The Assumptions and Math Behind WACC and APV Calculations

The Assumptions and Math Behind WACC and APV Calculations Richard Stanton U.C. Berkeley Mark S. Seasholes U.C. Berkeley This Version October 27, 2005 Abstract We outline the math and assumptions behind weighted average cost of capital WACC and adjusted present value APV calculations. We first derive a general formula for the discount rate of equity and beta of equity under minimal assumptions. We then take into consideration: i The existence or non-existence of taxes; ii Whether a firm has a constant amount of debt in dollar terms; iii Whether a firm targets a constant proportion of debt in its capital structure; and iv The frequency of debt rebalancing. These considerations give rise to well known results such as the Miles and Ezzell 1980 formula for WACC and differnt methods formulae for unlevering and re-levering betas. This document is intended for those who wish to understand the motivation behind valuing a firm with WACC and/or APV. Doctoral students and professors may find the document useful when teaching WACC and APV. In particular, this document helps answer questions like: Why does my firm use this formula to unlever beta, but you have taught us another formula? Understanding the assumptions behind both formulae turns out to be the key to answering such questions. Keywords: WACC, APV, Cost of Capital JEL Classification number: G32, A22, A23 This document is based on work titled WACC/APV Mathematical Details by Richard Stanton. Contact information: Mark S. Seasholes, U.C. Berkeley Haas School of Business, 545 Student Services Bldg., Berkeley CA 94720; Tel: 510-642-3421; Fax: 510-643-1420; email: mss@haas.berkeley.edu. c 2005. 1

1 Introduction This document is intended for students who wish to delve into the math behind WACC and APV calculations. The derivations in this document highlight many of the assumptions behind these valuation methods. This document can be used by students ranging from advanced undergraduates to advanced MBAs. However, only the most curious students will find it interesting and benefit from it. Doctoral students and professors may find the document useful when teaching WACC and APV. In particular, this document helps answer questions like: Why does my firm use this formula to unlever beta, but you have taught us another formula? Understanding the assumptions behind both formulae turns out to be the key to answering such questions. 2 Definitions We lay out the notation used throughout the document. There are a myriad of possible ways to denote the same economic quantity. Many are self explanatory. 2.1 Unlevered Firms We use the following notation to denote the market value of a firm and its equity without leverage in its capital structure: V U MV F,unlev Market enterprise value of an unlevered firm E U MV E,unlev Market value of the equity of an unlevered firm Notice that V U E U and we address this in Section 3. We also use the following definitions: F CF t After-tax cashflow of the unlevered firm at time t β A Beta of the FCFs could be written β U or β EU, but typically isn t r A Expected return of the unlevered firm s equity could be written r U or r EU 2

None of the results in this paper rely on a specific model of asset returns such as the CAPM. Section 3 only relies on the fact that the beta and return of a portfolio is the weighted average of its components. 2.2 Levered Firms For firms with leverage, we use the following notation: MV F,lev Market enterprise value of a levered firm MV E,lev Equity value of a levered firm D L MV D,lev Market value of a levered firm s debt We use β E to denote the beta of the levered firm s equity therefore it could be written β EL. We use r E to denote the expected return on the levered firm s equity could be r EL. For the levered firm s debt, β D is always used to denote the beta could be β, and r D is always used to denote the expected return of the debt could be r. If the levered firm is profitable, the existance of debt creates an interest tax shield. τ is the corporate tax rate; IT S t is the interest tax shield for year t; and S is the present value of all IT S. We discuss the correct discount rate r s,t later. For now, we simply note that it may not be constant over time. IT S t r D D L,t τ S P V IT S IT S t 1 + r s,t t t1 3 General Derivations For an unlevered firm, the value of the firm is the value of the equity: V U E U. For a levered firm, the value of the firm is the value of the equity plus debt: + D L. The 3

