19878 26w p001-009lated using this hamada-determined cost of equity and the new capital structure....

26C H A P T E R

26E-1

Web Extension: Comparison of Alternative Valuation Models

We described the APV model in Chapter 26 because it is easier to implement when the tar-get’s capital structure is changing than either the corporate value model or the free cash flowto equity model. In this extension we discuss the benefits and shortcomings of the two alter-native valuation models and show how the debt and interest projections were made. Youshould refer back to Chapter 26 when reading this extension, and remember that all threevaluation models give the same answers when implemented correctly using the sameassumptions. However, it can be difficult to do this when the capital structure is changing.The toolkit to this extension, IFM9 Ch 26 Tool Kit.xls, has detailed Excel models that show allof the calculations.

Corporate Valuation ModelAnalysts have used many versions of the corporate valuation model to value merg-ers. A “traditional” implementation involves the following steps, most of whichare the same steps we used in this chapter but with some important differences inthe details:

1. Forecast financial statements and free cash flows for a specified period of time.2. Estimate the long-term growth rate and target capital structure.3. Estimate the WACC at the horizon using the long-term capital structure.4. Calculate the horizon value using the constant growth model with the WACC

calculated above, the forecasted long-term growth rate, and the last projectedFCF.

5. Discount the horizon value and the specified free cash flows back to the pres-ent using the estimated WACC.

Note that the “traditional” approach to estimating the WACC at the horizon involvesusing the Hamada equation (Equation 15-8 in the text) to unlever the market-determined beta and then to relever it at the new target capital structure. This new

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26E-2 • Chapter 26 Web Extension: Comparison of Alternative Valuation Models

beta is used to calculate a levered required return to equity. The WACC is then calcu-lated using this Hamada-determined cost of equity and the new capital structure.

This approach may give incorrect results for several reasons. First, if the capi-tal structure is changing over time, the WACC will be changing over time, and it isdifficult to specify the correct discount rate. The changing capital structure alsoaffects the interest tax shield, and this too can lead to errors. Another, and moreserious, problem is using the Hamada equation to unlever and then relever theequity return. The Hamada equation assumes that both the debt and the interesttax shield are risk free. This means that as the equity return is relevered usingmore and more debt, all of the risk is concentrated on the equity. Because theinterest rate on corporate debt is always greater than the risk-free rate, clearly thedebt is not risk free. Furthermore, our discussion in Chapter 16 showed that theappropriate discount rate for the tax shield should be the unlevered cost of equity,not the cost of debt. Therefore, using Hamada to relever equity returns in calcu-lating the horizon WACC will give an incorrect WACC and thus an incorrect val-uation. The equation we use in this chapter to unlever and relever equity returns,Equation 16-15, assumes that the interest tax shield is risky and thus should bediscounted at the unlevered cost of equity.

To correctly implement the corporate valuation model when the capital struc-ture is changing you must take the following steps:

1. Project financial statements for a period of time, and calculate free cash flows.2. Project a long-term growth rate and a long-term target capital structure.3. Estimate the WACC at the horizon using the long-term expected capital struc-

ture, using Equation 16-15 to relever the equity return.4. Calculate the horizon value using a constant growth model with the WACC

calculated above, the growth rate assumed, and the last projected FCF.5. Calculate a new value of operations and WACC each year before the horizon

using the actual debt ratio for that year in Equation 16-15 for the WACCcalculation.

6. Discount the horizon value and the intermediate free cash flows back to thepresent using the individual WACCs estimated above.

Although this will, in principle, work, the APV is simpler to implement and shouldbe used when the capital structure is changing. Note also that simply using theWACC calculated in Step 3 to discount the free cash flows in the years before thehorizon will give an incorrect answer: Because the capital structure is changingeach year, the appropriate WACC is also changing each year. Discounting freecash flows at an incorrect WACC gives the wrong answer.

