In this note we reconsider in detail the proper discount rate for cash flows in perpetuity, the present value of tax savings and the calculation of terminal value. The note clarifies the use of real discount rates and concludes with a formulation that is inflation-neutral for a given assumption on the discount rate for the tax savings. We find that the only discount rate for tax savings that makes the value of the perpetuity inflation-neutral is Kd, the cost of debt. We also reconsider the intuitive approach to calculate the cost of capital for perpetuities from the nominal rates that compose that cost of capital, and then converting it into real cost of capital using Fisher relationship.

1. i
Summing Up About Growing and Non Growing Perpetuities: WACC, Levered and
Tax Savings Value. A Clarification
Ignacio Vélez–Pareja
Universidad Tecnológica de Bolívar
Cartagena, Colombia
ivelez@unitecnologica.edu.co
nachovelez@gmail.com
First Version: March 20, 2008
This version: March 21, 2008

2. ii
Abstract
In this note we reconsider in detail the proper discount rate for cash flows in
perpetuity, the present value of tax savings and the calculation of terminal value. The note
clarifies the use of real discount rates and concludes with a formulation that is inflation-
neutral for a given assumption on the discount rate for the tax savings. We find that the
only discount rate for tax savings that makes the value of the perpetuity inflation-neutral is
Kd, the cost of debt. We also reconsider the intuitive approach to calculate the cost of
capital for perpetuities from the nominal rates that compose that cost of capital, and then
converting it into real cost of capital using Fisher relationship.
JEL codes: D61, G31, H43
Key words or phrases: WACC, perpetuities, terminal value, tax savings

3. 1
In this note we reconsider in detail the proper discount rate for cash flows in
perpetuity, the present value of tax savings and the calculation of terminal value. The note
clarifies the use of real discount rates and concludes with a formulation that is inflation-
neutral for a given assumption on the discount rate for the tax savings. We find that the
only discount rate for tax savings that makes the value of the perpetuity inflation-neutral is
Kd, the cost of debt. We also reconsider the intuitive approach to calculate the cost of
capital for perpetuities from the nominal rates that compose that cost of capital, and then
converting it into real cost of capital using Fisher relationship.
Previous reports show that some of the real Weighted Average Cost of Capital,
WACC for perpetuities and their value were not inflation-neutral1
. In this note we correct
the reports regarding most of the not inflation-neutrality of WACC and value of the
perpetuity cases. On the other hand, others show that nominal cost of capital should be
derived from the real cost of capital inflating it in order to correct some distortions
regarding the true value of the perpetuity2
and granting the inflation-neutrality of the
perpetuity value.
The note is organized as follows: In Section One we present the value of a
perpetuity and the formulation for WACC. In Section Two we illustrate the calculation of a
perpetuity with a simple example. In Section Three we conclude. There is an Appendix
where we show the algebraic derivations of the formulations presented in Section One.
1
Vélez-Pareja (2004, 2006a, 2006b, 2007) and Vélez-Pareja and Tham (2006).
2
Bradlley and Jarrel (2003).

4. 2
Section One. The Value of a Perpetuity and the Formulation for WACC
We rescue what previous studies show regarding the value of a perpetuity3
. The
formulations are shown in Table 1:
Table 1. Formulation for Levered Value, Unlevered Value and Tax Savings Value*
ψ = Kd ψ = Ku
VL
VU
PVTS
PVTS
π = 0,
g = 0
FCF/w FCF/ku T×D%×FCF/w kd×T×D%×FCF/(w×ku)
π > 0,
g = 0
FCF/w FCF/ku Kd×T×D%×FCF/(w×Kd) Kd×T×D%×FCF/(w×Ku)
π = 0,
g > 0
FCF×(1+g)/w FCF×(1+g)/ku T×D%×FCF×(1+g)/w kd×T×D%×FCF×(1+g)/(w×ku)
π > 0,
g > 0
FCF×(1+g)/w FCF×(1+g)/ku =T×D%×FCF×(1+g)/w Kd×T×D%×FCF×(1+g)/(Ku×w)
*
In this table π is inflation rate, g is real growth rate for the FCF, FCF is free cash flow, w is real WACC, VL
is levered value, VU
is
unlevered value, ku is real cost of unlevered equity, Ku is the nominal cost of unlevered equity, T is corporate tax rate, ψ is the discount
rate of the tax shields, PVTS
is the value (present value) of the tax shield, D% is the leverage at perpetuity, kd is the real cost of debt and
Kd is the nominal cost of debt.
Observe that the calculation of the levered value, only requires the real and not the
nominal WACC. Using the basic value equation of the Adjusted Present Value, APV, we
derive the expressions for w, the real WACC. This basic equation says:
VL
= VU
+ PVTS
(1)
Observe that the formulation for levered, unlevered and tax shield values depends
on the assumption about ψ, π and g. The same is true for the real WACC. The formulations
for real WACC are shown in Table 2.
Table 2. Different Formulations for Real WACC, w, as a function of ψ and the
Inflation/Growth Scenario
ψ= Kd ψ= Ku
π = 0, g=0 ku×(1-T×D%) ku - kd×T×D%
π >0, g=0 ku×(1 - T×D%) ku×(1 – Kd×T×D%/Ku)
π =0, g>0 ku×(1 - T×D%) ku- kd×T×D%
π >0, g>0 ku×(1 - T×D%) ku×(1- Kd×T×D%/Ku)
3
Vélez-Pareja (2004, 2006a, 2006b, 2007) and Vélez-Pareja and Tham (2006). They show the use of a
plowback ratio to sustain growth in perpetuity.

