The real option value of cash E0

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 an
anonymous referee and the editor, Holger Mueller. Finally, I would like to thank seminar participants at the
Norwegian School of Management, New Lisbon University, the Norwegian School of Economics, Maastricht
University, the NFA 2011 in Vancouver, the FMA Asian Conference 2009 and the MFA 2009 in Chicago.
Any remaining errors are my own.

1

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 allequity 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
1

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

2


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 significant. 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

3


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

4

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 financing 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 Strebulaev (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
2
In another similar paper Asvanunt, Broadie, and Sundaresan (2011) compare a firm’s choice in managing
corporate liquidity between issuing costly equity, maintaining a cash balance or employing loan committments.
3
Using a representative agent framework, Eisfeldt and Rampini (2009) study level and dynamics of the
value of aggregate liquidity when external shocks occur. Similarly, the trade-off is between agency costs of
free cash flow and cost of external finance but the paper does not focus on corporate cash policy but on the
value of aggregate liquidity. Results show that the value of aggregate liquidity is highest when investment
opportunities are abundant but levels of current cash flow are low.

5

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, investment 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 concave 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 capital 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, Anderson 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
4

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

6

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,

7

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

8

(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 models, 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.

9

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 + σxdW Q

(1)

where dW Q 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 + σXdW Q 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
5

Another way to think about the assumption is that it is easier for management to steal from a dynamic
flow variable such as operating income than from a transparent stock variable such as cash holdings. This
should be specially true for large and complex business operations. The same assumption can be found in
Albuquere 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 stealing
are quadratic to outside shareholders.

10

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
(αx)2 + rC dt
2

(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 disadvantage 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,
6

Note that taxation is not included in this model. While there is a tax disadvantage of keeping cash
within the firm, it is also true that at the investor level, dividends are usually taxed at a higher rate than
capital gains. A meaningful calculation would therefore require one to specify the tax burden at the investor
level. To abstract from these practical complexities, this paper focuses on agency costs of free cash flow as
the opposing friction.
7
For more details see Proposition [1].
8
The 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) and
Hugonnier, Malamud, and Morellec (2012).

11

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

if C < IC,


0


e(C) =



γ + γ (IC − C) + γ (IC − C)2
 0
1
2

else

(3)

The specification of this function has been taken and adapted from Atinkilic and Hansen
9

The paper also differs with respect to Anderson and Carverhill (2011) who model earnings as a meanreverting process.
10
Boyle and Guthrie assume that prior to exercising the growth option the firm consists of assets in place
G and the cash account X. Assets in place generate an income stream equal to νGdt + φGdZ which directly
affects the cash account whose dynamics are given by dX = rXdt + νGdt + φGdZ.
11
The focus on equity financing is given for two reasons. First, from a theoretical perspective the model
analyzes whether cash has value irrespective of bankruptcy costs. Second, from an empirical perspective
Strebulaev and Yang (2012) show that being all-equity financed is not a short-run phenomenon but a rather
persistent event.

12

(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 = max (1 − α)x + (r − δ)xVx + (αx −
α

φ
(αx)2 + rC)VC + 1/2σ 2 x2 Vxx
2

(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.
13

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 ) =

K1 x∗
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
Q
future earnings are paid out as dividends such that V (xs , Cs ) = Es

∞ −r(t−s)
e
K1 xdt
s

+ Cs

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

Note that Cs = 0 if Cτ ≤ IC.

