growth, inflation and terminal value

Chair of Finance, 1 Prof. Dr. Bernhard Schwetzler

Prof. Dr. Bernhard Schwetzler

CCT Center for Corporate Transactions

Chair of Financial Management and Banking

Growth, Inflation and Terminal Value

Dahlem lectures on FACTS

12.11.2009


1. Motivation

2. Conventional wisdom: The Gordon-Shapiro model

3. A challenge to conventional wisdom: The Bradley/Jarrell model

4. Analyzing growth and inflation

5. Bonus track: Beta stability

Agenda


Motivation

Inflation obviously plays a role in corporate valuation, but there is no consensus how to properly take inflation into account

Recent claims: Valuation formulas and accounting numbers have to be adjusted

German Institute of CPA (IDW) proposes a stepwise procedure to combine retention – based growth and inflation – based growth

Bradley and Jarrell (2003, 2008) conclude that the well known Gordon-Shapiro model for terminal value calculation does not properly account for the effects of inflation

Our question: How should terminal values be calculated under inflation?


1. Motivation





Agenda


Earnings before Interest and Taxes (EBIT)*- Taxes

= Net Operating Profit less adjusted Taxes (NOPLAT)

+ Depreciation- Investment (fixed assets)± Change of Net Operating Working Capital

(- investments/+ divestments in operating current assets ± change of short-term liabilities)

± Change of pension reserves

= Free Cash Flow (Entity Approach)

* after interest on pension reserves

Starting point: NOPLAT, retention and free cash flow

• NOPLAT is the fictional profit less adjusted taxes of an all equity-financed firm

• In terminal value calculations retention determines the change in the firm´s assets

Retention as difference between earnings (NOPLAT) and free cash flow


resp. Int · RoI = g · NOPLATt

Growth requires net investments > 0!

In1 · RoI = ∆ NOPLAT2

NOPLATt = FCFt

t1 2 3 4 5

In1

∆NOPLAT2

NOPLATt,FCFt

NOPLATt

FCFt

Starting point = steady state:• Depreciation = Capex or net

investments (Int) = 0

• Net investments into net working capital ∆ NWC = 0

• ∆ Long-term reserves = 0

Steady state ⇒ No growth!⇒ Perpetuity Model

Required retentionin period 1= net

investment

Return on net investment

Additional NOPLAT from period 2…∞

Gordon-Shapiro (GS) model combines growth and return (1)

qt · RoI = gt

As Int/NOPLATt = retention rate q:


NOPLATt,FCFt

NOPLATt

FCFt

t1 2 3 4 5

In1,EC

∆ NOLPAT2

In2,EC

In3,ECIn4,EC

In5,EC

NOPLAT5

FCF5

∆ NOLPAT3

∆ NOLPAT4

∆ NOLPAT5

Retain in every future period:


q · RoI = g ∀ t

[ ]gWACC

q1NOPLATV 1T

T −−

= +

τ

gWACCRoIg1NOPLAT

V1T

T −

−

=τ

+


gWACCRoIg1NOPLAT

V1T

LT −

−

=τ

+

• The return on net investments RoI (Return on invested capital)

• The retention rate q

Along with the Free Cash Flow and the WACC , the perpetual growth rate and the Terminal Value is driven by two factors:

or [ ]RoIqWACC

q1NOPLATV 1TLT ⋅−

−= +

τ

q · RoI = g


τ


For a given growth rate g, RoI determines the fraction of NOPLAT that has to be retained and reinvested to "generate" g.

⇒ is the retention rate (portion of NOPLAT to be retained and invested)RoIg

Company A

g = 6 %; RoI = 20 %

⇒ %302.0

06.0RoIg

==

g = 6 %; RoI = 15 %

⇒ %4015.006.0

RoIg

==

1 2 3 4

NOPLATt 100 106 112.36 119.1

- Int = 0.3 ·NOPLAT -30 -31.8 -33.71 -35.73

FCFt 70 74.2 78.65 88.37

1 2 3 4

NOPLATt 100 106 112.36 119.1

- Int = 0.4 ·NOPLATt -40 -42.4 -44.94 -47.64

60 63.6 67.42 71.46

⇒ Both companies have identical growth rates g for NOPLAT and FCF.⇒ Company A has higher Free Cash Flows than Company B!

