Firm behaviour under the threat of liquidation

Journal

of Economic Dynamics and Control 20 (1996) 142771449

ELSEVIER

Firm behaviour under the threat of liquidation Alistair Milne”, Donald Robertson*.b aDepartment ofEconomics, Unicersitv of Surrey, Guildford GU2 SXH, UK ’ Frrcul~~of Economics, Cambridge Unicersi~, Cambridge (‘B3 YDD. UK (Received

April 1995; final version

received

October

1995)

Abstract

We study the optimal behaviour of a firm whose cash flow is determined by a diffusion process, facing liquidation if internal cash balances fall below some threshold. There is a conflict between the desire to pay dividends to satisfy shareholders and the need to retain cash as a barrier against possible liquidation. Stochastic optimal control characterises the value function and gives optimal decision rules for the firm. The firm has a precautionary motive for retaining earnings; the internal cost of funds and local risk-aversion are both decreasing functions of cash held, and, in natural extensions to our model,

output

Key words:

and

investment

are both

Liquidity constraints; E44

increasing

functions

of cash

balances.

Firm behaviour dynamics

JEL class$ication:

1. Introduction This paper considers the optimal dynamic decisions of a firm operating in a stochastic environment under an exogenous threat of liquidation (which we impose as a simple way of capturing credit market imperfections). The behaviour of such a firm differs markedly from the deterministic perfect market case. The firm exhibits local risk-aversion and there is a distortion to the cost of capital.

Part of this work was completed

whilst the authors were at the Centre for Economic Forecasting. London Business School. Financial support from the ESRC (grant W116251003) at that institution is gratefully acknowledged. We are grateful to an anonymous referee for helpful comments. We have also benefited from the reactions of Robert Chirinko, Avinash Dixit, Victor Hung, Robert Pindyck, Steve Satchell, and seminar participants at London Business School. the London School of Economics, and the Universities of Cambridge, Southampton, and Surrey.

0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1889(95)00906-C

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of Economic Dynamics and Con&o1 20 (1996) 1427-1449

The implications for firm behaviour of credit market imperfections in a static context, resulting from information asymmetries between borrower and lender, are now well understood (see, inter alia, Stiglitz and Weiss, 1981; Gale and Hellwig, 1985; Williamson, 1986; De Meza and Webb, 1987). In that literature it has been possible both to characterise the optimal form of financial contract (which depends on the informational assumptions and monitoring technology) and to discuss the efficiency of the firm’s decision making. However the extension of such models to a dynamic context resists explicit solution and the form of optimal contract in a multi-period context has not been fully analysed.’ In this paper we impose exogenous restrictions on the availability of external finance to a firm, in a manner similar to Greenwald and Stiglitz (1993) and Gross (1995), and investigate the optimal dynamic strategy of such a firm. Our analysis relates also to the many empirical studies which have appealed to static models of credit market imperfections as an explanation of various dynamic aspects of firm behaviour. This has produced some notable empirical findings with both micro- and macro-data. These include the role of cash flow and firm size as a determinant of fixed capital and inventory investment behaviour (beginning with Fazzari et al., 1988), differences in the cyclical behaviour of small and large firms (Gertler and Gilchrist, 1994), the cyclical behaviour of bank and nonbank credit (Gertler and Gilchrist, 1993), and the effect of loan supply on portfolio composition and investment (Kashyap, Stein, and Wilcox, 1993).’ A weakness of most of these studies is that because the assumed theory is static, the dynamic implications of the presence of credit market imperfections is not exploited in model specification. A further aim of this paper is to provide a plausible account of firm behaviour under credit market constraints that offers a more precise dynamic characterisation for empirical study. We start with a simplified model of a firm facing the threat of liquidation. The only decision is about the payment of dividends, and the firm seeks to maximise the expected discounted value of dividends. Like Greenwald and Stiglitz (1993), in the absence of any tractable analysis of the optimal financial contract, we appeal to general arguments about informational asymmetries as a justification for imposing exogenous constraints on the availability of finance for this firm. However, we then apply the theory of optimal control of continuous time

‘Gertler (1992) has solved one dynamic model of debt finance in which borrowers and lenders contract for several periods. This implies a value function for the firm with a shape similar to that obtained in this paper, and a distortion of firm decisions whenever net worth falls below a threshold level. Lucas and McDonald (1992) investigate a dynamic model of bank behaviour under asymmetric information. ‘For a recent survey of this empirical et al. (1994).

literature

and the related

theoretical

analysis

see Bernanke

A. Milne, D. Robertson / Journal of Economic Dynamics and Control 20 (1996) 1427-1449

1429

diffusion processes, rather than discrete time stochastic dynamic programming. This delivers clear results about the implications of financial constraints for firm behaviour. In particular, we demonstrate that the presence of a financial constraint influences firm behaviour well away from the liquidation threshold, that the presence of constraints increases the cost of capital and induces local risk-aversion, and that the degree of departure from unconstrained behaviour is greater the closer the firm is to the liquidation threshold. Our framework also allows straightforward extensions to different aspects of firm decision making, some of which we explore in the paper.3 In Section 3 we endogenise the output decision, showing that when output affects the rate of diffusion of cash balances, output itself depends on the degree of local risk-aversion. We show also that local risk-aversion is a decreasing function of cash held internally, that is, the firm is most risk-averse when it is closest to liquidation. In Section 4 we discuss the consistency of this latter finding with other standard models (including Greenwald and Stiglitz, 1993) which suggest the possibility of risk-loving behaviour when a firm is close to liquidation. Section 5 extends our model to include an investment decision.