14

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

B
B
rV B = x + (r − δ)xVxB + rCVC + 1/2σ 2 x2 Vxx

and is given by
15

(8)

V B (x, C) = C +
where B =

(K1 −1)x∗
B
δ

− IC − e(0)

1
x∗
B

β1

x
+ Bxβ1
δ

(9)

and x∗ denotes the corresponding optimal
B

trigger level, i.e.

x∗ =
B

β1
δ(IC + e(0))
(β1 − 1) (K1 − 1)

(10)

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

16

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)
where A =

(K1 −1)x∗
A
δ

− IC

1
x∗
A

(12)

β1

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

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

x∗ =
A

β1
δIC
(β1 − 1) (K1 − 1)

(13)

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 riskfree 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
13

The risk-free rate is similar to Gamba and Triantis (2008) and equal to Datastream’s historical monthly
Fed Funds data from 1955 to 2008. The volatility parameter is similar to Boyle and Guthrie (2003) and
Mauer and Triantis (1994). The agency cost parameter is taken from Eisfeldt and Rampini (2009).
14
It can be shown that the choice of K1 does not influence the relative gain from retaining cash as defined
by Equation [11] because the value of cash is scaled by the initial growth option such that the capacity
expansion factor cancels out. To make sure that other results are not driven by the choice of K1 , robustness
checks 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

Option Exercise and Cash Holdings
2.35
x*(C)
*
xB

2.3

x*
A

2.25

Investment Threshold

2.2

2.15

2.1

2.05

2

1.95

1.9
0%

10%

20%

30%

40%
50%
60%
70%
Cash Holdings / Investment Costs

80%

90%

100%

Figure 1: The Relation between Cash Holdings and Investment Thresholds. This figure
displays the optimal investment threshold, x∗ (C), as a function of the firm’s cash holdings which are
expressed relative to the total costs of the investment. The threshold is compared to the benchmark
case of full external financing (x∗ ) and to the first-best trigger level in case there are no financing
B
frictions (x∗ ). Results are shown for the following set of parameter values: r = 0.06, µ = 0.01,
A
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∗ , the trigger level of a firm which relies entirely on internal funds.
A
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∗ ), it can be seen that high (low) investment costs
B
make it less (more) attractive to further delay investment relative to the benchmark case.
Put differently, the intersection between x∗ (C) and x∗ shifts to the left (right) in case of
B
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
15

Put 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 volatility 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 financing 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∗ ) looks different than in the
A
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 accumulated 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

Option Exercise and Cash Holdings when IC = 40

Option Exercise and Cash Holdings when IC = 2.5

9.4

0.62
x*(C)
x*
B

9.2

x*(C)
x*
B

x*
A

x*
A

0.6

9
0.58


8.8

8.6

8.4

0.56

0.54

8.2
0.52
8
0.5
7.8

7.6
0%

10%

20%

30%

40%
50%
60%
70%

80%

90%

0.48
0%

100%

10%

20%

Option Exercise and Cash Holdings when σ = 40%

30%

40%
50%
60%
70%

80%

90%

100%


4.2
x*(C)
x*
B

x*(C)
x*
B

1.85

x*
A

4.1

x*
A

1.8
4



1.75
3.9

3.8

3.7

1.7

1.65

1.6

1.55
3.6
1.5
3.5

3.4
0%

1.45

10%

20%

30%

40%
50%
60%
70%

80%

90%

1.4
0%

100%

10%

20%

1.85

40%
50%
60%
70%

80%

90%

100%

1.75
x*(C)
x*
B

1.8

x*(C)
x*
B

1.7

x*
A

1.7

x*
A

1.65

1.6

1.75


30%


1.65

1.6

1.55

1.55

1.5

1.45

1.5

1.4

1.45

1.35

1.4
0%

10%

20%

30%

40%
50%
60%
70%

80%

90%

1.3
0%

100%

10%

20%

30%

40%
50%
60%
70%

80%

90%

100%

Figure 2: Robustness: The Relation between Cash Holdings and Investment Thresholds. This figure displays the optimal investment thresholds introduced in Figure [1]. The graph
in the upper left (right) increases (decreases) investment costs to 40 (2.5). The figure in the middle
left (right) increases (decreases) volatility to 40% (10%). The figure in the lower left (right) sets
volatility 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∗ is
B
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.
16

For details, see Figure [10].
Note 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 solves
the partial differential equation using a grid of different state points. However, one can use the optimal
cash retention and investment policy implied by the Crank Nicolson method and then employ Monte Carlo
simulation to simulate the evolution of cash holdings.
17

