Electronic copy available at: http://ssrn.com/abstract=1724082

The Real Option Value of Cash

Michael Kisser∗

Norwegian School of Economics

forthcoming in the Review of Finance

August 2012

Abstract

This paper focuses on the idea that cash has a real option value and it presents an

explicit valuation framework of cash holdings in the context of a capacity expansion

option. The model characterizes the optimal dynamic cash retention policy, the value

of internal funds and it provides a model implied regression specification based on

simulated data. Results imply that high cash flow volatility decreases the value of cash

and that optimal cash retention can actually delay investment relative to the case of

full outside financing. Both novel implications are confirmed by subsequent empirical

tests.

G31, G32, G35

∗∗I specially thank Engelbert Dockner, B. Espen Eckbo, Alois Geyer, Christopher Hennessy, Toni Whited,Jin Yu and Josef Zechner for their valuable, constant input and thoughtful advice. Also, I am grateful to ananonymous referee and the editor, Holger Mueller. Finally, I would like to thank seminar participants at theNorwegian School of Management, New Lisbon University, the Norwegian School of Economics, MaastrichtUniversity, the NFA 2011 in Vancouver, the FMA Asian Conference 2009 and the MFA 2009 in Chicago.Any remaining errors are my own.

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Electronic copy available at: http://ssrn.com/abstract=1724082

1 Introduction

Bates, Kahle, and Stulz (2009) document that U.S. industrial firms invest a substantial

fraction of their assets into cash. Retaining internal funds can be optimal as it avoids

incurring transaction fees and costs related to informational asymmetries when accessing

external capital markets. However, tax related disadvantages and agency conflicts between

management and shareholders may reduce the value of cash.

This paper focuses on the idea that cash has a real option value and it presents an

explicit valuation framework of cash holdings for all-equity financed firms in the context of

growth opportunities. The analysis of all-equity firms is particularly interesting because, as

I show, these firms have substantially increased their cash holdings from 9% of total assets

in 1980 to 33% in 2010. Moreover, in 2010 one in five U.S. industrial corporations has

been all-equity financed which underlines the economic magnitude of cash holdings for all-

equity financed firms. Finally, the focus on all-equity financed firms is a feature shared with

several real option models analyzing the interaction between cash holdings and investment,

see for example Hugonnier, Malamud, and Morellec (2012), Decamps, Mariotti, Rochet, and

Villeneuve (2011), Bolton, Chen, and Wang (2011) and Boyle and Guthrie (2003).

The objective of this paper is to investigate whether, conditional on being all-equity

financed, there is a value to holding cash.1 Specifically, I quantify the value of cash based on

a tradeoff between agency costs of free cash flow and costs of external finance in the context of

a capacity expansion option. The paper provides an interior solution for the firm’s dynamic

state contingent cash retention policy and it derives novel implications regarding the value

1The conditional analysis is motivated by the fact that existing capital structure models do not implythat it is optimal for a substantial fraction of firms not to employ leverage in their capital structure. Whiledynamic financing and investment models, see for example DeAngelo, DeAngelo, and Whited (2011), havebeen successfull in matching observed average industry leverage ratios to model implied moments, they donot imply that it is on average optimal to choose a zero leverage policy.

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Electronic copy available at: http://ssrn.com/abstract=1724082

of cash and cash flow volatility as well as the relation between cash holdings and optimal

investment policy. Both results are confirmed by subsequent empirical tests.

External financing fees entail both direct and indirect costs and are economically sig-

nificant. Hennessy and Whited (2007) structurally estimate external financing costs and

find that the variable cost component when accessing external equity markets is between

between 5% and 12%. Empirical studies by Lee, Lochhead, Ritter, and Zhao (1996) and

Lee and Masulis (2009) reveal equally significant magnitudes. On the other hand, Jensen

(1986) argues that saving cash is costly as management might be more likely to engage in

value-destroying ”empire-building” activities if cash reserves are abundant and thus have to

be monitored. Agency costs of free cash flow have been analyzed in several principal agent or

capital structure models, i.e. Eisfeldt and Rampini (2008) or DeMarzo and Sannikov (2006),

and empirical studies show that they are significant. In that regard, see for example Lang,

Stulz, and Walkling (1991) and Chen, Chen, and Wei (2011).

The paper derives several novel results. First, it provides an interior solution for the

firm’s dynamic state contingent cash retention policy. Results show that most of the time

it is optimal to retain only a fraction of each period’s cash flow which is consistent with

the empirically documented fact that firms simultaneously increase cash holdings and still

pay dividends. While the cash retention policy can be characterized in closed-form, the real

option model is then solved numerically to derive the optimal investment policy and the value

of cash. Second, the paper then provides a detailed analysis of the firm’s investment policy

and thereby finds that there is value to building cash reserves and that this time to build

may defer optimal investment. Moreover, the incentives to optimally retain cash and delay

investment are stronger in case (cash flow) volatility or investment costs of a project are low.

Third, the analysis further reveals that an increase in volatility generally reduces the value

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of internal funds. A simple way to think about the result is that low volatility allows the

firm to better plan the investment, retain cash more efficiently and thereby create additional

value. This result is of practical relevance because it implies that once cash no longer serves

as a buffer to reduce bankruptcy risk, its value might be substantially reduced in high

volatility states. The theoretical section concludes by employing Monte Carlo simulation to

generate artificial data and then estimating the value of internal funds using the regression

specification implied by the model. The regressions reveal that cash is valued at a premium

in the context of growth opportunities and that the premium is higher if the firm just started

to retain funds. Finally, the empirical section uses data from Compustat and estimates the

value of cash for the period from 1980 until 2010. Testing the main implications of the

model, I confirm that the value of cash and cash flow volatility are negatively related in the

context of growth options and I also show that there is a nonlinear relation between cash

holdings and investment policy.

The paper proceeds as follows. Section [2] provides a short review of the related literature

and Section [3] presents the model and the main valuation equations. Section [4] provides

a numerical analysis of the value of cash and the firm’s investment policy. Section [5] takes

the model to data and tests its main implications. Section [6] finally concludes.

2 Related Literature

The paper relates to a theoretical literature on the value of corporate cash holdings. Gamba

and Triantis (2008) focus on the value of financial flexibility in the context of a neoclassical

financing and investment model. In their setup, a firm retains cash for two reasons. First,

it helps the firm to avoid default in low profitability states as the cash on hand decreases

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its net debt exposure. Second, it allows the firm to prevent external financing costs when it

invests in high profitability states. Results show that the value of financial flexibility can be

substantial in case the firm has a low capital stock or when the firm is exposed to negative

income shocks.

Similarly, Asvanunt, Broadie, and Sundaresan (2010) analyze the relation between financ-

ing decisions and a firm’s investment policy. Focusing on a levered firm they investigate how

optimal investment policies differ depending on whether firm or equity value is maximized

and they show that firms with expansion options have lower leverage at the optimal capital

structure. Allowing the firm to save cash, they demonstrate that riskier firms have higher

optimal cash balances. This result has been also found by Acharya, Davydenko, and Stre-

bulaev (2012) who show empirically that there is a positive relation between cash holdings

and credit spreads.2

The approach presented in this paper differs from above in that it analyzes whether

cash has any significant economic value even when it does not serve as a liquidity buffer

reducing bankruptcy risk. Therefore, I am able to provide new insights regarding the relation

between the value of cash and volatility as well as the effect of cash retention on the optimal

investment decision. Furthermore, this paper focuses on agency costs of free cash flow in the

dynamic trade-off choice and I derive an interior solution for the optimal state-contingent

cash retention policy. This differs from above where the optimal saving policy is to retain the

entire cash flow or pay out a full dividend.3 The model presented in this paper is therefore

2In another similar paper Asvanunt, Broadie, and Sundaresan (2011) compare a firm’s choice in managingcorporate liquidity between issuing costly equity, maintaining a cash balance or employing loan committ-ments.

3Using a representative agent framework, Eisfeldt and Rampini (2009) study level and dynamics of thevalue of aggregate liquidity when external shocks occur. Similarly, the trade-off is between agency costs offree cash flow and cost of external finance but the paper does not focus on corporate cash policy but on thevalue of aggregate liquidity. Results show that the value of aggregate liquidity is highest when investmentopportunities are abundant but levels of current cash flow are low.

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consistent with the empirically documented fact that firms increase cash holdings while still

paying dividends

Other related theoretical work focuses on the interaction between cash holdings, invest-

ment and financing decisions. Decamps, Mariotti, Rochet, and Villeneuve (2011) analyze

how cash holdings impact firm value and stock prices. They show that firm value is a con-

cave function of a firm’s cash holdings and that, through this effect, the marginal value of

cash is negatively related to a firm’s stock price but positively to its volatility. Analyzing

a firm’s dividend boundary, they show that high volatility or low profitability increase the

likelihood that a firm retains cash.4 Hugonnier, Malamud, and Morellec (2012) investigate

the relation between cash holdings and investment decisions in case access to outside cap-

ital is uncertain. Their model reveals that cash holdings increase with cash flow volatility

and that negative supply side shocks decrease investment. Also, they show that sufficiently

large investment costs may make it optimal for a firm to decrease cash holdings and instead

finance externally. Put differently, higher cash holdings do not necessarily imply that a

larger fraction of a project is financed internally. Boyle and Guthrie (2003) analyze a firm’s

dynamic investment decision when the firm is allowed to save cash to relax an exogenously

given financing constraint resulting from asymmetric information. They show that due to

the possibility of future earnings shocks, a firm may be willing to exercise its growth option

prior to the benchmark case established by an otherwise unconstrained firm. Finally, Ander-

son and Carverhill (2011) present a dynamic trade-off model in which firms choose optimal

cash holdings, short-term debt and dividend policy under mean-reverting earnings. They

4Bolton, Chen, and Wang (2011) use a q-theoretic version of their model and show that investmentdepends both on marginal q, as well as the marginal value of liquidity. Modern q theory as introduced byLucas and Prescott (1971) argues that marginal adjustment costs of investing have to be equal to the shadowvalue of capital, coined marginal q. This shadow value measures the firm’s expectation of the marginal gainfrom investing. Further details on the q theory of investment can be found in Hayashi (1982) and Hennessy(2004).

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show that a firm’s target cash ratio is decreasing in profitability and that, once the target

ratio is reached, firms start paying dividends. The paper also implies that a firm adjusts the

net leverage ratio using its cash policy and that the traditional pecking order changes with

varying business conditions.

The model presented in this paper contributes to the literature by deriving an interior

solution for the optimal state-contingent cash retention policy whereas for the papers above,

the optimal payout policy is an all-or-nothing decision. Furthermore, this paper shows that

high investment costs generally reduce cash retention and that low volatility results in a

more extreme but efficient cash retention policy. As long as it is unlikely that the firm will

benefit from additional funds, a full dividend payout policy is optimal. Once the probability

of exercising the option is sufficiently large, the firm quickly raises the retention rate and

then retains the entire cash flow for a stable fraction of the state space. Moreover, I show

that cash retention may optimally lead to a delay in a firm’s investment decision. The result

shares the general notion with Hugonnier, Malamud, and Morellec (2012) that cash holdings

may have an ambiguous effect on investment, but the details are different. In their model,

the result is driven by a combination of capital supply uncertainty, fixed costs and sufficiently

high investment costs whereas in this paper, the result holds because there is a real option

value of cash. Firms postpone investment because there is an option value to build up even

more cash in the future and thereby to save on external financing costs. However, ceteris

paribus the incentives to delay investment are lower in case investment costs are high which

implies that the responsiveness of investment to cash holdings actually increases for this

scenario. Finally, I provide a detailed analysis of the value of cash and volatility and also

empirically test and confirm the main predictions of the model.

