International Research Journal of Finance and Economics ISSN 1450-2887 Issue 26 (2009) © EuroJournals Publishing, Inc. 2009 http://www.eurojournals.com/finance.htm
The Weighted Average Cost of Capital (WACC) for Firm Valuation Calculations
Fernando Llano-Ferro Universidad de Bogotá - Jorge Tadeo Lozano Economics Department Carrera 4 No.22-61 Bogotá- Colombia Abstract The Weighted Average Cost of Capital (WACC) is used in finance for several applications, including Capital Budgeting analysis, EVA® calculations, and firm valuation. WACC obtained by the standard formula leads to significant errors in Net Present Value of the Firm calculations; particularly in those that apply perpetual cash flow series. The present paper identifies the problem, and provides alternative, and accurate formulas to obtain WACC for Firm Valuation calculations.
Keywords: Weighted Average Cost of capital (WACC), Firm valuation JEL Classification Codes: G34
I. Introduction
The standard formula to obtain the Weighted Average Cost of Capital (WACC) is included in most undergraduate, and graduate Corporate Finance textbooks, and it is known by heart by teachers, students and practitioners around the world. Ei + D(1 − T )iD WACC1 = E (1) (E + D ) Where: E = Annual Free Cash Flow to Equity iE = Annual cost of equity D = Annual interest payments (before taxes) ID = Annual cost of debt T = tax rate Subscripts for WACC are used in this paper to identify different methods to calculate it. This formula is used in EVA®, Capital Budgeting Analysis, Firm valuation, etc.
II. Motivation
The value of a firm can be estimated by adding the Net Present Values of the Free Cash Flows to Equity, and the Free Cash Flows to Debt; each discounted with the appropriate interest rate. j =n FCFE j FCFD j NPVFIRM = NPVEQUITY + NPVDEBT = ∑ +∑ (2) j (1 + iD ) j j =1 (1 + iE ) An alternative procedure would be to calculate the Net Present Value of the Free Cash Flows to the firm; discounted with the WACC obtained through equation (1).
International Research Journal of Finance and Economics - Issue 26 (2009) NPVFIRM = ∑
j =1 j =n
149
(3) (1 + WACC ) j where FCFFj = FCFEj + FCFDj Unfortunately, the two results are not the same. It can be argued that the correct WACC is the one that makes the two results the same. The objective of this paper is to find mathematical formulae to calculate the “correct” WACC.
FCFFj
III. Numerical Example of Difference of the Firm Calculations
Let us calculate the value of a stable firm with E=$100, iE = 0.12 (12 %), D=$30, iD = 0.06 (6 %), and a tax rate of 0.4 (40 %). These values apply in perpetuity. Applying the formula of Net Present Value in perpetuity we obtain: C (4) NPV = i where: • C = Annual cash flow • i = Discount rate The Net Present Value of equity is NPVE = $100/0.12= $833.33, and the Net Present Value of Debt is NPVD = $30(1-0.4)/ 0.06=$300. Therefore, the Net Present Value of the Firm would be the sum, or NPVF = NPVE + NPVD = $1133.33. The WACC for our example, obtained by formula (1), is 0.1108. If we use this WACC to obtain the Net Present Value of the Firm, with formula (2), with the corresponding cash flows for the firm, the result is NPVF =112/0.1108 = $1065. The difference in the result of the two calculations of the value of the firm is about 6.5 %.
IV. Alternative Formulae to Calculate WACC
It is convenient and desirable to have a WACC that provides the same result, when discounting the combined stream of cash flows to the firm, instead of the cash flows to debt and cash flows to equity independently. To obtain the same result, WACC would have to be obtained from: D(1 − T ) E [(D(1 − T ) + E )] + = iD iE WACC2 or:
WACC2 =
iDiE [(D (1 − T ) + E )] [(iE D(1 − T ) + iD E ]
(5)
For our example WACC2 = 0.1041 would give the exact result. It would also give the correct result for calculations of Net Present Value of equal and perpetual cash flows. For the situation where annual Free Cash Flows to the Firm vary, from period to period, a third formula is required. To simplify somewhat the mathematics, I am going to use the continuously compounded equivalent interest rate. Thus:
(1 + i )n = ε i n
*
or:
i* = ln(1 + i )
(6)
150 from: FCFE j
International Research Journal of Finance and Economics - Issue 26 (2009) Then the WACC, that would provide accurate results under all circumstances, can be derived + FCFD j (1 − T ) =
[FCFE
j
+ FCFD j (1 − T )
]
ε
obtaining:
ji E
ε
ji D
ε
jWACC 3
WACC3 =
(7) In this case, FCFEj is the Free Cash Flow to Equity in period j, and FCFDj is the Free Cash Flow to Debt, before taxes, in period j. Please note that WACC3 is not constant. It varies from period to period. It decreases exponentially as a function of time.
1 ⎡ ε ni E ε ni D (FCFE + FCFD (1 − T ) ) ⎤ ln ⎢ ⎥ n ⎣ ε ni D FCFE + ε ni E FCFD (1 − T ) ⎦
(
)
V. Conclusion
We have three possible formulae to calculate WACC´s. The objective is to have a WACC that provides the same results with Value of the Firm calculations obtained through: j =n FCFE j FCFD j NPVFIRM = NPVEQUITY + NPVDEBT = ∑ +∑ j (1 + iD ) j j =1 (1 + iE ) or: j =n FCFFj NPVFIRM = ∑ j j =1 (1 + WACC ) WACC1, obtained from the traditional equation (1), which provides approximate results for Net Present Value of the Firm calculations of a few periods, say 6. We can use WACC2 (5) which gives accurate results for Net Present Value of the Firm calculations when applied to constant perpetual cash flows, and approximate results of calculations with nearly constant perpetual cash flows. Finally, we can use WACC3 (7), which provides precise results of Net Present Value of the Firm under all circumstances. WACC3 varies in every period, decreasing exponentially as a function of time.
References
[1] [2] [3] R.A. Brealey, S.C. Myers, F. Allen PRINCIPLES OF CORPORATE FINANCE. Eight Edition - McGraw-Hill Companies, Inc. 2006. McKinsey & Co. VALUATION. University Edition. Fourth Edition. John Wiley & Sons Inc. 2005 A. Damodaran (2005), “Acquisition Valuation”, Leonard N. Stern School of Business, New York University