24

Histogram of Actual Cash Levels at Investment

Histogram of Investment Thresholds

100

120

90
100
80

70
80

Frequency

Frequency

60

50

60

40
40
30

20
20
10

0

0

2

4

6

8
10
12
Actual Cash Level at Investment

14

16

18

0
1.9

20

1.95

2

2.05

2.1
2.15
2.2
Actual Investment Threshold

2.25

2.3

2.35

Figure 3: Implied Distribution for Cash Holdings and Exercise Levels. The left panel
displays a frequency distribution of actual cash holdings at the exercise time of the option. The
right 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 realizations 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
18

Clearly, the results depend on the magnitude of the agency costs, captured by the cost parameter φ. For
example, if one changes the value of φ to 0.025, then the relative gain from retaining cash evaluated at x0
and C0 increases to 8.55%. Even more so, if the firm does not suffer from agency costs of free cash flow at
all, then it would be able to increase the value of the capacity expansion option by 11.45% if it retains funds
within the firm.

25

Relation between S(x,C) and x for C=0
0.1

0.09

0.08

0.07

S(x,C)

0.06

0.05

0.04

0.03

0.02

0.01

0

0

0.5

1

1.5
x

2

2.5

3

Figure 4: The Relative Gain From Retaining Cash. This figure shows displays the relative
gain from saving as a function of operating cash flow (x), holding the level of cash holdings constant
at 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 , as
a 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
19

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

27

Figure 6: Optimal State-Contingent Cash Retention Policy. This figure shows the firm’s
optimal 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]
20

For example, Decamps, Mariotti, Rochet, and Villeneuve (2011) show that as long as a firm has not
accumulated sufficient cash reserves, the marginal benefit of cash ranges between unity and the marginal
cost of issuing shares. Because the costs of retaining funds are linear, this results in a payout decision which
can be characterized as all-or-nothing. As long as the firm has not accumulated sufficient cash reserves, the
marginal benefit of cash exceeds unity such that the firm retains the entire cash flow. Once the dividend
threshold 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 and
Triantis (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
21
Note that there are two regions for which it is optimal not to retain any cash. The first region is
characterized 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∞
Q
form solution for the second case, i.e. V (xs , Cs ) = Es s e−r(t−s) K1 xdt + 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 the
dynamic stochastic optimization problem posited in Proposition [1] which needs to be solved numerically.
22
For details, see Figure [11] in the Appendix.
23
Note that this result differs from Hugonnier, Malamud, and Morellec (2012) who find that high invest-

29

Cash Retention and Investment Thresholds
3
*

α (x,C) = 0
*

α (x,C) = 1
*

α*(x,C) = 0

2

x

2.5

x (C)

α*(x,C) = 1

1.5

α*(x,C) = interior

1

0.5

α*(x,C) = 0
0

0

1

2

3

4

5
C

6

7

8

9

10

Figure 7: Optimal Investment and Cash Retention Boundaries This figure shows the firm’s
different cash retention and investment boundaries. Specifically, α∗ (x, C) = 0 displays the boundary
below which it is optimal to pay out all funds as a dividend, α∗ (x, C) = interior, corresponds to
the region where an interior solution for the optimal cash retention policy exists. The variable
x∗ (C) corresponds to the optimal investment threshold above which it is optimal to pay out a full
dividend. Finally, for the region between α∗ (x, C) = 1 and x∗ (C) it is optimal to retain 100% of
the 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 a
project externally. In this paper, ceteris paribus high investment costs reduce incentives to delay investment
beyond 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.
24
Note that while Decamps, Mariotti, Rochet, and Villeneuve (2011) also display dividend boundaries for
different levels of cash flow volatility and profitability, the results are not directly comparable. The reason is
that in Decamps, Mariotti, Rochet, and Villeneuve (2011) the dividend boundary is expressed as the critical
level 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), higher
profitability 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 allows
for a less costly capacity expansion, thereby making cash valuable when profitability is high and volatility is
low.

