investment, uncertainty, and liquidity

Investment, Uncertainty, and Liquidity

GLENN W. BOYLE and GRAEME A. GUTHRIE n

ABSTRACT

We analyze the dynamic investment decision of a ¢rm subject to an endogen-ous ¢nancing constraint. The threat of future funding shortfalls lowers thevalue of the ¢rm’s timing options and encourages acceleration of investmentbeyond the ¢rst-best optimal level. As well as highlighting another way bywhich capital market frictions can distort investment behavior, this result im-plies that (1) the sensitivity of investment to cash £owcan be greatest for high-liquidity ¢rms and (2) greater uncertainty has an ambiguous e¡ect on invest-ment.

CORPORATEFINANCERESEARCHof the last 20 years has highlighted the role of marketfrictions in ¢rm investment and ¢nancing decisions. One important insight ofthis work is that when there are informational asymmetries (e.g., Greenwald, Sti-glitz, and Weiss (1984), Myers and Majluf (1984)) or agency problems (e.g., Stulz(1990), Hart and Moore (1994)), pro¢table projects may not proceed if the ¢rmhas insu⁄cient internal funds to pay for them.As a result, investment is less thanits ¢rst-best level.

In these models, projects are essentially ‘‘now-or-never’’ investments, so theinvestment decision is determined solely by the project’s current net presentvalue (NPV). In such a setting, the relationship between investment and liquidityis static: A shortage of internal funds reduces the number of states in whichthe ¢rm can invest and thus has an adverse e¡ect on today’s investmentopportunities. However, the investment-liquidity relationship can also have dy-namic features. For example, if projects can be delayed, then the trade-o¡between immediate and future investment may also be constrained by¢nancing considerations. In particular, delay of investment exposes the ¢rm tothe risk of an adverse cash £ow shock that eliminates its ability to ¢nance theproject.

To analyze the dynamic relationship between investment and liquidity, weintroduce a simple ¢nancing constraint into the investment timing model of

THE JOURNAL OF FINANCE � VOL. LVIII, NO. 5 � OCTOBER 2003

nBoyle is at the University of Otago and Guthrie is atVictoria University of Wellington. Forhelpful comments, we are grateful to Louis Ederington, John Handley, Jayant Kale, PeterMacKay, John Powell, and seminar participants at Otago, Melbourne, and the 2002 New Zeal-and Finance Colloquium. Valuable suggestions from Rick Green (the editor) and an anon-ymous referee also signi¢cantly improved the paper, while Peter Grundy provided usefulresearch assistance. Any remaining errors or ambiguities are our responsibility.

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McDonald and Siegel (1986).1 In that model, the ¢rm has perpetual rights to aproject and seeks to choose the investment date that provides the highest ex-pected payo¡. Because the project has an uncertain future value and an irrever-sible investment cost, the optimal policy is to invest only when the project’s NPVexceeds a positive threshold re£ecting the value of further delay.

The McDonald and Siegel (1986) model assumes that the investment decisioncan be made independently of the ¢nancing decision.2 Mauer and Triantis (1994)extend this setup to allow for the simultaneous determination of investment andthe debt/equity mix, but theydo not consider the possibility that the ¢rm may facefunding restrictions. In contrast, we consider the investment timing decision of a¢rm facing a capital market friction that constrains its investment choices.

We ¢nd that this friction a¡ects both the value of the project rights and theoptimal policy for exercising those rights. In particular, potential future ¢nan-cing restrictions encourage acceleration of investment beyond the ¢rst-best level.Thus, our model highlights another way by which ¢nancing constraints can dis-tort investment behavior:The threat of a future cash shortfall reduces the value ofa ¢rm’s timing options and leads to suboptimal early exercise of those options.This feature of our model is consistent with the empirical ¢nding of Whited(2002) that small (and presumably more ¢nanciallyconstrained) ¢rms invest moreoften than large ¢rms. It also ¢ts the standard folklore that smaller ¢rms aremore aggressive about entering newmarkets or launching new products than big-ger, safer, and less ¢nancially constrained ¢rms. Such a phenomenon is typicallyattributed to di¡erences in risk attitudes (i.e., more caution on the part of large¢rms) or to di¡erences in management and bureaucracy structure (i.e., slowerdecision making by large ¢rms), but our model suggests another explanation.

Our analysis identi¢es an additional determinant of the relationship between¢rm investment and liquidity. More cash not only loosens the restrictions on cur-rent investment, but also makes waiting less risky and thus increases the oppor-tunity cost of current investment.The ¢rst e¡ect encourages investment, but thesecond e¡ect does the opposite. Moreover, because the e¡ect on the risk of wait-ing is greatest for low-liquidity ¢rms, increases in cash £ow can have a greaterpositive e¡ect on the investment of high-liquidity ¢rms, consistent with the evi-dence of Kaplan and Zingales (1997) and Cleary (1999).

Our model also has implications for the relationship between investment anduncertainty. Although greater uncertainty about project value increases the va-lue of investment delayand lowers current investment, greater uncertaintyabout¢rm liquidity has the opposite e¡ect:More volatility in the ¢rm’s future cash £owdistribution raises the risk of future funding shortfalls, thereby lowering thevalue of waiting and increasing current investment. Most feasible measures of

1For examples of models similar to McDonald and Siegel (1986), see Bernanke (1983) andBrennan and Schwartz (1985). Majd and Pindyck (1987),Triantis and Hodder (1990), and Inger-soll and Ross (1992), among others, extend the McDonald and Siegel model in various ways.See Dixit and Pindyck (1994) for an excellent overview of this literature.

2 The same is true of more complex models that allow for partial reversibility, expandabil-ity, and industry structure e¡ects. See, for example, Dixit (1989), Dixit and Pindyck (1994), andAbel et al. (1996).

The Journal of Finance2144

uncertainty are likely to incorporate both types of volatility, so our model sug-gests one reasonwhyempirical research ¢nds little or no short-term relationshipbetween investment and uncertainty.3

Our paper builds on a long tradition in ¢nance research. Early work on thee¡ects of ¢nancing constraints (by, e.g., Charnes, Cooper, and Miller (1959) andBaumol and Quandt (1965)) uses mathematical programming techniques to ana-lyze the ¢rm’s optimal investment policy in the presence of an exogenous restric-tion on capital market funding, concentrating primarily on the appropriatechoice of objective function and discount rate when capital must be rationed. Inresponse, Elton (1970), Myers (1972), andWeingartner (1977) employ simple mod-els of economic equilibrium to show that the ¢rm’s investment objective is unaf-fected by a ¢nancing constraint: Any ¢rm that wishes to maximize the economicwelfare of its owners should maximize the market value of its assets subject tothe limitations imposed by the constraint.Moreover,Weingartner points out thata focus on market-imposed external funding limits is ill conceived in the absenceof plausible and explicitly modeled market frictions, so subsequent work byMyers and Majluf (1984) and others endogenizes the constraint as a function ofthe ¢rm’s reaction to speci¢c market frictions.

Continuing this strand of research, but utilizing more recently developedmodeling methods, Mello and Parsons (2000) derive the optimal operating andhedging policies of a ¢rm subject to a dynamic ¢nancing constraint. Like us, theyemphasize the role of the constraint in restricting the ¢rm’s options. However,because they focus on the operating policy of a ¢rm that has already invested,they do not consider the e¡ects of the ¢nancing constraint on the initial decisionto invest. Our focus on the investment timing decision therefore complementstheir analysis.

In the ¢rst section, we set out our model and o¡er a detailed description of theinvestment environment. In Section II, we compare the investment problemof theunconstrained ¢rmwith that of the ¢nancially constrained ¢rm. Section III out-lines the optimal investment policy for the constrained ¢rm. In Sections IVandV,we examine the respective e¡ects of liquidityand uncertainty on the constrained¢rm’s investment policy. SectionVI contains some concluding remarks.