Company B

The major impact factor is the return on net investments (RoI)


Profitability of net investments:

Perpetual Formula; Terminal Value is independent of g

⇒ WACCt = RoI Neutral growth; no excess returns

⇒ WACCt > RoI Growth destroys value: Company does not earn its cost of capital!The higher the growth rate g, the lower the terminal value

⇒ WACCt < RoI Growth increases value: Company generates excess returns! The higher the growth rate g, the higher the terminal value

τWACCNOPLATV 1TL

T+=

⇒ for RoI ⇒ ∞ NOPLAT and FCF grow without any growth in assets!

Gordon-Shapiro model allows sanity checks for growth assumptions


Is the (NOPLAT-related) retention rate necessary to achieve the growth rate g

Calculate the retention rate for the projected estimates of FCF and NOPLAT for phase II

1T

1T1T

NOPLATFCFNOPLATq+

++ −=

Link the projected growth rate and the retention rate to calculate the implied Return (RoI) on the net investments

qgRoI =

Cross-check for the profitability assumption behind the growth rate

Compare the implied RoI and the Cost of Capital (WACC) of the firm

1.

2.

3.

RoIg

q =

τWACCRoI ><

The Gordon-Shapiro model may be used to cross-check the assumptionsabout growth and profitability


Basic idea: Retention · Return = Growth

Analysis

Different combinations of retention and return create the same growth rate

I 50 % · 8 % = 4 %

II 25 % · 16 % = 4 %

Not all combinations create value

I RoIØ – WACC = 6 % > 0 %

II RoIØ – WACC = -2 % < 0 %

RoIØNet investments

Retention rate q25 % 50 %

8 %

10 %

16 %g = 4 %

I

WACC

Creating value

IIg = 4 %

Destroyingvalue

Using GS model for terminal value calculation allows for recombining growth by retention and return


For investments with limited lifetime there can be significant differences between RoI and IRR over the years

In € 0 1 2 3 4Cash Flows -900 450 350 450 350Dep. book value -225 -225 -225 -225Profit book value 225 125 225 125Residual book value 900 675 450 225 0RoI Book Value 25% 19% 50% 56%

In € 0 1 2 3 4Cash Flows -900 450 350 450 350Market Value IRR 900 708 561 272 0 Dep market value -192 -147 -289 -272 EVA 258 203 161 78 RoI Market Value 29% 29% 29% 29%

WACC = 10 %, NPV = 341,36 €, IRR = 28,67 % WACC = 10 %, NPV = 341,36 €, IRR = 28,67 %

0%

20%

40%

60%

1 2 3 4

RoI book value RoI market value

Caveat I: We should compare IRR and WACC, not RoI and WACC


Misleading conclusion when comparing average rates of return against WACC

RoIØ = 25 % > WACC = 12 % ⇒ Invest beyond 200!

RoI2 = 0 % < WACC = 12 % ⇒ Scale down investment to 100!

Example

Average

Marginal

Cash-Flowt = 1

Investment

WACC = 12 %RoI2 = 0 %

200

250

150

100

RoIØ = 25 %

RoI1 = 50 %

Caveat II: Important is the marginal rate of return, not the average rate of return


Analyzing investment strategies by comparing RoIØ against WACC allows for two different conclusions

Further industry expertise is needed No further expertise needed

Cash-Flow

Investment

RoIØ > WACC

WACC

RoIØ

J

I

II

RoIØ – WACC > 0

RoII – WACC < 0RoIII – WACC > 0

Depends on the shape of invest-ment curve

Investment

RoIØ < WACC

WACC RoIØ

J

RoIØ – WACC < 0

RoI’ – WACC < 0as RoI’ < RoIØ

Cash-Flow

Average:

Marginal:

Average:

Marginal:


RoI not only links earnings and assets, but also says something about the relation between assetgrowth and earnings growth

