General expression for the return to levered equity Ke and WACC Ignacio Vélez-Pareja Master Consultores Joseph Tham Duke University March 31, 2014 Variables and acronyms in this text1 PV Present value CF Cash flow DR Discount rate FCF Free cash flow TS Tax shields CFD Cash flow to debt CFE Cash flow to equity Un V Unlevered value of firm VTS Value of TS Using the basic tenet of finance, PVi+1 +CFi+1 PVi = 1+DR D EL Ku Ke Kd ψ WACCCCF WACCFCF RHS (LHS) Market value of Debt Levered Market value of equity Unlevered cost of equity Levered cost of equity Market cost of debt Discount rate for TS WACC for the CCF WACC for the FCF Right (Left) hand side (1) i+1 FCFi = VUni-1(1 + Kui) - VUni TSi =VTSi-1(1 + ψ i) - VTSi CFEi = ELi-1(1 + Kei) - ELi CFDi = Di-1(1 + Kdi) - Di From Modigliani and Miller (1958, 1963), we know that, FCFi + TSi = CFEi + CFDi = CCFi VLi = VUni + VTSi = Di + ELi To obtain the general expression for the Ke, substitute equations 2 to 5 into equation 6a, VUni-1(1 + Kui) - VUni + VTSi-1(1 + ψ i) - VTSi = ELi-1(1 + Kei) - ELi + Di-1(1 + Kdi) - Di (7a) Using 6b we simplify and obtain, VUni-1 + VUni-1Kui - VUni + VTSi-1+ VTSi-1 ψ i- VTSi = ELi-1+ ELi-1Kei - ELi + Di-1+ Di-1Kdi - Di (7b) We know that VUni + VTSi equals Di + ELi in 6b, hence they can be simplified, VUni-1 + VUni-1Kui + VTSi-1+ VTSi-1 ψ i = ELi-1+ ELi-1Kei + Di-1+ Di-1Kdi (7d) Un TS L Similarly V i-1 + V i-1 equal Di-1 + E i-1, hence these terms can be simplified too, VUni-1 + VUni-1Kui + VTSi-1+ VTSi-1 ψ i = ELi-1+ ELi-1Kei + Di-1+ Di-1Kdi (7e) VUni-1Kui + VTSi-1ψ i = ELi-1Kei + Di-1Kdi (8a) Solving for the return to levered equity, Ke and using the value equations from Modigliani and Miller, ELi-1Kei = VUni-1Kui + VTSi-1ψ I - Di-1Kdi (8b) In 6b, VUni + VTSi = Di + ELi, hence VUni = Di + ELi -VTSi, replace what is equivalent to VUni in 8b, ELi-1Kei = (ELi-1 + Di-1 - VTSi-1)Kui + VTSi-1ψ I - Di-1Kdi (9a) Collecting terms and rearranging, we obtain, ELi-1Kei = ELi-1Kui + (Kui - Kdi)Di-1 - (Kui - ψ i)VTSi-1 (10) Solving for the return to levered equity, we obtain, ୈ Ke୧ = Ku୧ + (Ku୧ − Kd୧ ) ైషభ − (Ku୧ − ψ ୧) షభ ై When ψ i = Kd షభ ୈ Ke୧ = Ku୧ + (Ku୧ − Kd୧ ) ైషభ − షభ ై షభ (6a) (6b) (11a) షభ షభ (2) (3) (4) (5) ൨ (11b) If cash flows are non-growing perpetuities then VTS = TD. ୈ Ke୧ = Ku୧ + (Ku୧ − Kd୧ )(1 − T) ైషభ షభ 1 (11c) This derivation is based on Tham, J. and Vélez-Pareja, I., 2004. Principles of Cash Flow Valuation. An Integrated MarketBased Approach. Boston: Academic Press. 1 When ψ i = Ku ୈ Ke୧ = Ku୧ + (Ku୧ − Kd୧ ) ైషభ (11d) షభ For convenience, when ψ i=Ke, we start from (10) ELi-1Kei = ELi-1Kui + (Kui - Kdi)Di-1 - (Kui - ψ i)VTSi-1 ELi-1Kei – VTSi-1ψ i = (ELi-1- VTSi-1) Kui + (Kui - Kdi)Di-1 When ψ i = Ke ELi-1Kei – VTSi-1Kei = [ELi-1- VTSi-1) Kui + (Kui - Kdi)Di-1] Solving for Kei Kei= [(ై షభ ି షభ )୳ ା (୳ ି ୢ )ୈషభ ] ై షభ ି షభ (10) (12a) (12b) = Ku୧ + (Ku୧ − Kd୧ ) ై ୈషభ షభ (12c) ି షభ WACC applied to the FCF From (6a) RHS VLi-1WACCFCFi = Di-1Kdi – TSi + ELi-1Kei From (6a) LHS VLi-1WACCFCFi = VUni-1Kui + VTSi-1ψ i – TSi In (6b), VLi = VUni + VTSi hence, VUni = VLi-VTSi and, similarly VUni-1=VLi-1 - VTSi-1. Replacing in (13) VLi-1WACCFCFi = (VLi-1 - VTSi-1)Kui + VTSi-1ψ i – TSi (14) VLi-1WACCFCFi = VLi-1 Kui - VTSi-1Kui + VTSi-1ψ i – TSi VLi-1WACCFCFi = VLi-1Kui - (Kui - ψ i)VTSi-1 – TSi Solving for WACCFCF in equation 15, we obtain, WACC୧େ = ై షభ ୳ ି(୳ ିψ )షభ – ୗ ై షభ When ψ i = Kd WACC୧େ = Ku୧ − (Ku୧ − Kd୧ ) షభ – ై When ψ i = Ku WACC୧େ = Ku୧ – When ψ i = Ke షభ = Ku୧ − (Ku୧ − ψ୧ ) షభ – ై షభ ୗ ై షభ ୗ (16c) షభ ୗ (16d) ై షభ WACC applied to the CCF CCFi = FCFi + TSi VLi-1WACCCCFi = VUni-1Kui + VTSi-1ψ i Applying the same as before, VUni-1=VLi-1 - VTSi-1, we replace again in (18) VLi-1WACCCCFi = VLi-1Kui - (Kui - ψ i)VTSi-1 Solving for the WACCCCF, we obtain, ై షభ ୳ ି(୳ ିψ )షభ When ψ i = Kd = Ku୧ − (Ku୧ − ψ୧ ) షభ ై షభ WACC୧େେ = Ku୧ − (Ku୧ − Kd୧ ) షభ ై When ψ i = Ku WACC୧େେ = Ku୧ When ψ i = Ke (15) (16b) ై షభ ୗ ై షభ (13) (16a) ై షభ WACC୧େ = Ku୧ − (Ku୧ − Ke୧ ) షభ – ై WACC୧େେ = (12) (17) (18) (19) (20a) (20b) షభ (20c) WACC୧େେ = Ku୧ − (Ku୧ − Ke୧ ) షభ ై (20d) షభ 2