I. The Investment Environment

A ¢rm owns the rights to an investment project and has the option to invest inthis project at any time. If the ¢rm exercises this option at time t, it pays a ¢xedamount I and receives a project worthVt. Project value follows the geometricBrownian motion process

dV ¼ mVdtþ sVde; ð1Þwhere m and s are constant parameters and e is aWiener process.The rights areperpetual: At each date, the ¢rm can exercise the rights and invest, or delayinvestment and retain the rights, or sell the rights to another ¢rm.

3Minton and Schrand (1999) report a negative long-run relationship between cash £owvolatility and investment.

Investment, Uncertainty, and Liquidity 2145

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In addition to the project rights, the ¢rm has a time t cash stock Xt and someexisting assets with perpetual life and market value G. For convenience, we as-sume that the existing assets do not reinvest any cash £ow, so G is a constant. Ateach date, investors can freely trade existing shares, and the ¢rm can sell newsecurities in order to fund investment in the project. However, the latter capacityis limitedby thewillingness of investors to provide additional capital at a reason-able price. For example, the ¢rm may be subject to credit and bank loan con-straints, restricted access to bond markets, or prohibitively expensive equitycosts.We do not derive from ¢rst principles the source of such restrictions (for arecent example of this approach, see Clementi and Hopenhayn (2002)), but in-stead focus on their e¡ects by assuming that investment is possible at date t ifand only if

I � Xt þGþ aVt; ð2Þ

for some constant aA[0, 1).The right side of equation (2) comprises the potentialsources of funds available to the ¢rm at date t: cash plus the realizable value of the¢rm’s assets.The friction embedded in equation (2) captures the idea that uncer-tainty about the ¢rm’s ability or willingness to extract full project value for out-side investors limits the amount of funding that these investors are prepared tosupply at a nonprohibitive price. For example, the moral hazard and agency pro-blems discussed by Jensen and Meckling (1976) and Stulz (1990) make it di⁄cultfor ¢rms to issue claims against the full project value.4 Alternatively, the frictionmight arise because of the contractual problems analyzed by Hart and Moore(1994): The inability to commit vital human capital to the project restricts thesupply of new funding. In this case, (1� a)Vt is the portion of project value attri-butable to the ¢rm’s unique human capital input. Regardless of the source of thefriction, the constraint is endogenous:The greater is project valueVt, the greaterthe ¢rm’s funding capacity.

The friction a imposes restrictions on the ¢rm’s investment policy. For (Xt,Vt)such that XtþGþ aVtoIoVt, the investment payo¡ Vt� I is positive, but the¢rm’s ¢nancial resources do not allow it to invest.5 The lower the ¢rm’s cash bal-ance, the more likely this is to occur, that is, the more severe is the ¢nancingconstraint. Conversely, as the cash balance becomes very large, the ¢nancingconstraint becomes immaterial.

Over time, the constraint can be relaxed by augmenting the initial cash bal-anceX.This occurs in twoways. First, if the ¢rm delays investment in the project,X is invested in riskless securities and earns the rate of return r. Second, the

4 Issuing debt encourages risk shifting from stockholders to bondholders, so investors arereluctant to lend beyond a certain point. Issuing equity dilutes the ownership stake of currentmanagers and owners, thereby reducing their respective incentives to work and monitor dili-gently. Investors may also ¢nd it di⁄cult to distinguish between good and bad projectswhen managers pursue their own objectives. These considerations limit the supply of equityfunding.

5When a¼1, the ¢rm can invest in any project with positive Vt� I so long as doing sowould not require it to liquidate, that is, so long as XtþGþVt� I� 0. See the discussionbelow equation (3).


¢rm’s existing physical assets generate uncertain operational cash £ow with dy-namics ndtþfdz, where n and f are constant parameters, and z is aWiener pro-cess with de dz¼ rdt.6 Thus, prior to investment in the project, the cash stockfollows the process

dX ¼ rXdtþ ndtþ fdz: ð3Þ

Note that (3) allows the ¢rm to issue protected debt to cover a cash de¢cit (i.e.,Xo0). However, this too is limited. If the de¢cit exceeds the realizable value ofthe ¢rm’s noncash assets, then the ¢rm must liquidate and sell the project rights.7

Once investment commences, the value of the project isV. Prior to investment,the value of the project rights is Fc(X,V).That is, Fc(X,V) denotes the expectedpresent value of the payo¡ from exercising or selling the project rights (at a datedetermined by the chosen investment policy) when the ¢rm’s current cash bal-ance is X and the project value is currentlyV. For those (X,V) where investmentis optimal

FcðX;VÞ ¼ V � I:

That is, if the ¢rm invests, the owners receive the payo¡V� I. Also

limX!1

FcðX;VÞ ¼ FuðVÞ; ð4Þ

where Fu(V ) is the value of the rights if these are held by a ¢rm that faces no¢nancing frictions. Equation (4) states that because the restrictions imposed bythe friction disappear as the cash stock becomes large, the value of the projectrights to a constrained ¢rm approaches the value available to an unconstrained¢rm.

As the ¢rm’s owners can freely trade their shares, they wish the ¢rm to adoptan investment policy that maximizes its market value. At each date prior to in-vestment, the market value of the ¢rm is equal to the sum of its cash X, the valueof its existing assets G, and the value of the project rights Fc(X,V).Thus, the op-timal investment policy maximizes Fc(X,V). Our primary interest is in charac-terizing the e¡ect of ¢nancial slack on this policy. To provide a benchmark, webegin by reviewing the case where the investment policy is unconstrained by¢nancial considerations.

6Although it plays no formal part in our analysis, it may be helpful to think of this frame-work as describing a ¢rm with a ‘‘lumpy’’ investment schedule. Additions or extensions to itsexisting stock of physical assets can take place only in indivisible units with extensive capitalrequirements, and while the ¢rm is waiting for su⁄cient ¢nancing capacity to accumulate,the existing cash stock is placed in short-term securities. In the meantime, the ¢rm’s existingassets continue to augment (or deplete) the cash stock.

7We assume that the project possesses some feature that is unique to the ¢rm, thereby en-suring that the rights are not fully transferable. Speci¢cally, we assume that the value of therights to an unconstrained ¢rm is Fu(gV) for some constant gA[0, 1). This makes selling lessattractive and ensures that the ¢rm must confront the investment policy consequences of alack of ¢nancial slack.


II. The Optimal Investment Policy:The Problem

A. The Unconstrained Firm

When the ¢rm has a su⁄ciently large cash balance, the ¢nancing friction isirrelevant and investment expenditure is unconstrained. In this case, the optimalinvestment policy can be found using the model developed byMcDonald and Sie-gel (1986) and simpli¢ed by Dixit and Pindyck (1994). Essentially, this modelsolves for Fu as a function of the investment policy, then chooses the policy thatmaximizes the value of this function.

With project value givenbyequation (1), the ¢rm invests if and only ifVexceedssome ¢xed threshold V̂Vu.The optimal investment policy consists of choosing thethreshold V̂Vu that maximizes Fu. Standard replication arguments imply that,prior to investment, Fu satis¢es the di¡erential equation

12s2V2Fu

VV þ ðr� dÞVFuV � rFu ¼ 0; ð5Þ

where subscripts denote partial derivatives, r is the riskless interest rate, and d isthe opportunitycost of cash £ows forgone due towaiting (henceforth the project’s‘‘dividend yield’’). Given the boundary conditions

Fuð0Þ ¼ 0; FuðV̂VuÞ ¼ V̂Vu � I;

equation (5) has the unique solution

FuðVÞ ¼ ðV̂Vu � IÞ V

V̂Vu

� �b

ð6Þ

where

b ¼ 12� r� d

s2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2rs2

þ 12� r� d

s2

� �2s

:

Maximizing (6) with respect toV̂Vu yields the optimal investment threshold

V̂Vu ¼ bIb� 1

;

and investment option value

FuðVÞ ¼ Ib� 1

� �1�b Vb

� �b:

It is straightforward to show that b41, so V̂Vu4I.That is, there are positive payo¡(V� I40) states inwhich the ¢rm does not invest, but instead chooses to wait. Indoing so, the ¢rm retains the opportunity to receive potentially higher payo¡sshould project value rise (or avoid losses if project value falls). This additional£exibility also ensures that Fu(V)�max{0,V� I}. However, these outcomes as-sume that project ¢nancing is guaranteed at all dates and thus ignore the possi-bility that the ¢rm’s timing £exibility may be restricted by its ¢nancingcapabilities.