value of a levered firm can also be written as the value of an unlevered firm plus the present value of interest tax shields: V U +S. If we equate these two different ways of expressing, substitute E U for V U, and reorganize we get: + D L V U + S V U D L + S E U D L + S 1 Equation 1 tells us that we can interpret as being equivalent to a portfolio of E U, D L, and S. Next, remember the beta of a portfolio P is the weighted average of its component betas. Weights are determined by market values: V P V 1 + V 2 + V 3 V1 β P β 1 + V P V2 V P V3 β 2 + β 3 V P This fact allows us to write the beta of in terms of the betas of E U, a short position in D L, and S. Following convention, we continue to write β A instead of β U or β EU, and obtain: EU S β E β A β D + β S 1 + D L S β A S β D + β S Equation 2 gives us a general relationship between β E, β A, β D and β S for an arbitrary debt policy. Later, we make assumptions about target leverage ratios which allows us to calculate S and β S. Not only is the beta of a portfolio equal to a weighted average of the betas of its components, but so is the expected return of a portfolio equal to a weighted average of the expected returns of its components. Following exactly the same logic as above, we obtain: r E 1 + D L S S r A r D + r S 3 Note that this result holds regardless of whether or not the CAPM is true. 2 3.1 No Taxes In the case of no taxes, the present value of tax shields is zero S 0 and Equation 2 becomes a familiar expression. Thus, we get our first results. In a world with no taxes and 4

arbitrary debt policy: β E β A 1 + D L β A β D or 4 EL β E + β D For the returns, Equation 3 gives us a similar expression remember S 0: r E 1 + D L r A r D r A or 5 EL r E + r D 3.2 Constant Amount of Debt Having a constant dollar amount of debt is the usual assumption in many APV calculations. If there is a constant dollar amount of debt outstanding D L, constant interest rate r D, and constant tax rate τ, the interest tax shield in every year is constant and we have: IT S t IT S r D D L τ The interest tax shields are known in advance in expectation every year, forever. Thus, we can calculate their present value using the perpetuity formula. The chance risk of not getting the interest tax shield in any given year is related to the debt rate. Therefore, we discount future interest tax shields to the present using r D. Discounting the interest tax shields by r D is based on the structure of cash flows when one assumes a firm issues a constant dollar amount of debt. The cash flow structure implies β S β D as the tax shield each period is proportional to the cash flow to bond holders. r S r D β S β D S IT S 1 + r D r DD L τ t r D t1 D L τ 5

Substituting into Equation 2 gives: β E 1 + D L S β A 1 + D L D L τ β A 1 + D ] L 1 τ β A S β D + β D + 1 τ β S τ ] β D β D 6 Substituting into Equation 3 gives: r E 1 + D ] L 1 τ r A 1 τ ] r D 7 Notice, under the following assumptions: i A constant amount of debt; ii A debt beta of zero; iii β S β D 0 and iv r S r D, we get a common formula used to unlever relever betas: β E β A 1 + D ] L 1 τ E L ] VL β A 1 τ 3.3 Constant Proportion of Debt Having a constant proportion of debt constant leverage ratio is the usual assumption behind WACC calculations. Suppose a firm changes the amount of debt outstanding each year to keep the proportion of debt constant. Split S into two pieces. The first piece is the present value P V of the first year s period s tax shield t 1. The second piece is the P V of all remaining years periods tax shields t > 1. The first part of the tax shield is a multiple of the first period s interest payment and has a beta of β D. The second part of the tax shield goes up or down in proportion to the value of the firm, and therefore has a beta of β A. Our assumptions are: β S β D and r S r D for the first period t 1. After the first period 6