Case 1: Asset growth = earnings growth

Case 2: Asset growth > earnings growth

Case 3: Asset growth < earnings growth

RoI

t

RoI

t

RoI

t

Implies a decreasingRoI over time

Implies a constantRoI over time

Implies an increasingRoI over time

Infinite time frame (TV calculation) & constant WACC Case 1 is reasonable

RoI

12 %

Operating assets

100 mill.€

Operating earnings

NOPLAT12 mill.€

Asset growth, earnings growth and rates of return RoI


RoI

12 %

RoI

12 %

Operating assets

100 mill.€

Operating earnings

NOPLAT12 mill.€

Operating assets

100 mill.€

4,8 mill. €

Operating earnings

NOPLAT12,576 mill.€

Retention rateq = 40 %

Payout ratio(1 – q) = 60 %

Retention rate q and RoI determine asset growthAssuming a constant RoI yields the identical earnings growth

Simplification: Check on validity of assumption „constant average return (RoI) @ growing asset base“

Asset growth =Op. assetst-1 * RoI * q

Op. assetst-1

RoI * q

12 % * 40 % 4,8 %

=

= =

Earnings growth =RoI*Op.ass.t-1*[RoI*q+1]

RoI * Op. assetst-1

RoI * q

12,576 / 12 -1 = 4,8 %

-1 -1

=

=

FCF 7,2 mill.€

There is a link between earnings and asset growth via the retention rate


• Standard GS model suggests that, given a significant retention rate, a firm can grow significantlyeven at moderate RoIs

q = 50 %, RoI = 12 % g = 6 % to infinity

• Standard argument against: „If your firm grows faster than the world economy, then at somepoint your firm will be the world economy.“

Let‘s check this: (mill. US dollars)Siemens NOPLAT 2008: 2.5131

World GDP 2008: 60.115.2202

How long is the wait?

It‘s 189 years!

It‘s not about quoting M. Keynes, but the contributionof Free Cash Flows beyond t = 100 (t = 50) tobusiness EV @ WACC = 12 % is just 3 % (16 %)

Assumptionsq = 50 %, g = 8 %g = 2 %

Siemens FCF

World GDP

1 Siemens annual report 2008, Income from continuing operations after income tax; exchange rate: 1,35185 $/€2 World Development Indicators database, World Bank, 15 September 2009

189Years

The killer argument: Firms becoming larger than world GDP


1. Motivation





Agenda


Bradley/Jarrell 2003/2008: „...misapplication of the constant-growth model under conditions ofinflation found throughout the finance and valuation literature.“ (2003, p.5)

Terminal Value ModelGordon / Shapiro

Adjusted Terminal Value ModelBradley / Jarrell

[ ]RoIqk

q1 NOPLATVT ⋅−

−⋅=

[ ]][ π⋅+⋅−

−⋅=

q)-(1 RoIqkq1 NOPLAT

VT

Growth earnings / cash flows:

g = q * RoI

Growth earnings / cash flows:

g = q * RoI + (1-q) *

Growth causedby retention and

investment

Growth causedby retention and

investment

Growth caused byinflation via

increase in asset‘sbook values

Requires inflation an adjustment of the TV-valuation formula?

π

π


The link between asset and earningsgrowth in the GS model

The link between asset and earnings growth in the BJ model

Old assets

100 mill.€

New assets

20 mill.€

Old assets

10 mill.€

New assets

2 mill.€RoI

RoI

BV assets Earnings

Growth in earnings / cash flows is purelycaused by new, additionally acquiredassets

Old assets

100 mill.€

New assets

20 mill.€

Old assets

10 mill.€

New assets

2 mill.€RoI

RoI

BV assets Earnings

AppreciationBV old assets

10 mill.€

Earnings on apprec. BV

1 mill.€

RoI

„...value of initial invested capital will growwith inflation and, with a constant real returnon invested capital, the firm‘s free cash flows... will grow at the same rate (2003, p.4)

Growth in earnings / cash flows is also caused by inflated asset base via appreciation

The additional growth factor is caused by inflation of the book values of the firm’s assets


1. Motivation





Agenda


• Basic version of the Gordon-Shapiro Model

– With q = retained funds:

– Using the link between RoI and retained funds, , we obtain:

• Bradley/Jarrell (2008) propose to adjust the GS-Model under inflation by

Is this adjustment justified?