B. The Constrained Firm

The problem facing the constrained ¢rm is more di⁄cult. Again, investment isjusti¢ed if and only ifVexceeds some minimum threshold. However, because Fc

is a function ofX aswell asV, the optimal threshold V̂Vc(X) must alsobe a functionofX, rather than a constant as in the case of the unconstrained ¢rm.

For those (X,V) where it is optimal to delay investment, Fc satis¢es the di¡er-ential equation (see theAppendix for details)

12s2V2Fc

VV þ rsfVFcXV þ 1

2f2Fc

XX þ ðr� dÞVFcV þ rðX þGÞFc

X � rFc ¼ 0: ð7Þ

The greater complexity of this equation means that an analytical solution for Fc

is unknown, as, therefore, is the investment threshold. In the next section, we¢rst o¡er an intuitive solution to this problem. Subsequently, we generate a nu-merical solution that bothveri¢es the intuitive version and provides some insightinto its economic signi¢cance.

III. The Optimal Investment Policy:The Solution

A. An Intuitive Solution

Because the friction a potentially subjects the ¢rm with ¢nite X to a binding¢nancing constraint, it is clear that

FcðV;XÞ � FuðVÞ:

The reason for this di¡erence is that the constrained ¢rm is forced to follow asuboptimal investment policy compared to that of the unconstrained ¢rm. First,there are states in which the unconstrained ¢rm would exercise the investmentrights, but the constrained ¢rm has insu⁄cient funding available and so mustcontinue to wait. Second, there are states inwhich the unconstrained ¢rmwouldchoose to wait, but the constrained ¢rm invests because the bene¢ts of delay areoutweighed by the risks of losing the ability to ¢nance the project.Thus, the addi-tional investment restrictions faced by the ¢nancially constrained ¢rm are two-fold: In some states it cannot begin investment when it wishes to do so; in otherstates it cannot a¡ord to delay investment when it wishes to do so.The potentialfor these outcomes means that the value of the project rights is lower if held by aconstrained ¢rm than if held by an unconstrained ¢rm.

What canwe sayabout the investment policy itself? Note ¢rst that equation (2)implies

V̂VcðXÞ � V̂VcmðXÞ � ðI �G�XÞ=a:

WhenX is low, V̂Vcm(X) is high as project value has to be high in order for the ¢rmto have su⁄cient funds for investment to proceed. But whenX is low, the ¢rm haslittle ¢nancial slack and so the risk of losing the ability to ¢nance the project ishigh. Consequently, there is little to be gained from delaying investment. This


suggests that, for low X

V̂VcðXÞ ¼ V̂VcmðXÞ:

By contrast, as X becomes higher, the risks of investment delay recede. Simulta-neously, V̂Vcm(X) falls.This suggests that for all X above some critical value X0

V̂VcðXÞ4V̂VcmðXÞ:Finally, we know that as X becomes very large, the ¢rm is e¡ectively uncon-strained.Thus

V̂VcðXÞ ¼ V̂Vu:

These observations imply the approximately ‘‘V-shaped’’ form for V̂Vc(X) illu-strated in Figure 1. For very low X, the risks of delaying investment are su⁄-ciently high that the optimal policy is to invest as soon as it is possible to do so.In this case, the threshold equals the minimum project value satisfying equation(2) and so is a decreasing function of X. As X rises above X0, the risk that the¢rmwill have insu⁄cient funds to ¢nance the project in the future falls, thereby

Figure 1. Optimal investment threshold for a constrained ¢rm.The unconstrained¢rm chooses to invest if (X,V) lies above the line labeled V̂Vu. For low X, the constrained¢rm is able to invest if and only ifV is su⁄ciently high, that is, when (X,V) lies above theline labeled V̂Vcm. However, once X rises above X0, the lower risk of future funding short-falls increases the value of investment delay, so the constrained ¢rmchooses to invest if (X,V) lies above the curve labeled V̂Vc. As a result, the ¢rm does not wish to invest in regionA,irrespective of whether it is constrained or unconstrained. In region C, project valueVexceeds both V̂Vc and V̂Vu, so it invests regardless. In region E, the ¢rm does not invest. InregionD, the ¢rm invests if unconstrained, but it cannot a¡ord to do so if constrained. Inregion B, the value of waiting is su⁄cient for the unconstrained ¢rm to delay investment,but the constrained ¢rm invests immediately because of the risk of future funding short-falls.


increasing the incentive to wait and raising the investment threshold. Initially,this e¡ect is strong as increases in X from a low level signi¢cantly reduce theprobability of future funding shortfalls.8 Eventually however, the risk of suchshortfalls becomes trivial, so further increases inX have little e¡ect and the con-strained ¢rm’s threshold converges on that of the unconstrained ¢rm.

Figure1also illustrates the e¡ect of the ¢nancingconstraint on the investmentdecision. In regions A, C, and E, the constraint does not alter the decision. Inregion A, project valueV is below both V̂Vc and V̂Vu, so the ¢rm does not wish toinvest, irrespective of whether it is constrained or unconstrained. In region C,project valueVexceeds both V̂Vc and V̂Vu, so it invests regardless. In region E, the¢rm does not invest: If unconstrained, it does not wish to do so; if constrained, itcannot a¡ord to do so. Bycontrast, the constraint leads the ¢rm tomake di¡erentdecisions in regionsB andD. In regionD, the ¢rm invests if unconstrained, but itcannot a¡ord to do so if constrained. In regionB, the value of waiting is su⁄cientfor the unconstrained ¢rm to delay investment, but the constrained ¢rm investsbecause the riskof subsequently entering regionD outweighs the value of waiting.

B. ANumerical Solution

We con¢rm the above intuitive solution by undertaking a concrete numericalanalysis that solves equation (7) using a procedure based on ¢nite di¡erencemethods, the details of which are provided in the Appendix.To do so, we use theset of parameter values appearing in Table I. Most of these are similar to thoseused by other authors, for example, Milne and Whalley (2000) and Mauer andTriantis (1994).The additional parameters are G, r, and f. The choice of G is ne-cessarily arbitrary, so we simply set it equal to the investment cost of $100.9 Set-ting the correlation between X and V to 0.5 is consistent with the investmentproject having similar, but not identical, characteristics to the ¢rm’s existing as-sets, thereby exposing the ¢rm to the risk of a cash shortfall when it wishes toinvest. Finally, given G¼100 and r¼ 0.03, f¼60 is chosen to correspond withactual corporate data.10

Table II provides an initial indication of the e¡ects of ¢nancing constraintson the investment timing decision. For various values of the cash balance X, we

8This occurs because higher X results in greater interest income, but has no e¡ect onfuture cash £ow volatility.

9 This means that ¢rms that must currently use external funds to invest (Xo100) expect toreceive a greater proportion of future increments to their cash stocks from the cash £ow gen-erated by their existing physical assets than from the interest return on their existing cashstocks. This seems reasonable insofar as these ¢rms should be motivated to improve the e⁄-ciency of their existing assets in order to break free of the ¢nancing constraint. For ¢rmswith higher liquidity (X�100), the primary expected contribution to their cash stocks is fromthe return on their existing cash. Again, this seems reasonable if, for example, ¢rms that haveaccumulated high X have done so by skimping on additions to their stock of physical assets.