2. The problem of cash management We consider the optimal policy for a firm receiving stochastic income, facing the threat of liquidation if cash-in-hand drops below some prescribed level. The firm seeks to maximise shareholder value defined as the value of a stream of dividend payments. With constraints on the availability of external finance the optimal policy is for the firm to make dividend payments when it has sufficient internal resources, and otherwise to build up internal cash. 2. I. Assumptions Let the cash-in-hand period dt the increment

held by the firm be denoted by X. Over a small time to .x is given by the Brownian motion:

d.x=(rx+p--_)dt+odz,

(1)

where r is the return on funds per unit time available to the firm, d is the (flow) dividend rate, and firm’s income over the time interval consists of two components, a deterministic part pdt (where p > 0 so the expected income flow is

3 We also point out parallels with the literature on consumption the discrete time analysis made by Deaton (1991) of optimal liquidity constraints.

and precautionary saving, especially consumption smoothing subject to

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of Economic Dynamics and Control 20 (1996) 1427-1449

positive) and a stochastic element adz where z is a Wiener process. Neither p nor c is here a function of the state x, but in subsequent sections we consider cases where p and rs are functions of further state and control variables. The value of the firm (V) at a time to is the expected value of the discounted flow of future dividend payments. If the payments at time t are d(t) and they are discounted at rate p, then I/ is given by

s a,

V=E

e-Qrd(tO + z)dz.

0

The firm chooses a dividend rule to maximise V. In general V will be a function of the current cash-in-hand x and current time to, V = V(x, to). For a timeinvariant problem such as we have here the time dependence is suppressed: v = V(x). We now make three assumptions that generate a demand to hold cash inside the firm as a hedge against possible liquidation. The first is an equity finance constraint. We assume that dividend payments must be nonnegative. This implies that the firm cannot raise new equity finance. Such a constraint can be justified as the consequence of shareholders and banks not being fully informed about the current state of the firm, so that dividend payments and capital issues take on a signalling role.4 The particular form of the equity finance constraint is not important. What is essential is that there is some restriction on new equity finance so as to motivate the key distinction between internal and external funds explored in this paper. Only the former are effective as a hedge against possible liquidation. This implies that the firm must balance two conflicting objectives: (i) keeping sufficient resources inside the firm to reduce the probability of future liquidation and (ii) paying out resources to satisfy shareholders. We propose this particular specification as a simple way of capturing this trade-off, rather than an entirely realistic model of dividend policy. Our second capital market constraint is an exogenously imposed liquidation trigger. If cash-in-hand x falls below a specified level (which we assume to be zero), the firm is liquidated and its value becomes zero from then on. This is effectively a cost of bankruptcy, in which the costs are the loss of future income.5 Again this constraint on the availability of debt finance is justified as a

4See Myers and Majluf (1984). Equity market imperfections are magnified if the management of the firm has objectives other than the maximisation of shareholder wealth, leading to the problems of agency cost discussed by Jensen and Meckling (1976). 5 We could instead assume (as do Greenwald and Stiglitz) that the crossing of the liquidation trigger invokes a costly monitoring process, which reveals the state of the firm to shareholders and allows it to make a one-off issue of new equity capital.

A. Milne, D. Robertson / Journal

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and Control 20 (1996) 1427-1449

1431

consequence of information asymmetries between the providers of finance and the management of the firm.‘j Finally we assume that the interest rate r paid on cash held within the firm is less than the shareholders’ discount rate, r < p.7 This provides the firm with an incentive to pay out dividends. Were this not true, the firm would maximise value by accumulating cash balances internally such that the threat of liquidation was effectively inoperative. This possibility is removed by assumption. The firm is thus unable to obtain external finance freely, faces an uncertain income and threat of liquidation if cash balances drop too low, yet any cash it receives it would like to pay out as dividends. It is the interaction of these conflicting forces that we study. The firm’s problem

is:

Choose d( ) to maximise V(x) subject to dx = (TX + p - d) dt + 0 dz and d 2 0, and V(x) = 0 if x I 0, and the problem stops. The optimal

dividend

rule will be a function

of cash-in-hand

alone d = d(x).

2.2. The optimal decision rule The Hamilton-Jacobi&Bellman equation for this problem value function V(x) and the optimal policy rule d(x) satisfy’ pI/=max{d+(rx+~-d)V,+ia2T/,,i.

implies

that the

(3)

In general the solution technique for such problems is to perform the maximisation in (3) to give the optimal policy rule. This can then be substituted back

“Instead of invoking liquidation it would be possible to work with a model without liquidation, but in which debt finance (x < 0) was charged an interest rate r’ > p. This penalty interest rate would induce a similar aversion to indebtedness as appears in our model. We choose to work with a liquidation threshold both because this is analytically more convenient and because it seems to be a better reflection of actual debt contracts. ‘The assumption is motivated either by appeal to agency or informational problems in a world of risk-neutral investors, or from the type of pricing formula derived in Rubinstein (1975) to value a stream of uncertain dividends. It also corresponds to the assumption of Deaton (1991) that consumers are impatient. ‘The problem is a discounted continuous time dynamic function bounded below by the constraint that d 2 0, and The assumption of a Wiener stochastic process rules out a unique functional solution which satisfies the Hamilton~ 1982: Whittle, 1983; Harrison, 1983; for discussion of the

stochastic programme, with the return eventual termination with a finite period. any problems of measurability. There is Jacobi-Bellman equation (see Malliaris. technical aspects of such problems).