( )π−+= q1RoIq'G

RoIqG ⋅=

( )GK

q1IncV 10 −

−=

GKRoIG1Inc

V1

0 −

−

=

Inflation, growth and the Gordon-Shapiro model


• The firm has access to one representative project in each period

• The nominal cash flows of the representative project are:

• The real cash flows of the representative project are:

• Link between nominal and real cash flows:

• Assumption: There is a unique internal rate of return:

( )T1 C,,C,b

( )T1 c,,c,b

( )ttt 1cC π+=

A model of the firm* – Cash flow definitions

( ) ( )π+π−= 1Rr

* See: Friedl / Schwetzler (2008): „Terminal Value, Accounting Numbers and Inflation“ available under: http://ssrn.com/abstract=1268930


• We capture growth by change of the size of the representative project

• Nominal growth:

• Note that the internal rate of return does not change with growth

Objective: Calculate terminal values

π+π+= ggG

A model of the firm – Growth definitions

• Real growth g as a percentage change of the size of the representative project (applies to all cash flows)


( )( ) ( ) ( )( ) ( )( ) 1TT

1211 g11cg11c1cg11bCF +−− +π++++π++π+++π+−=

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )11

1

2T2T

22

21

222

g11CF

g11c1cg11cg11bCF

+π+=

+π+++π+++π+++π+−= +−

( )GK

CFK1CFV 1

1t

tt0 −

=+= ∑∞

=

−

A model of the firm – Cash flows of the firm and terminal value calculation

3b−−

-3 -2 -1 0 1 2 33

1−C 3

2C− 33C−

21C− 2

2−C

11C− 1

3C−

01C

23C−

12C−

02C 0

3C

11C 1

2C 13C

2b−

2b−−

1b−−

0b−

1b−

21C 2

2C

Reference object in t=0


CFs (real)Project -3 -94,23 37,69 37,69 37,69Project -2 -96,12 38,45 38,45 38,45Project -1 -98,04 39,22 39,22 39,22Project 0 -100,00 40,00 40,00 40,00Project 1 -102,00 40,80 40,80 40,80Project 2 -104,04 41,62 41,62Project 3 -106,12 42,45Project 4 -108,24

Sum CF real 15,36 15,66 15,98 16,30 16,62Growth 2,00% 2,00% 2,00% 2,00%

CFs (nominal)Project -3 -86,24 35,53 36,60 37,69Project -2 -90,60 37,33 38,45 39,60Project -1 -95,18 39,22 40,39 41,60Project 0 -100,00 41,20 42,44 43,71Project 1 -105,06 43,28 44,58 45,92Project 2 -110,38 45,47 46,84Project 3 -115,96 47,78Project 4 -121,83

Sum CF nominal 15,36 16,13 16,95 17,81 18,71Growth 5,06% 5,06% 5,06% 5,06%

Assumptions:Cost of Capital(nominal): 9%Inflation: 3%Growth: 2%

Numerical example: Cash flows


• For the representative project, we define accounting earnings as:

• Total accounting earnings of the firm:

• Accounting earnings grow at nominal rate G:

• Book values grow at G as well:

• Return on Investment stays constant:0

1

1t

tt BV

IncBVIncRoI ==

−

( ) bd1cbdCInc tt

tttot −π+=−=

π+=−1t

tt BV

IncRoI

Bradley/Jarrell (2003)

A model of the firm – Accounting earnings and accounting returns

( ) 1t1t G1IncInc −+=

( )t0t G1BVBV +=

Project in t=0 Project in t=-1 Project in t= -T+1

( ) ( )[ ] ( ) 12

22111 G1bd1cbd1cInc −+⋅⋅−π++⋅−π+= ( )[ ] ( ) 1T

TT

T G1bd1c +−+⋅⋅−π++...


Book valuesProject -3 86,24 57,49 28,75 0,00Project -2 90,60 60,40 30,20 0,00Project -1 95,18 63,46 31,73 0,00Project 0 100,00 66,67 33,33 0,00Project 1 105,06 70,04 35,02 0,00Project 2 110,38 73,58 36,79Project 3 115,96 77,31Project 4 121,83

Sum Book values 193,66 203,45 213,75 224,57 235,93Growth 5,06% 5,06% 5,06% 5,06%

ProfitProject -3 0,00 6,78 7,85 8,95Project -2 0,00 7,13 8,25 9,40Project -1 0,00 7,49 8,66 9,88Project 0 0,00 7,87 9,10 10,38Project 1 0,00 8,26 9,56 10,90Project 2 0,00 8,68 10,05Project 3 0,00 9,12Project 4 0,00