10As G is the market value of the ¢rm’s existing assets, rG must equal the certainty-equiva-lent cash £ow generated by these assets. The choice of r¼ 0.03 and G¼100 yields certainty-equivalent cash £ow of $3. Assuming no systematic cash £ow risk, the choice of f¼ 60 impliesa ratio of cash £ow mean to cash £ow standard deviation of 1/20, approximately the valuefound for U.S. ¢rms listed in the COMPUSTAT database between 1995 and 1999.


calculate the investment thresholds and the values of the project rights for bothconstrained and unconstrained ¢rms.11 For the unconstrained ¢rm with para-meters as given inTable I, investment should be delayed until the current projectvalue of $100 reaches $222; given this policy, the value of the project rights is

Table IBaseline ParameterValues Used in the Numerical Solution Procedure

This table outlines the parameter values used to numerically solve equation (7) for the optimalinvestment timing policy in the presence of a ¢nancing constraint. Most values are similar toMilne andWhalley (2000) andMauer andTriantis (1994).The principal additional parametersFthe market value of existing assetsG and cash £ow volatility fFare chosen to correspond withCOMPUSTATdata.

Parameter Value

Project investment cost ($) I¼100Project value volatility s¼ 0.2Project dividend yield d¼ 0.03Riskless interest rate r¼ 0.03Project value-¢rm cash £ow correlation r¼ 0.5Cash £ow volatility ($) f¼ 60Market value of existing assets ($) G¼100Market friction a¼ g¼ 0.8

Table IIInvestmentThreshold andOptionValues for

Unconstrained and Constrained FirmsThis table reports the investment thresholds (V̂Vu and V̂Vc)and project rights values (Fu and Fc) for unconstrainedand constrained ¢rms, respectively. X denotes the ¢rm’scurrent cash stock. Parameter values used in generatingthe threshold and optionvalues are those given inTable I.In addition, for calculating Fu and Fc, we assume initialproject valueVequals 100.

Cash Stock Investment Thresholds OptionValuesX V̂Vu V̂Vc Fu F c

� 200 222 250 28.51 18.98� 100 222 168 28.51 21.07� 50 222 188 28.51 24.41

0 222 200 28.51 26.3050 222 208 28.51 27.35

100 222 212 28.51 27.92150 222 214 28.51 28.22200 222 216 28.51 28.37

11For the purposes of calculating the rights values, the initial project valueV is set equal tothe investment cost I¼100, so the project has signi¢cant waiting value.


$28.51. However, delay for the constrained ¢rm incurs the risk that the ¢rm’savailable ¢nancing will drop below $100, thereby making investment (tempora-rily, at least) impossible.This additional risk makes waiting less valuable, so theoptimal investment policy generally requires a lower threshold. For example,when the ¢rm has a moderate cash de¢cit (X¼ � 50), the threshold is $188, 15percent below that of an unconstrained ¢rm. Not surprisingly, this lowers thevalue of the investment rights, in this case to $24.41, a level some 14 percent belowits unconstrained value. For higher values ofX, the risks of delayare lower, so thedi¡erences are smaller, but the constrained ¢rm nevertheless adopts a lowerthreshold (and generates a lower rights value) than its unconstrained counter-part. By contrast, when the ¢rm has a signi¢cant cash de¢cit, the risk of losingthe ability to fund the project is so great that it cannot a¡ord to delay investmentat all. In this case, the project proceeds as soon as the ¢rm has su⁄cient externalfunding to make investment possible, so the threshold can be greater than itsunconstrained counterpart. For example, when X¼ � 200, the threshold is $250and the value of the rights is $18.98. Overall, by inducinga suboptimal investmentpolicy, the existence of a ¢nancing constraint can eliminate a signi¢cant propor-tion of a project’s potential value.

A more general picture of these e¡ects appears in Figures 2 and 3. Figure 2con¢rms the‘‘V-shaped’’ form for V̂Vc(X) illustrated in Figure 1. Overall, the ¢nan-cing constraint encourages the constrained ¢rm to invest earlier than the uncon-strained ¢rm. For very low X, the constrained ¢rm invests immediately uponobtaining the necessary funding, even in states where delay is optimal for theunconstrained ¢rm. For intermediate values of X, the constrained ¢rm adopts a

Figure 2. Cash £ow volatility and the constrained ¢rm’s investment thresholdfunction.The value of the constrained ¢rm’s investment threshold is plotted for di¡erentvalues of initial cash stock (X) and cash £ow volatility (f). Parameter values are given inTable I. For given f, a rise in the initial cash stock (beyond the minimum necessary topermit investment delay) decreases the risk that the ¢rm will have insu⁄cient funds to¢nance the project in the future, thereby increasing the value of waiting and raising theinvestment threshold. For givenX, a rise in cash £ow volatility increases the risk that the¢rmwill have insu⁄cient funds to ¢nance the project in the future, thereby decreasing thevalue of waiting and lowering the investment threshold.


lower threshold than the unconstrained ¢rm, so the former continues to invest instates where the latter would choose to wait. Only for high values of X, when the¢nancing constraint is immaterial, does the constrained ¢rm adopt the same in-vestment policy as the unconstrained ¢rm.

Figure 2 also illustrates the in£uence of cash £ow volatility f. For each X,greater cash £ow volatility increases the amount by which the constrained¢rm’sthreshold deviates from its unconstrained counterpart. The greater is f, thegreater the likelihood that adverse cash £ow shocks will eliminate the ¢rm’s abil-ity to ¢nance the project when it wishes to invest. In response, the ¢rm reducesits exposure to this risk by lowering the investment threshold.We can think ofthis as a formalized ‘‘bird-in-the-hand’’strategy; a relatively small payo¡ receivedsoon and with low risk is preferable to a potentially large payo¡ received later ifthere is signi¢cant risk of the latter payo¡ becoming zero due to a funding short-fall.

Figure 3 displays the relationship between the value of the project rights (Fc)and the ¢rm’s initial cash stock (X) for di¡erent project values (V). The e¡ect ofXon F c is not strictly monotonic, nor is it independent of V. In general, lower Xincreases the risk that the ¢rmwill be unable to ¢nance the project in the future,thereby decreasing the value of waiting and lowering Fc. However, whenX becomes very low, the ¢rm must liquidate and sell the project rights, so anyfurther changes in X have no impact on Fc. This explains the initial horizontal

Figure 3. The constrained ¢rm’s investment option value. The value of the projectrights for the constrained ¢rm (Fc) is plotted for di¡erent values of initial cash stock (X)and project value (V ). Parameter values are given in Table I. For givenV, a rise in X de-creases the risk that the ¢rm will have insu⁄cient funds to ¢nance the project in the fu-ture, thereby increasing the value of waiting and raising Fc. If V is low, the expectedwaiting time is high and so Fc increases monotonically with X. For intermediateV, anincrease in X raises Fc for values of X below that needed to permit investment.When Xonly slightly exceeds this minimum level, the risk that the ¢rmwill have insu⁄cient fundsto ¢nance the project in the future o¡sets the potential gains fromwaiting, so the projectrights are exercised and additional increments in X have no e¡ect on Fc until X is su⁄-ciently high to signi¢cantly reduce the risk of future funding shortfalls. IfV is above theunconstrained threshold, then immediate investment is optimal, so Fc increases sharplywithX for values ofX below theminimum required for investment, but is independent ofXthereafter.


component of each of the curves in Figure 3. Otherwise, for X su⁄ciently high topreclude the need for liquidation, the nature of the relationship between X andFc depends onV. IfV is low (the bottom curve in Figure 3), the expected waitingtime is long and so the funding shortfall risk is signi¢cant even if X is currentlywell above the investment cost I. In this case, Fc increases monotonically withXuntil it converges on the unconstrained optionvalue. Bycontrast, ifVexceeds theunconstrained investment threshold (the top curve in Figure 3), then immediateinvestment is optimal, soFc rises sharply withX for values ofX that take the ¢rmout of the liquidation zone (because additional cash reduces the expected subop-timal delay in investment), but is then independent ofXonce this rises su⁄cientlyto allow investment (because investment occurs and Fc¼V� I ).