t > 1 we assume β S β A. Finally, S PV1 st tax shield + PVRemaining tax shields: PV1 st tax shield r DD L,t1 τ 1 + r D PVRemaining tax shields S r DD L,t1 τ 1 + r D rd D L,t1 τ Sβ S 1 + r D β D + S r DD L,t1 τ β A 1 + r D Substitute the right-hand side of the equation directly above, for the Sβ S in Equation 2. Simplify and do the same with Equation 3 to obtain the following key formulae. Since the firm targets a constant proportion of debt, it is not necessary to carry the t 1 subscripts around. In other words, D L D L,t1,t1. β E r E 1 + D L 1 r ] Dτ β A 1 r ] Dτ β D 1 + r D 1 + r D 1 + D L 1 r ] Dτ r A 1 r ] Dτ r D 1 + r D 1 + r D 8 Note: These formulae do not explicitly depend on S, which cancels out of the calculations. They depend only on the relative values of D L and, so they are valid for any pattern of cash flows from the firm. In addition, the constant proportional debt assumption plays only one role after the first period the tax shields have a beta of β A. The exact rule for how the debt is adjusted each period is not important. The same result will, therefore, hold for any rule that yields this beta. For example, if a firm adjusts its debt each period to be proportional to that period s all-equity cash flow constant debt-coverage the same result holds. 3.4 Rebalancing Frequency Suppose the firm makes interest payments and adjusts its debt level more than once a year. As the adjustment frequency increases, the relationship between equity and asset betas eventually converges to Equation 4. To see this, consider a rebalancing frequency of k periods per year. If the debt rate r D is given as an effective annual rate, then we discount 7

the first term by 1+r D 1/k. More importantly, the tax shield over the first fractional period of the year depends on 1 + r D 1/k 1 ] τ. As the rebalancing frequency becomes higher and higher, the value of β E becomes the closer and closer to value we obtain in the absence of taxes! lim β E k β E 1 + D L 1 1 + D L β A 1 + rd 1/k 1 ] ] τ β 1 + r D 1/k A D L 1 1 + rd 1/k 1 ] ] τ β 1 + r D 1/k D 9 β D 10 Equation 10 makes it clear that continuous rebalancing in a world with taxes gives the same result as Equation 4 which comes from a world without taxes! Thus, students who believe that Equation 10 does not consider taxes because there is no τ term are not correct. We show the effect of different rebalancing periodicity in Table 1. In practice, using Equation 4 rather than Equation 9 usually gives economically similar results. In the example in Table 1, the tax rate is 35% per year, the asset beta is 0.8571, the company has $2,000 of debt outstanding, the costs of debt is 7.00%, the beta of debt is 0.2857, and the market value of equity is $6,760.40. Equation 9 gives a equity beta of 1.0223 with annual rebalancing. The equity beta increases to 1.0262 if the firm rebalances daily. The equity beta as calculated from Equation 4 is 1.0262 which is the same to four decimals as daily rebalancing. The economic difference between using 1.0223 and 1.0262 is small. Any estimation error of equity betas is typically orders of magnitude larger than this difference. 4 Weighted Average Cost of Capital WACC Assume that a firm maintains a constant leverage ratio: D L D L D L + Constant Remember F CF t refers to the all-equity firm s expected cash flow at time t i.e. the cash flow that would have occurred had the firm been all-equity financed. From earlier, we can write: + D L. The levered firm is a portfolio with two components and D L. 8