Sum Profit 24,68 25,93 27,24 28,62 30,07Growth 5,06% 5,06% 5,06% 5,06%

Numerical example: Book values and earnings


( )( ) ( )( )( )

+π+−+π+=− ∑

=

+−T

1j

1jj11 g11dg11bCFInc

( )( ) ( ) ( ) ( )( ) ( ) 1TT

T

12211

g11c

...g11c1c1g1bCF+−

−

+π++

+π++π++π++⋅−=–

Earnings/Income in t=1: Cash flow in t=1:

Retention:

Effect of investmentin t=1

Effect of depreciationof investments prior t=1

Requirement for zero net investment-case: ( )( ) 1g11 =+π+ (because ) ∑=

=T

1jj 1d

For the zero net investment case, real growth and inflationneed to neutralize each other. This yields also G = 0 .

The model facilitates the analysis of special cases

( ) ( )[ ] ( )( )[ ] ( ) 1T

TT

T

12

22111

G1bd1c

...G1bd1cbd1cInc+−

−

+⋅⋅−π++

++⋅⋅−π++⋅−π+=


1. Zero real growth and positive inflation: g* = 0, π > 0

In this case, nominal growth of income and cash flow equals the inflation rate: G* = π. Therefore, terminal value is:

2. Positive real growth and zero inflation: g* > 0, π = 0 In this case, terminal value is:

( ) ( )( ) 01d1bCFIncRETn

1j

1jj111 >

π+−π+=−= ∑

=

+−

Two special cases

( )*

1*

10 gK

q1IncGK

CFV−−

=−

=

( )π−−

=−

=K

q1IncGK

CFV 1*

10


• First, BV1 – BV0 = G BV0. Thus retention in period 1 is RET1 = G BV0.

• Second, absolute retention can be expressed by combining retention rate and accounting income in period 1: RET1 = q Inc1.

• Setting the two equations equal to each other, we get:

Hence, the presence of inflation does not affect the GS relation

RoIqBVIncq*GIncqGBV

0

11

*0 ==⇔=

General case: Arbitrary growth and inflation


• If the firm makes zero net investments, RETt = 0 ∀ t

• Retention is zero, if and only if , i.e. zero nominal growth

• In this case, Gordon-Shapiro simplifies to

• Bradley and Jarrell (2003, 2008) obtain a different representation of the Gordon-Shapiro model under inflation

Under inflation there is no need to adjust the Gordon-Shapiro model in the special case of zero net investments

( )( ) 11g1 * =π++

KIncV 1

0 =

The Gordon-Shapiro model for zero net investments


• We analyze an expansion of investment beyond the optimal scale and assume that the firm retains and additionally reinvests an amount of RET1 at its cost of capital K in t=1

• Again under positive net investments the absolute retention reflects the nominal growth in assets: RET1 = G´*BV0

• G´ denotes the nominal growth rate in assets under the sub-optimal investment program including the zero-NPV projects

• Linking retention and growth under the zero-NPV assumption requires an additional assumption ensuring that K = IRR = RoI is going to hold in any future period

• Except for the trivial case of a one period lifetime of the representative project, the condition RoI = IRR requires a special depreciation policy of the firm, the “IRR depreciation” for the representative investment that meets the requirement RoIt = IRR for t=1, ..., n

The Gordon-Shapiro model for zero net present value investments (1)


• Under this assumption period t=2 income is:

• Income growth between t=1 and t=2 is then:

Thus finally the Zero – NPV net investment case also yields the GS valuation equation as the correct result:

KqIncIncRETRoIIncInc 11112 ⋅⋅+=⋅+=

´GKq1IncInc

1

2 =⋅=−

The Gordon-Shapiro model for zero net present value investments (2)

( )K

IncKqKq1IncV 11

0 =−

−=

( )Kq1Inc1 ⋅+=


There is no need for adjustments of the valuation formula. The GS-valuation formula isbased on nominal future cash flows and nominal cost of capital and thus fully captures theimpact of inflation