If V is greater than I, but less than the unconstrained investment threshold(the middle curve in Figure 3), the relationship between Fc and X becomes morecomplex. First, in the range of X that is su⁄cient to allow the ¢rm to remain inbusiness, but insu⁄cient to permit investment, Fc is rapidly increasing in X be-cause each additional dollar reduces the probability that the ¢rm will face afunding shortfall when the optimal investment date arrives. Second, for X equalto or slightly greater than the minimum amount needed for investment, the po-tential bene¢ts of delaying investment are outweighed by the riskof subsequentlylosing the ability to ¢nance the project, so the ¢rm invests and additional incre-ments to X have no e¡ect on Fc (this is the dashed line component of theV¼180curve). Third, for X su⁄ciently above the minimum amount needed for invest-ment, the funding shortfall risk is small enough for investment delay to againbecome the optimal strategy. Additional increases in X then raise the value ofthe investment option until it converges on the unconstrained option value.

Finally, the e¡ect of cash on the optimal investment policy can also be depictedin terms of a hurdle rate outcome. For a project with perpetual life, letm denotethe hurdle rate margin, measured by the di¡erence between the project’s re-quired rate of return and its cost of capital. Thus, m represents the portion ofthe project hurdle rate attributable to investment timing considerations. In theAppendix, we show that

m ¼ dV̂VcðXÞ

I� 1

!:

The relationship betweenm andX has the same general form as the relationshipbetween V̂Vc andX: increasing strongly whenX is low, but leveling o¡ at higherXas the risk of future shortfalls becomes insigni¢cant. To give some idea of thequantitative nature of this relationship, we calculate the hurdle rate margin asa function of the ¢rm’s ability to ¢nance investment internally and summarizethe results in Table III.When the cash position is just su⁄cient to make waitingviable (which, for a ¢rm with the distributional parameters assumed in Table I,occurs when the cash coverage ratio X/I¼ �1.2), the hurdle rate premium is 1.6percent. At this point, a positive cash shock equal to the investment cost raisesthe hurdle rate premium by a further 1.26 percentage points. By contrast, whenthe cash position is equal to 50 percent of the investment cost, the hurdle rate


premium is 3.2 percent, but an additional cash shock raises this premium by only21 basis points. Overall, the di¡erence between the hurdle rate premium for anunconstrained ¢rm and that for a strongly constrained ¢rm is about two percen-tage points. Although the level of this di¡erence varies with parameter valuesand types of project, it is roughly constant in proportionate terms. For example,if the project has a ¢nite life, then the hurdle premia for constrained and uncon-strained ¢rms are greater than 1.6 percent and 3.6 percent respectively, but theratio is still approximately 1:2.

IV. Underinvestment, Accelerated Investment, and Liquidity

In an early analysis of the underinvestment problem, Myers and Majluf (1984)demonstrate that when ¢rm insiders have information that outsiders do not, thelatter interpret any attempt by the ¢rm to raise external funds as an indicationthat the ¢rm is overvalued. Consequently, they lower their estimate of ¢rm value,thereby raising the ¢rm’s cost of external ¢nancing and lowering the project’sNPV. As a result, a fall in internal cash increases the cost of capital and lowersinvestment. Similarly, Hart and Moore (1994) show that the inability of humancapital to credibly commit to a project can cause a ¢rm to forgo positive-NPVprojects when internal funds fall below a critical level. Again, a fall in internalcash lowers investment.

In our model, cash has two e¡ects on investment. First, similar to Hart andMoore (1994), a fall in internal cash increases the need for limited external ¢nan-cing and thus reduces the number of states in which investment can occur.That

Table IIICash and the Hurdle Rate Margin

This table calculates the e¡ect of cash on the investment timing decision in terms of a hurdlerate margin, measured by the di¡erence between the optimal investment hurdle rate and theunconstrained cost of capital for an in¢nite-life project owned by a ¢rm with parameters as inTable I.The cashcoverage ratio equals the ratio of cash to investment cost.The margin-cash £owsensitivity is the e¡ect (in percentage points) on the hurdle rate margin of a cash £ow injectionequal to the investment cost. For example, when the ¢rmhas acashoverdraft equal to 50 percentof the investment cost, the hurdle rate premium is 2.6 percentage points and a cash shock equalto the investment cost raises this by 0.6 percentage points.

Cash coverage ratio Hurdle rate margin Margin-cash £ow sensitivity

� 1.2 0.016 1.26� 1.0 0.020 1.02� 0.5 0.026 0.600.0 0.030 0.360.5 0.032 0.211.0 0.033 0.131.5 0.034 0.072.0 0.035 0.043.0 0.036 0.02


is, it raises V̂Vcm(X). Second, because the project does not evaporate if nottaken immediately, today’s cash position also has implications for the possibilityof future investment. In particular, a fall in internal cash today also makesit less likely that su⁄cient funding will be available in the future, thereby raisingthe risks of delaying investment.The ¢rst e¡ect causes ¢rms to forgo investmentin projects they would otherwise have taken; the second e¡ect causes ¢rms toinvest in projects earlier than they would otherwise have done. Figure 1illustrates this point: A fall in internal funds not only makes it more likely thatthe ¢rm will end up in region D, but also in region B. In static investmentmodels, where investment opportunities disappear if not taken today, onlythe ¢rst e¡ect operates. In a dynamic framework, by contrast, external ¢nancingrestrictions can distort investment decisions in two ways. On the one hand,they lead to underinvestment; on the other hand, they encourage acceleratedinvestment.

This observation has interesting implications for the recent debate onwhetheror not the sensitivity of investment to ¢rm liquidity is a useful measure of ¢nan-cing constraints. Beginning with an in£uential paper by Fazzari, Hubbard, andPetersen (1988), the standard approach has been to divide a sample of ¢rms intogroups re£ecting a priori rankings of likely ¢nancial constraints and then com-pare the investment-cash £ow sensitivities of these di¡erent groups. Most stu-dies ¢nd that the ¢rms deemed most likely to be ¢nancially constrained exhibitgreater sensitivity of investment to cash £ow, thereby suggesting that this sensi-tivity is indeed a useful measure of the severity of ¢nancing constraints. How-ever, this approach has been questioned by Kaplan and Zingales (1997, 2000)and Cleary (1999). Most tellingly, Kaplan and Zingales ¢nd that within the sam-ple of ¢rms Fazzari et al. argue are most likely to be ¢nancially constrained, the¢rms with signi¢cant liquidity problems exhibit a lower investment-cash £owsensitivity than the ¢rms that appear unlikely to have been ¢nancially con-strained. Fazzari et al. counter this by pointing out that ¢rms are likely to accu-mulate high liquidity precisely because they face signi¢cant ¢nancingconstraints, so liquidity may provide no information about the extent to whichthe ¢rm is constrained. However, as Dasgupta and Sengupta (2002) point out,this does not explain why the sensitivity of investment to cash £ow shocks isgreater for ¢rms with high liquidity. Kaplan and Zingales stress that it is impor-tant to understand the source of this result and speculate that it may be dueeither to a nonlinearity in the external ¢nance cost function or to currently un-known aversions to raising external ¢nance.