The expected return on is the weighted average return of its components: EL r V r E + r D + D L + D L or 11 r V EL r E + r D Consider buying both the equity and debt of a levered firm and D L today, holding them for one period, then selling both positions. In exchange for the initial cost of this investment,, we receive next period s free cash flow F CF t+1, next period s interest tax shield r D D L τ, and next period s value of the equity plus debt,t+1. If F CF t+1 is the cash flow of an all-equity firm at time t+1, then we can define the cash flow of the levered firm at t+1 C L,t+1 as the all equity cash flow plus the interest tax shield r D D L τ: C L,t+1 F CF t+1 + r D D L τ Here C L,t+1 is the total cash flow paid to debt and equity holders combined. The price we pay today must equal the present value of the future cash flows, so: F CF t+1 + r D D L τ +,t+1 1 + r V 12 C L,t+1 +,t+1 1 + r V Substituting repeatedly for,t+1,,t+2, etc., we obtain: C L,t+1 1 + r V + C L,t+2 1 + r V 2 + C L,t+3 1 + r V 3 +... 13 i.e. equals the present value of all future cash flows including tax shields, discounted at r V. This is not very surprising. However, there is a problem using Equation 13 in practice. We need to know the size of the tax shields each period. These depend on the amount of debt, which in turn and by assumption is a constant multiple of. However, is what we are trying to calculate. This is a way around this problem. Instead of adding the tax shields to the all-equity cash flows, we can value the firm instead by using only the all-equity cash flows whose value we do know in expectation, and discounting by an adjusted discount rate called the WACC or Weighted Average Cost of Capital. To see this, 9

rewrite Equation 12 using the fact that D L D L Collecting terms in, this becomes: to obtain: F CF t+1 + r D D L τ +,t+1 1 + r V F CF t+1 +,t+1 1 + r V D L r D τ 14 From Equation 11 we can substitute for r V to get: F CF t+1 +,t+1 1 + r E + D L r D D L r D τ F CF t+1 +,t+1 1 + r W ACC 15 where: r W ACC EL + D L EL r E + r E + r D 1 τ + D L r D 1 τ 16 Substituting repeatedly for,t+1,,t+2,... etc. gives: F CF t+1 1 + r W ACC + F CF t+2 1 + r W ACC 2 + F CF t+3 1 + r W ACC 3 +... Note: Using WACC works for any pattern of cash flows, as long as the firm maintains a constant leverage ratio. Equation 16 is the usual way WACC is defined, and is the most convenient way to calculate it given r E and r D. We can also and equivalently express WACC in terms of r A instead. To do so, start the same way as above, writing as the present value of the next period s payoff, but now write the present value of each of the pieces separately, as 1 F CF t+1 + r D D L τ +,t+1 17 1 + r A 1 + r D 1 + r A 1 r A is the correct discount rate for,t+1, since is always a constant multiple of V U. 10

This can be rewritten as: F CF t+1 + V L,t+1 1 1 + r A D L ] r V D τ L 1+r D Comparing this equation with Equation 15, we see that: ] D L V 1 + r W ACC 1 + r A 1 L r D τ 1 + r D r W ACC r A D Lr D τ 1 + ra 1 + r D 18 Equation 18 is the Miles and Ezzell 1980 formula for WACC. 5 Example: Valuing a Firm with Taxes In this section, we shall value one firm in a number of different ways. To highlight the robustness of our derivations, growth has been included. All results from this section are shown in Table 2. The firm has the following details: EBIT is $1,000 next year; i The expected operating income ii The growth rate of EBIT is constant g 3.00%; iii The tax rate is τ 35%; iv r A 11.00%; and v The firm rebalances its capital structure, if necessary, once a year. The firm s all-equity cash flow next year is: $1, 000 1 0.35 $650.00 and using the formula for a growing perpetuity, we get: V U 650 0.1100 0.0300 $8, 125.00 Now assume the firm currently has $2,000 of riskless debt r D 5.00%, and will keep a constant proportion of debt outstanding in future. 5.1 APV Using APV, we calculate the PV of the tax shields, then add this to V U. To calculate the PV of the tax shields, note that each tax shield is discounted back one year at r D, and 11