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to give a differential equation in I/. This can either be solved explicitly, or if, as is often the case for these types of problems, explicit solutions cannot be obtained, numerical methods can be used to characterise the behaviour of I/ (see Merton, 1971; Brennan and Schwartz, 1985; Pindyck, 1980,199l; Dixit, 1991; for examples and discussion). Here, since (3) is linear in the control variable d, the optimal control is bang-bang. From the first-order condition V, = 1 implies d is arbitrary, V, > 1 implies d = 0. Since V, is itself a function of x, this implies the dividend policy will be to switch between paying zero dividends (the minimum possible) for some values of x and to pay the maximum possible elsewhere. The assumption that Y< p implies that a policy of never paying a dividend would result in I/ = 0.9 If x > 0, the firm can do better by making some immediate dividend payout. This implies the firm will optimally choose to pay dividends for some finite value of x. Let x* be the smallest such value. In principle x* could be zero, in which case the optimal policy is to pay out all cash and go into liquidation. This would clearly be an optimal response to a very poor outlook for the future. This case does not, however, generate any further behaviour of interest. We assume that the parameters are such that x* is positive. At x* any increment to cash is paid immediately as a dividend (referred to as barrier control). Values of cash holdings above .x* cannot be attained because of the continuous sample paths of the assumed diffusion process. We sum up the above in the following: Proposition 1. For the problem described above in Eqs. (I) and (2), the optimal dividend rule takes the following form: 3x* subject to d(x) = 0 for o
x I x*

and barrier

control at x = x*,

where

It is the interaction of the liquidation threat and the nonnegativity of dividends that drives this solution. If dividends could be negative, the firm would borrow if it received a bad shock and pay out whenever it had cash. The optimal level of cash held internally would be zero, with dividends set to keep dx = 0.

9A formal proof of this statement can be obtained by considering the finite-horizon problem, where the firm operates for T periods and operates a policy of never paying a dividend (x* infinite). In this case the value of the firm satisfies the inequality:

where the right-hand side of this inequality represents the present discounted value of the terminal payout to shareholders, assuming that there is no liquidation trigger. In the limit, as T + co, this becomes V I 0.

A. Mike, D. Robertson /Journal

of Economic Dynamics and Control 20 (1996) 1427-1449

The liquidation threat causes the dividend value x* and cash to be held internally. 2.3. The characteristic

1433

policy to switch to d = 0 below the

shape of the value function

For the region 0 < x < x* the dividend is zero, so the maximand in the HamiltonJacobi~Bellman Eq. (3) yields the following differential equation for V:

pv =

VJrx

+ p) +

$J2vx,,

or equivalently, +cr2VXX+ V,(rx + 11)- pV = 0,

0 < x < x*.

(4)

Eq. (4) has no general analytical solution, but we can solve analytically in particular cases and characterise the form of the solution elsewhere.’ OThere are three boundary conditions: (i) Continuity of V at the liquidation threshold x = 0. This boundary condition obtains because the sample paths for x across the liquidation boundary are continuous (this condition is standard; see, for example, Whittle, 1983, Ch. 37, Thm. 4.1). (ii) V, = 1 at x*, i.e., continuity of‘ V, at the boundary x*, using that the value function,for x 2 x* is given by V(x) = V(x*) + x - x*. This condition emerges because control is instantaneous at the x* boundary (Dixit, 1991, contrasts the boundary conditions which emerge under the alternatives of impulse and instantaneous control). This boundary condition will apply whether or not x* is chosen optimally since diffusion paths across the boundary are continuous. (iii) V,, = 0 at x*, i.e., continuity of V,, at the boundary x*. This is a consequence of x* being chosen optimally. Otherwise the value function could be increased at x* by a small shift of X* in the direction that V, < 1.” These boundary P = 0 (appropriate

“‘Numerical

techniques

conditions determine the solution to (4). For the specific case if internal funds are held in zero interest accounts or as cash

can also be used to investigate

the solution;

see Appendix

1for an example.

‘lcondition (iii) is conventionally referred to as smooth pasting (though the origin of this term relates to problems of impulse, not instantaneous control; see Dixit, 1991). The combination of conditions (ii) and (iii) is what Dumas (1991) describes as supercontact.

1434

A. Milne, D. Robertson / Jouml

itself) we obtain:’

of Economic Dynamics and Control 20 (1996) 1427-1449

2

~02vx, + pv, - pv = 0,

0
<

x*,

(5)

which has the solution

V(x) = A exp(m,x)

0
+ Bexp(m2x),


(6)

where ml and m2 are the roots of +a2m2 + pm - p = 0,

(7)

i.e., m1,2 = { - /A + J(p’ + 2p02))/02. Th e values of A, B, and x* are then determined by the three boundary conditions.13 The solution is graphed in Fig. 1 for a particular choice of parameter values. The function slopes upwards from the liquidation point (zero) with gradient declining, meeting the 45” line at x*. When Y # 0, these same boundary conditions determine the solution, though it cannot be obtained explicitly. We can still derive key qualitative features of the solution over the range 0 < x < x*, from consideration of the stochastic co-state equation:r4

3g”K,, +

Vx,(rx + /l) -

pv, = 0,

0 < x <

x*.

(8)

This implies that limXrX* V,,, = p/is” > 0, where the limit is understood as x approaches x* from below (because V,,, is discontinuous at x*). From this we can derive the following: Proposition 2. For all values of x, 0 < x < x *, the value function satisjies (4) and (8), the boundary conditions (i), (ii), and (iii), and If,.,, > 0, V,, < 0, V, 2 1. ProoJ:

See Appendix

2, Theorem

1, and Corollary

1.

