cash flow multipliers and optimal investment decisions

Cash Flow Multipliers and OptimalInvestment Decisions

Holger KraftGoethe University, Department of Finance, Frankfurt am Main, GermanyE-mail: [email protected]

Eduardo SchwartzUCLA Anderson School of Management, Los Angeles, USAE-mail: [email protected]

Abstract

Valuation multipliers are frequently used in practice. By postulating a simplestochastic process for the firm’s cash flows in which the drift and the variance ofthe process depend on the investment policy, we develop a stylised model thatlinks the cash flow multiplier to the optimal investment policy. Our model impliesthat the multiplier increases with investment at a decreasing rate. On the otherhand, the multiplier is inversely related to discount rates. Using an extensive dataset we examine the implications of our model. We find strong support for thevariables postulated by the model.

Keywords: firm valuation, valuation multiples, real options

JEL classification: C61, G12, G13, M40

Valuation multiples are frequently used in practice since they offer a quick way to value afirm without estimating the whole series of future cash flows. This paper offersa parsimonious model that relates the value of a firm to its current cash flow. Postulating asimple stochastic process for the firm’s cash flows (before investment) in which the drift

We thank David Aboody, Saurabh Ahluwalia, Geert Bekaert, Michael Brennan, JudsonCaskey, Peter Ove Christensen, John A. Doukas (the editor), Mark Garmaise, DanielHoechle, Jack Hughes, Ralph Koijen, Christian Laux, Volker Laux, Hanno Lustig, KristianMiltersen, Claus Munk, Naim Bugra Ozel, Richard Roll, Geoff Tate, and an anonymousreferee for very helpful discussions, comments, and suggestions. We also thank seminarparticipants at Aarhus University, Georgia State University, Hebrew University Jerusalem,Koc University Istanbul, Universidad Carlos III in Madrid, Universidad de Valladolid(Simposio de Opciones Reales), the 1st World Finance Conference in Viana do Castelo(Portugal), the Copenhagen Business School, and the Financial Economics Conference atGraduate School of Economics of the Fundacao Getulio Vargas in Rio de Janeiro for manyhelpful comments and suggestions. All remaining errors are of course our own. Holger Kraftgratefully acknowledges financial support by Deutsche Forschungsgemeinschaft (DFG).

European Financial Management, Vol. 9999, No. 9999, 2014, 1–31doi: 10.1111/eufm.12047

© 2014 John Wiley & Sons Ltd

and the variance of the process depend on the investment policy of the firm, we derive aclosed‐form solution of the cash flow multiplier that takes into account optimalinvestment and how this investment affects future cash flows.We show how the cash flowmultiplier is (negatively) related to the discount rate and (positively related) to optimalinvestment. Furthermore, it is shown that the multiplier depends on optimal investment ina nonlinear way. Then this multiplier is decomposed into two parts: the first part reflectsthe firm value without investment, whereas the second part captures the option to investoptimally in the future.Using a data set comprised of more than 15,800 firms over 38 years we examine the

different predictions of our model using macro and firm‐specific explanatory variables.Both types of variables can be subdivided into variables that are part of our model andvariables that are used as controls. For instance, we include macro variables that affect thediscount rate such as the real short‐term interest rate, the slope of the term structure ofinterest rates, and a credit spread (spread of Baa bonds over Treasuries). Increases in all ofthese variables have a positive effect on the discount rate, and therefore should have anegative effect on the cash flow multiplier. Besides, we add inflation and the volatility ofthe S&P500 index to control for the state of the economy. As firm specific variables weinclude the proportion of cash flows invested that comes directly from our theoreticalmodel and should, if investment is optimal, be positively related to the cash flowmultiplier. We also run regressions where we include a squared term to check the modelprediction that the multiplier increases with investment at a decreasing rate. In most of theregressions we include size, leverage, and a dividend dummy as control variables as wellas firm and/or industry fixed effects. Besides, we study the effect of R&D expenses thatare part of a firm’s investment policy, but that are missing in about 50% of theobservations. We find that all the explanatory variables related to the discount rate ‐ thereal short term interest rate, the slope of the term structure, and the spread of Baa bondsover Treasuries ‐ have the correct sign and most of them are significantly negative. Theproportion of cash flow invested is always highly significant and positive as predicted bythemodel.We also provide empirical evidence that the relationship is nonlinear and relatethe cash flow multiplier to the average size of an industries’ investment policy.Furthermore, firms in certain industrial sectors require more investment because

obsolescence in the sector is faster, because the sector is more competitive, or because thesector is more heavily regulated. In our theoretical model the drift of the cash flow process(without investments) can proxy for this phenomenon. The smaller (or more negative) isthis drift without investments, the more investment will be required to keep or increase thelevel of future cash flows. This would imply that, even though the cash flowmultiplier fora given firm is positively related to the proportion of its cash flow invested, the multipliershould be negatively related to the average investment proportion of the industry to whichit belongs since it would be amore intensive investment industry.We find evidence of thisin the data as well.While some of our empirical results are consistent with simpler models, i.e. that the

multiplier is positively related to the reinvestment rate and negatively related to theinterest rate, in our model the reinvestment rate is endogenous and the relation betweenthemultiplier and the reinvestment rate is non‐linear. In addition, as mentioned above, ourmodel can address issues that simpler models cannot, such as the relation of the firmmultiplier to the average investment proportion of the industry to which it belongs.Our paper is related to several strands of the literature. First it contributes to an extensive

literature on multiples. Boatsman and Baskin (1981) and Alford (1992) analyse the


2 Holger Kraft and Eduardo Schwartz

accuracy of price‐earning multiples. Boatsman and Baskin (1981) document that for firmswhich have homogenous historical earnings growth the valuation errors are smaller. Alford(1992) studies empirically how valuation errors vary when comparable firms are selectedaccording to industry, risk, and earnings growth.Hefinds that pricing errors can be reducedif a more sophisticated industry definition is used. Baker and Ruback (1999) study how toestimate industry multiples and how to choose a measure of financial performance as abasis of substitutability. They find that EBITDA is a better single basis of substitutabilitythan EBIT or revenue. They analyse the valuation properties of a comprehensive list ofmultiples and also examine related issues such as the variation in performance acrossindustries and over time. Liu et al. (2002) analyse the valuation properties of acomprehensive list of multiples. They also examine related issues such as the variation inperformance across industries and over time. This analysis is extended by Liu et al. (2007).Bhojraj and Ng (2007) examine the relative importance of industry and countrymembership in explaining cross‐sectional variation in firm multiples. These papers,however, do not include macroeconomic variables in the analysis. Nissim and Penman(2001) develop a structural approach to financial statement analysis for equity valuation.They identify ratios as drivers of future residual earnings, free cash flows and dividends.In a series of papers, Ang and Liu (2001, 2004, 2007a) address theoretical issues related

to our paper. For example, Ang and Liu (2001) derive a model that relates firm value toaccounting data under stochastic interest rates, heteroskedasticity and adjustments for riskaversion. Ang and Liu (2004) develop a model that consistently values cash flows withchanging riskfree rates, predictable risk premiums, and conditional betas in the context ofa conditional CAPM. Finally, Ang and Liu (2007a) show theoretically that a givendividend process and any of the variables – expected return, return volatility, and theprice‐dividend ratio – determines the other two. Although they do not model investmentdecisions explicitly, they derive a partial differential equation for the price‐dividend ratiothat is also satisfied by the cash flow multiplier in our paper given that the correspondingfirm invests optimally. Therefore, our model can be thought of as an extension of theirmodel endogeniseng a firm’s investment decision. Furthermore, Brennan and Xia (2006)analyse the risk characteristics and valuation of assets in an economy in which theinvestment opportunity set is described by the real interest rate and the maximum Sharperatio. They study the betas of the securities and the discount rates of risky cash flows intheir model.Berk et al. (1999) develop amodel that values the firm as the sum of the present value of

its current cash flows and its growth options and is thus similar in spirit to the theoreticalmodel in our paper. But their main interest is to study the dynamics for conditionalexpected returns. In a related paper, Carlson et al. (2004) derive two theoretical modelsthat relate endogenous firm investment to expected return. Titman et al. (2004) documenta negative relation between abnormal capital investments and future stock returns.Anderson and Garcia‐Feijoo (2006) find in an empirical study over the time period from1976 to 1999 that growth in capital expenditures explains returns to portfolios and thecross section of future stock returns. Furthermore, Liu et al. (2009) and Li and Zhang(2010) study the cross section of stock returns in investment‐based asset pricingframeworks. Xing (2009) finds that in the cross‐section portfolios of firms with lowinvestment‐to‐capital ratios have significantly higher average returns than those with highinvestment‐to‐capital ratios.Our theoretical findings concerning the value of a firm’s option to invest are related to

the real options literature that started with the papers by Brennan and Schwartz (1985) and


Cash Flow Multipliers and Optimal Investment Decisions 3

McDonald and Siegel (1986). More recently, Grenadier (2002) and Aguerrevere (2009)show that in competitive markets the value of the option to invest can decreasesubstantially.Finally, our paper is related to the literature that looks at the investment sensitivity

of stock prices. There are a number of papers that have studied this sensitivity froma theoretical and empirical point of view, including Dow and Gorton (1997),Subrahmanyam and Titman (1999), and, more recently, Chen et al. (2007).The paper proceeds as follows. Section 1 develops the theoretical model. Section 2

solves for the optimal investment and the optimal cash flowmultiplier and illustrates someof the implications of the model. Section 3 introduces our benchmark free cash flowdefinition that is based on a firm’s income statement. We also provide an alternative freecash flow definition based on the firm’s cash flow statement. Section 4 describes the dataused in the empirical analysis and Section 5 presents the results of the panel regressions.We also include additional explanatory variables such as R&D expenses and leverage.Section 6 reports results of several robustness checks on the basic results. In particular, westudy the robustness of our results to the alternative free cash flows definitions. Section 7provides insights into the value of the option to invest using the data available. Finally,Section 8 concludes. Some of the proofs are given in Appendix A.

