web extension 24A FCF t t 1 TS t (1 r su ) t t 1

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1 The Adjusted Present Value (APV) Approachl 24A-1 web extension 24A The Adjusted Present Value (APV) Approach The corporate valuation or residual equity methods described in the textbook chapter work well when a company keeps a constant capital structure. However, in many situations, there will be a nonconstant capital structure in years immediately following the merger. For example, this often occurs if an acquisition is financed with a temporarily high level of debt that will be reduced to a sustainable level as the merger is digested. In such situations it is extremely difficult to correctly apply the corporate valuation model or the equity residual model because the cost of equity and the cost of capital are changing as the capital structure changes. Fortunately, the adjusted present value model is ideally suited for such situations, as we show below. Recall from Chapter 12 that interest payments are tax deductible. This means that the government receives less tax revenue from a levered firm than from an otherwise identical but unlevered firm, which leaves more money available for the levered firm s investors. More money for investors increases a firm s value, all else equal. In other words, the value of a levered firm is equal to the value of an unlevered firm plus an adjustment for tax savings. The adjusted present value (APV) approach explicitly employs this concept by expressing the value of operations as the sum of two components: (1) the unlevered value of the firm s operations (i.e., as though the firm had no debt), plus (2) the present value of the interest tax savings, also known as the interest tax shield: V Operations V Unlevered V Tax shield (24A-1) The value of an unlevered firm s operations is the present value of the firm s free cash flows discounted at the unlevered cost of equity, and the value of the tax shield is the present value of all of the interest tax savings (TS), discounted at the unlevered cost of equity r su : 1 V Unlevered a q t1 FCF t (1 r su ) t (24A-2) and V Tax shield a q t1 TS t (1 r su ) t (24A-3) To apply Equations 24A-2 and 24A-3, the FCF and TS must eventually stabilize at a constant growth rate. When they do so, we can use an approach similar to the ones we used for the nonconstant dividend model in Chapter 8 and the corporate valuation model in Chapter 23. In those approaches, we explicitly projected the years with nonconstant growth rates, found the horizon value at the end of the nonconstant growth period, and then calculated the present value of the horizon value and the cash flows during the forecast period. 1 Although some analysts discount the tax shield at the cost of debt or some other rate, we believe that the unlevered cost of equity is the appropriate discount rate for the interest tax savings. See Chapter 12 for a detailed explanation.

2 24A-2 Web Extension 24A The Adjusted Present Value (APV) Approach Here is a description of how to apply that approach in the APV model. 1. Calculate the target s unlevered cost of equity, r su, based upon its current capital structure at the time of the acquisition. In other words, you unlever the target s cost of equity. From Chapter 12, Equation 12-6 expresses a firm s levered cost of equity, r sl, as a function of its unlevered cost of equity, its cost of debt (r d ), and the amount of debt (D) and equity (S) in its capital structure: r sl r su (r su r d )(D>S) (24A-4) Because the weights of debt and equity in a capital structure, w d and w s, are defined as D/(D S) and S/(D S), the ratio of D/S can be expressed as w d /w s. We make this substitution in Equation 24A-4 and solve for the unlevered cost of equity: r su w s r sl w d r d (24A-5) Keep in mind that r sl, r d, w d, and w s are based upon the target s capital structure immediately before the acquisition. 2. Project the free cash flows, FCF t, and the annual interest tax savings, TS t. The tax savings are equal to the projected interest payments multiplied by the tax rate: 2 Tax savings (Interest expense)(tax rate) (24A-6) You must project enough years so that the FCF and the tax savings are expected to grow at a constant rate (g) after the horizon, which is at Year N. This means that the capital structure must become constant at Year N 1 to ensure that the projected interest payment at year N will grow at a constant rate after year N. Later in this Web Extension we provide a detailed explanation of how to project financial statements that reflect a constant capital structure. For the remainder of this, we will assume that your trusty assistant has made such projections. Notice that the APV approach does not require a constant capital structure in each and every year of the analysis, only that the capital structure must eventually become stable in the post-horizon period. 3. Calculate the horizon value of an unlevered firm at Year N (HV U,N ), which is the value of all free cash flows beyond the horizon discounted back to the horizon at the unlevered cost of equity. Also calculate the horizon value of the tax shield at Year N (HV TS,N ), which is the value of all tax shields beyond the horizon discounted back to the horizon at the unlevered cost of equity. Because FCF and TS are growing at a constant rate of g in the post-horizon period, we can use the constant growth formula: Horizon value of unlevered firm 1HV U,N 2 FCF N1 r su g FCF N11 g2 r su g (24A-7) and Horizon value of tax shield 1HV TS,N 2 TS N1 r su g TS N11 g2 r su g (24A-8) The unlevered horizon value is the horizon value of the company if it had no debt. The tax shield horizon value is the contribution the tax savings after year N make to the 2 The tax shield is based only on interest expense, not the net value of interest expense and interest income. This is because the impact of interest income is taken into account when the value of short-term investments is added later to the value of operations. Including the impact of interest income in the tax shield would be double counting. In other words, there are no side effects due to owning a short-term investment: The value of the investment to the company is just the reported value. This is in contrast to debt, which does have a side effect in the sense that the cost to the company is less than the reported value due to the tax shield provided by the debt.