31

The relation between volatility and the value of internal funds
0.25

0.2

R(x,C)

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5

σ

Figure 8: The Real Option Value of Cash and Volatility. This figure shows that the relation
between 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
25

Note that this result is specially interesting because it suggests that value is driven by the relative delay
in investment, as illustrated in Figure [2]. While lower volatility decreases the investment threshold for all
financing alternatives, it increases incentives to delay investment relative to the case of full external financing.
26
In 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

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

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

1.027

1.028

1.026

1.027

1.025

1.026

1.024
VC

1.028

1.029

VC

1.03

1.025

1.023

1.024

1.022

1.023

1.021

1.022

1.02

1.021

1.02

1.019

0

0.1

0.2

0.3

0.4

1.018

0.5

0

0.1

0.2

0.3

0.4

σ

0.5

σ

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

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

1.016

1.006

1.015
1.005
1.014

1.013
1.004

VC

VC

1.012

1.011

1.003

1.01
1.002
1.009

1.008
1.001
1.007

1.006

0

0.1

0.2

0.3

0.4

1

0.5

σ

0

0.1

0.2

0.3

0.4

0.5

σ

Figure 9: The Marginal Value of Cash and Volatility for Different Levels of Cash Holdings. This figure displays the marginal value of cash, VC , as a function of volatility and additionally
controls 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
27

Note that results are also robust with respect to previous robustness checks regarding the impact of
profitability, 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 different
investment 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
∞

e−r(s−t) xs dt

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

(14)

t+1

where xt denotes the level of current cash flow and the term E[

∞ −r(s−t)
e
xs dt]
t+1

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

∞ −r(s−t)
e
xs dt
t+1

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 + b1 xt + b2 Ct + b3 G(xt ) +

(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
28

Note 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 results
when 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) xs dt given that the value of
the benchmark firm can not be observed in practice. The observed value of the growth option is
defined using the closed form approximation provided in Proposition [2], i.e. Bxβ1 . Cash holdings
and operating cash flow are the state variables of the underlying model. The regression is estimated
using ordinary least squares (OLS), focusing on all firms which have not yet exercised the growth
option. Results are displayed for three cases: (i) the full sample, (ii) a subsample when the time
to build cash reserves is restricted to eight years and (iii) a subsample when the time to build cash
reserves is restricted to four years.
(1)
All
Coefficient
Growth Options
0.943∗∗∗
Cash Holdings
1.012∗∗∗
Cash Flow
1.123∗∗∗
∗

p < 0.05,

∗∗

p < 0.01,

∗∗∗

(2)
(3)
If Time < 8 years If Time < 4 Years
Coefficient
Coefficient
0.909∗∗∗
0.874∗∗∗
1.032∗∗∗
1.036∗∗∗
1.171∗∗∗
1.292∗∗∗

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 endogeneity 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
29

For 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 + β1 CFt + β2 Ct + β3 dCt + β4 dCt+1 + β5 GOt + β6 Xt +

(16)

I follow Section [4] and define the dependent variable analogously as the difference between 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 variable 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
30

The exact definition of the variables is given in Table [3].
It 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 of
non-cash assets in the definition of the dependent variable.
32
Examples include Hayashi (1982), Fazzari, Hubbard, and Petersen (1988), Erickson and Whited (2000)
and Hennessy (2004).
31

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
33

See 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.
34
Besides, for a substantial fraction of the sample R&D expenses equal zero or are missing which would
unnecessarily reduce sample size.
35
Note that trimming is done with respect to the full sample.
36
For 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 cash
holdings of all-equity financed firms. Specifically, it uses 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 excluded from the sample.
All-equity firms are defined as carrying neither short-term debt (mnemonic: dlc) nor long-term
debt (mnemonic: dltt) in their capital structure, cash holdings (mnemonic: ch) are stated relative
to the value of their assets (mnemonic: at).
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010

