General expression for the return to levered equity Ke

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,
WACC୧୊େ୊ =
౐౏
୚ై
౟షభ ୏୳౟ ି(୏୳౟ ିψ ౟)୚౟షభ – ୘ୗ౟
୚ై
౟షభ
When ψ i = Kd
୚౐౏
WACC୧୊େ୊ = Ku୧ − (Ku୧ − Kd୧ ) ୚౟షభ
–
ై
When ψ i = Ku
WACC୧୊େ୊ = 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
WACC୧୊େ୊ = Ku୧ − (Ku୧ − Ke୧ ) ୚౟షభ
–
ై
WACC୧େେ୊ =
(12)
(17)
(18)
(19)
(20a)
(20b)
౟షభ
(20c)
୚౐౏
WACC୧େେ୊ = Ku୧ − (Ku୧ − Ke୧ ) ୚౟షభ
ై
(20d)
౟షభ
2