Summing up about growing and non growing perpetuities wacc levered and tax savings value a clarification 2008 ignacio

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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

[email protected]

[email protected]

First Version: March 20, 2008

This version: March 21, 2008

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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

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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).

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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

V

L V

U PV

TS PV

TS

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.

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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, (V

U) 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

V

L V

L V

U PV

TS V

L PV

TS V

L

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

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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.

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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.

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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:

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APV = VL = V

U + PV

TS (4a)

PVTS

= VL - V

U

Where APV is Adjusted Present Value; VL is the levered value of a perpetuity; V

U

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)


w = ku - kd×T×D%


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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


w = ku - ku×T×D% = ku×(1-T×D%)


For ψ = Ku

PVTS

= Kd×T×D/Ku

PVTS

= Kd×T×D%×FCF/(w×Ku) = FCF/w – FCF/ku


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

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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%]


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


w = ku- kd×T×D%


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

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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


w = ku×(1- T×D%)


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)


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.

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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

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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