1. Model

This section develops a parsimonious model where firm value naturally arises as thepresent value of the firm’s free cash flows. Investment decisions are made endogenouslyaffecting the expected growth rate and the volatility of the cash flow stream. The firmvalue is given by1

V ðc; xÞ ¼ maxp

E

Z 1

0eR s

0RuduðCs � I sÞds

� �; ð1Þ

where C denotes the firm’s free cash flow before investments and I ¼ pC denotes thedollar amount of the cash flow that is invested.2 The variable p can be continuouslyadjusted by the firm and stands for the percentage of the cash flow that the firm invests.This paper is concerned with valuing a firm as present value of total cash flows and is notdirectly addressing the issue of the capital structure. However, if the percentage p isbigger than one, this can be interpreted as a situation where the firm uses externalfinancing (equity or debt). The variable x denotes the initial value of a state process X thatcaptures economic variables that impact the firm value, such as interest rates. The risk‐adjusted discount rate R is assumed to be a function of this state process, i.e. with a slightabuse of notation R ¼ RðX Þ. For illustration purposes, suppose for simplicity that X isequal to the default‐free short interest rate and that the risk‐adjusted discount rate is linearin the short rate, i.e.

R ¼ wþ cr ð2Þ

1An alternative representation using the pricing kernel is provided in Appendix C.2At this point, this assumption is without loss of generality. Formally, p is a an adaptedprocess. Therefore, one can always choose p ¼ f=C where f is an adapted process.



with w and c being constants. A simple model that would fit into this framework is thefollowing: Suppose that the risk‐adjusted discount rate is given by R ¼ r þ bl, where l isthe risk premium and b is the firm’s beta that is constant. If the default‐free interestrate predicts the risk premium, then the premium could be linear in the interest rate,l ¼ l þ lrr; with constants l and lr such that w ¼ bl and c ¼ 1þ blr in ourparametrisation (2). We assume the cash flow to follow the dynamics3

dC ¼ C½mðp;X Þdt þ sðp;X ÞdW �; Cð0Þ ¼ c; ð3Þ

where the expected growth rate and volatility, m and s, are functions of the state processand the percentage of the firm’s cash flow reinvested. The process W is a Brownianmotion. This specification implies the following result:

Proposition 1 (Linearity of Firm Value) Firm value is linear in the cash flow, i.e.

V ðc; xÞ ¼ f ðxÞc; ð4Þ

where f ðxÞ ¼ V ð1; xÞ.Notice that V=c is the ratio of the current firm value over the current cash flow (for

short: cash flowmultiplier), which will be central in our further analysis. One can think ofthe cash flow multiplier as the multiple by which the current cash flow is multiplied toobtain the current firm value. In the literature on the dividend‐discount model and itsgeneralisations, usually this multiplier is assumed to be beyond the control of the firm andthus to be exogenously given. In contrast, we explicitly model the firm’s opportunity tochange its risk‐return tradeoff by allowing the firm to control the expected growth rate andthe volatility of the cash flow stream by its investment policy.4 To illustrate our model andunless otherwise stated, we use the following specification of these parameters5

mðp; xÞ ¼ m0ðxÞ þ m1

ffiffiffip

p þ m2p; sðpÞ ¼ s0 þ s1ffiffiffip

p þ s2p; ð5Þ

where all coefficients except for m0 are constants and m0 is a linear function of the stateprocess,m0ðxÞ ¼ �m0 þ m0x. The functionm0 characterises the expected growth rate if thefirm does not invest at all (p¼ 0). If the firm does not invest at all, then projects in placewill depreciate. Therefore, we expectm0 on average to be negative. The parametersm1 andm2 capture the firm’s ability to control its future growth rate. To avoid explosion of themodel, m2 is assumed to be negative. Economically, this is justified since firms mighteventually realise negative NPV projects.6 As we show later on, representation (5) gives a

3This builds on the ideas of Merton (1974), Duffie and Lando (2001), and Goldstein et al.(2001), among others, who use lognormal models in which the firm cannot control forinvestment. Note that this representation does not allow for negative cash flows. In addition,this is true for all the literature dealing with multipliers.4 In general, the dependence of m and s might be involved. For instance, nonlinearspecifications can reflect frictions.5Any concave function for the drift would provide the same qualitative results. Thisparametrisation is chosen simply for computational convenience.6Notice that (5) comes from a quadratic function where the argument is replaced by the squareroot.



decomposition of the cash flowmultiplier comparable to the one for the firm value derivedby Berk et al. (1999).Furthermore, we allow the investment decisions to have an impact on the riskiness of

the firms cash flow stream and thus the volatility s can depend on p as well. For instance,if a firm invests in a new product, then both the firm’s expected growth rate and thevolatility of its cash flow stream might increase. Figure 1 illustrates two stylisedspecifications of the drift when the investment proportion is varied between zero and one.The drift starts below zero and then increases until it reaches its peak. For the lower curvethe peak is reached around p¼ 0.7, whereas for the upper curve the peak is reached forsome p that is greater than one.

2. Solving for the Optimal Cash Flow Multiplier

The firm’s decision problem (1) is a dynamic optimisation problem that can be solvedusing stochastic control methods. This is the first goal of this section. We assume thatthe state of the economy is characterised by the short interest rate that has Vasicekdynamics7

dr ¼ ðu � krÞdt þ hdWr; ð6Þ

Fig. 1. Functional forms of the expected growth rate.

The figure illustrates two different forms of the expected growth rate. In both cases, it is assumed thatm0 ¼ �0:03 and m1 ¼ 0:1. For the upper curve, we have m2 ¼ �0:03 and for the lower onem2 ¼ �0:06.

7A generalisation to two state variables can be found in Appendix D that is available from oneof the authors’ website at https://sites.google.com/site/hkraftfinance/publications.



https://sites.google.com/site/hkraftfinance/publications

whereWr is a Brownian motion that is correlated with the Brownian motionW that drivescash flows, i.e. d < W ;Wr >¼ rdt with constant correlation r. In this specification, theshort interest rate captures the randomness of the discount rate. For simplicity, we assume(2) and (5).8 In AppendixA, it is shown that the cash flowmultiplier satisfies the followingBellman equation

0 ¼ maxp

fð�m0 þ m0r þ m1

ffiffiffip

p þ m2pÞf þ 1� p� ðwþ crÞfþðu � krÞf r þ 0:5h2f rr þ rhðs0 þ s1

ffiffiffip

p þ s2pÞf rg:ð7Þ

Under the assumption that the Bellman equation is concave in p, which follows ifm1 > 0, the optimal investment strategy of the firm is given by

p� ¼ m1f þ rhs1f r2ð1� m2f � rhs2f rÞ

� �2

: ð8Þ

Substituting the optimal investment level back into the Bellman equation (7) leads to adifferential equation for the cash flow multiplier

0 ¼ ðw þ crÞf þ 1þ ðu þ rhs0 � krÞf r þ 0:5h2f rr þðm1f þ rhs1f rÞ2

4ð1� m2f � rs2hf rÞ; ð9Þ

where w ¼ �m0 � w and c ¼ m0 � c are constants. Notice that the last ratio in (9)disappears if the firm does not invest. In this case, the cash flowmultiplier has the explicitsolution

f 0ðrÞ �Z 1

0E e

R s

0wþcrudu

h ids ¼

Z 1

0eAðsÞ�BðsÞrds; ð10Þ

with A and B being deterministic functions of time. The expected value E½�� is taken underthe measure under which the short rate has the dynamics

dr ¼ ðu þ rhs0 � krÞdt þ hdW ð11Þ

with W being a Brownian motion under this measure. If the firm is investing optimally,then the last term in (9) can be thought of as an additional cash flow that the firm is able togenerate by doing so. From a real option perspective, this fraction can be interpreted as thefirm’s option to invest optimally at a particular time t in the future. Brealey, Myers, andAllen (2010) call this the net present value of growth opportunities. The present value ofthis continuous series of options is given by

Oðr; f Þ ¼Z 1

0E e

R s

0wþcrudu ðm1f ðrsÞ þ rhs1f rðrsÞÞ2

4ð1� m2f ðrsÞ � rs2hf rðrsÞÞ

" #ds ð12Þ

8Otherwise, the corresponding Bellman equation must be solved numerically.



such that the optimal cash flow multiplier becomes the sum of (10) and (12), i.e.

f ðrÞ ¼ f 0ðrÞ þOðr; f Þ: ð13Þ

The second argument in the definition ofO is added to emphasise thatO depends on f.The firm has a series of options to invest and the net present value of these options ispositive, O � 0. The option value O is however not explicit since it depends on theoptimal cash flowmultiplier fwhich is unknown and a part of the solution.9 Nevertheless,at least the first part of the representation (13) is explicitly known and equal to the solutionwithout investing. Berk, Green, and Naik (1999) derive a similar decomposition for thefirm value.10