Free Cash Flow to Equity (FCFE) ModelThe FCFE model has some intuitive appeal since the value of equity is directlyestimated from its cash flows rather than having to find the value of operationsfirst. However, its “traditional” implementation when the capital structure ischanging has some problems, just like the corporate valuation model. The first dif-ficulty is defining free cash flow to equity. The idea of FCFE is to identify all ofthe cash flows available to equity holders. In our framework FCFE is calculated as

Free cash flow� Interest expense� Interest tax shield� Increase in debt� Free cash flow to equity

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Chapter 26 Web Extension: Comparison of Alternative Valuation Models • 26E-3

The first three terms are clear; the equity holders have a claim on free cash flowless after-tax interest expense. The last term, increase in debt, is also a cash flowto equity since, by definition, free cash flow already includes any required rein-vestment. Additional debt over and above free cash flow is available for sharerepurchases or dividends and so is available to shareholders. This term, however,adds a wrinkle to the FCFE model that isn’t part of the other two models. TheFCFE model requires knowing both interest expense and debt level since changesin debt level affect FCFE.

A “traditional” implementation of the FCFE model would involve the follow-ing steps:

1. Project financial statements for a period of time, and calculate FCFE.2. Project the long-term growth rate and the long-term capital structure.3. Estimate the required rate of return to equity at the horizon using the long-

term expected capital structure.4. Calculate the horizon value using a constant growth model with the return to

equity calculated above, the growth rate assumed, and the last projectedFCFE.

5. Discount the horizon value and the intermediate FCFEs back to the presentusing the equity return estimated above.

As with the traditional implementation of the corporate valuation model, the traditional implementation of the FCFE model involves unlevering a market-determined beta and then relevering it using Hamada. This levered beta is thenused to calculate a levered return to equity, which is used as the discount rate.

These steps are subject to the same criticisms as the traditional implementa-tion of the corporate valuation model: (1) Using a single discount rate when the capital structure is changing results in incorrectly valuing the cash flows, and(2) using Hamada for unlevering and relevering assumes risk-free debt and usesthe risk-free rate to discount the tax shield, which we argue in Chapter 16 is incor-rect. However, this technique suffers from an even more serious problem.

The joint assumption at the horizon that the firm is at its target capital struc-ture and that FCFE is growing at a constant rate can only be true if the firm isalready at its target capital structure in the next-to-last projected year. This can beseen in the following example. Suppose a firm has the following financial projec-tions and is expected to grow at a constant steady-state rate of 6 percent after2008. Its tax rate is 40 percent, its cost of debt is 8 percent, its horizon target per-cent of debt is 20 percent, and its cost of equity at that debt ratio is 13 percent.This steady-state growth rate means free cash flow and debt will both grow at 6percent.

2006 2007 2008 2009 2010

FCF $ 3.00 $ 2.00 $ 2.12 $ 2.25Debt $15.00 20.00 15.00 15.90 16.85Interest 1.20 1.60 1.20 1.27Interest tax shield 0.48 0.64 0.48 0.51Change in debt 5.00 �5.00 0.90 0.95FCFE 7.28 �3.96 2.30 2.44Growth in FCF �33% 6% 6%Growth in debt �25% 6% 6%Growth in FCFE �154% nmf 6%

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As assumed, the growth rate of FCF and debt stabilize at 6 percent in 2009.This means, among other things, that a constant growth horizon value can be cal-culated using FCF2008. However, notice that growth in FCFE does not stabilizeuntil the following year, 2010. A constant growth horizon value cannot be calcu-lated using FCFE2008, and, in this case, since FCFE2008 is negative, the calculationwould be nonsensical. Instead, a horizon value must be calculated the year afterfree cash flows stabilize. The horizon value in 2009 is

A second, less tractable, problem with the horizon value is that we don’t actu-ally know what the level of debt should be in 2009. Based on our assumed debtratio of 20 percent, a horizon value of $34.83 gives a debt level of $8.71 in 2009,not the $15.90 we assumed. But if we, instead, assume a debt level of $8.71 in2009, FCFE2009 will be different (because the change in debt over 2009 will nolonger be 0.90) and so HVEquity2009 will no longer be $34.83. Thus, the jointassumptions that debt, and hence FCFE, will grow at a constant rate of 6 percentafter 2008, that the 2008 debt level is $15.00, and that the horizon debt ratio willbe 20 percent are inconsistent with each other.