5. 3
Exception made for the cases for ψ= Ku and π >0, g=0 and π >0, g > 0 all the
expressions are standard and well known formulations for WACC. These non standard
expressions for WACC are non inflation-neutral values. The amazing issue is that all of
them are real WACC and not nominal WACC as expected.
Section Two. An Example
Assume a FCF in perpetuity of 10, with a corporate tax rate T, of 40%, constant
perpetual leverage D% of 30%, a real cost of unlevered equity, ku of 10%, a real cost of
debt, kd of 8%, a real growth rate of 3% and an inflation rate π of 5%.
Using Fisher relationship we calculate the nominal values of some of these variables
as follows: Kd is 13.4% and Ku is 15.5%. With this information, we calculate the real
WACC, w for different scenarios of g, π and of ψ. This is shown in Table 3.
Table 3. Real WACC w, for different Scenarios
ψ= Kd ψ= Ku
π = 0, g=0 8.80% 9.04%
π >0, g=0 8.80% 8.96%
π =0, g>0 8.80% 9.04%
π >0, g>0 8.80% 8.96%
Observe that for ψ equal to Kd, w is constant, as expected. Also observe the effect
of inflation (π > 0) in w when ψ is Ku. This means that w for those cases is not inflation-
neutral.
In Table 4 we calculate the levered (VL
) and unlevered value, (VU
) for the
perpetuity and the PVTS
for each scenario.
Table 4. Value of Tax Shields, Levered and Unlevered Values for a Perpetuity
ψ= Kd ψ= Ku ψ= Kd ψ= Ku
VL
VL
VU
PVTS
VL
PVTS
VL
π = 0, g=0 113.6 110.6 100.0 13.6 113.6 10.6 110.6
π >0, g=0 113.6 111.6 100.0 13.6 113.6 11.6 111.6
π =0, g>0 117.0 113.9 103.0 14.0 117.0 10.9 113.9
π >0, g>0 117.0 114.9 103.0 14.0 117.0 11.9 114.9

6. 4
Observe that the APV and the calculation of the levered value using FCF and
WACC match for every assumption on ψ, the discount rate for the tax savings and for each
scenario for g and π.
Section Three. Concluding Remarks
In this note we have shown the different formulations for WACC for perpetuities.
Interesting features of this are the following:
1. Levered value of a perpetuity is calculated using real WACC instead of
nominal WACC. The formulation for the value of a perpetuity contradicts
the current literature that calculates it using nominal WACC.
2. The derivation of the WACC for perpetuity is done departing from real
formulation and not from nominal formulation and deflating it as intuitively
and usually is done.
3. The APV and the traditional approach of discounting the FCF with WACC
match.
4. When the risk of TS (the discount rate of TS) ψ, is Kd, real WACC, w is
inflation-neutral.
5. When the risk of TS is not Kd (Ku or any other value) w is not inflation-
neutral.
6. The meaning of the previous conclusion is that when the risk of TS is not Kd
and inflation is included in the analysis, we will find the counter evident
fact that inflation creates value.

7. 5
7. Further reflections have to be done to find if the approach to calculate first
the real discount rate and inflating it instead of using the nominal discount
rate, applies consistently to finite cash flows.

