Liquidity Risk and Distressed Equity

Mamdouh Medhata

Department of Finance and FRIC Center for Financial Frictions

Copenhagen Business School

Job Market Paper for the 2015 Academic Job Market

Preliminary draft—June 4, 2014

Abstract

I show theoretically and empirically that cash holdings used to offset liquidity risk can help rationalizethe anomalous returns of distressed equity. In my model, levered firms with financing constraints candefault because of liquidity or solvency, but firms seek to manage their cash to avoid the former. Aninsolvent but liquid firm has a large fraction of its assets in cash, which makes its equity beta low and helpsrationalize low expected returns. Using data on rated US firms between 1970 and 2013, I find empiricalevidence consistent with my theoretical predictions: i) the average insolvent firm holds cash that meets orexceeds its current liabilities; ii) firm-specific betas and risk-adjusted returns decline both as firms becomeless solvent and as cash levels decline; and iii) a portfolio long firm with high cash and short firms with lowcash has increasing returns for less solvent firms, while a portfolio long firms with high solvency and shortfirms with low solvency has increasing returns for firms with less cash. My results suggest that there is nodistress anomaly for insolvent but liquid firms.

Keywords: Distress risk, equity valuation, liquidity risk, cash holdings

JEL Classification: G12, G32, G33, G35

aI thank David Lando, Darrell Duffie, Lasse Heje Pedersen, Kay Giesecke, Sebastian Gryglewicz, Kristian Miltersen, and seminarparticipants at Copenhagen Business School, Lund University, and Stockholm School of Economics for helpful comments and discussions.I also thank Moody’s Investor’s Service for providing data on ratings and defaults. Financial support from the FRIC Center for FinancialFrictions (grant no. DNRF102) is gratefully acknowledged.

Correspondence: Copenhagen Business School, Solbjerg Plads 3A, 2000 Frederiksberg, Denmark. Telephone: +45 38 15 35 25. Email:mme.fi@cbs.dk.

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Introduction

Recent studies of distressed equity rely on capital struc-ture theory to rationalize why high default risk predictslow future returns, i.e. the “distress anomaly” docu-mented by Dichev (1998), Griffen and Lemmon (2002),Campbell, Hilscher, and Szilagyi (2008), and others.Garlappi and Yan (2011) show that potential share-holder recovery upon resolution of distress increases thevalue of the default option and lowers expected returnsnear the default boundary. Opp (2013) argues that fasterlearning about firm solvency in aggregate downturns hasthe same effect, while McQuade (2013) argues that dis-tressed equity hedges against persistent volatility riskand therefore commands lower expected returns.

In this paper, I argue that cash holdings used to offsetliquidity risk can help rationalize the anomalous returnsof distressed equity. In contrast to extant rationales ofthe anomaly, my model features levered firms with fi-nancing constraints that can default because of liquidityor solvency, but firms seek to manage their cash to avoidthe former. The model predicts that an insolvent but liq-uid firm has a large fraction of its assets in cash, whichmakes its equity beta low and helps rationalize low ex-pected returns. Using data on rated US firms between1970 and 2013, I find empirical evidence consistent withmy theoretical predictions.

The model features a representative levered firm thatgenerates uncertain earnings. The firm has commonequity and coupon-bearing debt in its capital structure.Because of capital market frictions, the firm has no ac-cess to external financing, and can, in particular, not is-sue additional equity to cover coupon payments. De-fault is costly and can occur because of liquidity or sol-vency. To reduce the risk of the former, the firm retainssome earnings as precautionary cash.1 How much ofearnings to retain and how much to distribute to share-holders is determined by the following trade-off: An ex-tra dollar in dividends increases equity value conditionalon survival, while an extra dollar in cash holdings in-

1There is ample empirical evidence suggesting that the precau-tionary motive is the most important determinant of corporate cashholdings and their secular growth since the 1980s—see, for instance,Opler, Pinkowitz, Stulz, and Williamson (1999), Bates, Kahle, andStultz (2009), Acharya, Davydenko, and Strebulaev (2012), and ref-erences therein.

creases the probability of survival. Equity holders havethe option to strategically declare insolvency and with-draw the firm’s cash holdings when its asset value fallssufficiently low relative to its liabilities.

It is optimal for the firm to aim at a target level ofcash that eliminates its liquidity risk (Proposition 1) andonly distribute dividends when its cash is at or above thetarget level (Proposition 2). The target cash level de-creases as the firm becomes less solvent because a lesssolvent firm can only survive smaller earnings-shortfallsand thus demands less cash. In equilibrium, cash isat its target level (i.e. the firm is liquid) and equity isthe sum of current cash holdings, expected excess earn-ings, and the option to declare insolvency (Proposition3). Equity holders optimally declare the firm insolventand withdraw its cash holdings when its asset value con-sists mostly of cash. In line with the model, I find formy sample of rated US firms that average cash levelsdecline as firms become less solvent, but that the aver-age insolvent firm still holds cash that meets or exceedsits current liabilities.

The paper’s main contribution is to shed new lighton the distress anomaly by characterizing the cross-sectional relation between cash levels and equity returnsas solvency varies.

The equilibrium expected return on the firm’s equityis determined by its conditional equity beta, i.e. equity’ssensitivity to systematic earnings risk, which dependson the target cash level and the probability of insolvency(Proposition 4). The model predicts that the equity betaof the liquid firm is hump-shaped in its probability ofinsolvency (Corollary 4.1). The intuition is as follows.When the liquid firm is solvent, its assets consist mostlyof expected earnings and are therefore sensitive to sys-tematic risk, which implies a high beta that increasesin the probability of insolvency. However, as the liquidfirm nears insolvency, its cash becomes an increasinglylarger fraction of its assets, and, because cash is insensi-tive to systematic risk, this implies a low equity beta thatdecreases in the probability of insolvency. This providesa theoretical rationalization of the distress anomaly forinsolvent but liquid firms.

The model produces the following testable predic-tions. Because the target cash level declines with theprobability of insolvency, the model predicts that equitybetas are not only hump-shaped in the probability of in-

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solvency, but also in the target cash level. Furthermore,because of these hump-shaped relationships, a portfo-lio long firms with high cash and short firms with lowcash (HCmLC) will have increasing expected returns asfirms become less solvent, and, conversely, that a port-folio long firms with high solvency and short firms withlow solvency (HSmLS) will have increasing expectedreturns as cash levels decline.

Consistent with the models predictions I find for mysample of rated US firms that i) firm-specific equity be-tas and risk-adjusted returns decline as firms becomeless solvent and as cash levels decline, ii) the HCmLCportfolio strategy has increasing risk-adjusted returns asfirms become less solvent, and ii) the HSmLS portfo-lio strategy has increasing risk-adjusted returns as cashlevels decrease.