“It is also possible to obtain an explicit solution for the case where p = 0 and r < 0; in this case, although the problem is undiscounted, liquidation eventually takes place. The problem is thus one of transient dynamic programming and the value function exists and has a finite value (see Whittle, 1983, for fuller discussion). 13The solution is x* = (m, - mJ

’ 210g(m,/ml)

and A = - B = (ml exp(mrw*) - mZ exp(m,x*))~

‘.

14First-order conditions for the optimality of the control variables imply that small changes to control variables have only second-order effects on the value function (an envelope property). Hence the costate equation is the same even when p and CJare functions of controls which are themselves a function of x.

A. Milne. D. Robertson 1 Journal of Economic Dynamics and Control .?(I (1996) 3427-1449

O 0.00

0.02

0.04

0.06 Cash

0.00

0.10

Holding

Fig. I. The value function

0.14

0.12

0.16

1435

0.10

of Firm of the firm.

Based on the following underlying parameters for the diffusion process and the optimisation problem: p = 0.14, I? = 0.005, p = 0.05, r = 0. The liquidation threshold is at zero. These parameters imply a value of Y* of 0.17872. If cash holdings are regarded as being measured in units of $100,000 and the time period as annual, then Eq. (1) describes a firm expecting to receive an income of $14,000 per annum (think of this as after expenses profit flow), with a standard deviation of $7,070. so that the firm expects to lose money approximately one year in forty. The firm’s shareholders are discounting the future at 5% per annum. With these parameters the firm will hold cash balances of about $18,000 before paying a dividend.

Remurk.

Proposition 2 implies the uniqueness of x*. Hence for x > x* we have V’(x) = V(x*) + x - x* (since control is then applied to bring the firm back to the dividend barrier by paying an instantaneous dividend of x - x*). Proposition 2 shows the general solution has the same shape as obtained when r = 0. Diagrams of this kind are characteristic of models with constraints on the availability of capital; similar diagrams are presented by Gertler (1992), Greenwald and Stiglitz (1993), and Gross (1995). The internal cost of capital (V,) is greater than the external cost whenever x < x*, and the premium on internal funds increases as the firm approaches the liquidation threshold.” This distortion to the cost of capital results in modification to all dimensions of firm decision making that involve intertemporal comparison of costs and benefits. “As in Deaton (1991) we also find that the presence away from the point at which the constraint binds.

of a liquidity

constraint

affects behaviour

well

1436

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3. The model with an endogenous output decision

The cash management model described above allows investigation of the problems associated with calculating solutions for stochastic optimal control models, and has already shown certain aspects of firm behaviour in these situations. To understand more fully the behaviour of optimising firms in stochastic environments facing the threat of liquidation we extend the model within the same overall framework. In this section we describe a simple extension to the model to include the optimal choice of output. This allows us to capture real effects from the financial imperfections. The firm now chooses the level of output and a dividend policy, and both expected cash flow and its variance are modeled as increasing functions of the level of output. We shall adopt a specific quadratic functional form (though similar conclusions could be drawn using other specifications) in which the cash evolution equation (1) becomes dx = [rx + ~(1 - (1 - y/y*)‘)

- d] dt + ay/y* dz.

(9)

Thus expected revenues are a convex quadratic function of the level of output y (normalized by y*, which is then the income-maximising choice of output for a nonstochastic income stream of (1 - (1 - y/~*)~)). The uncertainty of income now increases linearly with the level of output. This uncertainty can be reduced by choosing a lower level of output but this results in a loss of expected revenue. Cash management emerges as the special case in which y is fixed at the level y*. If output is reduced to zero, then both expected cash flow and its variance fall to zero. The firm’s problem is now to choose a level of output y and a dividend policy d to maximise the objective function V(x) as in (2). The HamiltonJacobi-Bellman equation for this problem implies that the value function satisfies the following differential equation at the optimal choices of d and y: pV = ryy {d +

[rx + ~(1

- (1 - y/y*)“) - d] V, + ia’(y/y*)”

if,,}.

(10)

Applying the results of Section 2 we note that (10) is linear in the control d, so the optimal dividend rule will still be bang-bang. We investigate the choice of output y in the region where no dividends are paid, d = 0. Eq. (10) then implies that y is chosen optimally as the solution to

Pv = max{Crx+ Al - (1 Y

Y/Y*)~)I

v,+ +a2(yly*YK,},

(11)

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in which the first-order condition for the maximisation condition for a maximum holds) implies

1431

over JI (the second-order

If the firm’s income were nonstochastic (a = 0) or if external finance was unconstrained, then the optimal choice of output is J = y*. Using the results on the boundary conditions derived the simple cash management case we see that at the upper boundary (where dividend payments start) V,, = 0, so that the optimal choice of output implied by Eq. (11) is also J*, the same as in the deterministic case. Let W = - V,,/ V, which we interpret as a measure of (local) risk aversion. As a consequence of Proposition 2 we know that W > 0 for .Y < .x*. We can then write the optimal choice of output as J’ = y*/i1

+ (C2/2/.4W}

< L‘*,

(12)

that is, the output level is reduced below the optimal level of the deterministic case, except at the boundary x*. Note that W depends on the level of cashin-hand .Y,so that the reduction in output is a function of X. Moreover we have the following: Proposition

3.

Over the entire region 0 I x I x*, W, < 0, so that W is declining

as x increases. Proof:

See Appendix

2, Theorem

2.