Our model suggests another source.When the ¢rm has limited access to exter-nal funding, greater cash £ow not only relaxes the constraint on current invest-ment, but also decreases the likelihood that future investment will beconstrained, thereby increasing the value of the timing option and thus the op-portunity cost of current investment. Consequently, the sensitivity of investmentto cash £ow is inversely related to the e¡ect of cash £owon the value of the timingoption. When the ¢rm is relatively cash rich, the risks of delay are small, sofurther increases in cash have little e¡ect on the option value. By contrast, whenthe ¢rm has a smaller amount of cash, the risks of delay are higher, so additional


cash leads to a larger increase in the value of the timing option.12 As a result, agiven cash injection can have a smaller positive impact on the investment of low-liquidity ¢rms than on the investment of high-liquidity ¢rms, consistent with the¢ndings of Kaplan and Zingales (1997) and Cleary (1999).13

Of course, this is unlikely tobe the full storyas current project value, and there-fore the attractiveness of immediate investment, may also be a¡ected by ¢rm li-quidity. For example, the models of Myers and Majluf (1984) and Greenwald et al.(1984) suggest that project value increases with liquidity because of lower ¢nan-cing costs: More cash reduces the need for costly external ¢nancing and therebylowers the cost of capital.The net e¡ect of a given cash change on investment thusdepends on the relative magnitude of the change in ¢nancing costs vis-a' -vis thechange in timing option value; an increase in cash encourages immediate invest-ment via the former, but discourages it via the latter. If the ¢nancing cost e¡ect issigni¢cantly greater than the timing option e¡ect, then the latter is unlikely to bea major source of the Kaplan and Zingales (1997) and Cleary (1999) results.

However, additional empirical evidence suggests that timing option e¡ects domatter for the relationship between investment and liquidity.To seewhy, considera group of high-liquidity ¢rms with similar investment opportunities and con-straints. As we have seen, the timing option e¡ect is minor for high-liquidity¢rms, so the ¢nancing cost e¡ect is likely to dominate and the investment-cash£ow sensitivity should increase with liquidity in this group of ¢rms. By contrast,for a similar group of low-liquidity ¢rms, the timing option e¡ect is relativelylarge, so the investment-cash £ow sensitivity should increase much more slowly,or even decline, with liquidity.These predictions are consistent with the resultsobtained by Huang (2002) in a study of 1,235 U.S. ¢rms listed on COMPUSTATbetween 1988 and 1998. He calculates the investment-cash £ow sensitivity of sub-groups of both high and low dividend payout ¢rms. For the high payout ¢rms, theinvestment-cash £ow sensitivity is higher for the more constrained subgroups,but the reverse is true for the low payout ¢rms.

Several other recent studies have also examined the theoretical relationshipbetween investment and liquidity. Almeida and Campello (2001) and Povel andRaith (2001) o¡er stories similar to ours, in that both identify o¡setting e¡ectsof liquidity on investment. Almeida and Campello show that more cash not onlydirectly allows more investment, but also allows it indirectly by increasing bor-rowing capacity and, moreover, that this indirect e¡ect is greater for ¢rms thatface smaller frictions (in our model, this corresponds to the case where a is anincreasing function of X). Povel and Raith point out that higher cash not onlyallows the constrained ¢rm to invest more, but also encourages it to do so: Moreinvestment generates greater revenues that alleviate the constraint. As the ¢rm’sliquidity positionworsens, the latter e¡ect becomes stronger, so investment has a

12This can also be seen in Figure 1. A decrease in X moves more projects into region Bwhen X is low than when X is high.

13 For very low X, a rise in X a¡ects investment only if it is large enough to allow the ¢rm toinvest, that is, move it to the V̂Vcm(X) line in Figure 1. This is consistent with the empiricalevidence that the investment of ¢rms with very low liquidity is insensitive to cash £ow.


relatively weak relationship with cash £ow. Moyen (2002) suggests that theinvestment of high-liquidity ¢rms is more sensitive to cash £ow changes for tworeasons. First, the value of future investment opportunities increases with cur-rent cash £ow, but less constrained ¢rms are better placed to take advantage ofthese opportunities. Second, some of the greater cash £ow is absorbed by divi-dends in constrained ¢rms, but this e¡ect is smaller in ¢rms with higher liquid-ity. Dasgupta and Sengupta (2002) examine the investment decision of a ¢rmsubject to moral hazard problems. More cash diminishes the extent of this pro-blem and thereby permits greater investment. In general, this e¡ect is strongerfor ¢rms with low liquidity, but over certain ranges of liquidity, higher-liquidity¢rms may bene¢t more.Thus, the relationship between investment and liquidityis not monotonic.Whited (2002) develops a model of dynamic investment decisionmaking in which the risk of future funding shortfalls is negligible (because, inthe terminology of our model, the correlation r between X and V equals one),but lower liquidity increases the ¢xed adjustment costs of changing the capitalstock. As a result, ¢rms with less slack have, on average, a greater length of timebetween large, indivisible, investment projects. Finally, Alti (2001) shows that anincrease in cash can lead to higher investment even in the absence of capital mar-ket frictions. In his model, changes in cash provide additional information aboutthe quality of a ¢rm’s investment opportunities. Moreover, this information e¡ectis stronger for low-cash ¢rms because such ¢rms are typically younger and havegreater ex ante uncertainty about their future prospects. Consequently, the sen-sitivity of investment to cash £ow is greater for low-cash ¢rms.

These models identify a variety of mechanisms by which cash £ow can in£u-ence investment. More cash can signal greater project quality or investment op-portunities, reduce moral hazard and capital stock adjustment costs, increaseborrowing capacity, and decrease investment bene¢ts. However, although allthese stories focus on dynamic investment issues, none explicitly considers thee¡ect of cashon the value of the ¢rm’s real options. Bycontrast, our model empha-sizes the value of cash for optimal investment timing: More cash lowers the costof delaying investment. Thus, none of these stories is incompatible with ours,although which, if any, best identi¢es the primary drivers of the investment-liquidity relationship remains an empirical question.14

V. Investment and Uncertainty

Our model also has implications for the relationship between investment anduncertainty.When investment is unconstrained, all uncertainty emanates fromthe stochastic evolution of project value, so greater uncertainty increases the va-lue of waiting and lowers investment. However, the empirical evidence for this

14Whited’s (2002) ¢nding that higher liquidity increases the investment hazard rate mightinitially seem inconsistent with our model, but the two are easily reconciled. Consider, forexample, two ¢rms with the sameV, one in region E of Figure 1, the other in region B. Theformer waits to invest, but the latter invests immediately. Thus, the less liquid ¢rm has thelower hazard, asWhited’s results suggest.


relationship is inconclusive; Ghoshal and Loungani (1996, 2000) ¢nd that invest-ment is a decreasing function of price or pro¢t uncertainty in industries withlarge numbers of small ¢rms, but not for other industry structures; Caballeroand Pindyck (1996) report a statistically signi¢cant relationship between invest-ment and the variance of the marginal revenue product of capital, but one that issubstantially smaller than predicted by the unconstrained ¢rm model. Our mod-el suggests one possible reason for these ambiguous results. In the presence of a¢nancing constraint, the ¢rm faces not only uncertainty about future project va-lue, but also uncertainty about its ability to ¢nance desirable investments.Thesehave opposite e¡ects on the investment threshold; payo¡ uncertainty raises thethreshold, while ¢nancing uncertainty lowers it.Thus, anyattempt to empiricallyidentify the relationship betweenuncertainty and investment will pick up o¡set-ting uncertainty e¡ects unless the exact nature of the uncertainty is carefullyidenti¢ed. For example, the uncertainty measures used in the above studies arebased on historical estimates of volatility in some aspect of ¢rm performance,and thus seem likely to include aspects of both payo¡ and ¢nancing uncertainty.Consequently, it is unsurprising that the estimated relationship between invest-ment and uncertainty is small or nonexistent.