8. 6
Appendix
Formulations for real WACC
In this Appendix we use the following variables: Ku = nominal cost of unlevered
equity; ku = real cost of unlevered equity; Kd = nominal cost of cost of debt; kd = real cost
of cost of debt; π = inflation rate; G = nominal growth rate of CFs; g = real growth rate of
CFs; W = nominal Weighted Average Cost of Capital, WACC; w = real Weighted Average
Cost of Capital, WACC. We show the derivation of the formulas from Table 1.
Fisher equation says:
1 + nominal rate = (1 + real rate)×(1 + π) (1a)
Where π is the inflation rate. From this expression we can derive the following that
will help the reader through the algebra:
Nominal rate – π = Real rate × (1 + π) (1b)
Real rate = (Nominal rate – π)/(1 + π) (1c)
Nominal rate A – Nominal rate B = (Real rate A – Real rate B)/(1+π) (2)
For instance,
Ku – G = (ku-g)×(1+ π) (3)
Now we examine the relationships between levered and unlevered values, value of
tax shields and discount rates. We consider four scenarios which combines real growth and
inflation. These scenarios are:
• No inflation, no growth.
• Non zero inflation, no growth.
• No inflation, growth.
• Non zero inflation, growth.
We use repeatedly the APV formulation:

9. 7
APV = VL
= VU
+ PVTS
(4a)
PVTS
= VL
- VU
Where APV is Adjusted Present Value; VL
is the levered value of a perpetuity; VU
is the unlevered value and PVTS
is the value of TS.
1. No inflation no growth
The FCF is constant.
VL
= FCF/w
APV = FCF/ku + PVTS
PVTS
= kd×T×D%×FCF/(w×ψ)
PVTS
depends on ψ.
For ψ = kd
PVTS
= kd×T×D%×FCF/(w×kd) = FCF/w - FCF/ku
FCF/ku = FCF/w - kd×T×D%×FCF/(w×kd)
Multiplying by w×ku/FCF and simplifying, we solve for w.
w = ku × (1-T×D%)
This expression for w is inflation-neutral
For ψ= Ku
PVTS
= kd×T×D/ku
PVTS
= kd×T×D%×FCF/(w×ku) = FCF/w - FCF/ku
FCF/ku = FCF/w - kd×T×D%×FCF/(w×ku)
Multiplying by w×ku/FCF and simplifying, we solve for w.
w = ku - kd×T×D%
This expression for w is inflation-neutral

10. 8
2. Non zero inflation no growth
The FCF grows only for inflation effects. There is no need for a perpetual
investment to sustain the inflationary growth.
VL
= FCF×(1+ π)/(W- π) = FCF×(1+ π)/(w×(1 + π)) = FCF/w
APV = FCF×(1+ π)/(Ku- π) + PVTS
= FCF/ku + PVTS
PVTS
= Kd×T×D%×FCF/(w×ψ)
PVTS
depends on ψ.
For ψ = Kd
APV = FCF/w = FCF/ku + Kd×T×D%×FCF/(w×Kd)
FCF/w - Kd×T×D%×FCF/(w×Kd)= FCF/ku
Multiplying by w×ku/FCF and simplifying, we solve for w.
w = ku - ku×T×D% = ku×(1-T×D%)
This expression for w is inflation-neutral
For ψ = Ku
PVTS
= Kd×T×D/Ku
PVTS
= Kd×T×D%×FCF/(w×Ku) = FCF/w – FCF/ku
Multiplying by w×ku/FCF and simplifying, we solve for w.
w = ku×(1 – Kd×T×D%/Ku)
This expression for w is not inflation-neutral
3. No inflation and growth
The FCF grows at a real rate g. There is need for a perpetual investment to sustain
the real growth. This perpetual investment is a function of g, the real growth. The fraction