To gain further insights, let us assume for the moment that the interest rate r is constant.In this case, it makes sense to simplify notations by setting m0 ¼ 0, w ¼ l ¼ const, andc ¼ 1. This implies that m0 ¼ �m0 ¼ const and w þ cr ¼ m0 � r � l ¼ const. The risk‐adjusted interest rate is the sum of the short rate and a risk premium, i.e. R ¼ r þ l.Furthermore, the optimal cash flow multiplier f is a constant and (13) simplifies into

f ¼Z 1

0eðm0�r�lÞs dsþ

Z 1

0eðm0�r�lÞs ðm1f Þ2

4ð1� m2f Þds|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼Oðf Þ

; ð14Þ

where the transversality conditionm0 � r � l < 0 is assumed to hold. Notice that, by (8),equation (14) can be rewritten as follows f ¼ f 0 þ f 0ð1� m2f Þp�

; where f 0 ¼R10 eðm0�r�lÞsds is the cash flow multiplier without investment. In this special case,one can see some parallels to Berk et al. (1999). The risk premium l captures thesystematic risk and

R10 eðm0�rÞsds can be interpreted as perpetual, riskless consol bond

where the payments on this bond depreciate at the a constant rate m0 < 0. One differencehowever is that in their model the firm has a series of given European options to invest,whereas in our paper the firms endogenously controls the fraction of cash‐flows to beinvested, which makes the representation of the cash flow multiplier implicit.Solving for the optimal multiplier f and taking logarithms yields

ln f ¼ ln f 0 þ lnð1þ p� Þ � lnð1þ f 0m2p

� Þ; ð15Þ

which establishes a nonlinear relationship between the cash flow multiplier and theinvestment strategy p� that we will use later on. The following proposition shows thatunder a mild condition a unique cash flow multiplier exists.

Proposition 2 (Cash Flow Multiplier under Constant State Process) If m21=4� m2ðm0�

r � lÞ < 0, then the optimal cash flow multiplier is uniquely given as the positive root ofthe quadratic equation

0 ¼ m21=4� m2ðm0 � r � lÞ�

f 2 þ ðm0 � r � l� m2Þf þ 1: ð16Þ

9From a formal point of view, (13) is a fixed point problem for f.10See equation (32) in their paper.



Notice that in the special case when the firm has no control over the expected growthrate of its cash flow stream (m1 ¼ m2 ¼ 0), relation (16) becomes a linear equation withsolution f ¼ 1=ðr þ l� m0Þ. This is a version of the Gordon growth model.Furthermore, due to the transversality condition, a necessary requirement for thecondition of Proposition 2 to hold is m2 < 0.To study the implications of Proposition 2, let us consider a numerical example. Similar

to the previous section, we choose m0 ¼ �0:03, m1 ¼ 0:1 and m2 ¼ �0:03. Besides, weassume that r¼ 0.04 and l¼ 0.03 such that the risk‐adjusted interest rate is R¼ 0.07. Thepositive root of (16) which is the cash flow multiplier equals 13.06. If the firmsuboptimally decides not to invest, then the option value in (14) is zero and the cash flowmultiplier is 10. Therefore, the option value equals O ¼ 3:06. Put differently, theopportunity to invest increases the cash flow multiplier by 30 percent. Let us consider asecond example where all parameters are the same as in the first example except for m0

which is assumed to be ‐0.05. As briefly discussed above, one reason for this lower valuemight be that the industry requires more investments to maintain its cash flow level. Thecash flow multiplier resulting from optimal investing is now 9.91, whereas the cash flowmultiplier without investing is 8.33. Therefore, the option value becomes O ¼ 1:58 or18% of the optimal cash flow multiplier. This suggests that in an investment intensiveindustry the real option to invest loses value both in absolute as well as in relative terms. Infact, we are able to show that this is in general true.

Theorem 3 (Value of the Option to Invest) If, in addition to the assumption ofProposition 2, condition m0 � r � l� m2 < 0 holds, then the optimal cash flowmultiplier f, the option value O, and the ratio O=f are increasing in m0.

Remark. The requirement m0 � r � l� m2 < 0 is a bit stronger than the transversalitycondition sincem2 < 0. Nevertheless, it is satisfied for reasonable parametrisations of themodel.

Put differently, the previous theorem says that the option’s absolute and relative valuesdecrease ifm0 becomesmore negative. This result puts some of the classical results on realoptions into perspective and it is related to Grenadier (2002) and Aguerrevere (2009): Ifthe firm is forced to invest for instance because competitors do the same, then the option toinvest loses (part of) its value. Hence, the cash flow multiplier decreases.Finally, we establish the relation of the cash flow multiplier and a stochastic short rate.

Then, the presence of the fraction in (9) turns the differential equation into a highlynonlinear equation, which makes solving the equation more challenging.11 Nevertheless,there is an explicit power series representation.

Theorem 4 (Optimal Cash Flow Multiplier under Stochastic Interest Rates) The cashflow multiplier has the following series representation

f ðrÞ ¼X1n¼0

X1i¼0

aðnÞi r � u

k

� �i

hn ð17Þ

where the coefficients aðnÞi satisfy an explicit recursion provided in Appendix B.

11At least this is so if there are two state variables since in this case the equation is a nonlinearpartial differential equation, a case treated in Appendix D that is available from one of theauthors’ website at https://sites.google.com/site/hkraftfinance/publications.



https://sites.google.com/site/hkraftfinance/publications

To illustrate the implications of the previous theorem we consider a particular firm,Coca Cola. We calculate the cash flow multiplier over the time period from 1971 to 2008,where the relevant information about Coca Cola comes fromCompustat.12We choose theparameters of the riskfree short rate process (6) to be k¼ 0.08, h¼ 0.015, and u¼ 0.004.This implies that the mean reversion level is u=k ¼ 0:05, which is about the sampleaverage of the one‐month Fama‐French riskfree rate as reported by CRSP.13 Firm value isdefined as the sum of the book value of the assets plus the difference between the marketvalue and book value of the equity, minus deferred taxes. Free cash flows (beforeinvestment) are defined as EBITDA plus asset sales minus taxes and the change inworking capital.14 For Coca Cola we have 38 observations of the cash flow multiplierobserved on December 31 of the particular year. Figure 2 plots the logarithms of theseobservations against the realisations of the Fama‐French one‐month riskfree rate.Obviously, both variables are negatively related. This figure also depicts the least‐squarefit of our model which is almost linear. Various numerical experiments have shown thatthis is a general feature of our model.

Fig. 2. Log cash flow multiplier of Coca Cola.

This figure depicts the logarithms of 38 observations of Coca Cola’s cash flowmultiplier over the periodfrom 1971 to 2008, as a function of the riskfree rate. It also shows the fit of our model, which is very closeto linear.

12We postpone a detailed description of the data and the definition of the variables to Sections3 and 4.13The reason for not formally estimating u and k over the period from 1971 to 2008 is thatthere are regime shifts during this period like the spike in 1979 that cannot be well calibratedby a one‐factor model.14A more detailed discussion about free cash flows is provided in the next section.



3. Free Cash Flows

One of the main objectives of our model is to estimate the total value of the firm. Since ourtheoretical model solves directly for the cash flow multiplier, we need an estimate of thefree cash flows (before investments) to be able to derive the total value of the firm.Moreover, in the empirical implementation of the model we need both the total value ofthe firm and the free cash flows (before investments) to calculate the multipier. Free cashflows are cash flows generated by a business, which might be different from those paid tothe firm’s claimants. This distinction is not the focus of our study since we are interested invaluing the whole firm.15 Free cash flows can be derived in two ways: First, usinginformation from a firm’s income statement, one can adjust earnings for interestpayments, investments and accruals. More precisely, these means16

Free Cash Flows ¼ Earningsþ Interest� Investments� Accruals ð18Þ

Second, a firm’s statement of cash flows gives direct access to information aboutcash flows. This statement was mandated by the Financial Accounting StandardsBoard in 1987 and is available in Compustat since 1988. To be able study the effect ofdiscount rates on cash flow multipliers over several business cycles, it is desirable tohave a long time series of cash flow multipliers. Therefore, we decided to use the firstdefinition of free cash flows for which enough data is available since 1971. In ourrobustness section, we also consider the alternative cash flow definition that is derivedfrom the cash flow statement. As will be seen, the results are very similar for bothdefinitions.To be more concrete, for our benchmark case we compute free cash flows (before

investments) on the basis of Table 1 that reports a standard derivation of free cash flowsusing a firm’s income statement.17 The third column provides the correspondingCompustat item. Since we are interested in studying free cash flows before investments,we start with EBITDA and adjust for taxes and changes in working capital and asset sales(see Table 2). In the light of relation (18), one can verify that accruals are proxied bychange in working capital, asset sales, and depreciation and amortisation.18 Tosummarise, in our benchmark regressions we use the following definition of free cashflows (before investment): We take the difference between EBITDA and taxes (oibdp

15See, e.g., Penman and Yehuda (2009) for a detailed discussion.16See Penman (2010), p. 130ff. Bowen et al. (1987) and Sloan (1996), among others, discussthe differences between cash flows and earnings. Barth et al. (1999) study the characteristicsof the accrual and cash flow components of earnings that affect their relation with firm valueusing the valuation framework in Ohlson (1999), in which the value relevance of an earningscomponent depends on its ability to predict future abnormal earnings incremental to abnormalearnings and the persistence of the component.17See, e.g., Welch (2011, chap 13).18Notice that earnings (syn. net income) can be calculated from EBIT by subtracting incometaxes and interest rate expenses. Free cash flows (as given in Table 1) can also be expressed interms of EBIT. Equating both relations leads to an equation for earnings. Comparing thisequation with (18) gives the result.