3 The Adjusted Present Value (APV) Approach 24A-3 horizon value of the levered firm. Therefore the horizon value of the levered firm is the sum of the unlevered horizon value and the tax shield horizon value. 4. Calculate the present value of the free cash flows and their horizon value. This is the value of operations for the unlevered firm, that is, the value it would have if it had no debt. Also calculate the present value of the yearly tax savings during the forecast period and the horizon value of tax savings. This is the value that the interest tax shield contributes to the firm. The sum of the value of unlevered operation and the value of the tax shield is equal to the value of operations for the levered firm. V Unlevered a N t1 FCF t 11 r su 2 t HV U,N 11 r su 2 N (24A-9) V Tax shield a N t1 TS t 11 r su 2 t HV TS,N 11 r su 2 N V Operations V Unlevered V Tax shield (24A-10) (24A-11) 5. To find the total value of the firm, add the value of operations to the value of any nonoperating assets, such as marketable securities. To find the value of equity, subtract the value of the debt before the merger from the total value of the firm. Unlevered value of operations Value of tax shield Value of operations Value of nonoperating assets Total value of firm Value of debt Value of equity To find the stock price per share, divide the value of equity by the number of shares. The APV technique is especially useful in valuing acquisition targets. Many acquisitions are difficult to value using the corporate valuation model because (1) acquiring firms frequently assume the debt of the target firm, so old debt at different coupon rates is often part of the deal, and (2) the acquisition is usually financed partially by new debt that will be paid down rapidly, so the proportion of debt in the capital structure changes during the years immediately following the acquisition. Thus, the debt cost and capital structure associated with a merger are generally more complex than for a typical firm. The easiest way to handle these complexities is to specify each year s expected interest expense and use the APV method to find the value of the unlevered firm and the interest tax shields separately, and then sum those values. Illustration of Valuation Using the APV Approach Using the Tutwiler illustration in Chapter 24, the APV approach requires an estimate of Tutwiler s unlevered cost of equity. Inputting Tutwiler s capital structure, cost of equity, and cost of debt, Equation 24A-5 can be used to estimate the unlevered cost of equity: r su w s r sl w d r d % % % (24A-5)

4 24A-4 Web Extension 24A The Adjusted Present Value (APV) Approach In other words, if Tutwiler had no debt, its cost of equity would be %. The horizon value of Tutwiler s unlevered cash flows (HV UL,2016 ) and tax shield (HV TS,2016 ) can be calculated using the constant growth formula with the unlevered cost of equity as the discount rate: 3 HV U,2016 FCF 2017 (r su g) FCF 2016(1 g) (r su g) HV TS,2016 TS 2017 (r su g) TS 2016(1 g) (r su g) $1.57(1.06) $28.7 million The sum of the two horizon values is the horizon value of operations, $153.1 million, which is the same as the horizon value calculation we reached with the corporate valuation model. Row 11 in Table 24-3 shows the projected free cash flows. The unlevered value of operations is calculated as the present value of the free cash flows during the forecast period and the horizon value of the free cash flows: $3.2 V Unlevered ( ) $3.2 ( ) $5.6 2 ( ) 3 $6.4 $6.8 $ ( ) ( ) 5 $88.7 million This shows that Tutwiler s operations would be worth $88.7 million if it had no debt. Next, the yearly interest tax shields are calculated. Row 6 in Table 24-3 shows yearly interest expense. Given the tax rate of 40%, the interest tax shield for 2012 is $ $1.2. Below are all the yearly interest tax shields Interest tax savings Interest (T) $1.2 $1.3 $1.4 $1.5 $1.57 The value of the tax shield is calculated as the present value of the yearly tax savings and the horizon value of the tax shield: $1.2 V Tax shield ( ) $1.3 ( ) $1.4 2 ( ) 3 $1.5 $1.57 $ ( ) ( ) 5 $21.4 million Thus, Tutwiler s operations would be worth only $88.7 million if it had no debt, but its capital structure contributes $21.4 million in value due to the tax deductibility of its interest payments. Since Tutwiler has no nonoperating assets, the total value of the firm is the sum of the unlevered value of operations, $88.7 million, and the value of the tax shield, $21.4 million, for a total of $110.1 million. The value of the equity is this total value less Tutwiler s outstanding debt of $27 million: $110.1 $27 $83.1 million. This is also the value we obtained using the corporate valuation model. Table 24A-1 summarizes all three cash flow valuation methods and their assumptions. $6,800(1.06) $124.4 million 3 Note that we report two decimal places for the 2016 tax shield even though Table 24-3 reports only one decimal place. All calculations are performed in Excel, which uses the full nonrounded values.