Yes, inflation does have an impact upon firm value. This impact is highly firm-specific(ability to pass price increases to customers etc.) and reflected in estimates of future cash flows

No, inflation does not have an impact on the valuation formula

(Our) conclusion)*:

• Additional earnings / cash flows should only be caused by returns on newly acquired assets

• Writing up the book value of an asset should not have any impact on the firm‘s cash flows

• Accounting standards usually do not allow for appreciations above historic costs

* See: Friedl / Schwetzler (2008): „Terminal Value, Accounting Numbers and Inflation“

Some points against the logic of the BJ valuation formula


1. Motivation





Agenda


The predictability of stock returns and the link of returns and risk associated with stocks are issues of major interest among academics and practitioners in the area of finance

A systematic explanation of these relations has been made by Sharpe (1964) and Lintner (1965) with the development of the capital asset pricing model (CAPM)

An important aspect of the model is the consistency of the risk-measuring factor

beta

Practical applications of the CAPM implicitly assume beta-factor stability over

the whole period under consideration

Various CAPM tests have been conducted, fuelled by the many necessary assumptions the model builds on:

Empirical research on CAPM validity and beta-stability yields contradictory results

Until 1992, many studies affirmed the explanatory power of the CAPM

In 1992, Fama/French published their landmark study, opposing the CAPM

• e.g. Black/Jensen/ Scholes (1972), Blume/Friend (1973), Fama/MacBeth (1973

• Fama/French (1992) do not find beta to be an explanatory factor for average stock returns

• They find size & BE/ME-ratio more appropriate

Subsequently, the interest in research on CAPM validity and beta-factor stability has remained

Controversy of results among the numerous studies

• Finance researchershave been keenlyinterested in exploringthe stability charact-eristics of the beta-factor

• German studies: Sauer/ Murphy (1992) & Elsas/ El-Shaer/Theissen (2005) con-firmed validity of the CAPM, while Ebner/ Neumann (2005) disapproved


Reliability of estimation Stability of estimation

„True“ betas can also change over time!

„True“ beta

Estimator

Instable

Stable

Stable

Instable

Reliable

Reliable

Unreliable

Unreliable

Only the estimatorcan be observed!

• Low beta stability is not necessarily a reason for a critique of the CAPM• During a „regime shift“ R2 decreases even if the model is true!

Stability of beta-factors (or rather of their estimators) is not necessarily a desirable attribute


Pos

itive

rj

rm

. . . . . . .

. . .

. . . . .

. .. .

rj

rm

. . . . . . . .. .

. . . . .. .

Changes in betaC

hang

esin

R2

Positive NegativeN

egat

ive

• Low beta• Low R2

• High beta• High R2

• High beta• Low R2

• Low beta• High R2

• Low beta• High R2

• High beta• Low R2

• High beta• High R2

• Low beta• Low R2

rj

rm

. . . . . . .

. . .

. . . . .

. .. .

. . . . .. . .

rj

rm

. . . . . . . .. . .

. . . . . .

. . .

. . . . .

. .. .

rj

rm

rj

rm

. . . . . . . .. .

. . . . .. .

rj

rm

. . . . .. . .

rj

rm

. . . . . . .. . .

Simultaneous changes of beta and R2 indicate a “regime shift”

There are four different types of changes and regime shifts

Type I Type II

Type IVType III


Beta

R2

Automobile 200 days

Automobile 200 days

Regime Shift Type IV

Effects of the technology bubble:

between 1999 and 2001

Decreasing betas

Decreasing goodness of fit

Betas and goodness of fit are subject to significant changes even on industry level: E.g. automobile


Beta

R2

Banks 200 days

Banks 200 days

Betas and goodness of fit are subject to significant changes even on industry level: E.g. banks

Regime Shift Type IV



Decreasing betas

Decreasing goodness of fit


Beta

R2

Media 200 days

Media 200 days

Betas and goodness of fit are subject to significant changes even on industry level: E.g. media

Regime Shift Type I



Increasing betas

Increasing goodness of fit


Beta

R2

Telco 200 days

Telco 200 days

Betas and goodness of fit are subject to significant changes even on industry level: E.g. telco

Regime Shift Type I



Increasing betas

Increasing goodness of fit

growth, inflation and terminal value

Documents