11. 9
of the FCF to be invested is known as plowback ratio. In this case we assume the plowback
ratio is g/w.
VL
= FCF×(1+g)×(1-g/w)/(w-g) = FCF×(1+g)/w
VU
= FCF×(1+g)×(1-g/ku)/(ku-g) = FCF×(1+g)/ku
APV = FCF×(1+g)/ku + PVTS
PVTS
= kd×T×D%× FCF×(1+g)/(w×ψ)
PVTS
depends on ψ
For ψ = kd
PVTS
= kd×T×D%×FCF×(1+g)/(w×kd) = T×D%×FCF×(1+g)/w
VL
= FCF×(1+g)/w = FCF×(1+g)/ku + T×D%×FCF×(1+g)/w
Multiplying by w×ku/(FCF×(1+g) and simplifying, we solve for w.
w = ku×[1 - T×D%]
This expression for w is inflation-neutral
For ψ = Ku
PVTS
= kd×T×D%×FCF×(1+g)/(w×ku) = FCF×(1+g)/w – FCF×(1+g)/ku
kd×T×D%× FCF×(1+g)/w×ku = FCF×(1+g)/w – FCF×(1+g)/ku
Multiplying by w×ku/(FCF×(1+g) and simplifying, we solve for w.
w = ku- kd×T×D%
This expression for w is inflation-neutral
4. Inflation and growth
The FCF grows at a nominal rate G. There is need for a perpetual investment to
sustain the real growth. This perpetual investment is a function of g, the real growth. This

12. 10
perpetual investment is a function of g, the real growth. The fraction of the FCF to be
invested is known as plowback ratio. In this case we assume the plowback ratio is g/w.
VL
= FCF×(1+G)×(1-g/w)/(W-G) = FCF×(1+g)/w
VU
= FCF×(1+G)×(1-g/ku)/(Ku-g) = FCF×(1+g)/ku
APV = FCF×(1+g)/ku + PVTS
PVTS
= Kd×T×D%× FCF×(1+g)/(w×ψ)
PVTS
depends on ψ
For ψ = kd
PVTS
= Kd×T×D%× FCF×(1+g)/(w×Kd) = T×D%× FCF×(1+g)/w
VL
= FCF×(1+g)/w= FCF×(1+g)/ku + T×D%× FCF×(1+g)/w
Multiplying by w×ku/(FCF×(1+g) and simplifying, we solve for w.
w = ku×(1- T×D%)
This expression for w is inflation-neutral
For ψ = ku
PVTS
= Kd×T×D%× FCF×(1+g)/(w×Ku)
VL
= FCF×(1+g)/w= FCF×(1+g)/ku + Kd×T×D%× FCF×(1+g)/(w×Ku)
Multiplying by w×ku/(FCF×(1+g) and simplifying, we solve for w.
w= ku×(1- Kd×T×D%/Ku)
This expression for w is not inflation-neutral.
In Table A1 we present a summary of the previous derivations.

13. 11
Table A1. Expressions for w (real WACC)
ψ= Kd IN*
ψ = Ku IN
π = 0, g=0 ku × (1-T×D%) Yes ku - kd×T×D% Yes
π >0, g=0 ku×(1-T×D%) Yes ku×(1 – Kd×T×D%/Ku) No
π =0, g>0 ku×[1 - T×D%] Yes ku- kd×T×D% Yes
π >0, g>0 ku×(1 - T×D%) Yes ku×(1- Kd×T×D%/Ku) No
*
IN means inflation-neutral.
Bibliographic References
1. Bradley, Michael H. and Jarrell, Gregg A., 2003, "Inflation and the Constant-Growth
Valuation Model: A Clarification" (February). Simon School of Business Working
Paper No. FR 03-04. Available at SSRN: http://ssrn.com/abstract=356540 or
DOI: 10.2139/ssrn.356540
2. Vélez-Pareja, Ignacio, 2007, Proper Valuation of Perpetuities in an Inflationary
Environment Without Real Growth November 17, Working paper at SSRN
http://ssrn.com/abstract=1030821
3. Vélez-Pareja, Ignacio, 2006a, A New Approach to WACC, Value of Tax Savings
and Value for Growing and Non Growing Perpetuities: A Clarification, Enero 4,
Working Paper en SSRN, Social Science Research Network.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=873686.
4. Vélez-Pareja, Ignacio, and Joseph Tham, 2005, Cash flows, WACC, Value of Tax
Savings and Terminal Value for Growing and Non Growing Perpetuities, Working
Paper at SSRN, Social Science Research Network, , August 22.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=789025.
5. Vélez-Pareja, Ignacio, 2006b, Conditions for Consistent Valuation of a Growing
Perpetuity, Working Paper at SSRN, Social Science Research Network, January 30.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=879515

14. 12
6. Vélez-Pareja, Ignacio, 2004, Proper Determination of the Growth Rate for Growing
Perpetuities: The Growth Rate for the Terminal Value Working Paper en SSRN,
Social Science Research Network, January.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=493782