This particular problem is avoided in both the corporate valuation model andthe APV model because the firm can recapitalize in 2009 to reach its target debtratio without affecting its free cash flows and, hence, horizon value.

Because of these difficulties, the FCFE model is actually quite difficult toimplement correctly into a spreadsheet when the capital structure is changing. Todo so you must take the following steps:

1. Project financial statements for a period of time, and calculate free cash flowto equity.

2. Project a long-term growth rate and a long-term target capital structure.3. Estimate the levered cost of equity at the horizon using the long-term expected

capital structure, using Equation 16-15 to relever the equity return.4. Calculate the horizon value using a constant growth model in the year after

steady state is reached with the cost of equity calculated above, the growthrate assumed, and the last projected FCFE.

5. Calculate a new value of equity and cost of equity each year before the hori-zon using the actual debt ratio for that year in Equation 16-15 for the cost ofequity calculation.

6. Pick a debt level in the horizon year so that the actual debt ratio is equal tothe projected debt ratio.

7. Discount the horizon value and the intermediate free cash flows back to thepresent using the individual costs of equity estimated above.

Again, although these steps are, in principle, implementable, the extra complexityis unnecessary. The APV is much easier and should be used when the capital struc-ture is changing.

Projecting Consistent Debt and Interest Expenses if Capital Structure Is ConstantRecall that the APV model and the FCFE model both require a projection of inter-est expense. If the projected interest expense is not consistent with the assumed

HVEquity2009 �2.30 (1.06)

0.13 � 0.06� $34.83

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constant capital structure, then the APV and FCFE models will produce incorrectanswers. This section will show how the debt levels and interest expenses in Table26-3 in the text were constructed in a manner consistent with the assumed con-stant capital structure. Keep in mind, though, that if the capital structure isassumed to be constant, then it is always easier to use the corporate valuationmodel rather than either the APV model or the FCFE model.

Line 9 in Table 26-3 in the text shows Tutwiler’s projected free cash flows,and Lines 13 and 19 show the projected interest expense and debt. Here are thesteps required to project the debt levels in Line 19:

1. Calculate the WACC that corresponds to the constant capital structure.2. Calculate the horizon value of operations using the corporate valuation model

horizon value formula.3. Calculate the value of operations in each year of the projections as the present

value of the next year’s value of operations and the next year’s free cash flows.4. Calculate the projected debt level by multiplying the value of operations by

the percent of debt in the assumed constant capital structure.

The projected interest expense in any year is the projected interest rate multipliedby the projected amount of debt at the beginning of the year, as calculated abovein Step 4. The results of Steps 1 through 4 are shown in Table 26E-1.

Step 1. WACC Calculation This is the same calculation we performed inChapter 26. Tutwiler will maintain its current capital structure consisting of 30.17percent debt and 69.83 percent equity. Tutwiler’s cost of equity was calculated tobe 13 percent, and its cost of debt is 9 percent. Tutwiler’s tax rate is 40 percent soits WACC is

WACC � wd(1 � T)rd � wSrS

� 0.3017(1 � 0.40)(9%) � 0.6983(13%) � 10.707%

Value of Operations, Debt, and Interest Calculations (Millions of Dollars)

Table 26E-1

1/1/07 12/31/07 12/31/08 12/31/09 12/31/10 12/31/11

FCF $ 3.2 $ 3.2 $ 5.6 $ 6.4 $ 6.8Horizon value 153.1Value of operations $110.1 118.7 128.2 136.3 144.5 153.1Value of debta 33.2 35.8 38.7 41.1 43.6 46.2Interest expenseb 3.0 3.2 3.5 3.7 3.9

aTutwiler has $27 million in debt before the acquisition. Once the acquisition is consummated, we are assuming Caldwellwill increase the debt to $33.2 million to maintain the 30.17 percent debt level. This additional debt is needed becausewe are assuming the capital structure will remain constant after the acquisition. The additional debt will be on the booksby the first day of 2007.bThe interest expense in 2007 will be based on the debt level at the start of 2007, which is $33.2 million.