5.1

Fraction All-Equity Firms Average Cash Holdings
0.06
0.09
0.06
0.09
0.07
0.08
0.08
0.09
0.07
0.08
0.07
0.10
0.08
0.12
0.09
0.14
0.09
0.23
0.09
0.24
0.09
0.24
0.11
0.28
0.12
0.25
0.13
0.24
0.12
0.21
0.13
0.26
0.13
0.29
0.14
0.29
0.14
0.29
0.14
0.31
0.14
0.31
0.15
0.32
0.16
0.31
0.18
0.33
0.19
0.32
0.19
0.31
0.19
0.31
0.19
0.33
0.19
0.33
0.19
0.33
0.19
0.33

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

ˆ
cross-sectional interdependence among the error terms. Denoting R as the estimated value
of cash and controlling for lead and lagged variables in the regression setup, it follows that

ˆ
ˆ
ˆ
ˆ
R = β2 + β3 − β4

(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
37
38

ˆ ˆ
ˆ
Standard errors are computed using the variance-covariance matrix of β2 , β3 and β4 .
For results, please see Table [8] in the Appendix.

41

Table 3: The Value of Cash. This table displays results when estimating the value of cash
for the baseline scenario. The regression uses 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. Subsequent variable definitions are based
on COMPUSTAT mnemonics. The dependent variable is defined according to Section [4] as the
difference between the total market value of the firm (prccf ∗ csho ) and the value of its non-cash
assets (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 and
capx
dCt+1 = Ct+1 − Ct . The definition of the growth option is given by GO = at . The analysis
accounts for control variables typically used in the literature and includes the level and changes
of R&D expenditures (xrd) and dividend payments (dvc + dvp) , the changes in the firm’s net
assets (at - ch) and operating cash flow and the lead change in total firm value to account for all
non-captured market expectations. The regression is estimated accounting for firm fixed effects and
by including time dummies. Standard errors are computed following Discroll and Kraay (1998) to
account for possible cross-sectional interdependence among the error terms.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.
Cash flow
Cash
dCt
dCt+1
Growth Option
Control Variables
Time Dummies
Observations
R2
∗

p < 0.05,

∗∗

p < 0.01,

Coefficient T-Statistics
-1.334∗∗
-3.24
2.252∗∗∗
5.15
0.939∗∗∗
4.80
2.475∗∗∗
9.49
1.051
1.44
yes
yes
5658
0.312
∗∗∗

42

p < 0.001

Table 4: The Value of Cash in the Context of Growth Options and Volatility. This
table displays results when estimating the value of cash in the context of growth opportunities and
cash flow volatility. Previously introduced variables follow the definition given in Table [3]. New
variables include cash flow volatility, σt , which is defined as the time-varying volatility of operating
cash flow. Volatility estimates are based on quarterly observations, initial volatility is calculated
for the period from 1980 to 1988 under the additional requirement of having at least 16 quarterly
observations. Each year the estimation window is extended by one year while the initial observation
period is held fixed at 1980. Results are presented for three cases: (i) the full sample, (ii) for firms
with 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].

Cash Flow
Cash
GO
GO x C
GO x C x σ
Observations
R2
∗

p < 0.05,

∗∗

(1)
(2)
(3)
Full Sample
Total Assets > $50mn
Coefficient T-Statistics Coefficient T-Statistics Coefficient T-Statistics
-1.132∗∗
-2.65
-0.325
-0.52
0.040
0.06
2.545∗∗∗
6.11
2.091∗∗∗
6.22
1.478∗∗
3.05
2.098
1.38
4.959∗∗∗
4.61
5.540∗∗∗
4.33
4.049
1.29
2.933
0.96
7.005
1.39
-0.265∗
-2.20
-0.280∗
-2.35
-0.253∗
-2.14
4463
2474
1803
0.343
0.406
0.380

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 + β1 CFt + β2 Ct + β3 Ct GOt + β4 Ct GOt σt + β5 dCt + β6 dCt+1
+b7 GOt + b8 Xt +

(18)

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

Volatility estimates are based on quarterly observations. Initial volatility is calculated for the period
from 1980 to 1988 and I follow Han and Qiu (2007) and Minton and Schrand (1999) in requiring at least 16
quarterly observations. Each year the estimation window is extended by one year while the initial observation
period 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 impact, 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 interaction 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 when
dropping the level of growth opportunities, i.e. GOt from Equation [18]. All variables and the
estimation procedure follow Table [4].