remaining years at r A. Why? The first year s tax shield is known today. However, all future tax shields depend on the value of the firm. Since we have assumed the firm will keep a constant proportion of debt outstanding in the future, the amount of debt outstanding scales with firm value. The growth in firm value is related to r A. Thus, the first year s tax shield is: The present value of all tax shields is: IT S t1 r D D L,t1 τ $35.00 0.05 $2, 000 0.35 S P V IT S 35 35 1.03 35 1.032 + + 1.05 1.05 1.1100 1.05 1.1100 +... 2 35 0.1100 0.0300 1.1100 1.0500 $462.50 From Equation 1, the value of equity in this levered firm is the value of unlevered firm, minus the value of debt, plus the present value of the interest tax shields: V U D L + S 8, 125.00 2, 000.00 + 462.50 $6, 587.50 Also from the derivation of Equation 1, the value of the levered firm is greater than the unlevered firm V U by exactly the present value of interest tax shields: + D L V U + S 6, 587.50 + 2, 000.00 8, 125.00 + 462.50 $8, 587.50 $8, 587.50 Using Equation 9 with k 1, we can calculate r E, which will be useful later: r E 1 + D L 1 r ] Dτ r A 1 r ] Dτ r D 1 + r D 1 + r D 1 + 2, 000 1 6, 587.50 ] 0.05 0.35 2, 000 0.1100 1 1.05 6, 587.50 ] 0.05 0.35 0.05 1.05 12.7913% 12

Note: APV is easy to use in this case. Everything is growing at a constant rate which makes it easy to calculate the value of the future tax shields. APV also works well in cases where we assume the amount of debt remains constant or when the amount changes deterministically over time. However, in cases where the firm s cash flows are changing over time, it is harder to calculate the size of future tax shields. 5.2 WACC Given the results above, we can calculate WACC: r W ACC r A r DD L τ 1 + ra 1 + r D 0.05 2, 000 0.35 0.1100 8, 587.50 10.5691%. 1.1100 1.05 The value of the entire levered firm is the growing perpetuity value of the free caseflows using r W ACC as the discount value: F CF t+1 r W ACC g 650 0.105691 0.0300 $8, 587.50 The value of equity in this levered firm is, again, the total firm value minus the value of debt D L : 8, 587.50 2, 000 $6, 587.50 Note: To calculate WACC based on a known amount of debt D L, one needs the value of to calculate. This is fine if we are valuing a marginal project, but less acceptable if we are trying to value the whole firm. We can get around this problem by solving iteratively using tools such as Excel s Solver. The idea is to: 1 get the value for ; 2 calculate WACC, discount the cash flows, and calculate a new value for, 3 Have a solver try different values for until the calculated value in step 2 equals the input value in step 1. 13

5.3 Discounting Total After Tax Cash Flow The total after-tax cash flow in year 1, including the tax shield, is $650 + $35 $685. This is expected to grow at 3% per year. The pretax weighted average cost of capital is: EL r V r E + r D + D L EL r E + 6, 587.50 8, 587.50 10.9767%. r D 0.127913 + + D L 2, 000.0 0.05 8, 587.50 Note that this is slightly lower than r A since the first tax shield has been discounted at r D not r A. So, from Equation 13 we use the growing perpetuity formula to get: C L,t+1 r v g 685 0.109767 0.0300 $8, 587.50 Note: Discounting total, after-tax cashflows shares the disadvantages of both APV and WACC discussed above. We need to forecast future tax shields like APV, and we also need r E and before we can calculate r V like WACC. 5.4 Discounting Cash Flows to Equity Only A common valuation methodology discounts the cash flows to equity holders only using the equity rate r E. This method has the advantage of not having to calculate and then subtracting the value of the debt. It does work, but you have to be very careful. We already know that r E 12.7913%. The total after tax cash flow to equity and debt holders combined in year 1 is $685, of which $2, 000 0.05 $100 goes to debt holders, leaving $585 for equity 14