Thus the financial constraints on the firm generate a value function where the degree of local risk-aversion increases as the firm moves closer to liquidation. The firm’s value function exhibits prudence in the sense of Kimball (1990)” This in turn implies that the level of output is an increasing function of cash holdings.

lbDeaton makes a similar point about the effect of liquidity constraints households to risk: ‘The precautionary motive for saving is strengthened liquidity constraints’ (Deaton, 1992, p. 197).

on the attitude by the existence

of of

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A. Milne. D. Robertson / Jouml

of Economic Dynamics and Con@01 20 (1996) 1427-1449

Finally note that the dividend threshold x*, which emerges when cash holdings evolve according to (9), is always less than the dividend threshold of the cash management problem discussed in Section 2 (which corresponds to y being maintained at the fixed level y*). This can be established by a simple argument from contradiction. First amend the notation so that V’ and x,* represent the value function and the dividend payment threshold under cash management (i.e., when output is restricted by y = y*) and VY and xz represent those when output is set according to (12). The opportunity to alter the level of output must increase the value function for all levels of cash holding x > 0, so V’ 2 I/’ for all x > 0. Moreover, substitution of the boundary conditions at x* into the Hamilton-Jacobi-Bellman equation shows the value functions at the dividend payment thresholds satisfy Vy(x:) = I”($) = p/p. Finally the uniqueness of x* implies that the value function for x > x* is given by V(x*) + x - x*. Suppose now that xr c xz. Evaluating the two value functions at x$ we have V’(xf) = V’(x,*) + (xz - x,*) = p/p + (x,* - x,*) > p/p = VY(xg) which contradicts I/’ 2 I/’ for all x > 0. Hence x*c 2 xF.17

4. The threat of liquidation and attitude towards risk In the last section we found that the threat of liquidation induces prudence (local risk-aversion increases as internal cash holdings fall) on the part of the firm. This is in apparent conflict with other studies (including Greenwald and Stiglitz, 1993) who find that firms close to bankruptcy may behave in a riskloving rather than a risk-averse manner. The different conclusions about behaviour towards risk rest on different assumptions about the stochastic disturbance to the cash flows the firm faces. Other studies have generally applied discrete time analysis, in which the shock (which corresponds to our adz) may alter firm cash balances by a discrete (noninfinitesimal) amount in each time period. In such cases an increase in the variance of the shock may increase the value of the firm, due to a convexity in the value function caused by the liquidation possibility. As a specific example, consider in the discrete time framework some action on the part of the firm that would increase the variance of a (symmetric) stochastic disturbance to cash holdings that occurs each period. Suppose also that the mean of the disturbance is on the threshold of liquidation (this ensures, since the distribution is symmetric, that the increase in variance has no effect on the probability of liquidation taking place). The widening of the left-hand tail of the

“In the special i-=0 explicit is possible. In this case case where solution XT = (a*/2~)ln(l + 2~*/02p) which is indeed less than the corresponding cash management threshold. See Milne (1995) for details of this derivation.

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1439

distribution (which lies entirely beyond the liquidation threshold) causes no cost to the firm. However, the increase in the right-hand tail is of benefit to the firm. Thus, close to liquidation the firm prefers a higher variance of the disturbance and, in situations where it has control over the variance, behaves in a risk-loving manner. In contrast, we have modelled the stochastic disturbance as a Brownian motion, so that all sample paths are continuous. Here firms diffuse gradually into liquidation rather than being pushed suddenly into liquidation by a discrete shock. Any increase in the variance of the disturbance to the cash balances of the firm is therefore costly, because it is always evaluated in terms of the local curvature of the value function (which exhibits local risk-aversion induced by the threat of liquidation); and the cost to the firm of an increase in the variance of the stochastic disturbance always rises when internal cash falls (because of the local prudence of the value function). Risk-loving behaviour could still arise in our set-up in response to a noncontinuous stochastic disturbance.‘* Consider an amendment to our model to include the threat of a natural hazard, such as a fire, that would result in a discrete (negative) shock to the cash balances of the firm.‘” The financially unconstrained firm operating in a world of perfect capital markets will be completely risk-neutral with regard to this threat and be uninterested in insurance.‘O The financially constrained firm, which we have studied operating under the threat of liquidation, will normally be averse to this risk (because the curvature of the value function induces an aversion to variation in cash balances). It will thus be interested in insurance against the occurrence of such natural hazards. However, when close to the liquidation threshold, the nonconvexity of the value function at that threshold means that the firm no longer worries about such risks and is thus not interested in insurance. It may therefore act in a risk-loving way towards this discrete stochastic disturbance, even though the value function is locally risk-averse. There is thus no real conflict between our findings that the value function is locally risk-averse and the threat of liquidation induces prudence on the part of the firm, and the examples of risk-loving behaviour provided by other studies. The attitude of the firm towards discrete shocks is affected both by local risk-aversion (over those regions of the value function where the shock does not result in liquidation) und by the nonconvexity at the liquidation threshold. For

‘*The standard technique for analysing such a noncontinuous disturbance in a continuous time framework is to treat them as a ‘jump process’ in which the probability of a change taking place over a small period dt is proportional to dr (the period until a jump takes place having a Poisson distributton). ‘“Assuming “‘This

that the risk is sufficiently

small, Proposition

wdl be the case even if shareholders

are risk-averse,

2 still apphes. assuming

the risk of fire is diversifahle.

