . In fact, if this were not the case, the market value of leverage would tend to decrease.

Let us now analyze the problems linked with the application of growth formulas to levered companies. We will start with the APV approach.

Refer back to our previous example and assume that:

• The operating cash flow (FCFO) reinvestment rate is equal to 20 percent.
• The new investment rate of return on the operating capital is 12 percent.

Also assume the following:

• Net financial debt: 5000
• Interest rate (): 5%

Based on these new assumptions, the company's income statement now reports the following results:

As we know, the levered value of the company is equal to the sum of its unlevered value and the value of the net tax shield:

• Unlevered value of the company in a growth scenario:

  

This value is equal to the one derived in the preceding paragraph.

• Tax shield (when ):

  

It is important to highlight that is equal to 5,000 because it is assumed that the company will make its first investment at the end of and it will take on debt at the same time. Therefore, the first tax shield refers to current debt (=5000)

• Levered company value:

  

As in the example discussed in the previous paragraph, the computed value can be divided into two parts: the base value and the net present value of growth opportunities. Using the data from the previous example, we have the following:

• Unlevered NPVGO:
• Tax shield value:

The unlevered NPVGO is equal to the difference between the total investment in and the net present value of the incremental flow (FCFO). The tax shield from the investment in is equal to the debt undertaken to finance the investment multiplied by the tax rate. This tax rate expresses the savings linked to the tax shield (in the case of a perpetuity: ).

The expected value of all growth opportunities is:

• Unlevered NPVGO =
• Tax shield =

To summarize, we⁹ can break up and reaggregate the components of the value of the company as determined by the APV approach:

Our previous discussion highlights a very important conclusion: the creation of value in a growth scenario is mostly due to the incremental tax shield linked to the increase (based on ) in debt. This is another crucial point to bear in mind when using the constant growth models. As a matter of fact:

• The rate leading to the exact value of the tax shield can be less than .
• It is not certain that the company can systematically use tax shields related to its interest payments.
• The hypothesis of using in order to discount the incremental tax shields is not at all prudent. In our previous example, if were to be determined using , the tax shield value would consistently decrease from 1707.7 to 291.1. As a matter of fact:
• The tax shield value from the first reinvestment is:

  

• The net present value of all tax shields instead equals:

  

10.3.3 Discount Rate Adjustment Methods

In the perpetual growth scenario, the use of the adjusted discount rate formulas requires the same caution as the nominal growth approach discussed in the previous subsection. Specifically:

• We should use the same formulas consistent with the tax shield discount rate chosen by the expert to determine and the WACC*.
• The debt ratio should be expressed based on the economic equity value, which should include the net present value of the incremental tax shields.

Going back to the value of equity computed through the APV method, it is possible to verify the use of the adjusted discount rate formulas when discounting the tax shields at :

Once is computed, we can determine with the usual adjusted discount rate process:

The¹⁰ value determined above corresponds to the one computed through the APV process. The same result can be obtained starting from the asset-side perspective by using the WACC* and the .

10.3.4 Steady-Growth Scenario: A Wrap-up

Exhibit 10.7 shows a summary of the valuation formulas consistent with a constant growth scenario.

ASSETS SIDE	APV APPROACH
	STANDARD APPROACH
EQUITY SIDE	STANDARD APPROACH

They are similar to those previously examined at the beginning of this chapter. However, the application of these formulae is more complex. To be precise, there could be issues linked to real growth scenarios as well as to the use of in the computation of the net present value of tax shields.

It is also necessary to check the methods used in the computation of the FCFO and the FCFE. In the case of real growth, they should be computed consistently with the reinvestment needed to sustain growth in real terms.

For convenience, we will recall the meaning of the above symbols:

• = operating capital value (asset value)
• = equity value
• = expected operating cash flow at time 1
• = expected equity cash flow at time 1
• = weighted average cost of capital when tax benefits are present, as determined by using the appropriate formulas and the chosen tax rate
• = equity cost of capital when there are tax benefits, as determined by the appropriate formulas and the chosen rate
• =equity cost of capital for an unlevered company
• = cash flow growth rate in real terms
• = tax rate
• = net financial debt at the beginning of the period
• = cost of debt
• = tax benefit discount rate

The formulas are the same as those in Exhibit 10.2 in which we assume, however, that the tax benefits are discounted using . This hypothesis can be modified even if we are in a steady-state scenario.