The paper also relates to empirical studies on the value of cash. In a multi-country study,

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Pinkowitz, Stulz, and Williamson (2006) estimate the impact of investor protection on the

value of cash holdings. Regressing firm value on different accounting variables including

either the level or changes in cash, they find that cash is valued lower in countries with less

protection. An alternative way has been suggested by Faulkender and Wang (2006) who

regress excess stock return on cash and different control variables to get an estimate of the

marginal value of cash. They find that the average marginal value of cash across all firms

equals $0.94 for the United States and that cash is generally more highly valued when the

existing cash holdings are low. Dittmar and Mahrt-Smith (2007) investigate the impact of

corporate governance mechanisms on the value of cash and use both approaches to estimate

the value of cash. They find that $1.0 of cash can be valued as low as $0.42 in case of poor

corporate governance.

This paper differs in that it employs a model implied regression specification which is

first tested on simulated data. Using data from Compustat for U.S. firms, I then show that

cash is valued on average at par, that is all-equity financed firms do not destroy value by

holding cash. Moreoever, subsequent tests reveal that the value of cash is negatively related

to cash flow volatility, thereby confirming one of the main implications of the model.

Bates, Kahle, and Stulz (2009) document a significant increase in average cash holdings

for the period from 1980 to 2006, specifically for non-dividend paying and riskier firms. Using

variants of the regression setup proposed by Opler, Pinkowitz, Stulz, and Williamson (1999)

they find that this is mostly due to changing firm characteristics. This paper shows that the

increase in cash holdings has been even greater for all-equity financed firms. In fact, cash

holdings of all-equity financed firms more than triple over the same sample period and, as

of 2010, constitute approximately one third of the average firm’s total assets.

Finally, focusing on the cash-flow sensitivity of cash, Almeida, Campbello, and Weisbach

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(2004) show that firms save operating cash flow if they are financially constrained. Han and

Qiu (2007) extend the model of Almeida, Campbello, and Weisbach (2004) by not allowing

the firm to hedge future cash flow risk. They are able to show that an increase in volatility

of cash flow leads to higher contemporary saving decisions. Riddick and Whited (2009)

question the results found in Almeida, Campbello, and Weisbach (2004) and argue that the

correlation is mainly due to measurement error in the market-to-book ratio which acts as a

proxy for marginal q. Finally, Denis and Sibilkov (2010) build upon Almeida, Campbello,

and Weisbach (2004) and show that cash allows constrained firms to invest more and thereby

it increases shareholder value. This paper indirectly contributes to this literature by showing

that the relation between cash holdings and investment thresholds is effectively non-linear

and decreasing. Put differently, only sufficiently large cash holdings accelerate investment

decisions.

3 The Model

Similar to Dixit and Pindyck (1991) and McDonald and Siegel (1986), I model a firm which

has the option to increase production capacity. Departing from traditional real option mod-

els, I focus on the question of how the expansion is financed. To make payout policy matter,

external financing is assumed to be costly due to reduced-form informational asymmetries

while retaining cash entails monitoring costs due to agency costs of free cash flow. The real

option value of cash is derived by comparing firm value under the optimal cash retention

policy to the case when the firm finances the project fully externally. For what follows, the

terms (real) option value of cash and value of internal funds will be used interchangeably.

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3.1 Basic Setup

Consider a firm which produces a single product and operates at some initial capacity level

K0 which, without loss of generality, is normalized 1. The cash flow produced by the firm is

risky and follows a Geometric Brownian Motion

dx = µxdt+ σxdWQ (1)

where dWQ is a standard Brownian motion under the risk neutral measure Q and µ and

σ are mean and volatility of the growth rate of x. I further assume that there exists a traded

asset being perfectly correlated with the firm’s cash flow which has the following dynamics

dX = rXdt+ σXdWQ where r > µ and δ ≡ r − µ.

The firm is all-equity financed such that all earnings accrue to shareholders either via

dividend payments or via capital gains. If the firm retains its earnings, it can put the money

on a cash account where it earns a riskless return r. However, following Jensen (1986) saving

cash is costly as management might be more likely to engage in value-destroying ”empire

building” when cash reserves are abundant. Shareholders therefore would want to monitor

the firm, which comes at a cost. I follow Eisfeldt and Rampini (2009) in assuming that

only the fraction of the operating cash flow which is retained within the firm is subject to

quadratic agency costs. The main intuition underlying this argument is that liquid funds

can then be allocated to a financial intermediary, i.e. a bank, such that each period only

the newly retained fraction of earnings has to be monitored.5 Letting C denote the cash

5Another way to think about the assumption is that it is easier for management to steal from a dynamicflow variable such as operating income than from a transparent stock variable such as cash holdings. Thisshould be specially true for large and complex business operations. The same assumption can be found inAlbuquere and Wang (2008) who model agency costs between inside and outside shareholders. Specifically,they assume that inside shareholders may steal a constant fraction of revenues and that the costs of stealingare quadratic to outside shareholders.

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account, α the retained fraction of cash flow and combining with the process for the cash

flow generation described above, we get that

dC =

{αx− φ

2(αx)2 + rC

}dt (2)

where φ is a parameter capturing the severity of agency costs of free cash flow.6 Firm

value is maximized by allowing the firm to choose its optimal cash retention policy, i.e. by

treating α as a stochastic optimal control variable.7

The explicit treatment of agency costs of free cash flow distinguishes the model from the

existing literature, see for example Asvanunt, Broadie, and Sundaresan (2010) and Gamba

and Triantis (2008), as saving becomes increasingly expensive the higher the fraction of

retained earnings. Specifically, Gamba and Triantis (2008) assume that there is a tax dis-

advantage of keeping the cash within the firm, thereby resulting in a linear treatment of

agency costs. Asvanunt, Broadie, and Sundaresan (2010) assume that the return on the

cash account is lower than the risk-free rate r, i.e. rx < r.8 On the other hand, quadratic

agency costs capture the intuition that if the firm is to receive a positive cash flow shock,

management is more likely to deduct part of the cash flow and use it for empire building

activities. To prevent management from doing so, shareholders thus have to incur higher

monitoring costs.

It is important to notice that the setup is also different from Decamps, Mariotti, Rochet,

6Note that taxation is not included in this model. While there is a tax disadvantage of keeping cashwithin the firm, it is also true that at the investor level, dividends are usually taxed at a higher rate thancapital gains. A meaningful calculation would therefore require one to specify the tax burden at the investorlevel. To abstract from these practical complexities, this paper focuses on agency costs of free cash flow asthe opposing friction.

7For more details see Proposition [1].8The same assumption can also be found in Bolton, Chen, and Wang (2011), Decamps, Mariotti, Rochet,

and Villeneuve (2011), Asvanunt, Broadie, and Sundaresan (2011), Anderson and Carverhill (2011) andHugonnier, Malamud, and Morellec (2012).

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and Villeneuve (2011), Hugonnier, Malamud, and Morellec (2012) and Boyle and Guthrie

(2003) who assume distinct dynamics for operating profits and the cash account. In these

models, cash flow is modeled as an Arithmetic Brownian Motion and cash holdings may also

serve to cover operational losses.9 The difference becomes most evident when compared to

Boyle and Guthrie (2003) who investigate the possibility of future financing shortfalls and

its implications for optimal exercise policy compared to an otherwise unconstrained firm.10

In this model, the focus is on another aspect. Starting with a firm which has to finance

the whole project externally, I analyze how much value the firm would add by not paying

out all the cash flow as dividends and instead optimally retaining part of it to reduce future

financing needs.

The firm has the option to increase production capacity to a level K1 > 1 by paying

some necessary investment costs, denoted as IC. However, if it lacks internal funds it has to

raise all or part of the missing amount externally. It is thus assumed that the firm can issue

costly outside equity to finance the project.11 Specifically, I consider the following general

cost function e(C) which equals

e(C) =

γ0 + γ1 (IC − C) + γ2 (IC − C)2 if C < IC,

0 else

(3)

The specification of this function has been taken and adapted from Atinkilic and Hansen

9The paper also differs with respect to Anderson and Carverhill (2011) who model earnings as a mean-reverting process.

10Boyle and Guthrie assume that prior to exercising the growth option the firm consists of assets in placeG and the cash account X. Assets in place generate an income stream equal to νGdt+φGdZ which directlyaffects the cash account whose dynamics are given by dX = rXdt+ νGdt+ φGdZ.

11The focus on equity financing is given for two reasons. First, from a theoretical perspective the modelanalyzes whether cash has value irrespective of bankruptcy costs. Second, from an empirical perspectiveStrebulaev and Yang (2012) show that being all-equity financed is not a short-run phenomenon but a ratherpersistent event.

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(2000) and Hennessy and Whited (2007) who structurally estimate external financing costs,

thereby capturing in a reduced form both costs stemming from informational asymmetries

as well as transaction costs. The overall costs of capacity expansion are therefore given by

the sum of investment costs and costs of external finance.

Total firm value depends on both state variables x and C and is given by the sum of

expected dividend payments and expected capital gains which include the cash retained

within the firm and the capital gain due to potential capacity expansion.

Proposition 1 Total firm value, denoted by V (x,C) is a function of both state variables x

and C and has to satisfy the following Hamilton-Jacobi-Bellman (HJB) equation under the

risk-neutral measure Q

rV = maxα

{(1− α)x+ (r − δ)xVx + (αx− φ

2(αx)2 + rC)VC + 1/2σ2x2Vxx

}(4)

where the first order condition implies that

α∗ =VC − 1

φxVC(5)

with the additional requirement that α∗ ∈ [0, 1].

Proof: See Appendix.

One can see that the optimal cash retention policy depends on different factors. When

agency costs of free cash flow converge to zero there will be an all-or-nothing type of solution.

As long as the marginal value of cash exceeds one, the firm would want to retain all earnings.

When there is no value premium of cash, it would instead pay out all proceeds as a dividend.

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The introduction of quadratic agency costs of free cash flow implies that there will be some

allocation of x and C such that it will be optimal to save a fraction of current earnings. In

line with general intuition, there is a positive relation between the severity of agency costs

of free cash flow and implied dividend payout ratio.

In order to determine total firm value, Equation [4] has to be solved with respect to the

following boundary conditions

V (0, C) = C

V (x∗, Cτ ) = K1x∗

δ+ Cτ − IC − e(Cτ )

Vx(x∗, Cτ ) = K1

δ

(6)

where x∗ is the investment threshold of the capacity expansion option and Cτ denotes the

amount of cash available at the time the option is exercised. The first condition states that

if the value of the cash flow hits zero, the firm is liquidated and is only worth the value of

the cash account, C. The second condition implies that at the time of exercising the option

the firm receives the payoff of the capacity expansion, pays the investment and financing

costs and retains a corporate cash account equal to Cτ . The last condition is the traditional

smooth-pasting condition ensuring optimal exercise policy.