Table 1

Free cash flows from income statement.

This table reports the cash flow definition used in this paper. The first column shows how free cash flowscan be calculated from the relevant accounting figures. The second column provides the correspondingnames in Compustat. The third column gives the Compustat items.

Accounting Figure Compustat Name Compustat Item

Sales do. SALE� Costs of goods sold do. cogs� Selling, General & Administrative

Expensesdo. xsga

¼ EBITDA Operating Income beforeDepreciation

oibdp

� Depreciation & Amortization do. dp¼ EBIT Operating Income after

Depreciationoiadp

� Taxes Income Taxes ‐ Total txtþ D Deferred Taxes and Tax Credit do. D txditc¼ NOPLATþ Depreciation & Amortization do. dp� Capital Expenditures do. capx� D Net Working Capital Working Capital Change ‐ Total wcapch (D wcap)þ Asset Sales Sale of Property sppe¼ Free Cash Flows

Table 2

Free cash flows (before investment) from income statement.

This table reports the cash flow definition (before investment) used in this paper. The first column showshow free cash flows before investment can be calculated from the relevant accounting figures. Thesecond column provides the corresponding names in Compustat. The third column gives the Compustatitems.

Accounting Figure Compustat Name Compustat Item

EBITDA Operating Income beforeDepreciation

Oibdp

� Taxes Income Taxes ‐ Total Txtþ D Deferred Taxes and Tax Credit do. D txditc� D Net Working Capital Working Capital Change ‐ Total wcapch (D wcap)þ Asset Sales Sale of Property Sppe¼ Free Cash Flows (before Investment)



minus txt plus D txditc), subtract the change in working capital (annual change in wcap)and add asset sales (sppe).19

As a robustness check, we calculate free cash flows using net cash flow from operations(see Table 3).20 To obtain this figure, we add Compustat items “Operating Activities –NetCash Flow (oancf)” and “Interest and Related Expense – Total (xint)”. Since our model isbuilt on cash flows before investment, we do not subtract capital expenses from this sum.To proxy for taxes, we will also run an additional robustness check. We assume a tax rateof 30% and multiply the interest expenses by 0.7. Then free cash flows are given as thesum of net cash flow from operations and 70% of the interest expenses.

4. Data

Our sample period covers 38 years ranging from 1971 to 2008. The data comes fromseveral sources. The first is the combined annual, research, and full coverage 2008Standard and Poor’s Compustat industrial files. The sample is selected by first deletingany firm‐year observations with missing data.21 As already discussed in the previoussection, in our main analysis we use the following definition of free cash flows (beforeinvestment) that is reported in Table 2:We take the difference between EBITDA and taxes(oibdp minus txt plus D txditc), subtract the change in working capital (annual change in

Table 3

Free cash flows (before investment) from cash flow statement.

The table reports an alternative definition of free cash flows derived from a firm’s cash flow statement. Toproxy for taxes, we also use an alternative definition where we add only 70% of the interest rate expenses(instead of 100% as reported in the table). The first column shows how this alternative definition can becalculated from the relevant accounting figures. The second column provides the corresponding names inCompustat. The third column gives the Compustat items.

Accounting Figure Compustat NameCompustat

Item

Net Cash Flow from Operations Operating Activities – Net Cash Flow Oancfþ Interest Rate Expense after Tax Interest and Related Expense – Total Xint¼ Free Cash Flows (before Investment)

19As a robustness check, we have also run the benchmark regressions with a free‐cash‐flowdefinition where we do not add asset sales, since asset sales could be interpreted as negativecapital expenditures. The regression results are however almost identical with very similarpoint estimates and significance levels. We thank a referee for pointing this out.20See, e.g., Penman and Yehuda (2009) or Richardson (2006).21The only exception is deferred taxes (Compustat txdb) that we set to zero if it is missing. Thereason is that deferred taxes are typically an insignificant part of firm value compared to thebook and market value of the assets (at, csho, prcc_f, and ceq) and we would have lost around10% of the observations if we had deleted them. To check the robustness of this assumption,we run our benchmark regression (6) that is reported in Table 7 excluding all observationswhere deferred taxes are missing. As expected, the results are virtually unchanged.



wcap) and add asset sales (sppe). Here the corresponding Compustat items are reported inbrackets. The cash flow multiplier is then defined as the ratio between firm value and freecash flows, where firm value is given by the sum of book value (at) plus the differencebetween market capitalisation (prcc_f� csho) and book value of equity (ceq) minusdeferred taxes (txdb), where we set deferred taxes equal to zero if they are missing. Theconcept of valuation multiples cannot be applied if the corresponding value driver isnegative.22 Therefore, we disregard negative realisations of the cash flow multipliers sothat our sample consists of 104,800 observations stemming from 15,830 firms.23

We use several macroeconomic explanatory variables: The one‐month Fama‐Frenchriskfree rate was obtained from CRSP. Since data on expected inflation is not easilyavailable over the time period from 1971 to 2008 (e.g. TIPS are traded since 1997 only),we use the realised inflation over the previous 12 months as reported by the Bureau ofLabor Statistics.24 However, as inflation is not only measuring price changes, but is alsorelated to the state of the economy (e.g. high inflation might create uncertainty), in theregressions we control for this second feature by regressing on inflation as well. The realriskfree rate is then approximated by the difference between the Fama‐French riskfree rateand this inflation rate. The Treasury yields and the corporate bond yields are from GlobalFinancial Data. The slope of the Treasury yield curve is defined as the difference betweenthe 14 year Treasury yield and the riskfree rate. The 14 year Treasury yield is obtained bylinearly interpolating between the 10 year and 20 year yields.We use this maturity since itcan be matched against the industrial Baa corporate bond yields reported by Moody’s tocalculate the 14 year yield spread between corporate bonds and Treasury bonds.25

Besides, we calculate the historical volatility of the stock market from the value weightedS&P 500 index as reported in CRSP. We use the version without dividends, but thevolatilities obtained from the version with dividends are almost identical. We include thelast 250 trading days to compute the volatility. Finally, we include the relative value of theS&P 500 index.More precisely, we use the log of the S&P 500 index and subtract its trendthat was observed from 1926 to the end of 1970. Table 4 presents the summary statistics ofthe macro variables measured at the last trading day of a particular month. All variablesare annualised and quoted in percentages. For instance, the average riskfree rate over theperiod from 1971 to 2008 was 5.565%. The maximum of 16.15% was reached in May of1980 and the minimum of 0.03% in November and December of 2008. The yield spreadreached its maximum of 5.788% in November of 2008 and its minimum of 0.754% inOctober of 1978. Finally, the average annualised historical volatility is about 14.8%.Since a year has about 250 trading days, we multiply the daily volatility by the square rootof 250 to obtain the annual volatility.In the empirical analysis to follow, we regress the logarithm of the cash flow multiplier

on several variables. The first three are closely related to the term structure and include thereal riskfree rate, the slope of the Treasury term structure, and the spread of Baa ratedbonds over Treasury bonds. We decided to include the real riskfree rate (instead of thenominal one) since one could argue that the model (1) is set up in real terms. Holding the

22Similar problems can occur with negative earnings that make payout ratios ill‐defined. See,e.g., Li and Zhang (2010).23We also drop financial firms, although our results hardly change if we include them.24More precisely, we use the 12 month consumer price index ‐ all urban consumers.25The average maturity of the bonds in the index provided by Moody’s is about 14 year.



other variables fixed, an increase in either of these variables increases the discount rate atwhich free cash flows are discounted. Since in our model the discount rate is negativelyrelated with the multiplier, we expect to observe negative relations between the multiplierand these variables. We also include the historical volatility of the S&P 500 as anexplanatory variable measuring aggregate equity market risk. We expect to observe anegative relationship between volatility and the multiplier. The same is true for inflationthat is also used as a control variable. As last macro variable that controls for the relativevalue of the stock market we include the detrended log‐value of the S&P 500 index. Weexpect the cash flow multiplier to be positively related to this measure.The first‐order condition (8) of our model suggests that the multiplier increases with the

proportion of the cash flows invested. To test this prediction empirically, we add a proxyfor this variable to the set of our explanatory variables. We measure the investmentproportion by the ratio of the annual capital expenditures (Compustat capx) over the freecash flows. This variable is winsorised at the 1% level. We will later study the sensitivityof the results to this procedure.There are other variables, such as size, dividend yield, and leverage, that are known to

affect firm valuation. Tomake sure that the variables suggested by ourmodel, in particularthe proportion of cash flow invested, has an effect on firm valuation beyond thesevariables, we need to control for their possible effect. To control for size effects, weinclude the logarithm of the real market capitalisation as an explanatory variable. Themarket capitalisation of a firm is defined as the product of the number of sharesoutstanding and the price per share (Compustat csho and prcc_f). Real marketcapitalisation is then calculated by dividing market capitalisation by the consumer priceindex (CPI). We also control for whether a firm pays a dividend by adding a dividenddummy. Finally, we take a firm’s leverage into account.We use the ratio of long‐term debt

Table 4

Summary statistics for the macro variables.