1440 A. Milne, D. Robertson /Journal of Economic Dynamics and Control 20 (1996) 1427-1449

discrete shocks where a substantial part of the probability mass lies beyond the liquidation threshold, the firm prefers higher to lower variance even though the value function is locally risk-averse for the range of cash balances where the firm operates. General statements about risk-aversion, independent of the distribution of the shocks, can be made only if the value function exhibits global risk-aversion, i.e., local risk-aversion for all values of cash, - co < x < co. This is not the case for our model: we have demonstrated local risk-aversion only for the range of values 0 I x I x*, and so statements about the attitude of the firm towards risk are dependent on the distribution of the relevant stochastic disturbance.

5. The model with an investment decision

In this section we describe an extension of the basic model to include an additional state variable other than cash holdings. Output is a concave function of a quasi-fixed factor of production which is costly to adjust. This we label k and think of as capital stock, though it could represent any factor of production with convex costs of adjustment. We assume costs to adjusting this productive input are quadratic in the flow of gross investment i. There are thus two state variables at time t,,: the cash-in-hand of the firm x and the inherited capital stock k. The controls are the dividend paid, d, and the level of investment, i, Y=

y(k),

(13)

dk = (i - Gk)dt, dx =

(YX +

(14)

py(k) - pii - i13i2/k - d)dt + opydz,

(15)

where p is the price of output and pi the price of investment. The evolution of the capital stock is governed by Eq. (14), increased by investment i and depreciating at a rate 6. This equation is deterministic, but could be made stochastic. Cash flow is given by the diffusion process (15). Again the firm seeks to maximise the present value of a stream of dividend payments as in (2). In a nonstochastic world the firm would choose the capital level such that the cost of incrementing the capital stock by one more unit, pi, is just matched by the increase in the value of the firm arising from that one unit, that is, if the value of the firm is V(x, k), then optimality requires V, = pi. For the stochastic case we have the Hamilton-Jacobi-Bellman equation, pl/ = y:X {d + v,[rX + py - pii - i&“/k + ~a2y2pZVx, + V,(i - 6k)).

- d]

(16)

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1441

Here we now have a region in (x, k) space in which no dividends are paid. Optimal dividend policy is again barrier control at an upper boundary x*(k), where the dividend payment threshold is now a function of the state variable k. The first-order condition for i in this region is I’, - {pi + Ui/k} V, = 0,

(17)

or i/k = Hi ‘pi{4 - 1)

where

4 = V,J(VXpi),

which expresses investment per unit of capital as a function of the ratio of the marginal increment to firm value from undertaking the investment to the price of the investment, as in the nonstochastic case, but here also deflated by the term V,, that is, the marginal valuation of cash balances to the firm. Substitution yields a nonlinear partial differential equation that cannot be solved analytically in this region21 The boundary conditions at x* are as discussed earlier, so that V, = 1 and V,, = 0. Thus on this boundary the formula for investment reduces to the standard q theory, with q = Vk/pi. It is shown in Appendix 2 (Theorem 1, Corollary 2) that Proposition 2 continues to apply to the q model of investment, as long as the firm has a level of capital stock which exceeds (or is only slightly less than) the level of capital stock, %, which results in zero net investment on the dividend paying frontier (i.e., iz is that level of capital which satisfies q(^k,x*) = 1 + es/pi). Proposition 2 in turn implies a larger reduction in investment the closer the firm is to the liquidation barrier. Once again internal cash resources have real effects on the firm’s decisions, but here these arise because the internal cost of capital exceeds the external cost of capital (V, > l), and this induces a greater discounting of the future benefits of investment.

6. Conclusions We analyse a model of firm behaviour in which internal funds follow a diffusion process, and the firm is subject to liquidation (and loss of all future revenue) if internal funds fall below an exogenous threshold. When the firm’s objective is maximisation of the expected present value of future dividends, it must balance the conflicting objectives of maintaining internal cash (as a barrier against

*‘Numerical solution to this partial differential discussed in Milne and Robertson (1995).

equation

and the associated

boundary

x*(k) is

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A. Mine, D. Robertson / Journal of Economic Dynamics and Control 20 (1996) 1427-1449

possible liquidation) and paying cash out to satisfy shareholders. The optimal policy is bang-bang control, with dividend payment only if internal funds are above some threshold x*. The value function satisfies a second-order nonlinear differential equation, with boundary conditions determined by the liquidation and dividend payment thresholds. We provide an explicit solution for one simple example and prove that the value function takes a characteristic shape, with a first derivative I’, (the cost of internal funds), which is greater than unity for x < x* and monotonically decreases from x = 0 until reaches the value V, = 1 at x = x*, and a second derivative V,,, which is negative and monotonically increases over the range 0 < x < x* until it reaches the boundary value V,, = 0 at x*. These results (which generalise to situations where there are other control variables, and with certain technical restrictions to additional state variables) imply that firm behaviour, subject to the threat of liquidation, becomes both locally more risk-averse and more impatient as the level of cash-in-hand falls. We provide two specific examples of how this can affect firm behaviour. If the level of output affects the variance of returns, exposing the firm to additional risk as output is increased, then shortage of internal funds increases riskaversion and leads to a reduction of output. With costs of adjusting the capital stock an amended q relationship arises, in which a shortage of internal funds effectively increases the firm’s rate of discount, lowers the value of q for any given level of capital stock, and reduces the level of investment. This framework provides an alternative to the static analysis extensively used in empirical work. The analysis predicts that a variety of firm decisions (including output and capital investment) depend upon the level of internal liquid assets. Moreover this dependency is likely to be increasingly nonlinear as the internal resources of the firm are reduced. The nonparametric estimation reported in Gross (1995) provides empirical support for this prediction. A second empirical prediction then follows. This is that, as in most static models of capital market imperfections, when financing constraints bind, there is a correlation between cash flow and real expenditure decisions. But, in contrast to the static analysis, this correlation emerges during endogenously determined time periods and not as a function of specific firm characteristics. Our analysis thus suggests that criteria which vary over time may be more successful than fixed criteria in determining periods in which cash flows are correlated with real decisions. Support for this second prediction (in the context of a panel study of inventory investment) is reported in Milne (1995). The nature of dividend behaviour implied by our analysis does not, however, correspond to observed behaviour, at least for publicly quoted companies who typically maintain a smooth ratio of dividends to income. Such stability can be explained by positing a signalling role for dividends providing an incentive to smooth dividends. This could be introduced into our model (in the manner of Gross, 1995) by altering the objective function to be a concave function of the dividend