Note that the value-matching condition reflects the fact that after exercising the option all

future earnings are paid out as dividends such that V (xs, Cs) = EQs

[∫∞se−r(t−s)K1xdt

]+Cs

where s > τ .12 This is optimal because the firm has then exhausted its growth option such

12Note that Cs = 0 if Cτ ≤ IC.

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that there is no marginal benefit of retaining additional cash. To avoid incurring agency

costs of free cash flow, the retention rate α is thus optimally set to zero.

3.2 The Real Option Value of Cash

The value of cash is derived by comparing total firm value under the optimal cash retention

policy to the case when all earnings are paid out as dividends. As such it quantifies the

maximum increase in firm value by optimally trading off costs of external finance against

agency costs of free cash flow.

Definition 1 The real option value of cash is defined as the change in total firm value from

having zero financial slack to following an optimal cash retention policy. Specifically, it is

given by

R(x,C) ≡ V (x,C)− V B(x,C) (7)

where V B(x,C), the benchmark case, denotes firm value of an all-equity firm which pays

out all earnings as a dividend to its shareholders and which finances the project completely

externally.

For the benchmark case, closed-form expressions for the value of the firm and the optimal

investment threshold exist and are summarized in Proposition [2].

Proposition 2 V B(x,C) satisfies the following partial differential equation

rV B = x+ (r − δ)xV Bx + rCV B

C + 1/2σ2x2V Bxx (8)

and is given by

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V B(x,C) = C +x

δ+Bxβ1 (9)

where B =(

(K1−1)x∗Bδ

− IC − e(0))(

1x∗B

)β1and x∗B denotes the corresponding optimal

trigger level, i.e.

x∗B =β1

(β1 − 1) (K1 − 1)δ(IC + e(0)) (10)

Clearly, if C0 = 0 then V B = V B(x).

Proof: See Appendix.

The value of internal funds, as introduced by Definition [1], gives an absolute answer to

the value of cash but it can not be used to judge whether the amount gained or lost from

not paying out dividends is economically significant. To overcome this problem, I introduce

the relative gain from retaining cash and compare the value of internal funds to the initial

value of the capacity expansion option for the benchmark firm.

Definition 2 The relative gain from retaining cash is defined by comparing the real option

value of cash to the value of the capacity expansion option of the benchmark case. Specifically,

it is defined as

S(x,C) ≡ V (x,C)− V B(x,C)

Bxβ1(11)

By construction S(x,C) captures the gain from saving by comparing the value of internal

funds to the value of the initial growth option and it quantifies by how much the firm can

relatively increase the value of its growth option if it follows an optimal cash retention policy.

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While the benchmark case has a closed-form solution, it turns out that Equation [4] can

not be solved analytically if subject to the boundary conditions given in Equation [6]. I

therefore choose to solve V (x,C) numerically by resorting to finite difference methods, i.e.

Crank Nicolson Scheme. Further details regarding the numerical solution can be found in

the Appendix.

However, it is possible to gain some intuition regarding the unknown functional form

for R(x,C) by investigating an extreme scenario. Suppose that the initial cash endowment

of the firm is larger than the required investment costs, i.e. C0 ≥ IC. Assume further the

firm decides to compute the real option value of its cash holdings. It is straightforward to

compute firm value and exercise threshold in case the firm uses the cash and finances the

project internally. Similarly, the solution is also analytically available if the firm does not

use the cash and instead finances externally, i.e. it is given by the benchmark scenario.

Proposition 3 The upper bound for the real option value of cash is given by RU(x) where

RU(x) = xβ1 (A−B) (12)

where A =(

(K1−1)x∗Aδ

− IC)(

1x∗A

)β1and B is as defined in Proposition [2]. The optimal

investment threshold for the case of full internal financing is given by

x∗A =β1

(β1 − 1) (K1 − 1)δIC (13)

Proof: See Appendix.

The subsequent numerical analysis will focus on the full model but we will make use of

the closed form solution for R(x) when needed. Specifically, we will compare the investment

17

threshold under endogenous cash retention to the threshold implied by the upper bound for

the real option value of cash.

4 Numerical Analysis

This section first investigates the effect of cash retention and cash holdings on the firm’s

investment decision. I then analyze the dynamics of optimal cash retention policy, compute

the value of internal funds and analyze its relation with respect to volatility. Finally, the

model is used to propose a regression specification which is then tested on simulated data.

Similar to many other financing and investment models, the problem studied in this

paper does not have a closed form solution. I therefore solve the model using numerical

techniques and illustrate the results using a simple example. For this purpose, the risk-

free rate is set to 6%, the drift rate µ to 1%, cash flow volatility to 19% and the agency

cost parameter φ is set equal to 0.05. These values are similar to both existing papers and

empirical observations.13 Assuming a starting value of the cash flow process of 1, i.e. x0 = 1,

it follows that the initial fundamental value of the firm equals 20. In order to make the

growth option economically relevant, I set the costs of the expansion option equal to 10

and assume that production can be increased by 50 percent, i.e. (K1 − 1) = 0.5.14 Finally,

external financing cost parameters are taken from Hennessy and Whited (2007) and are

set to their estimate for small firms to capture the effect of external financing constraints.

Specifically, the variable cost component γ1 is assumed to be 12% whereas the quadratic cost

13The risk-free rate is similar to Gamba and Triantis (2008) and equal to Datastream’s historical monthlyFed Funds data from 1955 to 2008. The volatility parameter is similar to Boyle and Guthrie (2003) andMauer and Triantis (1994). The agency cost parameter is taken from Eisfeldt and Rampini (2009).

14It can be shown that the choice of K1 does not influence the relative gain from retaining cash as definedby Equation [11] because the value of cash is scaled by the initial growth option such that the capacityexpansion factor cancels out. To make sure that other results are not driven by the choice of K1, robustnesschecks will lessen the impact of the capacity expansion option.

18

component γ2 equals 0.04%. Various robustness checks concerning the impact of investment

costs, cash flow volatility, the magnitude of the capacity expansion option, profitability and

external financing costs will be performed and discussed for each subsection.

4.1 Cash Holdings and Investment

In a recent paper, Denis and Sibilkov (2010) show that financially constrained firms benefit

from cash holdings as it enables them to pursue value increasing investment projects. This

section adds to the discussion by showing that the effect of cash holdings on investment is

effectively nonlinear and can both defer and accelerate investment relative to the case of

complete outside financing.

The green dashed line in Figure [1] depicts the benchmark case, i.e. the threshold x∗B

in case the project is financed fully externally. As expected, in case the firm only has an

explicit capacity expansion option but no freedom regarding the choice of the corresponding

financing strategy, the investment threshold does not depend on the level of cash or, more

precisely, on the level of cash relative to total investment costs. However, if the firm is allowed

to optimally retain cash, then the relation between cash holdings and investment becomes

nonlinear and more complex, as shown by the blue solid line. Specifically, it can be seen

that under the optimal retention policy the firm may invest later, i.e. at a higher investment

threshold, than under the benchmark case. This result is specially interesting as the costs of

investing into the expansion project can never by higher than under the benchmark case. It

is important to emphasize that this behavior is still optimal because the firm basically has

another option to exercise the project at a lower strike price in the future. Only if the level

of cash is sufficiently high, i.e. when accumulated cash holdings exceed approximately 70

percent of the investment costs, it becomes optimal for the firm to exercise its option earlier

19

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

Cash Holdings / Investment Costs

Investm

ent T

hre

shold

Option Exercise and Cash Holdings

x*(C)

x*

B

x*

A

Figure 1: The Relation between Cash Holdings and Investment Thresholds. This figuredisplays the optimal investment threshold, x∗(C), as a function of the firm’s cash holdings which areexpressed relative to the total costs of the investment. The threshold is compared to the benchmarkcase of full external financing (x∗B) and to the first-best trigger level in case there are no financingfrictions (x∗A). Results are shown for the following set of parameter values: r = 0.06, µ = 0.01,σ = 0.19, K1 = 1.5, IC= 10, γ1 = 12% and γ2 = 0.04%.

20

than in the case of complete external financing. In fact, in this case the exercise threshold

converges to x∗A, the trigger level of a firm which relies entirely on internal funds.

This result is important as it provides an alternative view on the impact of cash holdings

on investment. If cash holdings are low relative to investment costs and a firm actively retains

cash, then it is optimal to delay investment relative to the case of full outside financing. This

is because retaining cash has an additional option value to exercise the project at an even

lower price in the future. Only when cash holdings are sufficiently high, they have a strong

and positive impact on the investment decision.

To gain additional understanding regarding the incentives to delay investment, I further

investigate the dynamics of the exercise threshold under different scenarios. The upper part

in Figure [2] shows investment thresholds in case investment costs are increased (left graph)

or decreased (right graph) by a factor of four. Two issues are apparent from the graphs.

First, in case cash holdings are sufficiently low and the firm optimally retains cash, then it is

always optimal to delay investment relative to the case of full external financing.15 Second,

when focusing on the intersection between the investment threshold of the full model, x∗(C),

and the case of full external financing (x∗B), it can be seen that high (low) investment costs

make it less (more) attractive to further delay investment relative to the benchmark case.

Put differently, the intersection between x∗(C) and x∗B shifts to the left (right) in case of

high (low) investment costs. Thus, while this paper shares the general result with Hugonnier,

Malamud, and Morellec (2012) that low cash holdings do not always increase investment,

it differs with respect to the details. In Hugonnier, Malamud, and Morellec (2012), very

high investment costs reduce the value of cash in case cash reserves are low. This is because

capital supply is uncertain and costly such that the firm only holds a minimum cash reserve

15Put differently, in both cases there is always some region where x∗(C) > x∗B .

21

to reduce liquidation risk and therefore finances the project externally. In this model, cash

has value because it allows the firm to reduce financing costs when exercising the growth

option. This value makes it optimal to delay investment relative to the case of external

financing in order to accumulate cash. However, if investment costs are very high the firm is

less willing to retain cash in order to delay investment. This in turn increases the sensitivity

of investment to cash holdings.

The middle part of Figure [2] investigates the impact of volatility on the investment

thresholds. Specifically, the left (right) graph corresponds to an increase (decrease) of volatil-

ity by approximately a factor of two. The effect is as follows. First, higher (lower) volatility

increases (decreases) the investment thresholds of all financing alternatives. Second, in both

cases it makes sense to delay investment relative to the benchmark case of full external fi-

nancing in case cash holdings are sufficiently low. However, the incentive to delay investment

significantly increases in case of low cash flow volatility. Specifically, it can be seen that the

convergence pattern to the unconstrained exercise threshold (x∗A) looks different than in the

previous cases. This is because low volatility allows the firm to better plan its investment

such that it prefers to delay investment as long as possible and thereby accumulates more

cash. In fact, the lower part of Figure [2] illustrates investment thresholds in case volatility

is very low and it reveals an even more extreme investment behavior. For these cases it

becomes optimal to always delay investment relative to the case of full external financing

in case cash holdings are insufficient to fund the investment. Only when the firm has accu-

mulated all necessary cash reserves, it becomes optimal to exercise the option at the trigger

level of an unconstrained firm.