This table provides summary statistics for the macro variables over the period from 1971 to 2008.Riskfree denotes the one‐month Fama‐French riskfree rate. Inflation denotes the annualized inflation ratecalculated from the consumer price index (CPI). Real_riskfree is the difference between both variables.Slope denotes the difference between the 14 year yield on Treasury bonds and the riskfree rate. Thematurity of 14 year is chosen since it approximatelymatches thematurity of the Baa spread as reported byMoody’s. Baagov denotes the spread between the yield on Baa corporate bonds and Treasury bonds.Log_sp_notrend denotes the relative value of the S&P 500. This is calculated by taking the logarithm ofthe index value and subtracting its trend that has been observed from 1926 to 1970. Vol_sp denotes theannualized historical volatility of the S&P 500 calculated using index values of the last 250 trading days.

Mean Std. Dev. Min. Max. Median

Riskfree 5.565 2.856 0.03 16.15 5.107Real_riskfree 0.912 2.42 �8.184 6.641 1.014Inflation 4.653 2.992 0.091 14.756 3.645Slope 1.958 1.421 �2.726 6.511 2.086Baagov 1.956 0.628 0.754 5.788 1.818Log_sp_notrend 0.374 0.458 �0.473 1.319 0.369Vol_sp 14.839 5.575 7.547 40.697 13.426



(Compustat dd1 plus ddltt) to firm value, which is intended to measure the likelihood ofdistress. Roll et al. (2009), among others, argue that firms which have more leverage areriskier and therefore require a higher discount rate. This would lead to a negative sign ofthe regression coefficient. Alternatively, Jensen and Meckling (1976) argue that debtmight make managers more careful about investments, which would result in a positivesign of the regression coefficient.Table 5 presents the summary statistics of the firm specific variables and Table 6

summarises the correlations between all firm specific and all macro variables. Note thatthe highest correlation in the table is between volatility and the Baa spread (0.71); both ofthem represent some measure of global risk. The second highest correlation in the table isbetween the log of the cash flowmultiplier and the proportion of the cash flow invested,p,which are the two key variables of interest in our analysis.

5. Panel Regression Results

In this section we examine the determinants of the cash flowmultiplier by running severalpanel regressions that use all the information contained in the time‐series. The residuals ofthe cross‐sectional regressions are likely to be serially correlated. Furthermore, as we willdemonstrate later on, there might be cross‐sectional dependance as well. To overcomethese potential problems, we correct our t‐statistics using the approach outlined in Driscolland Kraay (1998). They assume an error structure that is heteroscedastic, autocorrelatedup to some lag, and possibly correlated between the units.26 The resulting standard errorsare heteroscedasticity consistent as well as robust to very general forms of cross‐sectionaland temporal dependence. In our robustness checks, we will discuss this point in moredetail.Our benchmark result (1) is a firm‐fixed‐effects regression presented in Table 7. As

postulated in the previous section, the real riskfree rate, the Baa spread, inflation, and

Table 5

Summary statistics for the firm specific variables.

This table provides summary statistics for the firm specific variables. Log_ratio stands for the logarithmof the cash flow multiplier. Pi denotes the investment proportion given by the ratio between the annualcapital expenditures and the free cash flows that is winsorized at the 1% level. Log_real_size denotes thelogarithm of the real market capitalization. This variable is inflation adjusted using the consumer priceindex (CPI). Leverage is the ratio of the debt’s book value over the market capitalization of the firm. Thestatistics are based on 104,800 observations.

Mean Std. Dev. Min. Max. Median

Log_ratio 2.65 1.067 �5.062 23.709 2.467Pi 1.03 1.587 0 10.523 0.605Log_real_size �0.348 2.231 �12.968 7.927 �0.466Leverage 0.178 0.171 0 0.948 0.137

26 In our regressions, the maximum lag is three years.



Table

6

Correlatio

nmatrix.

Thistablereportsthecorrelations

betweentherelevant

macro

andfirm

‐specificvariables.The

correlations

arecalculated

onthebasisof

104,800observations

from

1971

to2008.

Log_ratio

Real_riskfree

Inflation

Slope

Baagov

Log_sp_notrend

Vol_sp

Pi

Log_real_size

Leverage

Log_ratio

1.000

Real_riskfree

0.038

1.000

Inflation

�0.098

�0.479

1.000

Slope14

�0.009

�0.118

�0.263

1.000

Baagov14

�0.076

�0.128

�0.111

0.181

1.000

Log_sp_notrend

0.114

0.210

�0.675

�0.193

�0.062

1.000

Vol_sp

�0.066

�0.128

�0.146

0.095

0.708

0.149

1.000

Pi

0.580

0.025

0.023

�0.012

�0.017

�0.031

�0.023

1.000

Log_real_size

0.117

�0.037

�0.100

�0.031

�0.009

0.140

�0.026

�0.006

1.000

Leverage

�0.291

�0.058

0.141

�0.002

0.037

�0.153

0.027

�0.017

�0.092

1.000



Table

7

Benchmarkregressionsandregressionswith

excluded

variables.

The

tablereportstheresults

ofpanelregressionswith

Driscoll‐Kraay

errors

that

correctforavarietyof

dependencies

includingspatialdependencies.Onlyin

regressions(3)and(8)thestandard

errorsareclusteredatthefirm

level.Regression(1)isapooled

regression

with

firm

fixedeffects.Regressions

(2),(3),and(8)

includedummiesforthe

48Fam

a‐Frenchindustries,butno

firm

fixedeffects.Regressions

(4)–(9)exclude

someof

theexplanatoryvariablesandwith

theexception

of(8)have

fixedeffectsas

(1).Allregressionsarebasedon

104,800observations

stem

mingfrom

15,830

firm

s.The

timeperiod

is1971

to2008.T

het‐statisticsare

reported

inbrackets.The

significancelevelscorrespond

tothefollo

wingp‐values:� p

<0.05,��

p<0.01,��

� p<0.001.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Real_riskfree

�0.015

��0

.015

0.006

�0.014

��0

.014

��0

.004

(�2.38)

(�1.81)

(0.99)

(�2.04)

(�2.19)

(�0.46)

Inflation

�0.019

��0

.019

��0

.006

�0.021

��0

.016

�0.019

(�2.40)

(�2.25)

(�1.01)

(�2.16)

(�1.88)

(�1.55)

Slope

�0.001

�0.001

0.002

�0.003

�0.000

�0.003

(�0.14)

(�0.14)

(0.34)

(�0.33)

(�0.00)

(�0.29)

Baagov

�0.046

��0

.037

�0.023

��0

.088

��

�0.054

(�1.99)

(�1.65)

(�2.11)

(�3.65)

(�1.64)

Log_sp_notrend

�0.040

0.063

0.530�

��0

.069

�0.010

0.013

(�0.84)

(1.17)

(10.61)

(�1.22)

(�0.21)

(0.18)

Vol_sp

�0.007

��0

.008

��0

.001

�0.011

��

�0.013

��(�

2.82)

(�2.80)

(�1.17)

(�4.09)

(�3.03)

Pi

0.414�

��0.400�

��0.411�

��0.414�

��0.414�

��0.414�

��0.411�

��0.421�

��(50.14)

(54.50)

(144.79)

(50.53)

(49.94)

(49.65)

(144.66)

(49.32)

Log_real_size

0.195�

��0.066�

��0.124�

��0.198�

��0.194�

��0.203�

��0.126�

��(13.91)

(9.65)

(40.77)

(13.97)

(13.52)

(19.05)

(41.26)

Leverage

�0.608

��

�1.296

��

�0.902

��

�0.625

��

�0.615

��

�0.681

��

�0.909

��

Div_dum

my

�0.154

��

�0.159

��

�0.166

��

�0.153

��

�0.156

��

�0.166

��

�0.169

��

(�13.57)

(�13.63)

(�19.12)

(�12.77)

(�13.66)

(�9.74)

(�19.41)

Intercept

2.789�

��2.863�

��1.951�

��2.798�

��2.719�

��3.042�

��2.491�

��2.806�

��2.217�

��(31.01)

(26.56)

(16.11)

(26.71)

(29.64)

(22.01)

(101.66)

(42.56)

(97.08)

R2

0.514

0.489

0.518

0.513

0.514

0.016

0.508

0.494

0.459

Firm

fixedeffects

yes

nono

yes

yes

yes

yes

noyes

FFindustry

dummies

noyes

yes

nono

nono

yes

noYearfixedeffects

nono

yes

nono

nono

yes

no



volatility have significantly negative impacts on the cash flowmultiplier. The slope of theterm structure and the relative value of the S&P 500 are insignificant. The firm specificvariables are all very significant and have the expected signs. Besides, as predicted by themodel, the fraction of the cash flows invested, p, is very significant and positively relatedwith the multiplier. The same is true for the size measure, which serves as a controlvariable. On the other hand, in line with the literature, the remaining two control variables,leverage and the dividend dummy, have negative loadings.27