Cash Flow
Cash
GO x C
GO x C x σ
Observations
R2
∗

p < 0.05,

∗∗

(1)
Full Sample
-1.089∗
-2.49
2.417∗∗∗
5.57
7.228∗
2.54
-0.264∗
-2.17
4463
0.3431

p < 0.01,

∗∗∗

(2)
Total Assets > 50mn
-0.232
-0.36
1.766∗∗∗
5.74
9.827∗∗∗
3.36
-0.277∗
-2.29
2474
0.403

(3)
Total Assets > 100mn
0.143
0.22
1.075∗
2.49
16.041∗∗∗
3.89
-0.252∗
-2.12
1803
0.377

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 Volatility. This table displays results when estimating the value of cash in the context of growth opportunities 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 in
total assets and (iii) for firms with more than $100mn in total assets. The coefficients are estimated
analogously to Table [4] but lagged values for level and changes cash holdings, growth opportunities
and cash flow volatility are used.
(1)
Full Sample
Cash Flow
0.635
1.44
C (lagged)
1.033
1.83
GO (lagged)
1.906
1.29
GO x C (lagged)
4.680
0.89
GO x C x σ (lagged) -0.322∗∗∗
-3.81
Observations
3199
R2
0.283
∗

5.2

p < 0.05,

∗∗

p < 0.01,

∗∗∗

(2)
1.071
1.59
0.806
1.76
1.662
0.61
6.863
1.04
-0.336∗∗∗
-3.80
1828
0.337

(3)
1.189
1.75
1.058
1.40
-0.200
-0.06
9.573
1.00
-0.285∗∗
-2.73
1382
0.326

p < 0.001

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.
40

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

45

Table 7: The Impact of Cash Holdings on Investment. This table shows results when
estimating the impact of cash holdings on a firm’s investment decision. The dependent variable is
investment (capx), Ct is the level of cash holdings relative to property, plant and equipment net
2
of depreciation (ppent) and Ct is the squared value of C. The variable Y is a vector consisting
of several control variables such as operating cash flow (defined as in Table [3]) , size (logarithm
of sale), growth (two year growth rate of sale) and market-to-book ratio which is defined as (Vinvt)/at where V = prccf∗csho + dlt + dlc). All variables are cut off at the 1% level to reduce
the effect of outliers. The regression accounts for firm fixed effects and is estimated using OLS and
employing either simultaneous or lagged regressors and time dummies.

Cash
Cash2
Cash flow
MTB Ratio
Size
Growth Rate
Observations
R2
∗

p < 0.05,

∗∗

(1)
(2)
Simultaneous
Lagged
Coefficient T-Statistics Coefficient T-Statistics
-0.051498∗∗∗
-9.57
-0.008838∗
-2.22
∗∗∗
0.000210
9.02
0.000032∗
2.08
0.051170∗∗∗
4.15
0.045092∗∗∗
3.78
0.003753
0.20
0.246171∗∗∗
4.36
2.073676∗∗∗
7.54
1.932559∗∗∗
5.48
0.062725
0.64
-0.009631
-0.23
5155
3895
0.117
0.102

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 + β1 Ct + β2 Ct2 + β3 Y +

(19)

where Invt is capital expenditures, Ct is the level of cash holdings relative to property,
plant and equipment net of depreciation and Ct2 is the squared value of C. The variable Y is a
vector consisting of several control variables such as operating cash flow, size, growth, marketto-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 generate 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 volatility and the value of cash as well as the nonlinear relation between cash holdings and investment 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 underlines 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 + σxdW Q . 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 (θ1 V − θ2 X) and it
follows that the total return from holding the portfolio over a short time interval dt equals

dP = θ1 ((1 − α)xdt + dV ) − θ2 dX

(20)