In sum, my results suggest that cash holdings used tooffset liquidity risk can help rationalize the anomalousreturns of distressed equity, and show that there is nodistress anomaly for insolvent but liquid firms.

Related literature

The distress risk anomaly has been investigated empir-ically by Dichev (1998), Griffen and Lemmon (2002),Vassalou and Xing (2004), Campbell et al. (2008), andthe references therein.

The hump-shaped relationship between equity betaand probability of default was first directly documentedGarlappi, Shu, and Yan (2008). The models of Garlappiand Yan (2011), Opp (2013), and McQuade (2013) ra-tionalize the hump-shaped beta by means of shareholderrecovery, shareholder learning, and persistent volatil-ity risk. I complement these theories by arguing thatcash holdings used to offset liquidity risk reduce equitybetas for liquid firms nearing insolvency, which yieldsthe novel testable predictions for which I find empiricalsupport.

The equity valuation model presented in this paperbuilds on the classical time-homogenous framework ofe.g. Leland (1994), but augments it with a target cashlevel similar to Gryglewicz (2011), who studies the op-timal capital structure of a firm facing liquidity and sol-vency concerns. A related framework is considered byDavydenko (2013), who studies whether corporate de-faults are driven by insolvency or illiquidity.

Technically, Gryglewicz (2011) specifies thefirm’s cumulated earnings as an arithmetic Brown-ian motion—following the classical pure-liquiditymodels of e.g. Jeanblanc-Picque and Shiryaev (1995)and Radner and Shepp (1996)—but allows the driftcomponent of this process to be randomized over aknown two-point distribution. The added uncertaintyabout expected earnings implies a state-dependentassets value, which—contrary to the pure-liquiditymodels—allows for strategic insolvency.

My model deviates from the framework of Gry-glewicz (2011) by specifying cumulated earnings as ageometric Brownian motion instead of an arithmeticBrownian motion with randomized drift. This again im-plies a state-dependent value for the firm’s assets, whichensures strategic insolvency, but has several implica-tions for the model.

First, it simplifies the model’s information structure,which allows me to easily add a systematic risk com-ponent to the firm’s earnings process and therefore tostudy equity returns and their determinants. Second, Ishow that in my model, the firm’s asset value processis also a geometric Brownian motion, thus making myspecification of cumulated earnings consistent with theclassical asset-value based models of e.g. Black and Sc-holes (1973), Merton (1974), Black and Cox (1976), Le-land (1994), Fan and Sundaresan (2000), Goldstein, Ju,and Leland (2001), Duffie and Lando (2001), etc.

This paper is also related to the literature on the de-terminants and implications of corporate cash holdings.Relevant empirical studies include Opler et al. (1999),Bates et al. (2009), Acharya et al. (2012), and Davy-denko (2013). Relevant, theoretical studies, includeDecamps, Mariotti, Rochet, and Villenueve (2011), andBolton, Chen, and Wang (2011, 2013).

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1 An equity valuation modelwith liquidity risk

This section develops an equity valuation model for alevered firm with financing constraints that can defaultbecause of liquidity or solvency. The firm retains someof its earnings as cash to avoid the former. In equilib-rium, the firm only defaults because of solvency, and inthat case its equity beta is low because cash is a largefraction of its asset value.

1.1 A financially constrained firmI consider a levered firm generating uncertain cashflows in a continuous-time economy with infinite time-horizon, [0,∞). Corporate earnings are taxed at the rateτ ∈ (0, 1), with a full loss offset, and the instantaneousrisk-free interest rate, r, is assumed to be constant.

The firm’s liabilities consist of common equity stockand, due to tax benefits, consol (infinite maturity) bondswith total coupon rate k per time unit. The number ofshares outstanding and the total coupon level are as-sumed to be predetermined before time zero.

The firm’s assets are fixed throughout its lifetimeand continuously generate earnings (revenue net of ex-penses). For my purpose of modeling cash holdings, itis convenient to specify the stock rather than the flow ofearnings. The model’s main state variable is thereforethe firm’s cumulated earnings before interest and taxes(EBIT) up to time t, which I denote Xt. I assume that Xt,under a physical probability measure, P, is a geometricBrownian motion with dynamics

dXt = µPXtdt + σXtdWPt . (1)

Here, WPt is a standard P-Brownian motion that drivesthe firm’s total (idiosyncratic and systematic) earningsrisk. Instantaneous earnings (per dt) are thus given bythe increment dXt, which can be either positive (a profit)or negative (a loss), depending on the realization of thetotal earnings shock, dWPt .2 The drift parameter, µP, is

2Modeling instantaneous earnings as the increment—instead ofthe level—of a stochastic process is common in models of liquiditymanagement and related topics. See, for instance, Jeanblanc-Picqueand Shiryaev (1995), Radner and Shepp (1996), Demarzo and San-nikov (2006), Decamps et al. (2011), Gryglewicz (2011), Acharyaet al. (2012), and Bolton et al. (2011, 2013).

the P-expected growth rate of earnings, while the dif-fusion parameter, σ, is the volatility rate of earnings.3

After taxes and coupons, the firm’s instantaneous netearnings are (1 − τ)(dXt − kdt), which are at the discre-tion of equity holders until an eventual liquidation of thefirm. In liquidation, bond holders recover a fraction ofthe market value of the firm’s earnings-generating assets(which is derived below), while the remainder is lost tobankruptcy costs.

I assume that due to capital market frictions, the firmhas no access to external financing.4 Coupon paymentsafter time zero must therefore be internally financedthrough earnings. Consequently, the firm can be in dis-tress, and ultimately be liquidated, for two reasons: Ei-ther due to illiquidity, because its earnings are insuffi-cient to service a coupon payment, or due to insolvency,because the value of its earnings-generating assets doesnot outweigh future coupon payments.

In an economy without financing constraints, illiquid-ity never occurs because the firm finances any earningsshortfalls by issuing additional equity—specifically inthe form of negative dividends as in e.g. Black and Cox(1976) and Leland (1994). Since this is possible as longas equity value is positive, the firm is only in distressdue to insolvency. By contrast, when financing con-straints prohibit additional external financing, the firmhas a precautionary motive to reduce dividends and re-tain some earnings as cash holdings, i.e. non-productiveliquid assets, which serve to offset liquidity risk.

The managers of the financially constrained firm,which are assumed to act in the best interest of eq-

3In contrast to Gryglewicz (2011), I specify cumulated earn-ings in (1) as a geometric Brownian motion rather than an arithmeticBrownian motion with a randomized drift. This still implies a state-dependent value for the firm’s earnings-generating assets, which en-sures the existence of a strategic insolvency-trigger (see footnote 7),but simplifies the model’s information structure and allows me tofocus on systematic risk in my analysis of expected equity returns.Moreover, I show below that the implied value of earnings-generatingassets is again a geometric Brownian motion, which makes the modelconsistent with classical capital structure models (see footnote 8).