Column (2) in Table 7 reports results when we run a regressions with dummies for the48 Fama‐French industries (instead of firm fixed effects). The significance levels of thesignificant coefficients remain almost the same as before. However, the real riskfree rate isborderline significant at the 5% level now and the spread is significant at the 10% levelonly.Columns (3) and (8) report regression results when we control for industry dummies

and for year fixed effects. Column (3) shows the results including the macro variables28

and column (8) excluding them. The main results with respect to p and the firm specificvariables remain the same as before. As expected the year fixed effects takes a lot of thesignificance of the macro variables, to the point that the adjusted R2 is slightly largerwithout the macro variables.In order to test for the presence of firm‐specific fixed effects, we performed a robust

version of the Hausman test.29 The null hypothesis of no fixed‐effects is rejected at alllevels suggesting that there are fixed effects in the data. This is the reason why we havechosen regression (1) to be our benchmark regression. Unless otherwise stated, in thesequel we thus report the results of fixed‐effects regressions.Furthermore, we consider regressions where we exclude some of the explanatory

variables. The results are reported in columns (4)–(9) of Table 7. Comparing regressions(4) with regression (5) shows that volatility and the spread variable are substitutes: If wetake out one of them, then the remaining variable becomes highly significant.Furthermore, the explanatory power of the macro variables is small as regression (6)demonstrates. Only volatility is significant. On the other hand, the firm specific variablescan explain most of the variation in the cash flow multiplier. The loadings and theirsignificance levels remain almost unchanged if we take out themacro variables, which canbe seen in regression (7). In particular, the fraction of cash flows invested, p, explains alarge part of the variation in the multiplier (see (9)).Finally, we study the functional relationship between the cash flow multiplier and the

investment fraction p. Equation (15) can be approximated by applying the Taylorapproximation lnð1þ xÞ x� 0:5x2

ln f ln f 0 þ b1pþ b2p2; ð19Þ

where b1 is positive and b2 is negative. This means that the multiplier increases with p at adecreasing rate (decreasing marginal productivity of capital), i.e. there is a nonlinearrelationship between the multiplier and p. Regressions (10) and (12) of Table 8 confirmthis conjecture. Note that the nonlinear effect is small, but it is still very significant.

27See, e.g., Roll et al. (2009).28This specification was suggested by a referee.29See, e.g., Wooldridge (2002, p. 288ff).



The table also reports regression results whenwe cluster standard errors at the firm leveland control for industry dummies and for year fixed effects as in Table 7. Column (11)shows the results including the macro variables and column (13) excluding them. Themain results with respect to p, p2, and the firm specific variables remain the same asbefore. Again the year fixed effects takes some of the significance of the macro variables.

6. Robustness Checks

In this section, we report the results of several checks on the basic results. The testsconsider standard errors and additional explanatory variables. We first compare the

Table 8

Regressions with squared Pi.

The table reports regression results if we include the nonlinear term p2. Regressions (10) and (12) arefirm fixed effects regressions with Driscoll‐Kraay errors. Regressions (11) and (13) cluster standarderrors at the firm level and include industry and year dummies. The t‐statistics are reported in brackets.The significance levels correspond to the following p‐values: �p< 0.05, ��p< 0.01, ��p< 0.001.

(10) (11) (12) (13)

Real_riskfree �0.018�� 0.005(�2.63) (0.93)

Inflation �0.022�� 0.006(�2.95) (�1.03)

Slope 0.001 0.003(0.11) (0.41)

Baagov �0.040 �0.025�

(�1.78) (�2.26)Log_sp_notrend 0.012 0.541��

(0.25) (11.10)Vol_sp �0.007�� 0.001

(�2.79) (�0.84)Pi 0.730�� 0.710�� 0.749�� 0.742��

(23.45) (116.99) (22.80) (118.36)Pi2 �0.035�� 0.033�� 0.037�� 0.036��

(�13.26) (�46.01) (�13.06) (�48.75)Log_real_size 0.180�� 0.115��

(13.01) (37.35)Leverage �0.581�� 0.849��

(�14.18) (�33.48)Div_dummy �0.146�� 0.151��

(�12.75) (�17.34)Intercept 2.551�� 1.764�� 2.010�� 2.194��

(30.72) (14.85) (75.76) (30.89)R2 0.548 0.552 0.496 0.523Firm fixed effects yes no yes noFF industry dummies no yes no yesYear fixed effects no yes no yes



standard errors of regression (2) with the standard errors that obtain if we form clusters byfirm and year.30 The p‐values of our benchmark regression (1) are smaller than in thecorresponding regressions. This suggests that there is spatial correlation in the data.31

Besides, clustering by firm only clearly leads to overly optimistic standard errors. Thesame would be true if we run a fixed‐effects regression clustering by time only. Thissuggests that the standard errors of our benchmark regressions are conservative.Next, we study whether the level where we winsorise the investment fraction matters.

To analyse this question, we alternatively run regression where we winsorise p either atthe 5% level or setp to one if it is larger than one.32 The significance levels of the variablesare hardly affected. In particular, the fraction of the cash flows invested remains highlysignificant. The loadings on p however increase, since the upper bounds on p are smallerin these regressions.Our proxy for investments are capital expenditures (Compustat capx) that do not

include R&D expenses (Compustat xrd). The main reason for using this proxy is thatotherwise we would have lost more than 50% of our observations since xrd is oftenmissing in Compustat. To check whether including R&D expenses changes our results,we run regressions where we set R&D expenses to zero if they are missing. We then addR&D expenses to the capital expenditures whenever they are positive. In this case, thenumber of observations remains the same as in the benchmark regression. The results aregiven in column (14) of Table 9. The real riskfree rate and inflation are now significant atthe 10% level only, whereas the remaining significance levels are as before. Alternatively,one can define a second investment ratio for R&D expenses. Firstly, we set this ratio tozero whenever R&D expenses are missing. Column (15) shows that the results are almostidentical to our benchmark regression (1). Additionally, the investment fraction for R&Dexpenses is highly significant and positive. Finally, we have run regression (16) where wedisregard observations if R&D expenses are missing. This regression is based on 53,051observations coming from 9,076firms. Again both investment ratios are highly significantand have positive coefficients. Besides, the levels of the other firm specific variables arenot affected as well. For the macro variables the levels decrease so that the real riskfreerate is only borderline significant at the 10% level and inflation is not significant anymore.As alreadymentioned in Section 3, we compare our definition of free cash flows, which

is derived from a firm’s income statement, with a definition that draws upon the cash flowstatement. Table 10 reports the corresponding results where we either include firm fixedeffects or industry dummies for the 48 Fama‐French industries. Columns (10) and (20)provide regression results that use our benchmark cash flow definition (see Table 2) andthe same explanatory variables as the benchmark regressions (6) and (6), but includeobservations from 1988 onwards. This is because data for the cash flow definitionsderived from the cash flow statement is only available starting in 1988. Columns (17) and(19) use the alternative free cash definition given by the sum of net cash flow fromoperations and interest rate expenses (see Table 3), whereas columns (18) and (20) use the

30See, e.g., Pedersen (2009) and the references therein. The corresponding table is availableupon request.31To test for spatial correlation, we performed Pesaran’s test of cross sectional independenceon a subsample of firms with at least 30 observations. This test rejects independence at allsignificance levels, which suggest that Driscoll‐Kraay standard errors are more appropriate.32The results are available upon request.



alternative free cash definition given by the sum of net cash flow from operations and 70%of interest rate expenses, which takes into account that interest is tax deductable. First, itcan be seen that the differences between the regressions that take 100% of the interest rateexpenses into account and the regressions that take only 70% of the interest rate expensesinto account are small. Second, the signs and significance levels of the firm relatedvariables (Pi, Log_real_size, Leverage, Div_dummy) are the same for all regressions,which supports our finding with respect to the investment policy. For the firm fixed effect

Table 9

Regressions for different definitions of Pi.

The table reports regression results if we account for outliers in Pi in different ways. We include firmfixed effects. In regression (14), Pi is defined as the ratio between capital expenditures plus R&Dexpenses (if any) over free cash flows. In regressions (15) and (16), we include an investment fraction forR&D expenses. In regression (15) we set this ratio equal to zero if R&D investments are not reported,whereas in (16) we disregard the observations. Consequently, regression (16) is based on 53,051observations stemming from 9,076 firms, whereas regressions (14) and (15) are based on 104,800observations stemming from 15,830 firms. The t‐statistics are reported in brackets. The significancelevels correspond to the following p‐values: �p< 0.05, ��p< 0.01, ��p< 0.001.

(14) (15) (16)

Real_riskfree �0.010 �0.013� �0.012(�1.66) (�2.19) (�1.65)

Inflation �0.014 �0.017� �0.013(�1.65) (�2.14) (�1.41)

Slope �0.002 �0.001 �0.003(�0.16) (�0.13) (�0.22)

Baagov �0.049� �0.047� �0.061�

(�2.13) (�2.05) (�2.41)Log_sp_notrend �0.088 �0.057 �0.091

(�1.89) (�1.21) (�1.83)Vol_sp �0.007�� 0.007�� 0.006�

(�2.60) (�2.71) (�2.50)Pi 0.363�� 0.315��

(88.85) (67.56)Pi_total 0.251��

(147.59)Pi_rd 0.124�� 0.148��

(23.53) (33.59)Log_real_size 0.222�� 0.205�� 0.255��

(15.51) (14.49) (12.71)Leverage �0.514�� 0.556�� 0.524��

(�11.57) (�12.86) (�8.77)Div_dummy �0.159�� 0.150�� 0.162��

(�11.99) (�12.38) (�10.33)Intercept 2.808�� 2.763�� 2.850��

(29.06) (30.10) (28.72)R2 0.532 0.544 0.607



regressions the significance of the macro variables is lower. Notice, however, that we nowconsider a time span that has a length of only half the size as compared to our benchmarkregressions (1) and (2). Therefore, fixed effects are harder to identify (in particular if thepanel is unbalanced) and take out a lot of the variation. In regressions (10), (19) and (20)where we control for industry instead of firm fixed effects, the signs and significancelevels are the same not only for the firm related variables, but also for the macro

Table 10

Regressions for alternative cash‐flow definitions.