The con£icting e¡ects of payo¡ and ¢nancing uncertainty can also have impli-cations for the interpretation of other empirical results. Shin and Stulz (2000) ¢ndthat shareholder wealth is negatively related to equity price volatility (as a proxyfor cash £ow volatility), a result theyattribute to ¢nancial distress costs. However,our model makes it clear that such a result is also consistent with dynamic invest-ment decision making; greater cash £ow volatility reduces the value of the invest-ment option (and therefore shareholder wealth) because it increases the likelihoodof a future funding shortfall and therefore leads to suboptimal investment timing.

VI. Conclusion

When access to external funding is restricted, ¢rms become more reliant oninternal funds to ¢nance investment. Although this fundamental point has longbeen recognized, its implications for optimal investment timing have not pre-viously been analyzed. In this paper, we consider the implications of liquidityfor the investment policy of a ¢rm that owns the perpetual rights to a project.Our principal conclusions are as follows:

1. A ¢nancing constraint lowers the value of the rights because of the risk thatthe ¢rmwill face a funding shortfall when it wishes to invest.

2. The risk of future funding shortfalls lowers the optimal investment thresh-old, thereby resulting in suboptimal early investment.Thus, a ¢nancing con-straint may cause the ¢rm to sacri¢ce a signi¢cant proportion of a project’svalue not only by forcing it to forgo investment (as would be the case in sta-tic models), but also by encouraging it to accelerate investment.

3. Greater cash £ow permits more investment, but also raises the thresholdrequired to justify investment. Since the latter e¡ect is greatest for more


constrained ¢rms, such ¢rms can have lower investment-cash £ow sensitiv-ities than ¢rms that are less constrained. This contrasts with the conven-tional view that high sensitivities indicate strong constraints, but isconsistent with the evidence of Kaplan and Zingales (1997) and Cleary(1999).

4. Greater payo¡ uncertainty increases the threshold required to justify in-vestment, but greater ¢nancing uncertainty decreases it. Since most empiri-cal measures of volatility are likely to contain elements of both payo¡ and¢nancing uncertainty, their o¡setting e¡ects can help explain why the ob-served short-term relationship between investment and volatility is weak.

Our analysis has focused on single stand-alone projects that have no e¡ect onthe ¢nancing constraints faced by other projects. An obvious extension of ourwork would consider the more general situation where the ¢rm has a number ofcompeting projects, each of which has di¡erent implications for the ¢nancingconstraints faced byall others. For example, suppose a ¢rm has two projects withthe same positive NPVand cost I. If the ¢rm has insu⁄cient ¢nancial resourcesto launch both projects, then starting one of them now augments the future cashstock and thereby increases the likelihood of being able to subsequently launchthe other.Thus, there is an additional incentive for the constrained ¢rm to investnow, similar to the mechanism identi¢ed by Povel and Raith (2001). Bycontrast, ifthe payo¡s from the two projects are negatively correlated, the unconstrained¢rm has more incentive to delay investment in order to see which turns out best.In this case, the di¡erences that we have identi¢ed between constrained and un-constrained ¢rms seem likely to be accentuated, but other cases may yield di¡er-ent outcomes, so further analysis of this complex problem has the potential toyield additional insights.

APPENDIX

A. Derivation of Equation (7)

We assume that the risks inherent inV and X are spanned by the market ofexisting securities. Speci¢cally, suppose that there are traded assets or portfolioswith prices v and x that evolve according to

dv ¼ mvvdtþ svvde; ðA1Þ

dx ¼ mxxdtþ sxxdz: ðA2Þ

Then a long position in the investment option can be combined with short posi-tions of sVFV/(svv) units of asset v and fFX/(sxx) units of asset x to produce atotal return dR over the time interval dt such that (for shorthand, we drop the‘‘c’’ superscript on F since it is obvious that we are referring only to the


constrained ¢rm)

dR ¼ dF � sVFV

svv

� �dv� fFX

sxx

� �dx:

Using Ito“ ’s Lemma to obtain an expression for dF, substituting (A1) and (A2)for dv and dx, respectively, and simplifying, we obtain

dR ¼ 12s2V2FVV þ 1

2f2FXX þ rsfVFXV þ m� mvs

sv

� �VFV þ rX þ n� mxf

sx

� �FX

� �dt:

Since this return is riskless, the portfolio must earn the riskless rate of return.Therefore,

dR ¼ r F � sVFV

sv� fFX

sx

� �dt:

Equating this to the above expression for dRmeans that F satis¢es the di¡eren-tial equation

0 ¼ 12s2V2FVV þ 1

2f2FXX þ rsfVFXV

þ m� mvssv

þ rssv

� �VFV þ rX þ n� mxf

sxþ rf

sx

� �FX � rF: ðA3Þ

Further simpli¢cation can most readily be obtained if we assume the expectedreturns mv and mx are given by some equilibrium model such as the CAPM. If thelatter holds, then

mx ¼ rþ rxmsxl;

mv ¼ rþ rvmsvl;

where rxm(¼ rxm) and rvm(¼ rvm) are the correlation coe⁄cients of the marketreturn with dx and dv respectively, and l is the market price of risk. If d � mv� mis the project’s dividend yield, then

mþ d ¼ rþ rvmsvl:

Hence, the (A3) coe⁄cient onVFV, m� (mvs/sv)þ (rs/sv), becomes

m� rvmsl ¼ r� d:

Now letG denote the market value of a claim to the future cash £ow generatedby the ¢rm’s existing physical assets. Clearly G is independent of X and V, sodG¼ 0 over any time interval dt.Thus, the return on a long position inG consistsonly of the current cash £ow (ndtþfdz). Hence, using (A2), a long positionin G combined with a short position in f/(sxx) units of asset x yields a totalreturn of

ndtþ fdz� fsxx

� �dx ¼ n� fmx

sx

� �dt:


Since this return is riskless, we must have

n� fmxsx

¼ r G� fsx

� �;

which implies that the (A3) coe⁄cient on FX, rXþ n� (mxf/sx)þ (rf/sx), is equalto r(XþG). Making this substitution back into (A3) yields (7).

B. Numerical Solution Procedure for Equation (7)

The partial di¡erential equation is solved on a grid with nodes {(Xk, Vj):j¼1, y, J, k¼1, y, K}, where Xk�Xk�1¼dX and Vj¼ j dV. At node (Xk,Vj),the resulting di¡erence equation can be written in the form

0 ¼ ajFj�1;k þ bjFj;k þ cjFjþ1;k þ dkFj;k�1 þ ekFj;kþ1

þ fjðFjþ1;kþ1 þ Fj�1;k�1 � Fj�1;kþ1 � Fjþ1;k�1Þ;

where

aj ¼s2V2

j

2dV2 �ðr� dÞVj

2dV;

bj ¼� f2

dX2 �s2V2

j

dV2 � r;

cj ¼s2V2

j

2dV2 þðr� dÞVj

2dV;

dk ¼f2

2dX2 �rðXk þGÞ

2dX;

ek ¼f2

2dX2 þrðXk þGÞ

2dX;

fj ¼rfsVj

4dXdV;

and Fj,k¼F(Xk, Vj). This equation is de¢ned whenever 2� j�J�1 and2� k�K�1. We extend it to the edges of the grid using four boundary condi-tions. (i) When j¼1, we use F0,k¼ 0, since F(X, 0)¼ 0. (ii) When j¼J, we useFJþ1,k¼ 2FJ,k�FJ�1,k, motivated by the observation that

FðX;V þ dVÞ ¼ 2FðX;VÞ � FðX;V � dVÞ þOðdV2Þ:

(iii) When k¼1, we suppose that the liquidation constraint is binding, so thatFj,0¼Fu(gVj). (iv) When k¼K, we use Fj,Kþ1¼Fu(Vj), or the value of the projectif there is no ¢nancing constraint.