Additional robustness checks vary the impact of external financing costs, the magnitude of

the growth option and firm profitability. The corresponding results are qualitatively similar

22

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%7.6

7.8

8

8.2

8.4

8.6

8.8

9

9.2

9.4

Cash Holdings / Investment Costs

Inve

stm

ent

Thre

shold

Option Exercise and Cash Holdings when IC = 40

x*(C)

x*

B

x*

A

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

Cash Holdings / Investment Costs

Investm

en

t T

hre

shold

Option Exercise and Cash Holdings when IC = 2.5

x*(C)

x*

B

x*

A

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

Cash Holdings / Investment Costs

Inve

stm

en

t T

hre

sho

ld

Option Exercise and Cash Holdings when σ = 40%

x*(C)

x*

B

x*

A

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

Cash Holdings / Investment Costs

Inve

stm

en

t T

hre

sho

ld

Option Exercise and Cash Holdings when σ = 10%

x*(C)

x*

B

x*

A

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

Cash Holdings / Investment Costs

Inve

stm

ent

Th

resh

old

Option Exercise and Cash Holdings when σ = 8%

x*(C)

x*

B

x*

A

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

Cash Holdings / Investment Costs

Inve

stm

ent

Th

resh

old

Option Exercise and Cash Holdings when σ = 5%

x*(C)

x*

B

x*

A

Figure 2: Robustness: The Relation between Cash Holdings and Investment Thresh-olds. This figure displays the optimal investment thresholds introduced in Figure [1]. The graphin the upper left (right) increases (decreases) investment costs to 40 (2.5). The figure in the middleleft (right) increases (decreases) volatility to 40% (10%). The figure in the lower left (right) setsvolatility equal to 8% (5%).

23

and are displayed in the Appendix.16 Lower external financing costs reduce the impact

of cash holdings and thereby decrease incentives to optimally delay investment. Higher

profitability, on the other hand, increases the probability that the option will be exercised in

the future and thereby raises the incentive to retain cash and delay investment. Changing

the magnitude of the growth option increases the investment threshold under all financing

alternatives but has little differential impact, i.e. the intersection between x(C)∗ and x∗B is

largely unaffected.

Returning to the base scenario, one can use Monte Carlo Simulation to look at how much

cash a firm would save until it exercises the option.17 The left panel in Figure [3] displays

the distribution of the firm’s cash holdings just prior to exercising the growth option. The

firm will have saved on average 87% of the investment costs when it is about to exercise

the option. We can further see that in the majority of cases it will have more than half

of the necessary investment costs available as internal funds. Alternatively, one can look

at the actual investment threshold under the optimal cash retention policy. It turns out

that, on average, the firm will exercise the option if the cash flow equals 2.09 which is below

the threshold of the benchmark case. The implied distribution of investment thresholds, as

shown in the right panel in Figure [3], is skewed to the left. This means that there is a

high chance that the firm will be able to internally finance the investment at the unstrained

trigger level of 1.95, thereby underlining the importance of internal funds for the investment

decision.

16For details, see Figure [10].17Note that the model is solved using finite difference methods, i.e. the Crank Nicolson method. By

definition, the finite difference approach can not be used determine optimal cash holdings as it just solvesthe partial differential equation using a grid of different state points. However, one can use the optimalcash retention and investment policy implied by the Crank Nicolson method and then employ Monte Carlosimulation to simulate the evolution of cash holdings.

24

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80

90

100

Actual Cash Level at Investment

Fre

qu

ency

Histogram of Actual Cash Levels at Investment

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.350

20

40

60

80

100

120

Actual Investment Threshold

Fre

qu

ency

Histogram of Investment Thresholds

Figure 3: Implied Distribution for Cash Holdings and Exercise Levels. The left paneldisplays a frequency distribution of actual cash holdings at the exercise time of the option. Theright panel shows a frequency distribution of corresponding investment thresholds.

4.2 Optimal Retention Policy and the Value of Cash

As a next step, I investigate by how much the value of the initial growth option increases

if the firm follows an optimal corporate saving policy. Applying definition [2], the value

maximizing policy leads to an increase of S(x0, C0) to 6.6%. In other words, if the firm

starts with no cash at hand, then it is able to increase the value of the capacity expansion

option by approximately 7 percent.18

Figure [4] shows that this relative gain from retaining cash varies with different real-

izations of operating cash flow and therefore with different probabilities of exercising the

growth option. Holding the initial cash level constant at zero, we can see that the firm can

increase the value of the growth option by as much as 9% if the current realization of the

operating cash flow is around 0.15 units. However, if x then approaches zero, the relative

gain from retaining cash drops off precipitously as the probability that the option will ever

18Clearly, the results depend on the magnitude of the agency costs, captured by the cost parameter φ. Forexample, if one changes the value of φ to 0.025, then the relative gain from retaining cash evaluated at x0and C0 increases to 8.55%. Even more so, if the firm does not suffer from agency costs of free cash flow atall, then it would be able to increase the value of the capacity expansion option by 11.45% if it retains fundswithin the firm.

25

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

x

S(x

,C)

Relation between S(x,C) and x for C=0

Figure 4: The Relative Gain From Retaining Cash. This figure shows displays the relativegain from saving as a function of operating cash flow (x), holding the level of cash holdings constantat C0 = 0.

get exercised is very low. On the other hand, increasing x above 0.15 also decreases the

relative gain from saving as the firm approaches the exercise threshold and therefore is left

with less time to build up the necessary cash reserves.

Cash increases the value of the growth option because it allows the firm to save on

external financing costs and because the firm mitigates the impact of agency costs of free

cash flow by choosing the optimal cash retention policy. Recalling from the previous section

that the retention rate α is determined optimally by setting

α∗ =VC − 1

φxVC

it is evident that the optimal cash retention policy is driven by the severity of agency

costs of free cash flow, current operating cash flow and VC , the marginal value of cash.

26

Figure 5: The marginal value of cash. This figures displays the marginal value of cash, VC , asa function of cash holdings (C) and operating cash flow (x).

Investigating the marginal value of cash at the optimal payout policy, Figure [5] shows that

cash is valued at par in case existing cash holdings are substantial and/or the probability of

exercising the option is low, i.e. x is close to zero. Raising x increases the marginal value of

internal funds until the option gets exercised immediately and the benefit from holding cash

equals the marginal costs of external finance.19

Figure [6] illustrates that for values of x close to zero the firm chooses to pay out most

cash flow as dividends. The reason is that agency costs of free cash flow dominate as

the probability of exercising the option is low. However, for slightly higher values of x it

becomes optimal to retain some fraction of the cash flow in order to reduce future financing

costs. Moreover, by simultaneously increasing C we can see that the optimal retention ratio

19Note that because Figure [5] is based on an optimal trade-off between internal and external financingcosts, the marginal value of cash can not be less than one as otherwise firm value could be improved bychanging the cash retention policy.

27

Figure 6: Optimal State-Contingent Cash Retention Policy. This figure shows the firm’soptimal cash-retention policy α(x,C) as a function of its cash holdings (C) and operating cash flow(x).

increases to as much as 100%. Once the firm reaches the investment threshold, it exercises the

growth option and then optimally sets the retention rate to zero. This result stands in stark

contrast to the existing literature in which the optimal retention policy is an all-or-nothing

decision.20

The previous results suggest a clear relation between investment and cash retention policy.

Specifically, it has been shown that for sufficiently low cash holdings, it becomes optimal

to delay investment relative to the benchmark case of full external financing. Figure [7]

20For example, Decamps, Mariotti, Rochet, and Villeneuve (2011) show that as long as a firm has notaccumulated sufficient cash reserves, the marginal benefit of cash ranges between unity and the marginalcost of issuing shares. Because the costs of retaining funds are linear, this results in a payout decision whichcan be characterized as all-or-nothing. As long as the firm has not accumulated sufficient cash reserves, themarginal benefit of cash exceeds unity such that the firm retains the entire cash flow. Once the dividendthreshold is reached, the entire cash flows is paid out as a dividend. For different examples, see Bolton, Chen,and Wang (2011), Hugonnier, Malamud, and Morellec (2012), Anderson and Carverhill (2011), Gamba andTriantis (2008), Asvanunt, Broadie, and Sundaresan (2010) and Boyle and Guthrie (2003).

28

shows the exact relation between the investment threshold, cash retention boundaries and

the firm’s cash holdings. The blue solid line depicts the no-cash-retention boundary up to

which it is optimal not to retain any additional cash. As shown before, if the probability

of exercising the growth option is low (i.e. x is low) and/or existing cash holdings are

sufficiently high, then it is optimal to choose a full-payout policy. In addition, it can be seen

that this boundary is close to zero when existing cash holdings are low. At the same, this is

precisely when it becomes optimal to delay investment as much as possible, as indicated by

the investment threshold (the red dotted line). Note that after exercising the growth option

it becomes optimal not to retain any additional funds.21 In between these two boundaries,

the optimal strategy is to retain a positive fraction of each period’s cash flow. Moreover,

as illustrated by the green dashed line, at some point the firm chooses to retain all newly

generated cash flow. This is the case when the firm has already accumulated sufficient cash

holdings and is close to exercising the growth option.

The results are robust with respect to the previous robustness checks. For reasons of

brevity, only a brief discussion of the corresponding relation between investment and cash

holdings is provided.22 First, if investment costs are high (low), then it is optimal to retain

less (more) cash. In fact, for high investment costs it is never optimal to retain as much as the

entire cash flow whereas in case of low investment costs, the full cash retention region is large.

This is consistent with the previous finding that high (low) investment costs reduce (increase)

incentives to delay investment.23 Second, low volatility reduces uncertainty and allows the

21Note that there are two regions for which it is optimal not to retain any cash. The first region ischaracterized by a low probability of needing the additional funds (i.e. the area below the blue solid line)whereas for the second region the growth option has been exhausted. While it is possible to derive a closed-form solution for the second case, i.e. V (xs, Cs) = EQs

[∫∞se−r(t−s)K1xdt

]+ Cs where s > τ , this is not

possible for the first case. The reason is that the cash retention rate α is dynamic and thus the result of thedynamic stochastic optimization problem posited in Proposition [1] which needs to be solved numerically.

22For details, see Figure [11] in the Appendix.23Note that this result differs from Hugonnier, Malamud, and Morellec (2012) who find that high invest-

29

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

α*(x,C) = 1

Cash Retention and Investment Thresholds

α*(x,C) = 0

α*(x,C) = 1

x*(C)

Figure 7: Optimal Investment and Cash Retention Boundaries This figure shows the firm’sdifferent cash retention and investment boundaries. Specifically, α∗(x,C) = 0 displays the boundarybelow which it is optimal to pay out all funds as a dividend, α∗(x,C) = interior, corresponds tothe region where an interior solution for the optimal cash retention policy exists. The variablex∗(C) corresponds to the optimal investment threshold above which it is optimal to pay out a fulldividend. Finally, for the region between α∗(x,C) = 1 and x∗(C) it is optimal to retain 100% ofthe cash flow.

30

firm to better plan the investment. This results in a more extreme cash retention policy.