The table reports the results of panel regressions for alternative cash‐flow definitions. All regressions usedata from 1988 to 2008. In regressions (10) and (20) free cash flows (before investment) are defined asEBITDA minus taxes minus change in working capital plus asset sales. They involve involve 66,119observations. In regressions (17) and (19) free cash flows (before investment) are defined as net cash flowfrom operations plus interest rate expenses, whereas in regressions (18) and (20) free cash flows (beforeinvestment) are defined as net cash flow from operations plus 70% of interest rate expenses. Regressions(17) and (19) involve 82,285 observations. Regressions (18) and (20) involve 81,142 observations.Regressions (10), (17), and (19) are pooled regressions with firm fixed effects. Regressions (20), (19), and(20) are pooled regressions with dummies for the 48 Fama‐French industries. The t‐statistics are reportedin brackets. The significance levels correspond to the following p‐values: �p< 0.05, ��p< 0.01,��p< 0.001.

(10) (17) (18) (20) (19) (20)

Real_riskfree �0.039�� 0.004 0.001 �0.073�� 0.033�� 0.028��

(�3.60) (�0.39) (0.08) (�7.20) (�4.35) (�3.74)Inflation �0.027� �0.009 �0.006 �0.049�� 0.031�� 0.028�

(�2.32) (�0.91) (�0.56) (�3.91) (�2.58) (�2.45)Slope �0.008 0.009 0.014 �0.044�� 0.027�� 0.022�

(�0.74) (0.88) (1.32) (�4.14) (�2.82) (�2.37)Baagov �0.141�� 0.070�� 0.068�� 0.179�� 0.084�� 0.076��

(�5.47) (�2.84) (�2.61) (�9.04) (�3.60) (�3.24)Log_sp_notrend �0.082�� 0.044 0.049 �0.046 0.067 0.073�

(�3.27) (1.57) (1.82) (�1.41) (1.85) (2.06)Vol_sp �0.000 �0.004 �0.004 0.002 �0.004 �0.005

(�0.21) (�1.79) (�1.81) (1.20) (�1.48) (�1.88)Pi 0.432�� 0.375�� 0.352�� 0.413�� 0.353�� 0.333��

(53.16) (59.03) (82.79) (50.19) (47.04) (51.13)Log_real_size 0.224�� 0.176�� 0.176�� 0.072�� 0.061�� 0.062��

(11.13) (10.47) (11.28) (9.11) (6.47) (6.56)Leverage �0.471�� 0.332�� 0.193�� 1.255�� 0.900�� 0.742��

(�7.42) (�6.04) (�3.87) (�26.47) (�19.27) (�18.02)Div_dummy �0.153�� 0.076�� 0.081�� 0.152�� 0.211�� 0.212��

(�11.21) (�6.34) (�6.27) (�7.63) (�14.35) (�14.11)Intercept 2.952�� 2.629�� 2.628�� 3.341�� 3.101�� 3.116��

(31.86) (29.49) (30.22) (28.29) (28.73) (28.60)R2 0.516 0.487 0.487 0.485 0.450 0.443Firm fixed effects yes yes yes no no noFF industry

dummiesno no no yes yes yes



variables.33 We have also run robustness checks for the other regressions of the previoussections and found very similar results, which for the sake of space are not reported here.Finally, notice that the number of observations for our benchmark cash flow definition issmaller than for the alternative cash flow definitions, since it involves more Compustatitems that could potentially be missing. In particular, this is so for the change in workingcapital (annual change in wcap) and asset sales (sppe).We thus ran the regressions also onthe subset of observations where both definitions could be calculated and find that theresults for the alternative definitions of free cash flows become even more similar.

7. Value of the Option to Invest

We have shown that the cash flow multiplier consists of two parts (see, e.g., equation(13)): Whereas the first part is exogenous, the second part is endogenous and captures thefirm’s real option to invest, the so‐called present value of growth opportunities. Besides,Theorem 3 proves that the option value is increasing with m0. This parameter equals theexpected cash flow growth if the firm does not invest at all. We expectm0 to be on averagesmaller when the firm operates in an industry that is more investment intensive. Theinvestment intensity of an industry is measured by the average fraction of cash flows thatis reinvested, i.e. by the average p of a particular industry. To test this hypothesis, we runregressions where this average is included as an additional explanatory variable. We havealready seen that the cash flow multiplier increases with p. Following our line ofargument, the opposite should be true for the mean of the industry. There are two ways ofcalculating an industry mean. Firstly, one can calculate the mean over the whole sampleperiod leading to a constant. Secondly, one can compute the mean for every year of thesample period, which provides us with 48 time series of means for the 48 Fama‐Frenchindustries. In the first case, it clearly makes no sense to include firm dummies or fixedeffects since otherwise the coefficients of the average p cannot be identified. But also inthe second case dummies would absorb a lot of the variability that we expect to becaptured by the industry means of p. For this reason, we run pooled regressions withoutdummies and report the results in Table 11. Regression (21) includes the sameexplanatory variables as the benchmark regression (1). The variable Av_pi denotes theaverage p of the corresponding industry over the whole sample period of 38 years. Incontrast, Av_pi_annual denotes the average p of the corresponding industry calculatedevery year leading to 48 time series. It can be seen that in all regressions the coefficients onthe average p are significantly negative, which supports our line of argument above. Thesignificance levels of the other variables are almost unchanged. The real riskfree rate andinflation are however only significant at the 10% level and the spread variable is onlyborderline significant at the 5% level in regression (22). The results of regression (23) aresimilar except that the spread variable is significant again, but inflation becomesinsignificant.

8. Conclusion

We develop a simple discounted cash flow valuation model with optimal investment. Themodel predicts a positive relation between the cash flowmultiplier and a firm’s investment

33The only exception is the relative value of the S&P 500 that has a different sign, but thevariable itself is not significant (or only borderline significant).



policy and a negative relation between themultiplier and discount rates. These predictionsare confirmed in our empirical analysis where we include additional macro and firmspecific control variables. Our panel regression results indicate that the explanatoryvariables have the correct sign and for the most part are highly significant. Our model alsoimplies that the relation between the multiplier and investment is nonlinear. We provideempirical evidence that this is the case. Besides, we decompose the multiplier into two

Table 11

Regressions including average industry investments.

The table reports regression results if we include the average ps of the Fama‐French industries asadditional explanatory variables. These are pooled OLS regressions with Driscoll‐Kraay errors where weneither include fixed effects nor Fama‐French industry dummies. Regression (21) includes the sameexplanatory variables as the benchmark regression (1). In regression (22), the variable Av_pi denotes theaverage p of the corresponding industry over the whole sample period from 1971 to 2008. In regression(23), Av_pi_annual denotes the average p of the corresponding industry calculated every year, i.e. theseare 48 time series. As the benchmark regression (1), all regressions are based on 104,800 observationsstemming from 15,830 firms. The t‐statistics are reported in brackets. The significance levels correspondto the following p‐values: � p< 0.05, ��p< 0.01, ��p< 0.001.

(21) (22) (23)

Real_riskfree �0.017� �0.014 �0.013(�2.11) (�1.71) (�1.75)

Inflation �0.020� �0.018� �0.013(�2.25) (�2.07) (�1.42)

Slope �0.001 0.001 �0.007(�0.14) (0.07) (�0.76)

Baagov �0.049� �0.045 �0.056��

(�2.21) (�1.95) (�2.82)Log_sp_notrend 0.076 0.069 0.000

(1.43) (1.26) (0.00)Vol_sp �0.007� �0.008�� 0.008��

(�2.51) (�2.76) (�3.16)Pi 0.385�� 0.393�� 0.395��

(54.87) (56.17) (55.72)Log_real_size 0.056�� 0.063�� 0.060��

(7.27) (8.63) (7.98)Leverage �1.591�� 1.301�� 1.450��

(�28.01) (�29.32) (�28.12)Div_dummy �0.163�� 0.130�� 0.147��

(�11.20) (�9.87) (�11.23)Av_pi �1.111��

(�20.30)Av_pi_annual �0.511��

(�8.99)Intercept 2.906�� 3.407�� 3.170��

(29.03) (30.38) (33.67)R2 0.439 0.473 0.456



parts: the first part reflects the firm value without investment, whereas the second partcaptures the option to invest optimally in the future. We provide empirical evidence thatthe cash flow multiplier is strongly negatively related to the average investment policy ofthe particular industry.Since the cash flow multiplier depends on observable and relatively easily obtainable

variables, the approach taken in this paper could be easily used in practice. Even though itis based on a discounted cash flow model it does not require the estimation of expectedfuture cash flow and an appropriate risk‐adjusted discount rate.