A. Milne, D. Robertson / Journal of Economic Dynamics and Control XI (1996) 1427-1449

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paid, or more properly, by an explicit agency model of equity finance. The robustness of our conclusions to such amendments of the basic model merits further investigation.

Appendix 1 Numerical

solution jtir nonzero Y

We are unable to provide explicit solutions to the Hamilton-Jacobi-Bellman equations in the general case of nonzero Y.In this appendix we present numerical solutions to the differential equation for such cases. The solutions for V(x) show that, as r varies between zero and p, the general shape of V(x) does not change that much. The numerical solutions were computed by the program Mathematic’a. The naure of the boundary conditions in the HJB equation (4) is such that we know the values of V’(x) and V”(x) (and hence V(x)) at x* but don’t know .Y*, it is identified by the further condition V(0) = 0. We therefore use a repeated bisection search to discover x* for each r. That is, nominate x*, calculate the numerical solution V(x), and check the sign of V(0). If it is positive, raise x* and vice versa. Using this method we can identify the numerical solution to any desired degree of accuracy. Fig. 2 graphs for each r the solution from zero to the implied x* for that r. 3.5r r=0.04

32.5: 2-

:;I’, 0.05 Fig. 2. The value function

_ 0.1

0.15

0.2

for r = O.OO,O.Ol. 0.02, 0.03.0.04.

Appendix 2 Throughout value function

this appendix we shall assume such smoothness properties of the V( . ) as required. In particular, derivatives of sufficient order exist

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A. Milne, D. Robertson 1 Journal of Economic Dynamics and Control 20 (I 996) 1427-1449

and are continuous except possibly at points where control regimes shift. Note that in the interior of control regions any order differentiability of the value function will follow by repeated differentiation of Eq. (8), so long as the forcing term has derivatives of the required order and the controls (other than dividend payment) are unconstrained. Derivatives (and partials) are denoted by subscripts throughout. We first demonstrate sufficient conditions for Proposition 2. These are derived in a general situation in which there are further state variables, in addition to cash, that may or may not be controlled. We assume these other states are nonstochastic, although the condition under which Proposition 2 holds with stochastic states would be similar to that derived below. Suppose the state variables are cash x and a vector of other states s. The value function is then I/ = V(x,s). Write the state equations as ds = r/dt, dx = (p(x,s) - d)dt + a(x,s)dz, where q may be controlled, but we assume it does not depend on x. The firm’s problem is the maximisation of the expected dividend stream d (Eq. (2) of the text). Theorem 1.

Proposition

2 holds if

NB This condition is sufficient, but not necessary. Necessary conditions will be problem-specific, and in particular will depend on sgn( V,,) which cannot generally be allocated a priori. The optimal control of the dividend process is bang-bang. no-dividend-paying region we have the state and costate equations

Proof

3fJ2v,,+ pLI/, + qv,- pv = 0,

0 < x <

In the

x*,

with boundary conditions as in the text. We begin with the following: Lemma 1.

limx+,* V,,, > 0.

Proof This follows from second-order conditions for the optimal choice of the dividend payment threshold x*.

A. Milne, D. Robertson 1 Journal of Economic Dynamics and Control 20 II#(i)

1427-1449

1445

We now show by contradiction that, when (A.l) holds, the same inequality applies throughout the range 0 < x < x*. We have at x*: V,.,=O

and

V,=l.

(Note that these conditions imply:

dejne

3 region ‘31= (x* - 6,x*),

x*.) Continuity

6 > 0

st.

of V, and V,, and the lemma

V,,, > 0 in ‘Jz

5 V,, increasing

with x in 93,

so V._(x* - 6,s) < VXX(x*,s) = 0 in 6%

=3 V, decreasing

with x,

so VJx* - cS,s) > VX(x*,s) = 1 in $3.

Note that x* will in general depend upon the state vector s, so that the region $31will be s-dependent. We now show that 5%contains the interval (0,x”). Suppose not, let 2 = sup{x: E (0, x*) s.t. V,,, I 0) (the largest value of x not in ‘93). At .? we know V, > 1 and V,., < 0 because V,,, > 0 in (X”,.X*)and continuity of the derivatives. Substitute in the costate equation

Now for x* to exist we have (p - pX) > 0. Otherwise cash is valued more highly in the firm than paid as dividends, and the optimal control policy is nonpayment of dividends and cash accumulates in the firm. Given pV,, + vV,, 2 0, then

which contradicts the definition of x”. So no such 1 exists, thus (0,x*) c ‘3. The rest of Proposition 2 then follows immediately. Corollary

1.