5 Projecting Consistent Debt and Interest Expenses 24A-5 TABLE 24A-1 Summary of Cash Flow Approaches Approach Corporate Valuation Free Cash Flow to Equity Model Model APV Model Cash flow FCF NOPAT FCFE FCF Interest (1) FCF definition: Net investment in expense Interest tax shield (2) Interest tax savings operating capital Net change in debt Discount rate: WACC r sl Cost of equity r su Unlevered cost of equity Result of present Value of operations Value of equity due to (1) Value of unlevered value operations operations calculation: (2) Value of the tax shield. Together, these are the value of operations. How to get Value of operations Value of equity due to Value of operations equity value: Value of nonoperating operations Value of Value of nonoperating assets Value of debt nonoperating assets assets Value of debt Assumption Capital structure is Capital structure is None about capital constant. constant. structure during forecast period: Requirement for No interest expense Projected interest expense Interest expense analyst to projections needed must be based on the projections are project interest assumed capital structure. unconstrained. expense: Assumption at FCF grows at constant FCFE grows at constant FCF and interest tax savings horizon: rate g. rate g. grow at constant rate g. Projecting Consistent Debt and Interest Expenses Projecting financial statements for a merger analysis requires explicit assumptions regarding the capital structure in the postmerger years. This section shows how to project debt and interest expenses that are consistent with the capital structure assumptions. Refer to the worksheet Web 24A in the file Ch 24 Tool Kit.xlsx for all calculations. See Ch 24 Tool Kit.xlsx at the textbook s website for all calculations. Projecting Consistent Debt and Interest Expenses When Capital Structure Is Constant Recall that the FCFE model and the APV model (explained earlier in this Web Extension) both require a projection of interest expense. If the projected interest expense is not consistent with the assumed constant capital structure, then the APV and FCFE models will produce incorrect answers. This section will show how the debt levels and interest expenses in Table 24-3 in the text were constructed in a manner consistent with the assumed constant capital structure. Keep in mind, though, that if the capital structure is assumed to be constant, then it is always easier to use the corporate valuation model rather than either the APV model or the FCFE model.

6 24A-6 Web Extension 24A The Adjusted Present Value (APV) Approach TABLE 24A-2 Constant Capital Structure: The Value of Operations, Debt, and Interest Expense (Millions of Dollars) 1/1/12 12/31/12 12/31/13 12/31/14 12/31/15 12/31/16 FCF $3.2 $3.2 $5.6 $6.4 $6.8 Horizon value Value of operations $ Value of debt a Interest expense b a Debt w d (V op ). b The interest expense is based on the amount of debt at the beginning of the year: Interest expense in Year t r d (Debt t-1 ). Here are the steps required to project debt levels that are consistent with the assumed constant capital structure: 1. Use the techniques of Chapter 5 to project the operating items on the financial statements needed to calculate free cash flows. Notice that these projections don t depend on the capital structure because they are for operating items and not financial items. 2. Calculate the WACC that corresponds to the constant capital structure. 3. Calculate the horizon value of operations using the corporate valuation model horizon value formula. 4. 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. 5. Calculate the projected debt level by multiplying the value of operations by the percentage of debt in the assumed constant capital structure. The projected interest expense in any year is the projected interest rate multiplied by the projected amount of debt at the beginning of the year. Step 1. Project Operating Items The worksheet Web 24A in the file Ch 24 Tool Kit.xlsx shows the projected financial statement items related to Tutwiler Controls operations and its projected free cash flows. The free cash flows are shown here in the first row of Table 24A-2. The following sections explain the other rows of Table 24A-2. Step 2. WACC Calculation This is the same calculation we performed in Chapter 24. Tutwiler will maintain its current capital structure consisting of 30.17% debt and 69.83% equity. Tutwiler s cost of equity was calculated to be 13%, and its cost of debt is 9%. Tutwiler s tax rate is 40%, so its WACC is WACC w d 11 T2r d w s r s % % % Step 3. Horizon Value of Operations Tutwiler s free cash flow in 2016, FCF 2016, was projected to be $6.8 million with an expected growth rate of 6%. In Chapter 24, we calculated the horizon value, HV 2016, to be HV 2016 FCF 2016(1 g) WACC g $6.8(1.06) $153.1 million This horizon value is shown in the second row of Table 24A-2. Step 4. Calculate the Value of Operations Each Year Tutwiler s value of operations at the end of 2016 is simply the horizon value of operations, $153.1 million. The value of operations at the end of 2015 is the present value of all of the cash flows to be received after 2015, discounted back to This is equal to the present value of the value of operations in 2016 plus the 2016 free cash flow, discounted back 1 year:

7 Projecting Consistent Debt and Interest Expenses 24A-7 V op 2015 V op 2016 FCF WACC $153.1 $ $144.5 million Similarly, V op 2014 V op 2015 FCF WACC The value of operations for each year is shown in the third row of Table 24A-2. Step 5. Calculate the Amount of Debt Each Year We assumed that Tutwiler s capital structure will remain constant each year, with debt set at 30.17% of the value of operations. Thus in 2016 debt will be $153.1(0.3017) $46.2 million, and in 2015 debt will be $144.5(0.3017) $43.6 million. Interest expense is equal to the debt level at the start of the year, which is the debt level at the end of the previous year, multiplied by the interest rate on debt. The interest rate on debt is 9%, so in 2016 interest expense is $43.6(0.09) $3.9 million. The interest expenses for 2012 through 2015 are calculated similarly and are shown in Table 24A-2. The debt level in 2011 and the interest expense in 2012 deserve comment. In 2011, prior to the merger, Tutwiler has $27 million in debt, and this comprises 30.17% of its capital structure based on its premerger value. However, if the merger goes through, then Tutwiler s value will increase because of synergies with Caldwell, and, to maintain the assumed 30.17% of debt, Tutwiler will immediately 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 in Tutwiler s capital structure by the start of 2012 and will therefore contribute to its interest expense in Thus, Tutwiler s projected 2012 interest expense is $33.2(0.09) $3.0 million. Debt levels are shown in the fourth row of Table 24A-2. Projecting Consistent Debt and Interest Expenses When Capital Structure Is Nonconstant $144.5 $ $136.3 million In some situations, the capital structure is assumed to change during the forecast period prior to becoming constant at the horizon. Neither the corporate valuation model nor the FCFE model is appropriate because the discount rates vary during the forecast period. The APV is the appropriate approach, but it is necessary to project the interest expense at the horizon in a manner that is consistent with the assumed post-horizon constant capital structure. In this section we show how the interest expense at the horizon is calculated for the case in which Tutwiler s capital structure changes during the forecast period before becoming constant at the end of the horizon. To ensure correct calculations of the horizon value of the unlevered firm and the horizon value of the tax shield, the company must be at its long-term constant capital structure in the last year of projections, in this case This means the debt level at the end of 2015 must be consistent with the assumed long-term capital structure so that the interest expense in 2016 is also consistent with the long-term capital structure. The steps to project a consistent debt level for 2015 are similar to those described in the previous section: 1. Use the techniques of Chapter 5 to project the operating items on the financial statements needed to calculate free cash flows. Notice that these projections don t depend on the capital structure because they are for operating items and not financial items. 2. Calculate the levered cost of equity and WACC that will prevail in the post-horizon period when the capital structure has become constant. 3. Calculate the horizon value of operations using the corporate valuation model horizon value formula. 4. Calculate the value of operations in the last 2 years of the forecast period. 5. 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 of 2015, at which point its debt level must be consistent with a long-term capital structure consisting of 50% debt. The results of these calculations appear in Table 24A-3.