holders. But: 585 0.127913 0.03 $5, 974.69??? This is not the right answer. What s wrong is that we forgot to include part of the cash flow to equity holders. Next year, we expect the amount of debt outstanding to increase by 3%, from $2,000 to $2,060. The $60 raised by issuing additional debt is paid as a dividend to shareholders, so their total cash flow is actually: $685 $100 + $60 $645. Thus we get the same value for E U as before: 585 + 60 0.127913 0.03 $6, 587.50 Note: Discounting cashflows to equity only is generally harder to use in practice than APV or WACC. Not only do we need to forecast the future tax shields, but we also need to forecast the amount of new debt issued each period. 5.5 Risky Debt The valuation example above assumes the firm issues $2, 000 of riskless debt. We re-do the analysis but assume the firm issues $2, 000 of risky debt. The yield on the debt is 7.00% and it has a 0.2857 beta. The results of the valuation exercise with risky debt are shown in Table 3. Note the total firm value is now $8,760.40 which is higher than the total firm value of $8,587.50 from the example in Table 2. The increase in value is due to the higher tax shields that result from higher interest expense. 6 Conclusion This paper outlines the math and assumptions behind weighted average cost of capital WACC and adjusted present value APV calculations. We first derive a general formula for the discount rate of equity and beta of equity under minimal assumptions. We then take into consideration: i The existence or non-existence of taxes; ii Whether a firm has a constant amount of debt in dollar terms; iii Whether a firm targets a constant proportion of debt in its capital structure; and iv The frequency of debt rebalancing. One result shows a firm s equity beta is that same in a world with no taxes as it is in a world with taxes under the assumption the firm continuously rebalances its debt to a constant 15

proportion of capital structure. We highlight how changing assumptions, changes formulas for equity betas and discount rates not surprisingly. We end with a valuation example that provides consistent results for a firm with growing operating income. References 1] Miles, J. and R. Ezzel, 1980, The Weighted Average Cost of Capital, Perfect Capital Markets and Project Life: A Clarification, Journal of Financial and Quantitative Analysis, 15, 719-730. 16

Table 1 Rebalancing Frequency This table shows the effect of rebalancing on the equity beta β E. The firm rebalances its capital structure periodically to maintain a constant proportion of debt. The equity beta is calculated using Equation 10 from the paper. The tax rate τ is 35%; the asset beta β A is 0.8571; the company currently has $2,000 of debt outstanding but this number is expected to change; the cost of debt r D is 7.00%; the beta of debt β D is 0.2857; and the value of the firm s equity is $6,760.40. Periods Per Year Rebalancing Frequency β E 1 Annually 1.0223 2 Semi-An 1.0242 4 Quarterly 1.0252 12 Monthly 1.0259 52 Weekly 1.0261 365 Daily 1.0262 8,760 Hourly 1.0262 17

Table 2 Valuation Example with Riskless Debt This table shows results of a valuation exercise. The firm is expected to have an operating profit EBIT of $1,000 next year. The operating profit is expected to grow at 3% per year after that. The tax rate τ is 35%; the asset beta β A is 0.8571; the company has $2,000 of debt outstanding; the cost of debt r D is 5.00%; the beta of debt β D is 0.0000 and it is riskless. Section of Paper Calculations Symbols Result 5.0 Unlevered Firm V U 8,125.00 5.1 APV ITS t1 35.00 PVITS 462.50 6,587.50 r E 12.79% 8,587.50 5.2 WACC r WACC 10.57% 8,587.50 6,587.50 5.3 Total A/T CF r V 10.98% 8,587.50 5.4 CF to Equity 6,587.50 18

Table 3 Valuation Example with Risky Debt This table shows results of a valuation exercise. The firm is expected to have an operating profit EBIT of $1,000 next year. The operating profit is expected to grow at 3% per year after that. The tax rate τ is 35%; the asset beta β A is 0.8571; the company has $2,000 of debt outstanding; the cost of debt r D is 7.00%; the beta of debt β D is 0.2857 as this example considers risky debt. Calculations Symbols Result Unlevered Firm V U 8,125.00 APV ITS t1 49.00 PVITS 635.40 6,760.40 r E 12.16% 8,760.40 WACC r WACC 10.42% 8,760.40 6,760.40 Total A/T CF r V 10.98% 8,760.40 CF to Equity 6,760.40 19