4This could, for instance, be due to the debt-overhang problemof Myers (1977) or the information asymmetry problems of Lelandand Pyle (1977) and Myers and Majluf (1984). The assumption ofno external financing can be replaced by the milder assumption ofsufficiently high issuance costs, but this does not alter the qualitativenature of the results as long as i) liquidation of the firm is costly andii) only a fraction of future earnings can be pledged as collateral forexternal financing. See Acharya et al. (2012) for a further discussion.

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uity holders, determine the firm’s optimal cash-dividendpolicy by the following trade-off: An additional dol-lar in dividends increases equity value conditional onsurvival, while an additional dollar in cash holdings in-creases the probability of survival. Under an optimalpolicy, the firm’s liquidity risk will be eliminated, but itmay be strategically liquidated if equity holders deem itinsolvent. I assume that the firm’s managers have fulldiscretion over costless distribution of cash holdings toequity holders at any time before liquidation, and thatthere are no bond covenants limiting such payouts. Thismakes equity holders’ strategic insolvency-decision in-dependent of the firm’s cash holdings. In reality, it maybe difficult for bond holders to implement—let aloneenforce—covenants restricting payouts.

1.2 Optimal cash-dividend policyI now derive the financially constrained firm’s optimalpolicies for holding cash and paying out dividends.

Conditional on survival, the firm receives earningsand pays bond holders the tax-deductible coupon. Netearnings can then be paid out as dividends to equityholders or, to reduce liquidity risk, be retained as cashholdings. I assume cash holdings earn the risk-free rate,r, for instance through investment in short-term mar-ketable securities with a tax credit, and equity holderschoose payouts whenever they weakly prefer so.5

Let Ct be the firm’s cash holdings at time t and let Dt

be its cumulated dividend payouts up to time t. Then

dDt = (1 − τ)(dXt − k dt) − dCt + rCt dt. (2)

Because the firm has no access to external financing,Ct and Dt are nonnegative for all t > 0.6 Illiquid-ity occurs when cash holdings hit zero, i.e. at timeτC = inft>0{Ct ≤ 0}. Since expected earnings dependon Xt, equity holders will strategically declare the firm

5The assumption that cash earns r simplistically implies a zerocarry-cost of holding cash—such costs are, however, easily intro-duced in the present model without qualitatively altering the results.

6Usually, negative negative cash holdings are interpreted as thefirm drawing on a credit line, while negative dividend payouts areinterpreted as capital injections by equity holders. Since I assume noexternal financing beyond time zero, the firm has access to neither,and therefore its cash holdings and dividend payouts have to remainnonnegative at any time prior to liquidation.

insolvent when Xt falls below an optimally chosen trig-ger, X? ≥ 0, i.e. at time τX = inft>0{Xt ≤ X?}.7

Managers’ optimization problem

The firm’s managers choose the cash-dividend policythat maximizes the market value of equity. To calculatemarket values, I assume that the economy’s stochasticdiscount factor, Λt, is given by the dynamics

dΛt = −rΛtdt − ηΛtdZPt . (3)

Here, ZPt is a standard P-Brownian motion that drivessystematic earnings risk and has correlation ρ with thefirm’s total earnings risk, WPt , as given in (1). The con-stant η is then the market price of systematic earningsrisk. In the appendix, I detail how (3) defines a risk-neutral pricing measure, Q, under which Xt is geometricBrownian motion with dynamics

dXt = µQXtdt + σXtdWQt , (4)

where µQ = µP − ηρσ is the risk-neutral growth rate ofearnings and WQt = WPt + ηρt is a standard Q-Brownianmotion. Assuming r > µQ > 0, (4) implies that the time-t market value of the firm’s earnings-generating assets is

A(Xt) = EQt

[∫ ∞

te−r(u−t)(1 − τ)dXu

]= (1 − τ)

µQXt

r − µQ,

which is again a geometric Brownian motion.8In the fol-lowing, I focus on the case of A(Xt) > (1−τ)Xt, which isequivalent to the parameter restriction µQ > 1

2 r. Other-wise, it would be optimal for equity holders to liquidatethe firm at time zero—see the discussion of “asset shift-ing” due to too low earnings in Acharya et al. (2012).

7The existence of the strategic insolvency trigger, X?, followsfrom the assumption that Xt is a geometric Brownian motion. If Xtwere an arithmetic Brownian motion with constant drift µ, earnings-generating asset value would be constant and insolvency would beexogenously determined at time zero by the sign of µ − k.

8By Ito’s Lemma and (4), A(Xt) has Q-dynamics

dA(Xt) = µQA(Xt)dt + σA(Xt)dWQt .

This makes the specification of cumulated earnings in (4) (and, equiv-alently, in (1)) consistent with standard capital structure models as-suming that the firm’s asset value is a geometric Brownian motion—e.g. Black and Scholes (1973), Merton (1974), Black and Cox (1976),Leland (1994), Fan and Sundaresan (2000), Goldstein et al. (2001),Duffie and Lando (2001), etc.

5

The market value of the firm’s equity is the Q-expected, discounted value of future dividend payoutsuntil either illiquidity or strategic insolvency, plus a liq-uidation payout of any remaining cash holdings—thelatter is a limiting consequence of the assumption of un-restricted payout before liquidation. Given the strategicinsolvency trigger, X?, and for current (time t) values Xand C of cumulated earnings and cash holdings, equityvalue is thus given by

E(X,C) = supDEQt,X,C

[∫ τ

te−r(u−t) dDu + e−r(τ−t)Cτ

], (5)

where τ = τC ∧ τX is the firm’s liquidation time. Thesupremum is taken over all dividend payout policies, D,which are nonnegative, non-decreasing, satisfy (2), andare adapted to the filtration generated by the cumulatedearnings-process in (4).

Target cash holdings

To solve the manager’s optimization problem, note thathigher cash holdings lower lower liquidity risk but alsodividend payouts. I therefore conjecture that an opti-mal cash-dividend policy involves a target level of cashholdings, denoted C(X), which is the smallest amountof cash large enough to eliminate liquidity risk whencumulated earnings are at the level X.

To characterize C(X), note that the cumulated divi-dend process in (2) is positive and non-decreasing at allt > 0 if and only if i) the drift of Dt is nonnegative andii) the volatility of Dt is zero. By these requirements,C(X) satisfies a differential equation with a lower boundto all its solutions. The following proposition gives thesmallest solution. (All proofs are in the appendix.)