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Step 2. Horizon Value of Operations Tutwiler’s free cash flow in 2011,FCF2011, was projected to be $6.8 million with an expected growth rate of 6 per-cent. In the text, we calculated the horizon value, HV2011, to be

Step 3. Calculate the Value of Operations Each Year The value of opera-tions at the end of 2011 is simply the horizon value of operations, $153.1 million.The value of operations at the end of 2010 is the present value of all of the cashflows to be received after 2010, discounted back to 2010. This is equal to thepresent value of the value of operations in 2011 plus the 2011 free cash flow, dis-counted back one year:

Similarly,

Step 4. Calculate the Amount of Debt Each Year We assumed that thecapital structure will remain constant each year, with debt set at 30.17 percent ofthe value of operations. Thus in 2011 debt will be $153.1(0.3017) � $46.2 mil-lion, and in 2010 debt will be $144.5(0.3017) � $43.6 million. Interest expense isequal to the debt level at the start of the year, which is the debt level at the end ofthe previous year, multiplied by the interest rate on debt. The interest rate on debtis 9 percent, so in 2011 interest expense is $43.6(0.09) � $3.9 million. The inter-est expenses for 2007 through 2010 are calculated similarly and are shown inTable 26E-1.

The debt level in 2006 and the interest expense in 2007 deserve comment. In2006, prior to the merger, Tutwiler has $27 million in debt, and this comprises30.17 percent of its capital structure based on its premerger value. However, if themerger goes through, then Tutwiler’s value will increase because of synergies withCaldwell, and, to maintain the assumed 30.17 percent of debt, Tutwiler willimmediately issue an additional $6.2 million in debt, for a total of $27.0 � $6.2 �$33.2 million in debt outstanding. This additional $6.2 million in debt will be inTutwiler’s capital structure by the start of 2007 and will therefore contribute to itsinterest expense in 2007. Thus, Tutwiler’s projected 2007 interest expense is$33.2(0.09) � $3.0 million. Debt levels and their corresponding interest expenseappear in Table 26E-1.

Projecting the Interest Expense at the Horizon When Using the APV ApproachIn some situations, the capital structure is assumed to change during the forecastperiod prior to becoming constant at the horizon. Neither the corporate valuationmodel nor the FCFE model is appropriate because the discount rates vary duringthe forecast period. The APV is the appropriate approach, but it is necessary to

VOps2009 �VOps2010 � FCF2010

1 � WACC�

$6.4 � $144.51 � 0.10707

� $136.3 million

VOps2010 �VOps2011 � FCF2011

1 � WACC�

$6.8 � $153.11 � 0.10707

� $144.5 million

HV2011 �FCF2011 (1 � g)

WACC � g�

$6.8 (1.06)

0.10707 � 0.06� $153.1 million

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project the interest expense at the horizon in a manner that is consistent with theassumed post-horizon constant capital structure.

In this section we show how the interest expense at the horizon is calculatedfor the case in which Tutwiler’s capital structure changes during the forecast period.To ensure correct calculations of the horizon value of the unlevered firm (usingEquation 26-2) and the horizon value of the tax shield (using Equation 26-3), thecompany must be at its long-term constant capital structure in the last year of pro-jections, in this case 2011. This means the debt level at the end of 2010 must beconsistent with the assumed long-term capital structure so that the interestexpense in 2011 is also consistent with the long-term capital structure. The stepsto project a consistent debt level for 2010 are the same as in the previous section:

1. Calculate the WACC that corresponds to the projected long-term capitalstructure.

2. Calculate the horizon value of operations using the corporate valuation modelhorizon value formula.

3. Calculate the value of operations in the last two years.4. Calculate the projected debt level by multiplying the value of operations by

the percent of debt in the assumed constant capital structure.

In this example, Tutwiler will have a varying amount of debt until the end of2010, at which point its debt level will be consistent with a long-term capitalstructure consisting of 50 percent debt. The results of these calculations appear in Table 26E-2.