As long as it is unlikely that the firm will benefit from the additional funds, it optimally

chooses not to retain any cash. Once the probability of exercising the option increases, the

firm raises the retention rate and retains the entire cash flow for a relatively stable fraction of

the state space. Third, higher profitability generally increases cash retention as it raises the

likelihood that the investment threshold will be reached such that the firm is more willing to

incur agency costs of free cash flow. Fourth, low external financing costs reduce the potential

impact of retaining cash and therefore decrease the optimal cash retention rate. In fact, it

is never optimal to retain 100 percent of the cash flow. Finally, reducing the impact of the

growth option also reduces the potential payoff which in turns increases the relative impact

of agency costs of free cash flow. This results in a lower optimal cash retention rate which

is always below unity.24

An important finding of the preceding analysis is that volatility has a strong impact

on the optimal investment behavior and cash retention policy. Therefore, I now provide

a detailed analysis of the impact of volatility on the real option value of cash. Figure [8]

illustrates that the value of internal funds is negatively related to cash flow volatility. While

the result sounds surprising at first, the intuition is straightforward. Previous results have

ment costs can make it optimal for a firm to reduce cash holdings to a minimum level and instead finance aproject externally. In this paper, ceteris paribus high investment costs reduce incentives to delay investmentbeyond the case of full external financing. This is reflected in a lower cash retention ratio but, as seen before,this increases the responsiveness of investment to cash holdings.

24Note that while Decamps, Mariotti, Rochet, and Villeneuve (2011) also display dividend boundaries fordifferent levels of cash flow volatility and profitability, the results are not directly comparable. The reason isthat in Decamps, Mariotti, Rochet, and Villeneuve (2011) the dividend boundary is expressed as the criticallevel of cash holdings whereas in this paper, the boundary is measured as the critical level of cash flow.That being said, the predictions seem to differ. In Decamps, Mariotti, Rochet, and Villeneuve (2011), higherprofitability and lower volatility reduce the dividend boundary and decrease the likelihood to retain cash.For this paper, the effect is the opposite. The reason relates to the underlying need for cash. In Decamps,Mariotti, Rochet, and Villeneuve (2011), cash avoids inefficient liquidation whereas in this paper it allowsfor a less costly capacity expansion, thereby making cash valuable when profitability is high and volatility islow.

31

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

σ

R(x

,C)

The relation between volatility and the value of internal funds

Figure 8: The Real Option Value of Cash and Volatility. This figure shows that the relationbetween R(x,C) and cash flow volatility (σ) when x = 1 and C = 0.

shown that in case volatility is low, the firm optimally delays investment and accumulates

cash. Thus, when exercising the option the firm on average has saved more cash than in case

volatility was high. Put differently, low volatility allows the firm to finance a larger fraction

of the project internally and thereby it generates value.25

An alternative and complementary explanation is given by using definition [1]. Higher

volatility increases the value of the capacity expansion option but the increase in volatility

is more beneficial for the benchmark case of full external financing as this option is more

out-of-the money. Loosely speaking, retaining cash enlarges the area of downside risk which

is why an increase in volatility is not necessarily value enhancing.26 Moreover, it can be

25Note that this result is specially interesting because it suggests that value is driven by the relative delayin investment, as illustrated in Figure [2]. While lower volatility decreases the investment threshold for allfinancing alternatives, it increases incentives to delay investment relative to the case of full external financing.

26In this loose sense, the result can be interpreted as the opposite of the risk-shifting problem.

32

shown that this result is not a consequence of the endogenous cash account but that it even

holds more generally if one compares the value of an investment option with and without

external financing costs. For more details and the full closed-form expression for the upper

bound of the real option value of cash, please see the Appendix.

To make sure that results are not driven by defining the value of internal funds as the

difference between two growth options, I also show how the marginal value of cash changes

with different levels of volatility. Figure [9] plots VC as a function of σ for the case when cash

flow is held constant at x0 and the level of cash is set to 0.005, 1, 5 and 7.5 respectively. It is

interesting to note that for moderate levels of volatility, i.e. 25 to 50 percent, the marginal

value of cash is decreasing in volatility across low to medium endowments of cash whereas

the decrease only starts for volatility levels of 40% and higher in case cash holdings equal

75% of the investment costs.

The finding in this paper differs from the conventional wisdom that cash has value in case

uncertainty is high. While this perception has been accepted in the literature, there is little

detailed analysis on the impact of volatility on the value of cash holdings. In Gamba and

Triantis (2008), cash has value because it allows the firm to exercise growth opportunities in

case of superior operating performance and also, because it helps to avoid costly asset sales

in low states of nature. While the paper summarizes that high volatility enhances the value

of cash, it does not provide a detailed analysis of the impact of volatility on the value of

financial flexibility. Decamps, Mariotti, Rochet, and Villeneuve (2011) analyze the impact

of cash holdings on firm value and stock prices and show that there is a positive relation

between the value of cash and stock price volatility. The paper also shows that the marginal

value of cash is generally lower in case cash holdings are high and, in this context, find that

the effect of volatility on the marginal value of cash might be ambiguous. However, the

33

0 0.1 0.2 0.3 0.4 0.51.02

1.021

1.022

1.023

1.024

1.025

1.026

1.027

1.028

1.029

1.03

σ

VC

The relation between volatility and the marginal value of cash for C=0.005

0 0.1 0.2 0.3 0.4 0.51.018

1.019

1.02

1.021

1.022

1.023

1.024

1.025

1.026

1.027

1.028

σ

VC

The relation between volatility and the marginal value of cash for C=1

0 0.1 0.2 0.3 0.4 0.51.006

1.007

1.008

1.009

1.01

1.011

1.012

1.013

1.014

1.015

1.016

σ

VC

The relation between volatility and the marginal value of cash for C=5

0 0.1 0.2 0.3 0.4 0.51

1.001

1.002

1.003

1.004

1.005

1.006

σ

VC

The relation between volatility and the marginal value of cash for C=7.5

Figure 9: The Marginal Value of Cash and Volatility for Different Levels of Cash Hold-ings. This figure displays the marginal value of cash, VC , as a function of volatility and additionallycontrols for the level of existing cash holdings. The upper left (right) figure sets C = 0.005 (C = 1).The lower left (right) figure sets C = 5 (C = 7.5).

paper does not provide a direct and detailed analysis of the impact of cash flow volatility on

the value of cash. This paper fills the gap by providing a detailed analysis between the value

of cash and volatility and, by showing that, in case cash does not serve as a buffer against

bankruptcy risk, the relation is mostly negative.27

27Note that results are also robust with respect to previous robustness checks regarding the impact ofprofitability, financing costs and the magnitude of the growth option. For details, please see Figure [12]in the Appendix. Further unreported results show that the impact is also robust with respect to differentinvestment costs.

34

4.3 Simulated Data and Regression Analysis

This section uses regression analysis and relates observable model implied variables to overall

firm value in order to obtain an alternative estimate of the shadow value of cash. Specifically,

the model has shown that total firm value is given by

V (xt, Ct) = xt + Ct +G(xt, Ct) + E

[∫ ∞t+1

e−r(s−t)xsdt

](14)

where xt denotes the level of current cash flow and the term E[∫∞t+1

e−r(s−t)xsdt] represents

the discounted value of all future cash flows. Total firm value thus consists of the cash flow

generated today, the amount of cash the firm has retained, the value of the growth option

and the expected value of all future discounted cash flows based on the firm’s assets in place.

Using definition [1], I approximate R(x,C) = V (xt, Ct)−E[∫∞

t+1e−r(s−t)xsdt

]due to the

fact that the value of the benchmark firm can not be observed in praxis. The same holds

true for the valuation of the growth option and I therefore replace G(xt, Ct) with the closed

form approximation provided in Proposition [2]. This implies the following testable equation

R(xt, Ct) = a+ b1xt + b2Ct + b3G(xt) + ε (15)

Employing Monte Carlo Simulation, I calculate firm value, cash holdings, fundamental

firm value and then compute the approximate value of the growth option contingent on the

realization of the cash flow process. Furthermore, I set the length of the time step dt equal

to 1/250, the time horizon equal to 20 years and the number of replications equal to 1, 000.28

Table [1] shows corresponding results when Equation [15] is estimated for the entire set of

firms which have not yet exercised the growth option and for two subsamples, i.e. when the

28Note that results are robust to setting the time step equal to monthly or quarterly data.

35

Table 1: Regressions based on the Simulated Value of Cash. This table displays resultswhen estimating the value of cash based on the simulated dataset of Section [4]. The dependent

variable is approximated using R(x,C) = V (xt, Ct)− E[∫∞t+1 e

−r(s−t)xsdt]

given that the value of

the benchmark firm can not be observed in practice. The observed value of the growth option isdefined using the closed form approximation provided in Proposition [2], i.e. Bxβ1 . Cash holdingsand operating cash flow are the state variables of the underlying model. The regression is estimatedusing ordinary least squares (OLS), focusing on all firms which have not yet exercised the growthoption. Results are displayed for three cases: (i) the full sample, (ii) a subsample when the timeto build cash reserves is restricted to eight years and (iii) a subsample when the time to build cashreserves is restricted to four years.

(1) (2) (3)All If Time < 8 years If Time < 4 Years

Coefficient Coefficient CoefficientGrowth Options 0.943∗∗∗ 0.909∗∗∗ 0.874∗∗∗

Cash Holdings 1.012∗∗∗ 1.032∗∗∗ 1.036∗∗∗

Cash Flow 1.123∗∗∗ 1.171∗∗∗ 1.292∗∗∗

∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

time to save is limited to eight or four years. To reduce the impact of the initial starting

condition, I drop the first 100 realizations of each replication.

When the analysis is made for the entire simulated dataset, the average value premium

of cash equals 1.2 percent. Thus, even without conditioning on whether some firms already

have enough cash to internally finance the investment, cash is valued at a slight premium

to its notional amount. However, if the analysis is restricted to firms during the time of

building up the cash reserves, the estimated value of cash more than doubles to 3.2 and 3.6

percent respectively. In other words, by excluding firms which already saved a lot of cash

but still have not exercised their growth options, we can see that the marginal value of cash

increases substantially.

36

4.4 Main Testable Implications

The model has shown that cash is valuable in the context of growth opportunities and that

the actual value depends on the specific combination between cash holdings and cash flow.

In general, if there is a realistic probability that the growth option will be exercised, the firm

will optimally retain some fraction of its cash flow to save for future investment outlays.

The model makes two empirical predictions which are of first order importance. First,

there is a negative relation between the value of cash and volatility. Second, cash retention

has an ambiguous effect on a firm’s investment policy. Sufficiently low cash holdings defer

investment whereas higher values are expected to have the opposite effect.

5 Empirical Analysis

The objective of this section is to use the regression setup implied by the theoretical model

and test whether the two main implications regarding the relation between volatility and

the value of cash and between cash holdings and investment are rejected by empirical data.

Focusing only on the (partial) correlation between the variables of interest makes it possible

to empirically test the model in reduced form without the need to address potential endo-

geneity issues stemming from reverse causality.29 Nevertheless, as a final robustness check

I show that the results with respect to volatility and investment are not driven by reverse

causality and also hold true with lagged explanatory variables.