Proposition 1 (Target cash holdings). Suppose the firmhas no access to external financing after time zero—i.e.it cannot issue additional bonds and its cumulated divi-dend process in (2) is nonnegative and non-decreasing.Given the coupon level, k, and the strategic insolvencytrigger, X?, the smallest level of cash holdings largeenough to eliminate liquidity risk when cumulated earn-ings are at the level X is given by

C(X) = (1 − τ)[X − X? + k

r

]. (6)

The target cash level is the after-tax earnings abovestrategic insolvency trigger, (1 − τ)(X − X?), plus thepresent value of future after-tax coupons, (1 − τ)k/r.9

Note that since X ≥ X? and cash holdings earn r, theinterest on the target level is always sufficient to covera coupon payment. The target level is thus the smallestamount of cash large enough to ensure both nonnegativedividends as well as continued coupon payments even ifan earnings shock brings X down to X?. Consequently,by holding at least C(X) in cash as X varies, the finan-cially constrained firm’s liquidity risk eliminated and itis only liquidated if equity holders deem it insolvent.10

Note that C increases in X but decreases in X? be-cause as the firm becomes less solvent (lower X orhigher X?), it can only withstand a smaller earningsshock before being declared insolvent, so the optimalpolicy is to reduce cash holdings accordingly. By thesame logic, C decreases as excess earnings, X − X?, de-crease. Consequently, a firm with cash at the target leveland cumulated earnings close to the point of strategicinsolvency holds as little cash as possible.

Finally, the target cash level increases with thecoupon rate, k, because higher coupons imply a higherliquidity risk, and decreases with the risk-free rate, r,because higher interest implies higher compounding ofcurrent cash holdings and future retained earnings.

Optimal dividend payouts

Given the target cash level of Proposition 1, I conjecturethat the optimal dividend policy adjusts cash holdings,Ct, towards the target level, C(Xt), as Xt fluctuates.

9An inspection of the proof of Proposition 1 reveals that the formin (6) is independent of the assumption that the cumulated earningsprocess, Xt , is a geometric Brownian motion (see (1) or (4)). In fact,the form of (6) is for a general cumulated earnings process, as long assuch a process is consistent with the existence of a strategic solvencytrigger, X?. As argued in footnote 7, this is in particular the case whenXt is a geometric Brownian motion, but not if it were an arithmeticBrownian motion with constant drift.

10An intuitive way to derive (6) is to note that for C(Xt) to elimi-nating liquidity risk for any Xt ≥ X?, it is reasonable to assume thatrC(Xt) dt ≥ (1−τ)k dt. Given this, (2) implies that dDt is nonnegativefor all 0 < t ≤ τX if and only if dC(Xt) ≤ (1 − τ)dXt , implying

C(Xt) ≥ C(XτX ) + (1 − τ)[Xt − XτX

]≥ (1 − τ) k

r + (1 − τ)[Xt − X?

].

Choosing the smallest level satisfying this restriction gives (6).

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To see this, note that if Ct < C(Xt), it is suboptimalfor the firm to pay out dividends, as this would makeit vulnerable to liquidity risk, and all earnings shouldbe retained until cash holdings again reach the targetlevel. Conversely, if Ct > C(Xt), dividend payouts aretoo low, in that cash holdings are above the target level,and it is optimal to distribute the residual Ct − C(Xt) toequity holders. The conjectured dividend payout policyis therefore given by

dD?t =

0 if Ct < C(Xt)[rC(Xt) − (1 − τ)k

]dt if Ct = C(Xt)

Ct −C(Xt) if Ct > C(Xt).

(7)

The intermediate case follows by applying Ito’s Lemmato the target cash level (6) and using the dynamics of thedividend payout process in (2), and may also be written

dD?t = r(1 − τ)(Xt − X?) dt. (8)

From (7), when Ct = C(Xt), instantaneous dividendsare the interest earned on cash holdings net of an after-tax coupon. From (8), this is equivalent to the inter-est earned on after-tax excess earnings. It follows fromeither form that dividends equal zero at the strategicinsolvency-trigger, X?.

The following proposition asserts that the dividendpolicy conjectured in (7) (which depends on the targetcash level of Proposition 1) does maximize equity value.

Proposition 2 (Optimal dividend payouts). Suppose thefirm has no access to external financing after time zeroand that its equity value function in (5) is twice continu-ously differentiable. Then the dividend payout policy in(7) is optimal, in that it attains the supremum in (5).

Intuitively, the dividend policy in (7) maximizes eq-uity value because it optimally exploits that an extra dol-lar in cash holdings is, al else equal, at least worth itsface value to equity holders: ∂E(X,C)

∂C ≥ 1.11 From (7),earnings are retained whenever an additional dollar in

11This is because any excess cash holdings above the level dictatedby an optimal dividend policy can be paid out as a one-time dividend.More formally, under an optimal dividend policy, the value of equitywith C in cash must be greater than or equal to the value of equity withC − ε in cash plus a dividend payout of ε: E(X,C) ≥ E(X,C − ε) + ε.

Rearranging and letting ε → 0 gives ∂E(X,C)∂C ≥ 1.

cash decreases liquidity risk (i.e. when ∂E(X,C)∂C > 1),

while earnings are paid out whenever an additional dol-lar in cash does not further reduce liquidity risk (i.e.when ∂E(X,C)

∂C = 1).

1.3 Equilibrium equity valueThe cash-dividend policy of Propositions 1 and 2 im-plies that when cash holdings are at or above the tar-get level, liquidity risk is eliminated, and the firm isonly liquidated when equity holders strategically deemit insolvent. In the following, I study the correlationbetween the equity value and target cash holdings forvarying levels of solvency.

To this end, I focus on the equilibrium case wherethe firm’s cash holdings are at the target level, i.e. whenCt = C(Xt). Indeed, if Ct , C(Xt), the optimal dividendpolicy in (7) dictates that all variations in cash holdingsare due to the adjustment towards the target level.

In this case, equity is a claim on the firm’s assets (bothearnings-generating and liquid) that pays the continuousflow of dividends in (8) until strategic insolvency. LetE(X) = E(X,C(X)) be the value of equity under thisdividend policy. Using the Q-dynamics of Xt in (4),an application of Ito’s Lemma to the discounted gainsprocess of equity gives that E(X) solves the ordindarydifferential equation

rE(X) = 12σ

2X2 EXX(X) + µQX EX(X)+ (1 − τ)r(X − X?). (9)

As earnings approach infinity, the value of the firm’stotal assets—i.e. cash holdings plus earnings-generatingassets plus the tax-benefit of debt—converges towardsthe sum of risk-free equity and debt. This implies theasymptotic value matching condition

E(X)↗ C(X) + (1 − τ) µQX

r−µQ + τ kr −

kr (10)

for X ↗ ∞. Here, the first three terms in the limit arethe firm’s total assets and the fourth is risk-free debt.