Proposition

2 holds for the cash management

problem of Section 2.

Proqf We have g(x) = rx + p > 0, pX = r < p, b’,, = 0, as there are no other states. Hence condition (A.l) is satisfied for the optimal cash management problem of Section 2. Corollary 2. Proposition 2 holds for the optimal investment problem of Section 4. at least,fbr a range of values of the capital stock described below.

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A. Milne, D. Robertson /Journal

of Economic Dynamics and Control 20 (1996) 1427-1449

Proof: The states are cash x and capital k. We denote the expected increase per unit time of x and k by p and n. These are given by p = rx + py(k) - pii - i&*/k

and

rate of

q = (i - ok).

x* is defined by the relation VX(x*, k) = 1 and I/x.(x*, k) = 0. These implicitly define a dividend-payout boundary which is a function of the state k, x* = x*(k). At x*, V&X*, k) = 0 (since a small perturbation of x has no effect on future dividend payments). Thus condition (A.l) of Theorem 1 holds on x*. Lemma 2. There exists a value of k, k, at which n = 0 on x* (investment is at replacement level on the dividend-paying threshold) and where n < 0 for all k > k and x = x*(k). Proof From Section 5 investment is given by i/k = g-lpi{ Vk/‘(Vxpi) - l}, On x*, V, = 1 and this becomes i/k = O- ‘pi{ V,/pi - l}. For large k, V, is small (a consequence of declining marginal product of capital) and i/k -=c6 (implying n < 0). Also, at k = 0, y 2 0. Moreover, V, on x* falls as k increases, since the effect of any change in k along x* on V, can be written: dk = V&dk, as the term in VkXequals 0. Hence there is a unique value of k(k) for which y = 0 on x* and for which q < 0 for greater values of k. We interpret k as the stable value of k on the x* boundary (it is not a steady state since the steady state of this model corresponds to a probability distribution rather than a single value of k). Lemma 3. For x < x*, V,, > 0. This can be shown by considering a small + ve perturbation to the stock of cash. For a subset of the space of sample paths this perturbation will be sufhcient to prevent future liquidation, increasing the (marginal) value of capital. For all other sample paths the marginal value of capital is unaffected. Hence V,, > 0. We now show that Proposition Proof:

The proof proceeds

2 applies

by establishing

for all values of k 2 k. (A.l) for all values

k 2 k. As shown already, (A.l) holds on x *. Moreover

of x < x*(k),

on x* for k 2 k, n c 0 and

A. Milne, D. Robertson f Journal ofEconomic

Dynamics and Control 20 (1996) 1427-1449

1447

,U > 0. The proof now applies a similar contradiction argument to the proof of Proposition 2 itself. Continuity of V,, V,,, V,, and V,, imply that for each k 2 ^k: 3 region % = (x* - 6, x*),

(5 > 0

s.t.

k/ < 0,

/1 > 0.

and hence (A.l) applies

V,, decreasing

i increasing

with x in %,

so (by Lemma

in ‘9

3) q_X> 0 in $3

with x in ‘37.

We now show that ‘9I contains the interval (0,x”). Suppose not, let 5Z= sup{x E (0,x*) s.t. q = 0) (the largest value of x not in 3). At .? we know p > 0 (since the reduction of q is reflected in an increase in p for x < x*). Hence (A.l) still applies at 3, V,, < 0 and q*_> 0 and q declines with x in (2, x*). But this contradicts our assumption that ye= 0 at .Z. So no such 1 exists. thus (0, x*) c 5%. We have therefore that, for all k 2 ^k, q and hence investment increases with the cash holdings of the firm. Theorem

2.

For the model of Section 3, W = - V,, / V, is monotone decreasing.

Proof We allow for arbitrary control variables other than cash .Y. Let W = - V,,/V,. This implies

variables

affecting p and r~ but no state

Here with no further states (11= 0) and for the model of Section 3 (so that CL,= r) rearrangement of the state and costate equations (as given in the proof of Theorem 1) gives

w = ($7’)-‘{ - pv/v, + p),

(A.3)

V,,,l V,, = W-

(‘4.4)

Summing W,/W

’ {(P - 4 Kl V,, - P}.

(A.3) and (A.4) and substituting = ($17~)~‘{(p

- r)vJV,,

in (A.2) we get

- pV/V,}

< 0.

(A.5)

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A. Milne. D. Robertson / Journal of Economic Dynamics and Control 20 (1996) 1427-1449

The last inequality is a consequence of Theorem 1, which implies that V,/ V,, < 0 and V/V, > 0. The result follows on noting that W > 0. Theorem 2 can be extended to the case where there is a vector of state variables. In this case (A.3)-(AS) become

w = Wr’( K,,/~,

=

- PVIV, + P + VlYdK}, (P)-‘{(P - r)I/,lI/,,

W,lW = (+a’)-‘{(~

-P

-

(A.37 (A.4’)

rll/,xll/,x),

- r)I/,/C/,, - PVIV, + rCv,lv,

- v,JKJ>,

(A.5’)

where the other state variables obey ds = r]dt, with v](possibly) a controlled vector, where if controlled it may be set optimally (this ensures envelope properties apply). Then W declines monotonically if ye[ V,/ V, - V,,. V,,] < 0. This holds under the conditions of Theorem 1 if F’,and V,, are both positive in regions where the state variables s are being decumulated. Note V, > 0 if the value of the firm is increased by additions to the states, i.e., they are valuable. Note V,, > 0 if at higher values of the other state variables the insurance value of cash held internally rises.

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