We start by setting Fj,k¼ 0 ifXkþGþ aVjoI orVjoI, and set Fj,k¼Vj� I at allother nodes, and then solve the system using the method of Successive Over Re-laxation. During each iteration of this method, we solve the di¡erence equationat each node (Xk,Vj) in turn, replacing the calculated value of Fj,kwith Fu(gVj) ifXkþGþFu(gVj)o0, and withVj� I at any node for which XkþGþ aVj� I and


Fj,koVj� I.We stop iterating when the largest change in any Fj,k, measured rela-tive to its value at the end of the preceding iteration, is less than I/10000.

C. The Project’s Hurdle Rate

Suppose that the project has a perpetual life and generates cash £owof zt¼ dVt

if it is operating at date t. These assumptions ensure that the value of the com-pleted project equalsVt at date t, and are consistent with our interpretation inSection II of d as the project’s dividend yield. At time t, the project’s internal rateof return (IRR) is the number it such that

0 ¼Z 1

0e�itsEt½ztþs�ds� I ¼ zt

it � m� I ¼ dVt

it � m� I:

The project’s IRR is therefore

it ¼ mþ dVt

I:

Investment should occur at date t if and only ifVt� V̂Vc(Xt), that is, if and only if

it � mþ dV̂VcðXtÞI

:

Since the project’s cost of capital equals mþ d, the margin between the project’shurdle rate and its cost of capital is

mt ¼ mþ dV̂VcðXtÞI

!� ðmþ dÞ ¼ d

V̂VcðXtÞI

� 1

!:

REFERENCES

Abel, Andrew, Avinash K. Dixit, Janice C. Eberly, and Robert S. Pindyck, 1996, Options, the value ofcapital, and investment, Quarterly Journal of Economics 111, 753^778.

Almeida, Heitor, and Murillo Campello, 2001, Financial constraints and investment-cash £ow sensi-tivities: New research directions,Working paper, Publisher: NewYork University.

Alti, Aydogan, 2001, How sensitive is investment to cash £ow when ¢nancing is frictionless? Workingpaper, Carnegie Mellon University.

Baumol,William, and Richard Quandt, 1965, Investment and discount rates under capital rationing^aprogramming approach, Economic Journal 75, 317^329.

Bernanke, Ben, 1983, Irreversibility, uncertainty, and cyclical investment, Quarterly Journal of Eco-nomics 98, 85^106.

Brennan, Michael, and Eduardo Schwartz, 1985, Evaluating natural resource investments, Journal ofBusiness 58, 135^157.

Caballero, Ricardo, and Robert Pindyck, 1996, Uncertainty, investment, and industry evolution, Inter-national Economic Review 37, 641^662.

Charnes, Abraham,WilliamW. Cooper, and Merton H. Miller, 1959, Applications of linear program-ming to ¢nancial budgeting and the costing of funds, Journal of Business 31, 20^46.

Cleary, Sean, 1999,The relationship between ¢rm investment and ¢nancial status, Journal of Finance54, 673^692.


Clementi, Gian, and Hugo Hopenhayn, 2002, A theory of ¢nancing constraints and ¢rm dynamics,Working paper, Carnegie Mellon University.

Dasgupta, Sudipto, and Kunal Sengupta, 2002, Financial constraints, investment and capital struc-ture: Implications from a multi-period model,Working paper, University of Sydney.

Dixit, Avinash, 1989, Entry and exit decisions under uncertainty, Journal of Political Economy 97,620^638.

Dixit, Avinash, and Robert Pindyck, 1994, Investment Under Uncertainty (Princeton University Press,Princeton, NJ).

Elton, Edwin J., 1970, Capital rationing and external discount rates, Journal of Finance 25, 573^584.Fazzari, Steven, Glenn R. Hubbard, and Bruce Petersen, 1988, Finance constraints and corporate in-

vestment, Brookings Papers on Economic Activity 141^195.Ghoshal, Vivek, and Prakash Loungani, 1996, Product market competition and the impact of price

uncertainty on investment: Some evidence from US manufacturing industries, Journal of Indus-trial Economics 44, 217^228.

Ghoshal,Vivek, and Prakash Loungani, 2000,The di¡erential impact of uncertainty on investment insmall and large businesses, Review of Economics and Statistics 82, 338^343.

Greenwald, Bruce, Joseph Stiglitz, and Andrew Weiss, 1984, Informational imperfectionsin the capital market and macroeconomic £uctuations, American Economic Review 74,194^199.

Hart, Oliver, and John Moore, 1994, A theory of debt based on the inalienability of human capital,Quarterly Journal of Economics 109, 841^879.

Huang, Zhangkai, 2002, Financial constraints and investment-cash £ow sensitivity,Working paper,University of Oxford.

Ingersoll, Jonathan E., and Stephen A. Ross, 1992, Waiting to invest: Investment and uncertainty,Journal of Business 65, 1^29.

Jensen, Michael C., andWilliam H. Meckling, 1976,Theory of the ¢rm: Managerial behavior, agencycosts and ownership structure, Journal of Financial Economics 3, 305^360.

Kaplan, Stephen N., and Luigi Zingales, 1997, Do investment-cash £ow sensitivities provide usefulmeasures of ¢nancing constraints? Quarterly Journal of Economics 112, 169^215.

Kaplan, Stephen N., and Luigi Zingales, 2000, Investment-cash £ow sensitivities are not valid mea-sures of ¢nancing constraints, NBERWorking Paper 7659.

Majd, Saman, and Robert S. Pindyck, 1987, Time to build, option value and investment decisions,Journal of Financial Economics 18, 7^27.

Mauer, David C., and Alexander J.Triantis, 1994, Interactions of corporate ¢nancing and investmentdecisions: A dynamic framework, Journal of Finance 49, 1253^1277.

McDonald, Robert, and Daniel Siegel, 1986, The value of waiting to invest, Quarterly Journal of Eco-nomics 101, 707^727.

Mello, Antonio S., and John E. Parsons, 2000, Hedging and liquidity, Review of Financial Studies 13,127^153.

Milne, Alistair, and Elizabeth A.Whalley, 2000,Time to build, option value and investment decisions:A comment, Journal of Financial Economics 56, 325^332.

Minton, BernadetteA., and Catherine Schrand, 1999,The impact of cash £ow volatility on discretion-ary investment and the costs of debt and equity ¢nancing, Journal of Financial Economics 54,423^460.

Moyen, Nathalie, 2002, Investment-cash £ow sensitivities: The trade-o¡ model versus the internal-funds model,Working paper, University of Colorado at Boulder.

Myers, Stewart C., 1972, A note on linear programming and capital budgeting, Journal of Finance 37,89^92.

Myers, Stewart C., and Nicholas S. Majluf, 1984, Corporate ¢nancing and investment decisions when¢rms have information that investors do not have, Journal of Financial Economics 13, 187^221.

Povel, Paul, and Michael Raith, 2001, Optimal investment under ¢nance constraints: The roles ofinternal funds and asymmetric information,Working paper, University of Minnesota.

Shin, Hyun-Han, and Rene¤ Stulz, 2000, Shareholder wealth and ¢rm risk, Ohio State University DiceCenterWorking Paper No. 2000^19.


Stulz, Rene¤ , 1990, Management discretion and optimal ¢nancing policies, Journal of FinancialEconomics 26, 3^27.

Triantis, Alexander J., and James E. Hodder, 1990,Valuing £exibility as a complex option, Journal ofFinance 43, 549^565.

Weingartner, Martin H., 1977, Capital rationing: n authors in search of a plot, Journal of Finance 32,1403^1432.

Whited, Toni M., 2002, External ¢nance constraints and the intertemporal pattern of intermittentinvestment,Working paper, University of Iowa.


investment, uncertainty, and liquidity

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