On the other hand, as earnings approach the pointof strategic insolvency, the value of earnings-generatingassets disappears and equity is only a claim on the firm’scash holdings. This, combined with the assumption ofunrestricted payout before strategic insolvency, implies

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the asymptotic limited liability condition

E(X)↘ C(X?) for X ↘ X?. (11)

The boundary conditions (10)–(11) imply the so-lution to the differential equation (9) given in thefollowing proposition.

Proposition 3 (Equilibrium equity value). Suppose r >µQ > 0 and that the firm’s cash holdings are at the targetlevel. Given the coupon, k, and the insolvency-trigger,X?, the market value of equity when cumulated earningsare at the level X is given by

E(X) = C(X) + (1 − τ)[µQXr−µQ −

kr

]+(1 − τ)

[kr −

µQX?

r−µQ

]πQ(X), (12)

where πQ(X) =(X/X?)φ− and where φ− is given by

φ− =σ2 − 2µQ −

√(σ2 − 2µQ)2 + 8rσ2

2σ2 < 0.

The equilibrium value of equity is the sum of threeterms. The first is the face value of the target cashlevel. The second is the present value of expected earn-ings (i.e. the market value of earnings-generating assets)net of future coupon payments. The third term is thevalue of the limited liability put option to default on thefirm’s debt when cumulated earnings fall to the strate-gic insolvency-trigger. Finally, the factor πQ(X) goes to0 as X approaches infinity and goes to 1 as X approachesX?, and may thus, similar to Garlappi and Yan (2011),be interpreted as the firm’s instantaneous Q-probabilityof insolvency.

Strategic insolvency decision

The strategic insolvency-trigger, X?, is the value of cu-mulated earnings solving the smooth-pasting condition,

∂E(X)∂X

∣∣∣∣∣X=X?

=∂C(X)∂X

∣∣∣∣∣∣X=X?

,

which follows from the limited liability condition in(11). This states that equity holders strategically declare

the firm insolvent at the point where an extra dollar inearnings increases equity value by no more than the in-crease in cash holdings. The solution is

X? =kr

r − µQ

µQφ−

φ− − 1. (13)

Note that since φ−

φ−−1 ∈ (0, 1), it follows by the expres-sion for the target cash holdings in (6) that

C(X?) = (1 − τ)kr> (1 − τ)

µQX?

r − µQ= A(X?). (14)

Hence, equity holders strategically declare the firm in-solvent when cash holdings make up a greater frac-tion of total asset value than earnings-generating assets.This, as will be emphasized in the analysis of expectedequity returns, reduces the exposure of the firm’s equityto systematic risk when the firm approaches insolvency.

1.4 Expected returns and target cashIn this section, I study the effects of target cash on thefirm’s expected equity returns. Specifically, I derive anexpressions for the firm’s equity beta and study how itis affected by target cash holdings for varying levels ofsolvency.

Systematic risk and equity returns

Applying Ito’s Lemma to the equilibrium equity value,E(Xt), and using the differential equation (9), the excessreturn on equity, under Q, is given by

dE(Xt) + dD?t

E(Xt)− rdt = Xt

EX(Xt)E(Xt)

σdWQt .

Using the translation WQt = WPt + ηρt, it follows that,conditional on Xt, the P-expected, instantaneous excessreturn on equity may be expressed as

rEt − r = ψE

t ηρσ, (15)

where ψEt = Xt

EX (Xt)E(Xt)

=∂ log E(Xt)∂ log Xt

is the earnings sensi-tivity (or elasticity) of equity. The relation (15) read-ily implies that for a given level of correlated earningsvolatility, ρσ, and systematic volatility, η, higher earn-ings sensitivity implies higher expected return.

8

To characterize the market price of systematic risk, η,I assume that there exists a traded, diversified portfoliowith value Mt, subject only to the systematic risk com-ponent of the stochastic discount factor (3). The returnon this portfolio, under P, is then given by

dMt

Mt= rM dt + σM dZPt .

Since ZQt = ZPt + ηt and since the Q-expected, instanta-neous return on the portfolio has to be the risk-free rate,it follows that η = rM−r

σM , i.e. the expected excess returnon the portfolio relative to its volatility, or its Sharperatio. Combining this with (15) gives the relation

rEt − r = ψE

t βX (rM − r), (16)

where βX =ρσσM is the the correlated volatility of earn-

ings relative to the volatility of the diversified portfolio,i.e. the firm’s (unlevered) asset beta.

The relation (16) is a conditional capital asset pric-ing model (CAPM), stating that, given Xt, the expectedexcess return on the firm’s equity is proportional to theexpected excess return on the diversified portfolio. Theproportionally factor is the asset beta, βX , scaled bythe earnings sensitivity, ψE

t . In this sense, ψEt β

X is thismodel’s conditional equity beta at time t.

The following proposition gives an expression for thesensitivity, ψE

t , that highlights the effects of the firm’scash holdings and probability of insolvency on expectedequity returns.

Proposition 4 (Earnings sensitivity of equity). Giventhe coupon, k, and the time-t value of cumulated earn-ings, Xt, the earnings sensitivity of equity, ψE

t , can bewritten as

ψEt = 1 −

C(Xt)E(Xt)︸︷︷︸

Cash

+ (1 − τ)Xt

E(Xt)︸ ︷︷ ︸Earnings

+ (1 − τ)k/r

E(Xt)︸ ︷︷ ︸Leverage

− (1 − τ)(1 − φ−)[

kr −

µQX?

r−µQ

]πQ(Xt)E(Xt)︸ ︷︷ ︸

Insolvency

.

Proposition 4 benchmarks equity’s earnings sensitiv-ity to 1, which corresponds to the earnings sensitivity of

an unlevered firm (i.e. when k = 0) and would implyan equity beta as in the unconditional CAPM. Further-more, to measure the deviations from this benchmark,the sensitivity is decomposed into four terms.

The term labeled “Cash” is the firm’s cash-to-equityratio, and represents the percentage value of target cashto equity holders (relative to current value of their initialequity investment). Intuitively, the term is negative andreflects a discount (relative to the benchmark, i.e. theunconditional CAPM beta) because higher target cashmeans, all else equal, lower sensitivity to earnings risk.By (16), the isolated effect of a higher target cash is thuslower beta and lower expected returns.

The term labeled “Earnings” is a proxy for the firm’stotal earnings yield (i.e. the inverse of the price-to-earnings ratio) from the time of initial investment untiltime t. It represents the percentage value of the earningsclaim owned by equity holders. Intuitively, the term ispositive and reflects a premium (relative to the bench-mark) because a higher claim on the firm’s earningswill, all else equal, imply higher earnings sensitivity.The isolated effect of a higher earnings yield will thusimply higher beta and higher expected returns.