Applying Ito’s Lemma leaves us with
1
dP = θ1 (1 − α)xdt + Vx dx + VC dC + σ 2 x2 Vxx dt − θ2 dX
2
For θ1 = 1 , it immediately follows that θ2 equals

Vx x
X

rP dt. Combining above and using the fact that P = (V −

rV = (1 − α)x + (r − δ)xVx + (αx −

(21)

which then implies that dP =
xVx
X)
X

we obtain that

φ
(αx)2 + rC)VC + 1/2σ 2 x2 Vxx
2

(22)

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

rV = max (1 − α)x + (r − δ)xVx + (αx −
α

φ
(αx)2 + rC)VC + 1/2σ 2 x2 Vxx
2

Taking the FOC with respect to α implies that

49

(23)

α∗ =

VC − 1
φxVC

(24)

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

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

B
B
rV B = x + (r − δ)xVxB + rCVC + 1/2σ 2 x2 Vxx

(25)

which has to be solved with respect to

V B (0, Ct ) = Ct
V B (x∗ , Cτ ) =

K1 x∗
δ

+ Cτ − IC − e(0)

VxB (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
δ(IC + e(0))
(β1 − 1) (K1 − 1)

where β1 is the positive root of the fundamental quadratic

50

(27)

1
β(β − 1) + µβ − r = 0
2
It follows that V B (x, C) =

x
δ

+ Bxβ1 + C where B =

(28)
(K1 −1)x∗
B
δ

− IC − e(0)

1
x∗
B

β1

.

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 ) =

K1 x∗
δ

+ Cτ − IC

VxA (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
δIC
(β1 − 1) (K1 − 1)

(30)

where β1 is the positive root of the same fundamental quadratic as in Equation [28]. It
follows that V A (x, C) =

x
δ

+ Axβ1 + C where A =

(K1 −1)x∗
A
δ

− IC

1
x∗
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)xVx + (rC − C)VC + 1/2σ 2 x2 Vxx

(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 suﬃcient to observe that
x ∂β1
∂Axβ1
= Axβ1 log ∗
∂σ
x ∂σ
as

∂Axβ1 ∂x∗
∂x∗ ∂β1

(34)

equals 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

∂Rh (x)
∂β1
=
∂σ
∂σ

we get that

Axβ1 log

x
− Bxβ1 log
∗
x

x
x∗
2

(35)

Using the fact that x∗ = x∗ (1+γ) where γ = (γ1 +γ2 IC) and that Bxβ1 = Axβ1 (1+γ)1−β1 ,
2
we can rewrite the equation as
∂Rh (x)
∂β1
=
∂σ
∂σ

Axβ1 log

x
− Axβ1 (1 + γ)1−β1 log
x∗

x
+ γ)

(36)

x∗ (1

which again can be rewritten as
∂Rh (x)
∂β1
= Axβ1
∂σ
∂σ

log

x
− (1 + γ)1−β1 log
x∗

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

x
+ γ)

(37)

x∗ (1
x
x∗

> 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 =
Vxx =

1
2

1
2

Vi−1,j+1 −Vi−1,j−1
2dx

+

Vi−1,j+1 −2Vi−1,j +Vi−1,j−1
(dx)2

VC =

Vi,j+1 −Vi,j−1
2dx

+

Vi,j+1 −2Vi,j +Vi,j−1
(dx)2

Vi,j −Vi−1,j
dC

(38)

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

−aj Vi−1,j−1 − (bj − di,j )Vi−1,j − cj Vi−1,j+1 = aj Vi,j−1 + (bj + di,j )Vi,j + cj Vi,j+1 + ej

where

54

(39)

aj =

σ 2 j 2 −µj
4

bj = − σ
cj =
di,j =

2 j 2 +r

2

σ 2 j 2 +µj
4

αjdx−φ/2(αjdx)2 +ridc
dc

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 ) = VxA (x∗ , Cτ )
(41)