Because real-life data are more noisy than simulated inputs, Equation [15] is extended

by including a vector of control variables to make sure that results do not suffer from an

omitted variable bias. Also, I include both the lagged and lead change in a firm’s cash

29For a detailed discussion of this issue see Riddick and Whited (2009).

37

holdings. This is done because the results found in the previous section have shown that the

value of internal funds depends on the saving process and also on the intended use of cash.

Taken together, this implies the following regression setup

Rt = α + β1CFt + β2Ct + β3dCt + β4dCt+1 + β5GOt + β6Xt + ε (16)

I follow Section [4] and define the dependent variable analogously as the difference be-

tween the total market value of the firm and the value of its non-cash assets. Operating

cash flow is denoted as CFt, Ct captures cash holdings, dCt and dCt+1 are the lagged and

lead change in cash holdings and GOt is the proxy for the growth option. Finally, the vari-

able Xt is a vector of control variables commonly used in the literature and includes the

level and changes of research and development (R&D) expenditures and dividend payments,

the changes in the firm’s net assets and operating cash flow and the lead change in total

firm value to account for all non-captured market expectations.30 To avoid that results are

dominated by the largest firms in the sample, all variables are scaled by the book value of

assets.31

An important empirical issue concerns the choice of a proxy for the growth opportunity.

Traditionally, the literature on financing and investment decisions uses the ratio between

the market value of a firm’s physical assets and its replacement costs as a proxy for the

value of growth opportunities.32 However, given that the market value enters the numerator

of the dependent variable, one has to employ another proxy variable. Potential candidates

30The exact definition of the variables is given in Table [3].31It is important to notice that this specification is different from Pinkowitz, Stulz, and Williamson (2006)

as it allows for a joint inclusion of both the level and changes of cash while still accounting for the level ofnon-cash assets in the definition of the dependent variable.

32Examples include Hayashi (1982), Fazzari, Hubbard, and Petersen (1988), Erickson and Whited (2000)and Hennessy (2004).

38

used in the finance and accounting literature include R&D expenses and capital expenditures

(CAPEX).33 As this paper focuses on an investment project in tangible assets, i.e. a capacity

expansion option, I use CAPEX for the subsequent analysis.34 Specifically, I follow Goyal,

Lehn, and Racic (2002) and use the ratio between CAPEX and the book value of assets to

control for the presence of growth opportunities.

The study uses accounting data from COMPUSTAT and includes firm year observations

from 1980 to 2010. Financial firms (6000 ≤ SIC ≤ 6999), utilities (4900 ≤ SIC ≤ 4999) and

firms not incorporated in the United States are deleted from the sample and all variables

are cut-off at the 1% level to reduce the effect of outliers.35 The analysis is restricted to

all-equity financed firms by requiring that the firm carries no short and long-term debt. Due

to the inclusion of lead and lagged variables this requirement has to be fulfilled for three

consecutive years. This leaves a total of 5,658 firm year observations.

Table [2] displays the fraction of all-equity financed firms and the corresponding cash

holdings relative to the book value of assets for the sample period. It can be seen that both

the fraction of all-equity financed firms and their cash holdings have increased substantially.

Moreover, as of 2010 one in five firms has zero leverage and holds cash equal to one third of

the book value of assets.36

33See for example Stowe and Xing (2006), Pinkowitz and Williamson (2004), Goyal, Lehn, and Racic(2002), Lang, Ofek, and Stulz (1996), Gaver and Gaver (1993), Skinner (1993) and Smith and Watts (1992)among others.

34Besides, for a substantial fraction of the sample R&D expenses equal zero or are missing which wouldunnecessarily reduce sample size.

35Note that trimming is done with respect to the full sample.36For a detailed analysis of the increasing importance of all-equity financed firms and their corresponding

firm characteristics, see Strebulaev and Yang (2012).

39

Table 2: Average Cash Holdings of All-Equity Firms. This figure displays average cashholdings of all-equity financed firms. Specifically, it uses data from COMPUSTAT and includesfirm year observations from 1980 to 2010. Financial firms (6000 ≤ SIC ≤ 6999), utilities (4900≤ SIC ≤ 4999) and firms not incorporated in the United States are excluded from the sample.All-equity firms are defined as carrying neither short-term debt (mnemonic: dlc) nor long-termdebt (mnemonic: dltt) in their capital structure, cash holdings (mnemonic: ch) are stated relativeto the value of their assets (mnemonic: at).

Year Fraction All-Equity Firms Average Cash Holdings1980 0.06 0.091981 0.06 0.091982 0.07 0.081983 0.08 0.091984 0.07 0.081985 0.07 0.101986 0.08 0.121987 0.09 0.141988 0.09 0.231989 0.09 0.241990 0.09 0.241991 0.11 0.281992 0.12 0.251993 0.13 0.241994 0.12 0.211995 0.13 0.261996 0.13 0.291997 0.14 0.291998 0.14 0.291999 0.14 0.312000 0.14 0.312001 0.15 0.322002 0.16 0.312003 0.18 0.332004 0.19 0.322005 0.19 0.312006 0.19 0.312007 0.19 0.332008 0.19 0.332009 0.19 0.332010 0.19 0.33

5.1 Estimating the Value of Cash

Equation [16] is estimated accounting for firm fixed effects and by including time dummies.

Standard errors are computed according to Discroll and Kraay (1998) to account for possible

40

cross-sectional interdependence among the error terms. Denoting R̂ as the estimated value

of cash and controlling for lead and lagged variables in the regression setup, it follows that

R̂ = β̂2 + β̂3 − β̂4 (17)

When interpreting results, I further impose the null hypothesis that the true value is

equal to one, i.e. that cash is valued at par in a world without financing frictions.37

Table [3] displays results for the baseline model. It turns out that by plugging the

coefficients of Ct, dCt and dCt+1 into Equation [17] the estimated value is equal to 0.72 with

a corresponding t-statistic of -1.64. Thus, the average estimated value of corporate cash

holdings for all-equity financed firms is not statistically different from its notional amount.

Surprisingly, it can be seen that cash flow has a negative effect on our dependent variable

which is contrary to the results in Section [4]. It turns out that the negative coefficient is

driven by small firms experiencing negative cash flows while at the same time their valuations,

and thereby the dependent variable, increase. Estimating Equation [16] for firms with total

assets of more than $50 ($100) million reveals that the coefficient of cash flow is not different

from zero while leaving all other results unchanged. In fact, the corresponding t-statistics

for testing whether the value of cash is different from its notional amount increase to -0.82

and -0.59.38

The main interest concerns the fact whether volatility has a negative effect on the value

of cash in the context of growth opportunities. To answer this question, I extend the baseline

model by including two interaction terms. The first term interacts cash holdings with growth

opportunities to proxy for the value of cash in the context of growth opportunities, while

37Standard errors are computed using the variance-covariance matrix of β̂2, β̂3 and β̂4.38For results, please see Table [8] in the Appendix.

41

Table 3: The Value of Cash. This table displays results when estimating the value of cashfor the baseline scenario. The regression uses data from COMPUSTAT and includes firm yearobservations from 1980 to 2010. Financial firms (6000 ≤ SIC ≤ 6999), utilities (4900 ≤ SIC ≤4999) and firms not incorporated in the United States are deleted from the sample and all variablesare cut off at the 1% level to reduce the effect of outliers. Subsequent variable definitions are basedon COMPUSTAT mnemonics. The dependent variable is defined according to Section [4] as thedifference between the total market value of the firm (prccf ∗ csho ) and the value of its non-cashassets (at - ch). Operating cash flow is defined as CF = ib + dp - ∆ NWC where NWC = (act- ch) - lct and ∆ NWC = NWCt − NWCt−1. Cash holdings are C = ch, dCt = Ct − Ct−1 anddCt+1 = Ct+1 − Ct. The definition of the growth option is given by GO =

capxat . The analysis

accounts for control variables typically used in the literature and includes the level and changesof R&D expenditures (xrd) and dividend payments (dvc + dvp) , the changes in the firm’s netassets (at - ch) and operating cash flow and the lead change in total firm value to account for allnon-captured market expectations. The regression is estimated accounting for firm fixed effects andby including time dummies. Standard errors are computed following Discroll and Kraay (1998) toaccount for possible cross-sectional interdependence among the error terms.The analysis is restrictedto all-equity financed firms by requiring that the firm carries no short and long-term debt. Due tothe inclusion of lead and lagged variables this requirement has to be fulfilled for three consecutiveyears. This leaves a total of 5,658 firm year observations.

Coefficient T-StatisticsCash flow -1.334∗∗ -3.24Cash 2.252∗∗∗ 5.15dCt 0.939∗∗∗ 4.80dCt+1 2.475∗∗∗ 9.49Growth Option 1.051 1.44Control Variables yesTime Dummies yesObservations 5658R2 0.312∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

42

Table 4: The Value of Cash in the Context of Growth Options and Volatility. Thistable displays results when estimating the value of cash in the context of growth opportunities andcash flow volatility. Previously introduced variables follow the definition given in Table [3]. Newvariables include cash flow volatility, σt, which is defined as the time-varying volatility of operatingcash flow. Volatility estimates are based on quarterly observations, initial volatility is calculatedfor the period from 1980 to 1988 under the additional requirement of having at least 16 quarterlyobservations. Each year the estimation window is extended by one year while the initial observationperiod is held fixed at 1980. Results are presented for three cases: (i) the full sample, (ii) for firmswith more than $50mn in total assets and (iii) for firms with more than $100mn in total assets.The coefficients are estimated analogously to Table [3].

(1) (2) (3)Full Sample Total Assets > $50mn Total Assets > $100mn

Coefficient T-Statistics Coefficient T-Statistics Coefficient T-StatisticsCash Flow -1.132∗∗ -2.65 -0.325 -0.52 0.040 0.06Cash 2.545∗∗∗ 6.11 2.091∗∗∗ 6.22 1.478∗∗ 3.05GO 2.098 1.38 4.959∗∗∗ 4.61 5.540∗∗∗ 4.33GO x C 4.049 1.29 2.933 0.96 7.005 1.39GO x C x σ -0.265∗ -2.20 -0.280∗ -2.35 -0.253∗ -2.14Observations 4463 2474 1803R2 0.343 0.406 0.380∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

the second one interacts cash holdings with growth opportunities and volatility to infer the

marginal effect of volatility on the value of cash. Specifically, I estimate

Rt = α + β1CFt + β2Ct + β3CtGOt + β4CtGOtσt + β5dCt + β6dCt+1

+b7GOt + b8Xt + ε (18)

where σt is the time-varying volatility of operating cash flow.39 Table [4] displays cor-

responding results when Equation [18] is estimated for the full sample, for firms with total

assets of more than $50 million and for firms larger than $100 million in total assets.

39Volatility estimates are based on quarterly observations. Initial volatility is calculated for the periodfrom 1980 to 1988 and I follow Han and Qiu (2007) and Minton and Schrand (1999) in requiring at least 16quarterly observations. Each year the estimation window is extended by one year while the initial observationperiod is held fixed at 1980.