The “Leverage”-term is a proxy for the firm’s debt-to-equity ratio, and represents the percentage value ofthe firm’s financial obligations. Intuitively, the term ispositive and reflects a premium because higher couponpayments will, all else equal, imply more sensitivity toearnings risk. The isolated effect of more leverage istherefore higher beta and higher expected returns.

Finally, the term labeled “Insolvency” is a proxy forthe percentage value of the put option to default on thefirm’s debt at the strategic insolvency trigger, X?. Itrepresents the value of the limited liability provisionimbedded in the equity claim. Intuitively, it is nega-tive and reflects a discount because as the firm becomesless solvent, the value of its earnings-generating assetdecline and the equity claim approaches the value of thefirm’s cash (cf. (11)), which, all else equal, lowers eq-uity’s sensitivity to earnings risk. The isolated effect ofa higher value for the default option is thus lower equitybeta and lower expected returns.

9

Cash holdings and earnings sensitivity

The above considerations give an assessment of themodel’s isolated effects on equity beta and expected re-turns. More interestingly, however, is the relationshipbetween target cash holdings and the earnings sensitiv-ity (i.e. the behavior of equity beta and expected returns)for varying levels of solvency.

Corollary 4.1 (Target cash and earnings sensitivity).Suppose r > µQ > 1

2 r, let πQ(X) =(X/X?)φ− be the

Q-probability of insolvency as in Proposition 3, and letA(Xt) = (1 − τ) µ

QXt

r−µQ be the market value of earnings-generating assets. Then the target cash holdings, C(Xt),and the earnings sensitivity of equity, ψE

t , have the fol-lowing properties:

i) As πQ(Xt) → 0, cash holdings are worth less thanearnings-generating assets, while earnings sensi-tivity is higher than for an unlevered firm and in-creases in the probability of insolvency:

C(Xt) < A(Xt), ψEt > 1,

dψEt

dπQ(Xt)> 0.

ii) As πQ(Xt)→ 1, cash holdings are worth more thanearnings-generating assets, while earnings sensi-tivity is lower than for an unlevered firm and de-creases in the probability of insolvency:

C(Xt) > A(Xt), ψEt < 1,

dψEt

dπQ(Xt)< 0.

The first part of the corollary states that when the firmis solvent, its earnings-sensitivity is high (relative to anunlevered firm) and an increase in the probability of in-solvency will make the equity claim more sensitive toearnings risk. This is because for high levels of sol-vency, the firm’s earnings-generating assets are a largerfraction of its total asset value than its cash holdings,which magnifies the effect of an increase in the probabil-ity of insolvency on the equity claim. By (15) and (16),expected returns and equity beta are thus high and in-creasing in the probability of insolvency when the firmis solvent.

By contrast, the second part of the corollary statesthat when the firm has a high probability of insolvency,

its earnings-sensitivity is actually low (relative to an un-levered firm) and declining for a further decrease in sol-vency. The reason for this is the presence of cash in thefirm’s capital structure. As insolvency becomes increas-ingly apparent, the value of the earnings-generating as-sets declines and cash holdings become an increasinglylarger fraction of total asset value. This implies that forlow levels of solvency, equity will to a greater degreebe a claim on the firm’s cash, which decreases its sen-sitivity to earnings risk. Therefore, (15) and (16) implythat expected returns and equity beta are low and de-creasing in the probability of insolvency for low levelsof solvency.

Because target cash holdings, C(X), monotonicallydecrease in the probability of insolvency, πQ(X), Corol-lary 4.1 has the following implications:

• The earnings-sensitivity, ψEt , is also hump-shaped

in the target cash level: For high levels of targetcash, ψE

t is greater than 1 and increasing in C(X),while for low levels of target cash, ψE

t is less than1 and decreasing in C(X).

• A portfolio strategy long firms with high cash andshort firms with low cash (hereafter a HCmLCportfolio) will have negative (positive) expected re-turns for high (low) levels of solvency.

• A portfolio strategy long firms with high solvencyand short firms with low solvency (hereafter aHSmLS portfolio) will have negative (positive) ex-pected returns for high (low) levels of cash.

The first implications follows from the inverse rela-tionship between target cash and probability of insol-vency. The performance of the HCmLC strategy fol-lows because for high levels of solvency, equity betaincreases as cash decreases, so the long leg of HCmLCwill have lower expected returns than the short leg. Thesituation is, however, reversed for low levels of sol-vency, where equity beta decreases as cash decreases,so the long leg of HCmLC will have higher expected re-turns than the short leg. The same logic applies to theHSmLS strategy.

10

Probability of Insolvency

EBIT

-gen

erat

ing

asse

ts a

nd c

ash

0.0 0.2 0.4 0.6 0.8 1.0

4060

80100

120

Lower boundary of cash holdings

Insolvency trigger

Probability of Insolvency

Earn

ings

sen

sitiv

ity o

f equ

ity

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1.0

1.2

Unlevered

Probability of Insolvency

Earn

ings

sen

sitiv

ity o

f equ

ity

0.0 0.1 0.2 0.3 0.4

0.95

1.00

1.05

1.10

1.15

1.20

Unlevered

High cash

Low cash

High cash

Low cash

Cash holdings (reversed)

Earn

ings

sen

sitiv

ity o

f equ

ity

160 140 120 100 80 60

0.8

0.9

1.0

1.1

1.2

Unlevered

Lower boundary of cash holdings

High solvency

Low solvency

High solvency

Low solvency

Figure 1. Numerical illustration: Earnings-sensitivity of equity. This figure shows a numerical illustration of the earnings-sensitivity of equity.Top left: Earnings-generating asset value (black curve) and cash holdings (purple curve) as a function of the probability of insolvency with anindication of the strategic insolvency trigger (black dashed line) and the lower boundary of cash holdings (dashed purple line). Top right: Earningssensitivity of equity as a function of the probability of insolvency with an indication of the sensitivity of an unlevered firm at 1 (dashed line). Bottomleft: Earnings sensitivity of equity as a function of the probability of insolvency (from 0 to 0.4) with an indication of the high-cash minus-low cashstrategy (dashed purple line segments). Bottom right: Earnings sensitivity of equity as a function of cash holdings with an indication of the highsolvency minus low solvency strategy (dashed purple line segments). The parameters are µQ = 0.04, σ = 0.15, r = 0.06, τ = 0.15, k = 4.5, andX0 = 58.82

Numerical illustration

Figure 1 illustrates the results of Corollary 4.1. I con-sider a representative firm given by the following pa-rameters:

µQ = 0.04, σ = 0.15, k = $4.5, τ = 0.15, r = 0.06.