8 24A-8 Web Extension 24A The Adjusted Present Value (APV) Approach TABLE 24A-3 Nonconstant Capital Structure during Forecast Period: The Value of Operations, Debt, and Interest Expense at the End of the Forecast Period (Millions of Dollars) FCF $3.2 $3.2 $5.6 $ 6.4 $ 6.8 Horizon value Value of operations Value of debt 87.3 Interest expense 8.3 Step 1. Project Operating Items The worksheet Web 24A in the file Ch 24 Tool Kit.xlsx shows the projected financial statement items related to Tutwiler s operations and its projected free cash flows. The free cash flows are shown here in the first row of Table 24A-3. The following sections explain the other rows of Table 24A-3. Step 2. Calculate the Unlevered Cost of Equity and WACC at Post-Horizon Target Capital Structure Earlier in this Web Extension we calculated Tutwiler s unlevered cost of equity based on the pre-merger capital structure and pre-merger costs of debt and equity: r su w s r sl w d r d (13%) (9%) % Under the proposed 50% debt capital structure for the post-horizon period, the interest rate on the debt will increase to 9.5%. The cost of equity, r sl, will also increase due to the increased leverage. This post-horizon cost of equity can be calculated with Equation 24A-4, using the post-horizon capital structure cost of debt: r sl r su 1r su r d 21D>S % % 9.5%210.50> % The new WACC can then be calculated from this new r sl and r d : WACC w d 11 T2r d w s r sl % % % This is the WACC that should persist at the horizon and thereafter. Step 3. Calculate the Horizon Value of Operations The horizon value of operations at the new WACC is HV 2016 FCF 2016(1 g) WACC g $6.8(1.06) $185.1 This value is shown in the second row of Table 24A-3. Step 4. Calculate the Value of Operations in the Last Year and the Prior Year The value of operations at the end of 2016 is simply the horizon value, $185.1 million. The value of operations at the end of 2015 is the present value of the value of operations in 2016 and the free cash flow in 2016:

9 Projecting Consistent Debt and Interest Expenses 24A-9 V op 2015 V op 2016 FCF WACC $185.1 $6.8 $174.6 million The values of operations for 2015 and 2016 are shown in the third row of Table 24A-3. Step 5. Calculate the Debt Level in the Year Prior to the End of the Horizon The debt level in 2015 is now easy to calculate. It is the post-horizon target percentage of debt multiplied by the value of operations in 2015: Debt ($174.6) $87.3 million and the interest in 2016 is simply the debt at the end of 2015 multiplied by the interest rate: Interest 2016 $87.3(9.5%) $8.3 million This is the interest used to calculate the horizon value of the interest tax shield in the text. The debt levels and interest tax shields during the prior years need not conform to a constant capital structure. As long as the interest expense in the last projected year is expected to grow at a constant rate, which our calculations guarantee, the APV approach may be applied. There is a shortcut when calculating the APV if you don t need to know the separate values of the unlevered firm and the value of its tax shields. First, use the corporate valuation model s horizon value calculation to calculate the horizon value based on the WACC that will persist in the long term and the last year s projected free cash flows. Second, calculate the interest 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 any particular long-term debt policy. Third, add the interest tax shields, the horizon value, and the free cash flows together for each year. Fourth, discount these cash flows at the unlevered cost of equity. For example, Table 24A-4 shows the free cash flows from Table 24A-3 and the horizon value that we calculated for the case in which Tutwiler s capital structure was nonconstant during the forecast period but stabilized with 50% debt in the post-horizon period. The interest tax savings in the horizon year are calculated from Table 24A-3, while the other tax savings are provided in the file Ch24 Tool Kit.xlsx. These values are summed on the last row of Table 24A-4. The value of operations is their present value when discounted at the unlevered cost of equity: V op $5.2 $5.6 $8.4 $9.4 $195.2 $ TABLE 24A-4 Shortcut APV Calculation FCF $3.2 $3.2 $5.6 $6.4 $ 6.8 Horizon value Interest tax saving FCF, tax saving, and HV $5.2 $5.6 $8.4 $9.4 $195.2

10 24A-10 Web Extension 24A The Adjusted Present Value (APV) Approach This gives the value of the firm s operations, without separating out the unlevered value and the value of the tax shield. These calculations are simpler because a final interest expense consistent with the long-term capital structure need not be calculated, nor must separate unlevered values and tax shield values be calculated. This simplified calculation is also called the compressed adjusted present value model. 4 4 See S. N. Kaplan and R. S. Rubak, The Valuation of Cash Flow Forecasts: An Empirical Analysis, Journal of Finance, September 1995, pp , for a discussion of the compressed adjusted present value model.