43

Focusing on the coefficients for the full sample, it can be seen that the effect of volatility

on the value of cash is negative and statistically significant. As an important robustness

check, we can observe that although size has an impact on the coefficient and statistical

significance of operating cash flow, the negative relation with respect to volatility is not

affected by it. In fact, even for firms with a market capitalization of more than $100 million,

the effect of volatility on the value of cash is negative and statistically significant.

Interestingly, while the the growth option has a positive and statistically significant im-

pact, the coefficient of the interaction term between cash holdings and the growth option is

statistically insignificant. Intuitively, this suggests that most of the information regarding

the impact of growth opportunities is captured by the level of growth opportunities itself.

Dropping the variable GOt from Equation [18] and re-estimating reveals that the interac-

tion term is positive and statistically significant while still preserving the negative impact of

volatility on the value of cash. Full results can be seen in Table [5].

Table 5: Robustness: The Value of Cash and Volatility: This table presents results whendropping the level of growth opportunities, i.e. GOt from Equation [18]. All variables and theestimation procedure follow Table [4].

(1) (2) (3)Full Sample Total Assets > 50mn Total Assets > 100mn

Coefficient T-Statistics Coefficient T-Statistics Coefficient T-StatisticsCash Flow -1.089∗ -2.49 -0.232 -0.36 0.143 0.22Cash 2.417∗∗∗ 5.57 1.766∗∗∗ 5.74 1.075∗ 2.49GO x C 7.228∗ 2.54 9.827∗∗∗ 3.36 16.041∗∗∗ 3.89GO x C x σ -0.264∗ -2.17 -0.277∗ -2.29 -0.252∗ -2.12Observations 4463 2474 1803R2 0.3431 0.403 0.377∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

While the general interpretation of the regression coefficients follows Riddick and Whited

(2009), i.e. the coefficients are estimates of a partial correlation between the dependent

variable and the regressors, the subsequent robustness check shows that reverse causality

44

between market values and cash flow volatility does not drive results. Specifically, Table

[6] displays results in case lagged values for the level and changes of cash holdings, growth

opportunities and cash flow volatility are used. It can be seen that while the impact of cash

holdings and growth opportunities on firm value is statistically insignificant, the impact of

cash volatility on the value of cash is still negative and statistically significant.40

Table 6: Robustness: The Value of Cash in the Context of Growth Options and Volatil-ity. This table displays results when estimating the value of cash in the context of growth opportu-nities and cash flow volatility. Previously introduced variables follow the definition given in Table[4]. Results are presented for three cases: (i) the full sample, (ii) for firms with more than $50mn intotal assets and (iii) for firms with more than $100mn in total assets. The coefficients are estimatedanalogously to Table [4] but lagged values for level and changes cash holdings, growth opportunitiesand cash flow volatility are used.

(1) (2) (3)Full Sample Total Assets > $50mn Total Assets > $100mn

Coefficient T-Statistics Coefficient T-Statistics Coefficient T-StatisticsCash Flow 0.635 1.44 1.071 1.59 1.189 1.75C (lagged) 1.033 1.83 0.806 1.76 1.058 1.40GO (lagged) 1.906 1.29 1.662 0.61 -0.200 -0.06GO x C (lagged) 4.680 0.89 6.863 1.04 9.573 1.00GO x C x σ (lagged) -0.322∗∗∗ -3.81 -0.336∗∗∗ -3.80 -0.285∗∗ -2.73Observations 3199 1828 1382R2 0.283 0.337 0.326∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

5.2 The Relation between Cash Holdings and Investment

The theoretical model implies a nonlinear relation between cash holdings and investment.

Specifically, low cash holdings relative to investment costs lead to a delay in investment

compared to the benchmark case of full external financing whereas sufficiently high cash

holdings have the opposite effect.

40As a final robustness check, I analyze whether the negative relation with respect to volatility is alsorobust to employing the setup of Pinkowitz, Stulz, and Williamson (2006). It turns out that when usingtheir level regression, the effect of volatility on the value of cash is negative and statistically significantwhereas the effect in the changes regression is statistically indifferent from zero. For an explanation of theirmodel and the terms level and changes regression, please see Appendix.

45

Table 7: The Impact of Cash Holdings on Investment. This table shows results whenestimating the impact of cash holdings on a firm’s investment decision. The dependent variable isinvestment (capx), Ct is the level of cash holdings relative to property, plant and equipment netof depreciation (ppent) and C2

t is the squared value of C. The variable Y is a vector consistingof several control variables such as operating cash flow (defined as in Table [3]) , size (logarithmof sale), growth (two year growth rate of sale) and market-to-book ratio which is defined as (V-invt)/at where V = prccf∗csho + dlt + dlc). All variables are cut off at the 1% level to reducethe effect of outliers. The regression accounts for firm fixed effects and is estimated using OLS andemploying either simultaneous or lagged regressors and time dummies.

(1) (2)Simultaneous Lagged

Coefficient T-Statistics Coefficient T-StatisticsCash -0.051498∗∗∗ -9.57 -0.008838∗ -2.22Cash2 0.000210∗∗∗ 9.02 0.000032∗ 2.08Cash flow 0.051170∗∗∗ 4.15 0.045092∗∗∗ 3.78MTB Ratio 0.003753 0.20 0.246171∗∗∗ 4.36Size 2.073676∗∗∗ 7.54 1.932559∗∗∗ 5.48Growth Rate 0.062725 0.64 -0.009631 -0.23Observations 5155 3895R2 0.117 0.102∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

To briefly test whether there is a nonlinear impact of cash holdings on investment, I

estimate the following regression

Invt = α + β1Ct + β2C2t + β3Y + ε (19)

where Invt is capital expenditures, Ct is the level of cash holdings relative to property,

plant and equipment net of depreciation and C2t is the squared value of C. The variable Y is a

vector consisting of several control variables such as operating cash flow, size, growth, market-

to-book ratio. Regression [19] is estimated using either simultaneous or lagged regressors

and time dummies.

Table [7] displays corresponding results. It can be seen that low levels of cash holdings

have a negative effect on investment whereas for sufficiently high values, the effect becomes

positive. This result holds true irrespective of whether simultaneous or lagged regressors are

46

used and it confirms that there is indeed a nonlinear relation between cash holdings and

investment.

6 Conclusion

This paper focuses on the idea that cash has a real option value and thereby proposes an

explicit valuation framework for the value of internal funds which is based on a tradeoff

between agency costs of free cash flow and costs of external finance. Specifically, I model the

value of cash for an all-equity financed firm in the context of a capacity expansion problem.

The paper contributes to the existing literature on several fronts.

First, the model implies a closed-form solution for the optimal state-contingent cash

retention policy. Results show that most of the time it is optimal to retain only a fraction

of each period’s cash flow and are therefore consistent with the empirically documented fact

that firms increase cash holdings while still paying dividends.

Second, the paper provides a detailed analysis of the impact of optimal cash retention

on the firm’s investment policy. Specifically, it shows that for sufficiently low cash holdings

it becomes optimal to delay investment and retain more cash. Moreover, the incentives to

retain cash and delay investment are stronger in case cash flow volatility or investment costs

of a project are low.

Third, I further show that an increase in cash flow volatility generally reduces the value

of internal funds. A simple way to think about the result is that low volatility allows the firm

to better plan the investment, retain cash more efficiently and thereby generate additional

value. This result has important practical implications as it suggests that once cash does

not serve as a buffer against bankruptcy risk it is less valuable in high volatility states.

47

Fourth, the theoretical section concludes by employing Monte Carlo simulation to gener-

ate artificial data. The real option value of cash is then estimated using a regression setup

implied by the theoretical analysis. Results show that cash is valued at a premium to its

notional amount in the context of growth opportunities and that the premium is higher if

firms just started to retain funds.

Finally, the main predictions of the model regarding the negative relation between volatil-

ity and the value of cash as well as the nonlinear relation between cash holdings and invest-

ment are confirmed using data on U.S. public corporations between 1980 and 2010. The

paper shows that all-equity firms have increased cash holdings substantially and thus un-

derlines the relevance of the research question. In 2010, one in five U.S. industrial firms on

COMPUSTAT has zero leverage and holds cash equal to roughly one third of the book value

of assets.

48

Appendix A: Proofs

Proof of Proposition [1]

[Proof] Using the fact that µ = r − δ we can write that dx = (r − δ)xdt + σxdWQ. Let’s

suppose we construct a risk-free portfolio by holding θ1 units of the firm and shorting θ2

units of the traded asset. The long position of the portfolio entitles us to an instantaneous

dividend payment θ1(1 − α)x. The value of the portfolio P is given by (θ1V − θ2X) and it

follows that the total return from holding the portfolio over a short time interval dt equals

dP = θ1 ((1− α)xdt+ dV )− θ2dX (20)

Applying Ito’s Lemma leaves us with

dP = θ1

((1− α)xdt+ Vxdx+ VCdC +

1

2σ2x2Vxxdt

)− θ2dX (21)

For θ1 = 1 , it immediately follows that θ2 equals(VxxX

)which then implies that dP =

rPdt. Combining above and using the fact that P = (V − xVxXX) we obtain that

rV = (1− α)x+ (r − δ)xVx + (αx− φ

2(αx)2 + rC)VC + 1/2σ2x2Vxx (22)

The only missing step is to treat α as a stochastic optimal control by imposing that

rV = maxα

{(1− α)x+ (r − δ)xVx + (αx− φ

2(αx)2 + rC)VC + 1/2σ2x2Vxx

}(23)

Taking the FOC with respect to α implies that

49

α∗ =VC − 1

φxVC(24)

with the additional requirement that α∗ ∈ [0, 1].

�

Proof of Proposition [2]

[Proof] By assumption α is set to 0 such that the PDE in Equation [4] simplifies to

rV B = x+ (r − δ)xV Bx + rCV B

C + 1/2σ2x2V Bxx (25)

which has to be solved with respect to

V B(0, Ct) = Ct

V B(x∗, Cτ ) = K1x∗

δ+ Cτ − IC − e(0)

V Bx (x∗, Cτ ) = K1

δ

(26)

Assuming that V B(x,C) = νC + Bxβ + γx and solving the PDE with respect to the

boundary conditions implies that

x∗B =β1

(β1 − 1) (K1 − 1)δ(IC + e(0)) (27)

where β1 is the positive root of the fundamental quadratic

50

1

2β(β − 1) + µβ − r = 0 (28)

It follows that V B(x,C) = xδ

+Bxβ1 + C where B =(

(K1−1)x∗Bδ

− IC − e(0))(

1x∗B

)β1.

Proof of Proposition [3]

To derive the upper bound, we first compute firm value under full internal and external

financing. Focusing first on the case of full internal financing, we have that α = 0 as

C0 > IC. It suffices to solve the PDE given in [8] with respect to

V A(0, Ct) = Ct

V A(x∗, Cτ ) = K1x∗

δ+ Cτ − IC

V Ax (x∗, Cτ ) = K1

δ

(29)

Assuming that V A(x,C) = νC + Axβ + γx and solving the PDE with respect to the

boundary conditions implies that

x∗A =β1

(β1 − 1) (K1 − 1)δIC (30)

where β1 is the positive root of the same fundamental quadratic as in Equation [28]. It

follows that V A(x,C) = xδ

+ Axβ1 + C where A =(

(K1−1)x∗Aδ

− IC)(

1x∗A

)β1.