The initial earnings level is set at X0 = $58.82, so thatinitial earnings-generating asset value is A(X0) = $100.Insolvency is then triggered when cumulated earningshit X? = $30.10 (corresponding to A(X?) = $51.17).Initial equity value is then E(X0) = $125.36 while initialcash holdings are C(X0) = $88.65.

The top panels shows the firm’s earnings-generating

assets, target cash holdings, and earnings-sensitivity ofequity as a function of the probability of insolvency.The bottom panels show the earnings-sensitivity of eq-uity as a function of probability of insolvency (rightpanel) and target cash (left plot) with an indication ofthe HCmLC and HSmLS strategies.

The top panel illustrates the mechanism behind thehump-shaped equity beta in the probability of insol-vency. When the firm is solvent, it has a high targetcash level, but its earnings-generating assets are largerthan target cash. Consequently, equity’s earnings sensi-tivity is above 1 (i.e. above the earnings-sensitivity of anunlevered firm) for high levels of solvency. However, asthe firm approaches insolvency, target cash holdings de-

11

cline, but earnings-generating asset value falls as welluntil it declines below target cash, in accordance withthe relation in (14). Hence, the earnings sensitivity ofequity falls below that of an unlevered firm for high lev-els of insolvency probability. By the relation in (16),the firm’s equity beta and its expected returns are higherthan those of an unlevered firm when it is solvent, butare, in fact, lower than those of an unlevered firm whenit approaches insolvency.

The lower left panel shows the performance of theHCmLC portfolio strategy for varying levels of sol-vency. Because earnings-sensitivity initially increasesin the probability of insolvency, the short leg of HCmLChas higher expected returns than the long leg for highlevels of solvency, so HCmLC has negative expectedreturns for high levels of solvency. This is, however, re-versed for low levels of solvency, where the long leghas higher expected returns than the short leg. Thelower right panel shows the expected returns on theHSmLS portfolio strategy. Because earnings-sensitivityincreases as cash decreases from high levels, the shortleg of the strategy has higher expected returns than thelong leg for high levels of cash. The situation is, how-ever, reversed for low levels of cash.

1.5 Discussion and testable predictionsCorollary 4.1 effectively predicts that equity betas andexpected returns should be hump-shaped in both sol-vency and liquidity. As discussed in the paper’s intro-duction, such a relation was first derived by Garlappiand Yan (2011) in a model based on shareholder recov-ery in default—that is, when equity holders anticipate arenegotiation of debt-terms or even a violation the ab-solute priority rule after a default. In their model, thefirm is not financially constrained, and therefore doesnot face liquidity risk, while the recovery value thatultimately implies the hump-shape is specified exoge-nously.

By contrast, the driving force behind Corollary 4.1 isthe precautionary motive for holding cash, which im-plies that it is optimal for the financially constrainedfirm to offset its liquidity risk. Equity value in my modelwill thus not only be determined by the firm’s dividendpayouts, but also by the endogenously determined targetcash held by the firm, which becomes a larger fraction

of total asset value as the firm becomes less solvent, thuslowering equity beta.

While the mathematical mechanisms behind thehump-shape is the same in both models—namely thatequity value is determined by a non-zero underlyingasset close to default—the economic mechanisms aredifferent, and, thus, yield different testable predictions.In Garlappi and Yan (2011), equity risk declines closeto default because equity holders anticipate a substitu-tion of levered equity with unlevered asset value. In mymodel, equity risk declines because cash holdings re-duce the systematic risk of the underlying. The novelprediction of my model are thus that cash holdings de-crease equity betas and expected returns as solvencydecreases; that the HCmLC portfolio earns increas-ing average returns as solvency decreases; and that theHSmLS portfolio earns increasing average returns asliquidity decreases.

2 An empirical study of liquidityrisk and distressed equity

This section presents an empirical study of the model’spredictions for the effects of cash used to offset liquid-ity risk on the returns of distressed equity. I use firm-level data on US stocks prices, accounting numbers, andcredit ratings to estimate firm-specific betas and calcu-late cross-sectional portfolio returns.

2.1 Data

I search for data in the intersection of industrial firmswith stock prices in the CRSP database, accounting fun-damentals in the Compustat North American database,and credit ratings or default records in the Moody’s DRS(Default Risk Service) database.

For every US debt issuer in DRS’ “industrial” cate-gory with an available third party identifier, I search forthe corresponding security-level PERMNO-identifiersin the daily CRSP file and in the quarterly and yearlyCompustat files, taking name changes, mergers, accusa-tions, and parent-subsidiary relations into account, andexcluding issuers which I cannot reliably match. I onlyinclude common stocks (CRSP’s SHRCD 10-11) and I

12

exclude utilities and financial firms (CRSP’s SIC codes4900-4999 and 6000-6999). The final sample has 3,947unique firms, spanning 15,079,329 firm-days (720,371firm-months) over the period from January 1970 to De-cember 2013.

Because distress risk may ultimately result in a de-fault or a bankruptcy, I track these events for the firmsin the sample. I identify a default or bankruptcy event ifit is recorded in either DRS, CRSP (DLSTCD 400-490or 574, or SECSTAT ‘Q’), or Compustat (DLRSN 2-3 orSTALTQ ‘TL’), and I count multiple events for the samefirm occurring within a month as a single event. Thisresults in a total of 874 events incurred by 683 firms,of which 529 events were identified solely throughDRS, 134 solely through CRSP, and 137 solely throughCompustat—the remaining 87 events were identified si-multaneously by two or more sources.

I use the stock data to calculate market equity val-ues, ME (the product of CRSP’s PRC and SHROUT,adjusted by their cumulative adjustment factors), and Iaccumulate daily log-returns (ln of 1 plus CRSP’s RET)over a 20 trading day rolling window to obtain monthlyreturns. I require at least 10 trading days to calculatea monthly return, and I use delisting returns (CRSP’sDLRET) whenever possible.

When possible, I substitute yearly accounting num-bers for missing quarterly accounting numbers. Theseare then used to calculate quarterly book equity, BE =

AT − LT (Compustat’s total assets, ATQ, minus total li-abilities, LTQ), cash-ratios measuring balance sheet liq-uidity, proxies for solvency, and, finally, regression con-trols like firm size and dividends.

I align the quarterly accounting data and the dailystock data as follows: On a given trading day, the corre-sponding accounting numbers are the latest ones avail-able prior to that day. All raw variables and ratios arewinsorized at the 1% and the 99% quantiles to removethe influence of near-zero divisions, recording errors,and outliers.

2.2 Liquidity and solvency

The model predicts that cash holdings used to offset liq-uidity risk correlate with equity prices and expected re-turns in manner that depends on solvency. In this sub-

section, I present the variables which I employ to mea-sure liquidity and solvency.