55

Appendix C: Additional Robustness Checks
Option Exercise and Cash Holdings when γ1=5.3% and γ2=0.02%

Option Exercise and Cash Holdings when µ = 4%
2.1

2.1
x*(C)
x*
B

2.05

x(C)
xB

2.08

x*
A

xA

2.06

2

2.04


1.95

1.9

1.85

2.02

2

1.98

1.8
1.96
1.75

1.94

1.7

1.65
0%

1.92

10%

20%

30%

40%
50%
60%
70%

80%

90%

1.9
0%

100%

10%

20%

30%

40%
50%
60%
70%

80%

90%

100%

Option Exercise and Cash Holdings when K1=1.15
8
x*(C)
x*
B
x*
A

7.8


7.6

7.4

7.2

7

6.8

6.6

6.4
0%

10%

20%

30%

40%
50%
60%
70%

80%

90%

100%

Figure 10: Additional Robustness Check: The Relation between Cash Holdings and
Investment Thresholds. This figure displays the optimal investment thresholds introduced in
Figure [1] for three additional robustness 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.

56

Cash Retention and Investment Thresholds when IC=40

Cash Retention and Investment Thresholds when IC=2.5

11

0.7
*

*

α (x,C) = 0

*

α (x,C) = 0

10

α (x,C) = 0

α*(x,C) = 0

*

x (C)

α*(x,C) = 1
x*(C)

0.6
9
8

0.5

α*(x,C) = 1

7
0.4
x

x

6
5

0.3

*

α (x,C) = interior

4

*

α (x,C) = interior

0.2

3
2

*

*

α (x,C) = 0

0.1

α (x,C) = 0
1
0

0

5

10

15

20
C

25

30

35

0

40

0

0.5

1

1.5

2

2.5

C

Cash Retention and Investment Thresholds when σ=5%

Retention and Investment Thresholds when µ=4%
2.5
α*(x,C) = 0

2

α*(x,C) = 0

α*(x,C) = 1

*

α (x,C) = 0

α*(x,C) = 1

*

α (x,C) = 0

*

x*(C)

x (C)

1.8

2
1.6

*

α (x,C) = 1
1.4

*

α (x,C) = 1
1.5

x

x

1.2
1
1

α*(x,C) = interior

0.8
0.6

*

α (x,C) = 0

*

α (x,C) = 0

0.2
0

α*(x,C) = interior

0.5

0.4

0

1

2

3

4

5
C

6

7

8

9

0

10

0

1

2

3

4

5
C

6

7

8

9

10

Retention and Investment Thresholds when K1=1.15

Retention and Investment Thresholds when γ1=5.3% and γ2=0.02%
10

2.5

α*(x,C) = 0

α*(x,C) = 0
x*(C)

α*(x,C) = 0

x*(C)

9

*

α (x,C) = 0

8

2

7

6

1.5

x

x

*

α (x,C) = interior

5

α*(x,C) = interior

4

1

3

2

0.5

*

α (x,C) = 0
0

0

1

2

3

4

5
C

6

7

8

α*(x,C) = 0

1

9

0

10

0

0.1

0.2

0.3

0.4

0.5
C

0.6

0.7

0.8

0.9

1

Figure 11: Additional Robustness Check: Optimal Investment and Cash Retention
Boundaries. This figure shows optimal investment thresholds, as introduced in Figure [7] for
different 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 volatility
equal to 5%, (iii) profitability: the figure in the middle right sets µ = 4%, (iv) financing costs: the
figure in the lower left sets γ1 = 5.3% and γ2 = 0.02% and (vi) magnitude of growth option: the
figure in the lower right sets K1 = 1.15.

57

The real option value of cash E0

The real option value of cash E0

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