On the other hand, if the firm decides to pay out the initial cash balance and all future

51

earnings as dividends, then the dynamics of the cash account are given by the following

equation

dC = (rC − C)dt (31)

Using similar arguments as when deriving the PDE in Equation [4] we obtain that

rV ′ = x+ C + (r − δ)xV ′x + (rC − C)V ′C + 1/2σ2x2V ′xx (32)

Because of the full payout assumption it follows that total costs of exercising the option

are given by IC + e(IC). Assuming that the solution is given by V ′(x,C) = νC +Bxβ + γx

it directly follows that V ′(x,C) = V B(x,C) such that the solution is given by

Rh(x,C) = xβ1 (A−B) (33)

The Impact of Volatility on RU(x).

Concerning the partial derivative of any growth option with respect to volatility, it is suffi-

cient to observe that

∂Axβ1

∂σ= Axβ1 log

( xx∗

)∂β1∂σ

(34)

as ∂Axβ1∂x∗

∂x∗

∂β1equals zero. Given that the positive solution to the fundamental quadratic is

characterized by the same parameters for both the constrained and unconstrained firm, we

only need to know that ∂β1∂σ

< 0. Further details can be found in Dixit & Pindyck Dixit and

52

Pindyck (1991). Applying above to ∂Rh(x)∂σ

we get that

∂Rh(x)

∂σ=∂β1∂σ

{Axβ1 log

( xx∗

)−Bxβ1 log

(x

x∗2

)}(35)

Using the fact that x∗2 = x∗(1+γ) where γ = (γ1+γ2IC) and that Bxβ1 = Axβ1(1+γ)1−β1 ,

we can rewrite the equation as

∂Rh(x)

∂σ=∂β1∂σ

{Axβ1 log

( xx∗

)− Axβ1(1 + γ)1−β1 log

(x

x∗(1 + γ)

)}(36)

which again can be rewritten as

∂Rh(x)

∂σ= Axβ1

∂β1∂σ

{log( xx∗

)− (1 + γ)1−β1 log

(x

x∗(1 + γ)

)}(37)

Due to the fact that x < x∗ < x∗(1 + γ) we know that log(xx∗

)> log

(x

x∗(1+γ)

). The

question whether the expression in the bracket is positive or negative will depend on (1 +

γ)1−β1 which will lie between 0 and 1 for different values of γ and β1.

53

Appendix B: Numerical Solution

The PDE is solved on a grid with nodes (xj, Ci) : j = 1, ...,M, i = 1, ..., N where xj = jdx

and dC = Ci − Ci−1. Partial derivatives are approximated by

Vx = 12

(Vi−1,j+1−Vi−1,j−1

2dx+

Vi,j+1−Vi,j−1

2dx

)Vxx = 1

2

(Vi−1,j+1−2Vi−1,j+Vi−1,j−1

(dx)2+

Vi,j+1−2Vi,j+Vi,j−1

(dx)2

)VC =

Vi,j−Vi−1,j

dC

(38)

which implies that the resulting difference equation at node (xj, Ci) can be formulated

as

−ajVi−1,j−1 − (bj − di,j)Vi−1,j − cjVi−1,j+1 = ajVi,j−1 + (bj + di,j)Vi,j + cjVi,j+1 + ej (39)

where

54

aj = σ2j2−µj4

bj = −σ2j2+r2

cj = σ2j2+µj4

di,j = αjdx−φ/2(αjdx)2+ridcdc

ej = (1− α)jdx

(40)

Equation [39] is defined for 2 ≤ j ≤ M and 2 ≤ i ≤ N . It has been shown that if

C ≥ IC, firm value has a closed form solution. The PDE is thus solved by employing the

solution to Proposition [3] as a boundary condition. As long as x < x∗ we know that for

C ≥ IC value-matching and smooth-pasting conditions are given by

V (x∗, Cτ ) = V A(x∗, Cτ )

Vx(x∗, Cτ ) = V A

x (x∗, Cτ )

(41)

55

Appendix C: Additional Robustness Checks

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

2.05

2.1

Cash Holdings / Investment Costs

Inve

stm

ent

Thre

shold

Option Exercise and Cash Holdings when µ = 4%

x*(C)

x*

B

x*

A

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%1.9

1.92

1.94

1.96

1.98

2

2.02

2.04

2.06

2.08

2.1

Cash Holdings / Investment Costs

Investm

ent

Thre

sho

ld

Option Exercise and Cash Holdings when γ1=5.3% and γ

2=0.02%

x(C)x

B

xA

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

8

Cash Holdings / Investment Costs

Inve

stm

ent T

hre

sh

old

Option Exercise and Cash Holdings when K1=1.15

x*(C)

x*

B

x*

A

Figure 10: Additional Robustness Check: The Relation between Cash Holdings andInvestment Thresholds. This figure displays the optimal investment thresholds introduced inFigure [1] for three additional robustness checks: (i) a highly profitable firm (i.e µ = 4%), (ii) afirm facing low financing costs (i.e. γ1 = 5.3% and γ2 = 0.02% and (iii) a firm with a smallergrowth option, i.e. K1 = 1.15.

56

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10

11

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

Cash Retention and Investment Thresholds when IC=40

α*(x,C) = 0

x*(C)

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

α*(x,C) = 1

Cash Retention and Investment Thresholds when IC=2.5

α*(x,C) = 0

α*(x,C) = 1

x*(C)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

α*(x,C) = 1

Cash Retention and Investment Thresholds when σ=5%

α*(x,C) = 0

α*(x,C) = 1

x*(C)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

α*(x,C) = 1

Retention and Investment Thresholds when µ=4%

α*(x,C) = 0

α*(x,C) = 1

x*(C)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

Retention and Investment Thresholds when γ1=5.3% and γ

2=0.02%

α*(x,C) = 0

x*(C)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

C

x

α*(x,C) = 0

α*(x,C) = interior

α*(x,C) = 0

Retention and Investment Thresholds when K1=1.15

α*(x,C) = 0

x*(C)

Figure 11: Additional Robustness Check: Optimal Investment and Cash RetentionBoundaries. This figure shows optimal investment thresholds, as introduced in Figure [7] fordifferent robustness checks, (i) investment costs: the figure in the upper left (right) increases(decreases) investment costs to 40 (2.5), (ii) volatility: the figure in the middle left sets volatilityequal to 5%, (iii) profitability: the figure in the middle right sets µ = 4%, (iv) financing costs: thefigure in the lower left sets γ1 = 5.3% and γ2 = 0.02% and (vi) magnitude of growth option: thefigure in the lower right sets K1 = 1.15.

57

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

σ

R(x

,C)

The relationship between volatility and the value of internal funds when µ = 4%

0 0.1 0.2 0.3 0.4 0.51.03

1.035

1.04

1.045

1.05

1.055

1.06

1.065

σ

VC

The relationship between volatility and the marginal value of cash for C=0.005 and µ = 4%

0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

σ

R(x

,C)

The relationship between volatility and the value of internal funds when γ1=5.3% and γ

2=0.02%

0 0.1 0.2 0.3 0.4 0.51.01

1.011

1.012

1.013

1.014

1.015

1.016

1.017

1.018

1.019

1.02

σ

VC

The relationship between volatility and the marginal value of cash for C=0.005 and γ1=5.3% and γ

2=0.02%

0 0.1 0.2 0.3 0.4 0.50

0.005

0.01

0.015

0.02

0.025

σ

R(x

,C)

The relationship between volatility and the value of internal funds when K1=1.15

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51

1.001

1.002

1.003

1.004

1.005

1.006

1.007

σ

VC

The relationship between volatility and the marginal value of cash for C=0.005 and K1=1.15

Figure 12: Additional Robustness Check: Relation between Volatility and the Value ofCash. This figure displays the relation between the value of cash and cash flow volatility usingboth R(x,C) and VC as the definition of the value of cash. Results are presented for three differentrobustness checks: (i) a highly profitable firm (i.e µ = 4%), (ii) a firm facing low financing costs(i.e. γ1 = 5.3% and γ2 = 0.02% and (iii) a firm with a smaller growth option, i.e. K1 = 1.15.

58

Table 8: Additional Robustness Check: The Value of Cash for Large Firms. This tablepresents results when estimating the value of cash for the benchmark case. Variable definitionsfollow Table [3] and the sample is restricted to (i) firms with total assets larger than $50 mn and(ii) for firms with total assets larger than $100 mn. The coefficients are estimated analogously toTable [3].

(1) (2)Total Assets > $50mn Total Assets > $100mn

Coefficient T-Statistics Coefficient T-StatisticsCash flow -0.152 -0.24 0.317 0.52Cash 1.952∗∗∗ 5.98 1.450∗∗ 3.04dCt 1.065∗∗∗ 5.06 1.064∗∗∗ 3.86dCt+1 2.258∗∗∗ 5.49 1.730∗∗∗ 6.85Constant 0.605∗∗∗ 3.30 0.636 1.78Control Variables yes yesTime Dummies yes yesObservations 2865 2024R2 0.401 0.366∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

59

Table 9: Additional Robustness Check: The Value of Cash in the Context of GrowthOptions and Volatility. This table presents results in case the setup of Pinkowitz, Stulz, andWilliamson (2006) is applied to estimating the relation between volatility and the value of cash inthe context of growth opportunities. The dependent variable is the value of the firm, scaled bythe book value of total assets. All other variables are defined analogously to Table [3]. Similar toPinkowitz, Stulz, and Williamson (2006), the level regression only includes the level of cash (plus thecorresponding interaction terms) without simultaneously considering the changes in cash holdings.Analogously, the changes regression only includes the changes in cash (plus the correspondinginteraction terms) without simultaneously considering the level in cash holdings. The coefficientsare estimated analogously to Table [3].

(1) (2)PSW (2006) Level PSW (2006) Changes

Coefficient T-Statistics Coefficient T-StatisticsCash 0.672∗ 2.07GO x C 4.670∗∗∗ 3.64GO x C x σ -0.284∗ -2.45E 0.428 1.01 -0.540 -1.19GO x dC 1.825 1.14GO x dC x σ -0.102 -0.77Observations 4452 4452R2 0.257 0.290∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Pinkowitz, Stulz, and Williamson (2006) propose the following two specifications, coined

changes and level regressions, to test for the value of cash

VtAt

= α + β1dCtAt

+ β2dCt+1

At+ β3

dNAtAt

+ β4dNAt+1

At+ β5

YtAt

+ β6dYtAt

+ β7dYt+1

At+ β8

dVt+1

At+ εt (42)

VtAt

= α + β1CtAt

+ β2dNAtAt

+ β3dNAt+1

At+ β4

YtAt

+ β5dYtAt

+ β6dYt+1

At+ β7

dVt+1

At+ εt (43)

where for simplicity I introduce the vector Yt to summarize earnings, interest payments,

dividends and research and development expenses (with dYt and dYt+1 corresponding to the

60

lagged and lead changes in the underlying variables). Note that the level of non-cash assets

do not appear in either of the equations. Extending the Pinkowitz, Stulz, and Williamson

(2006) framework to include the impact of cash flow volatility shows that volatility has a

negative effect on the value of cash which is statistically significant at the 5% level for the

level regression but insignificant for the changes regression.

61

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