Step 1. Calculate the WACC at the New Target Capital Structure InChapter 26 we calculated the unlevered cost of equity based on the premergercapital structure and premerger costs of debt and equity:

rsU � wsrsL � wdrd

� 0.6983(13%) � 0.3017(9%) � 11.793%

Under the proposed 50 percent debt capital structure, the interest rate on the debtwill increase to 9.5 percent. The cost of equity, rsL, will also increase due to theincreased leverage. This new cost of equity can be calculated using Equation 16-15and the new debt and equity levels and the new cost of debt:

rsL � rsU � (rsU � rd)(D/S)� 11.793% � (11.793% – 9.5%)(0.50/0.50) � 14.086%

Value of Operations, Debt, and Interest Calculations (Millions of Dollars)

Table 26E-2

2007 2008 2009 2010 2011

FCF $3.2 $3.2 $5.6 $ 6.4 $ 6.8Horizon value 185.1Value of operations 174.6 185.1Value of debt 87.3Interest expense 8.3

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The new WACC can then be calculated from this new rsL and rd:

WACC � wd(1 � T)rd � wsrsL

� 0.50(1 � 0.40)(9.5%) � 0.50(14.086%) � 9.893%

This is the WACC that should persist at the horizon and thereafter.

Step 2. Calculate the Horizon Value of Operations The horizon value ofoperations at the new WACC is

Step 3. Calculate the Value of Operations in the Next to Last Year Thevalue of operations at the end of 2011 is simply the horizon value, $185.1 million.The value of operations at the end of 2010 is the present value of the value ofoperations in 2011 and the free cash flow in 2011:

Step 4. Calculate the Debt Level in the Next to the Last Year The debtlevel in 2010 is now easy to calculate. It is the target percent of debt multiplied bythe value of operations in 2010:

Debt2010 � 0.50($174.6) � $87.3 million

and the interest in 2011 is simply the debt at the end of 2010 multiplied by theinterest rate:

Interest2011 � $87.3(9.5%) � $8.3 million

This is the interest used to calculate the horizon value of the interest tax shield inthe text.

Note that this procedure suggests a shortcut when calculating the APV valueif you don’t happen to need to know separately the value of the unlevered firmand the value of its tax shields. Table 26E-3 shows the shortcut calculations.

In the shortcut, first use the corporate valuation model’s horizon value calcu-lation to calculate the horizon value based on the WACC that will persist in thelong term and the last year’s projected free cash flows. Second, calculate the inter-est tax shields that will result from the assumed debt levels prior to the horizon.These assumed debt levels prior to the horizon need not be consistent with anyparticular long-term debt policy; however, in Table 26E-3 we have used the taxshields that we projected earlier. Third, add the interest tax shields, the horizon

�$6.8 � $185.1

1 � 0.09893� $174.6 million

VOps2010 �VOps2011 � FCF2011

1 � WACC

�$6.8 (1.06)

0.09893 � 0.06� $185.1

HV2011 �FCF2011 (1 � g)

WACC � g

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value, and the free cash flows together for each year. Fourth, discount these cashflows at the unlevered cost of equity.

This gives the value of the firm’s operations, without separating out the unleveredvalue and the value of the tax shield. Notice that this is the same value of opera-tions we calculated in Chapter 26; however, the calculations are simpler because afinal interest expense consistent with the long-term capital structure need not becalculated, nor must separate unlevered values and tax shield values be calculated.This simplified calculation is also called the compressed adjusted present valuemodel.1

VOps �$5.2

1.11793�

$5.6

1.117932�

$8.4

1.117933�

$9.4

1.117934�

$195.2

1.117935� $133.0

Shortcut APV CalculationTable 26E-3

2007 2008 2009 2010 2011

FCF $3.2 $3.2 $5.6 $6.4 $ 6.8Horizon value 185.1Interest tax shield 2.0 2.4 2.8 3.0 3.3FCF, tax shield, and HV $5.2 $5.6 $8.4 $9.4 $195.2

1See S. N. Kaplan and R. S. Rubak, “The Valuation of Cash Flow Forecasts: An Empirical Analysis,” Journal of Finance, Sep-tember 1995, pp. 1059–1093, for a discussion of the compressed adjusted present value model.

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19878 26w p001-009lated using this hamada-determined cost of equity and the new capital structure....

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