Appendix A Proofs

Proof of Proposition 1. We set Y t ¼ Ct=c such that Y 0 ¼ 1. Then, problem (1) can berewritten as

V ðc; xÞ ¼ maxp

E

Z 1

0e�

R s

0RuduðcY s � pscY sÞds

� �

¼ cmaxp

E

Z 1

0e�

R s

0RuduðYs � psY sÞds

� �¼ cV ð1; xÞ:

ð20Þ

This implies Vcðc; xÞ ¼ V ð1Þ ¼ const and Vccðc; xÞ ¼ 0, which shows that V is linearin c. &

Proof of the Hamilton‐Jacobi‐Bellman equation (7). The firm value satisfies the Hamilton‐Jacobi‐Bellman equation

0 ¼ maxp

fðmðp; rÞcVc þ c� pc� RðrÞV þ ðu � krÞcV r þ 0:5h2cV rr þ rhsðpÞcV crg:

Applying the separation (4) yields (7). &

Proof of Proposition 2. The equation (16) follows from (14). ByVieta’s formulas, the twosolutions, f1 and f2, satisfy

f 1 f 2 ¼1

m21=4� m2ðm0 � r � lÞ :

implying that there exists a unique positive cash flow multiplier if m21=4� m2ðm0�

r � lÞ < 0, i.e. if the parabola defined in (16) has a maximum. &

Proof of Theorem 3.We set K ¼ m21=4� m2ðm0 � r � lÞ. By assumption of Proposition

2,K is negative. Due to the transversality condition, this implies thatm2 < 0.We interpret(16) as the implicit definition of f as a function ofm0. For this reason, we interpret the right‐hand side of (16) as a function F of f and m0. Then

f 0 ¼ df

dm0¼ � @F=@m0

@F=@f¼ �m2f

2 þ f

2Kf þ m0 � r � l� m2> 0;



since, by assumption, m0 � r � l� m2 < 0. Next recall that

O ¼ �1

m0 � r � l

m21 f

2

4ð1� m2 f Þ:

Therefore,

O0 ¼ dO

dm0¼ 1

ðm0 � r � lÞ2m21 f

2

4ð1� m2 f Þþ �1

m0 � r � l

m21

4

ff 0ð2� m2 f Þð1� m2 f Þ2

> 0:

Finally,O

f¼ O

�1m0 � r � lþO

. Consequently,

d

dm0

O

f¼ �1

m0 � r � l

O0 � 1m0 � r � lO

f 2> 0;

since O0 � 1

m0 � r � lO ¼ �1

m0 � r � l

m21

4

f f 0ð2� m2f Þð1� m2f Þ2

> 0: This completes the proof

of Theorem 3. &

Appendix B Series Expansion of Theorem 4

We firstly provide a representation of the coefficients in the series expansion (17) and then

prove Theorem 4). We define u ¼ u=k, ~w ¼ w þ cu, and Hk;ni;j ¼ ~aðkÞi bðnÞj þ 0:25cðkÞi cðnÞj ;

where

~aðnÞi ¼ 1fn¼i¼0g � m2aðnÞi � rs2ðiþ 1Þaðn�1Þ

iþ1 ;

bðnÞi ¼ 1fn¼i¼0g þ ~waðnÞi þ caðnÞi�1 � kiaðnÞi þ rs0ðiþ 1Þaðn�1Þiþ1 þ 0:5ðiþ 2Þðiþ 1Þaðn�2Þ

iþ2

cðnÞi ¼ m1aðnÞi þ rs1ðiþ 1Þaðn�1Þ

iþ1 :

ð21Þ

Then the coefficients are given by the following explicit recursion

aðnÞ0 ¼ �Pn�1

k¼1Hk;n�k0;0 þ RðnÞ

0

D0;

aðnÞm ¼ �P

ði;kÞ2I Hk;n�ki;m�i þ RðnÞ

m

Dm;

ð22Þ



where

RðnÞm ¼ ð1� m2a

ð0Þ0 Þ 1fm¼n¼0g þ caðnÞm�1 þ rs0ðmþ 1Þaðn�1Þ

mþ1 þ 0:5ðmþ 2Þðmþ 1Þaðn�2Þmþ2

h iþð1þ wað0Þ0 Þ 1fm¼n¼0g � rs2ðmþ 1Þaðn�1Þ

mþ1

h iþ 0:5m1rs1ðmþ 1Það0Þ0 aðn�1Þ

mþ1 ;

Dm ¼ ð1� m2að0Þ0 Þð~w� mkÞ � m2ð1þ ~wað0Þ0 Þ þ 0:5m2

1að0Þ0 ð23Þ

and I ¼ f0; 1;…;m� 1;mg � f0; 1;…; n� 1; ngnfð0; 0Þ; ðm; nÞg is an index set.34

We emphasise that this recursion is explicit and all equations (22) do not involve aðnÞm onthe right‐hand side. The only exception is the equation for að0Þ0 where að0Þ0 appears on theleft‐ and right‐hand side. This leads to the following quadratic equation.

0 ¼ ð0:25m21 � m2~wÞðað0Þ0 Þ2 þ ð~w� m2Það0Þ0 þ 1:

For reasonable parametrisations, numerical experiments suggest that this equation hasone positive and one negative root.

Proof of Theorem 4. We firstly multiply equation (9) by 4ð1� m2f � rs2hf rÞ. Then wesubstitute the representation (17) into the resulting equation. This leaves us with severalproducts of power series. Expanding these products and rearranging, we can rewriteequation (9) as follows:

X1n¼0

X1m¼0

Xnk¼0

Xmj¼0

~aðkÞj bðn�kÞm�j þ 0:25cðkÞj cðn�kÞ

m�j

( )ðr � uÞmhn ¼ 0:

Since the representation of a power series is unique, we conclude that f…g ¼ 0 for allðn;mÞ 2 N0 �N0. This gives a series of equations for the coefficients aðnÞm . Solving theseequations yields (22). &

Appendix C Pricing Kernel Formulation

This part of the appendix provides an alternative derivation of our results. As in the bodyof the paper, we consider a model with cash flow dynamics

dC ¼ C½mðp;X Þdt þ sðp;X ÞdWc�; Cð0Þ ¼ c: ð24Þ

For simplicity, we assume that the state process is one‐dimensional where the factor isthe short rate, X¼ r, possessing the dynamics

dr ¼ ðu � krÞdt þ hdWr: ð25Þ

34Therefore, the differencePn

k¼0

Pmi¼0 … has twomore elements than

Pði;kÞ2I…, namely the

elements with indices ði; kÞ ¼ ð0; 0Þ and ði; kÞ ¼ ðm; nÞ.© 2014 John Wiley & Sons Ltd


We can rewrite the cash flow dynamics in terms of two independent Brownian motionsWr and W c:

dC ¼ C½mðp; rÞdt þ sðp; rÞðrdWr þffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

pdW cÞ�; ð26Þ

where < Wr;Wc>t ¼ rt. Following Ang and Liu (2007b), among others, the pricingkernel (syn. deflator, stochastic discount factor) of the economy is assumed to be of theform

dL ¼ �L½rdt þ jrdWr þ jcdW c�; ð27Þ

where jr and jc are market prices of risk. Notice that jc can be zero if Wc modelsidiosyncratic shocks. We can now rewrite the deflator as

Lt ¼ e�R t

0rsdsZt; ð28Þ

where Z is the density of the risk‐neutral measure Q and has the dynamics

dZ ¼ �Z½jrdWr þ jcdW c�: ð29Þ

Notice that under Q

dWQr ¼ dWr þ jrdt and dW Q

c ¼ dW c þ jcdt ð30Þ

are Brownian increments. Having specified a deflator, the initial firm value is given by

V ðc; rÞ ¼ maxp

E

Z 1

0LsðCs � I sÞds

� �: ð31Þ

Changing the measure leads to

V ðc; rÞ ¼ maxp

EQ

Z 1

0e�

R s

0ruduð1� psÞCsds

� �; ð32Þ

where we use that I ¼ pC. The cash flow dynamics under Q are given by

dC ¼ C½ðmðp; rÞ � sðp; rÞfrjrðrÞ þffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

pjcðrÞgÞdt þ sðp; rÞdWQ

c �; ð33Þ

where dWQc ¼ rdWQ

r þffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

pdW Q

c is a Q‐Brownian increment. Recall that in thebody of the paper it is assumed that

mðp; rÞ ¼ m0ðrÞ þ m1

ffiffiffip

p þ m2p; sðpÞ ¼ s0 þ s1ffiffiffip

p þ s2p: ð34Þ



Now, assume that the market prices of risk are constant35 such that �j � rjrðrÞþffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

pjcðrÞ ¼ const. Then the cash flow dynamics become

dC ¼ C½ðmQ0 ðrÞ þ m

Q1

ffiffiffip

p þ mQ2 pÞdt þ sðpÞdWQ

c �; ð35Þwhere m

Q0 ðrÞ � m0ðrÞ � s0

�j, mQ1 � m1 � s1

�j ¼ const, and mQ2 � m2 � s2

�j ¼ const.Therefore, the structure of our model is preserved under the risk‐neutral measure Q, i.e.we can formulate the HJB etc. in terms of the risk‐neutral measure. In particular, ifm0ðrÞ ¼ �m0 þ m0r, then

mQ0 ðrÞ ¼ �mQ

0 þ m0r; ð36Þwhere �mQ

0 � �m0 � s0�j ¼ const.

If interest rates are constant, then �j ¼ jc and m0 ¼ 0.We can then apply the result fromTheorem 3 of the paper to conclude that the relative option value decreases if �mQ

0decreases. This can now happen in three ways: Either �m0 decreases or the market price ofrisk jc or the volatility s0 increase (given that both are positive).

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cash flow multipliers and optimal investment decisions

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