Liquidity measures

I use three cash-based variables to measure a firm’s liq-uid assets relative to its current liabilities. First, I usethe current ratio, CA/CL (Compustat’s current assets,ACTQ, divided by current liabilities, LCTQ), whichmeasures a firm’s total liquid holdings as a fraction ofits short-term liabilities. Second, I use the the quickratio, QA/CL (Compustat’s current assets, ACTQ, mi-nus inventories, INVTQ, the difference divided by cur-rent liabilities, LCTQ), which measures assets that can“quickly” be converted into cash in order to pay off

short-term liabilities. The current- and quick ratios aresimilar, but the quick ratio is more conservative, in thatit excludes inventories from current assets. In eithercase, a ratio below 1 indicates that the firm’s liquid as-sets are insufficient to meet short-term liabilities, i.e.illiquidity. Finally, I use the ratio of working capital tototal assets (Compustat’s current assets, ACTQ, minuscurrent liabilities, LCTQ, the difference divided by totalassets, ATQ), measuring the firm’s net liquid assets asa percentage of total book assets. A negative workingcapital ratio thus indicates illiquidity.

Solvency measures

To measure solvency, I use Moody’s senior unsecuredlong-term credit ratings (provided in DRS) as well astwo balance-sheet based variables: Leverage and inter-est coverage. The ratings give a categorical measureof solvency, based on an overall assessment of firm’sability to honor its financial obligations with an origi-nal maturity of one year or more. I measure leverageusing either the book leverage ratio, LT/AT , or the mar-ket leverage ratio, LT/(LT + ME), which are a stockvariable measuring total liabilities as a fraction of eitherbook or market assets. Finally, the interest coverage ra-tio, OI/IX (Compustat’s EBITDA-variable, OIBDPQ,over interest expense, XINTQ), is a flow variable mea-suring the firm’s ability to generate earnings in excessof its interest expense.

13

Current Ratio and Interest Coverage

Interest Coverage Ratio (deciles)

Cur

rent

Rat

io

High 9 8 7 6 5 4 3 2 Low

1.0

1.5

2.0

2.5

3.0

Current Ratio and Leverage

Book Leverage (deciles)

Cur

rent

Rat

io

Low 2 3 4 5 6 7 8 9 High

1.0

1.5

2.0

2.5

3.0

3.5

Current Ratio and Rating

Rating

Cur

rent

Rat

io

Aaa Aa A Baa Ba B Caa Ca C D

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Quick Ratio and Interest coverage

Interest Coverage Ratio (deciles)

Qui

ck R

atio

High 9 8 7 6 5 4 3 2 Low

1.0

1.5

2.0

Quick Ratio and Leverage

Book Leverage (deciles)

Qui

ck R

atio

Low 2 3 4 5 6 7 8 9 High

1.0

1.5

2.0

2.5

Quick Ratio and Rating

Rating

Qui

ck R

atio

Aaa Aa A Baa Ba B Caa Ca C D

0.6

0.8

1.0

1.2

1.4

Working Capital and Interest Coverage

Interest Coverage Ratio (deciles)

Wor

king

Cap

ital t

o To

tal A

sset

s

High 9 8 7 6 5 4 3 2 Low

0.00

0.10

0.20

0.30

Working Capital and Leverage

Book Leverage (deciles)

Wor

king

Cap

ital t

o To

tal A

sset

s

Low 2 3 4 5 6 7 8 9 High

0.0

0.1

0.2

0.3

0.4

Working Capital and Rating

Rating

Wor

king

Cap

ital t

o To

tal A

sset

s

Aaa Aa A Baa Ba B Caa Ca C D

0.00

0.05

0.10

0.15

0.20

Figure 2. Liquidity measures sorted across solvency measures. This figure shows the three liquidity measures (current ratio, CA/CL, quick ratio,QA/CL, and working capital, WC/AT ) plotted against the three solvency measures (deciles of interest coverage, OI/IX, deciles of book leverage,LT/AT , and Moody’s credit rating). Solid lines indicate means within groups while dashed lines indicate medians. In all panels, a horizontal moveto the right corresponds to lower solvency, while a vertical move downwards corresponds to lower liquidity.

Liquidity and solvency

Figure 2 illustrates how liquidity varies with solvency.In the leftmost column, the three liquidity measures

are plotted against the interest coverage ratio. Firmsgenerating the highest earnings relative to their interestexpense hold liquid assets that are 2-3 times their cur-rent liabilities and their working capital is about 35%of book asset. Liquidity declines as interest coverage

declines, except for the firms with lowest interest cover-age, where liquid assets again rise to 1.5-2 times currentliabilities and working capital again rises to around 20%of book assets. Importantly, liquid assets exceed currentliabilities and working capital is positive across all lev-els of interest coverage. This corroborates the findingsof Acharya et al. (2012) and supports the model’s as-sumption that levered firms hold cash to offset the riskof an earnings-shortfall.

14

The middle column shows the liquidity measuresacross book leverage (the plots are essentially identicalfor market leverage). Firms with the lowest leveragehold 2.5-3.5 times as much liquid assets as their cur-rent liabilities and have a working capital ratio of about40%. As leverage increases, liquidity decreases mono-tonically. Still, the most levered firms have liquid assetsthat equal or are slightly above their current liabilitiesand a positive working capital ratio just short of 10%.This indicates that, on average, even the most leveredfirms are still liquid.

Finally, the rightmost column shows the liquiditymeasures across credit ratings. The upper investment-grade firms (from Aaa to Aa) hold liquid assets thatare 1-1.5 times their current liabilities and have a work-ing capital ratio of around 15%. With such high rat-ings, these firms need to worry less about financing con-straints, which is reflected in their relatively modest re-serves of liquid assets. As ratings move away from in-vestment grade, liquidity increases and reaches its high-est level for upper speculative grade firms (from Ba toB), who hold liquid assets in the range of 1.3-2 timestheir current liabilities and have a working capital ratioof around 20%. For these firms, access to external fi-nancing is likely to be somewhat constrained, which isreflected in their larger reserves of liquid assets. Whileliquidity generally declines as ratings deteriorate intolower speculative-grade (from Caa to C) and default(D), even the lowest rated firms have liquid assets thateither slightly exceed or just fall short of their current li-abilities, once more corroborating the impression fromthe middle column that even the riskiest firms are, onaverage, liquid.

In sum, Figure 2 indicates that firms use liquidity re-serves to offset the risk of an earnings-shortfall; that liq-uidity generally declines with leverage, but that even themost levered firms are liquid; and that firms in the mid-dle of the rating scale are the ones who hold the mostliquid assets, but that even the firms with lowest ratingshave liquid asset that either slightly exceed or just fallshort of their current liabilities.

2.3 Liquidity risk and equity returnsIn this section, I test the model’s prediction regardingthe correlation between balance-sheet liquidity and eq-

uity returns. The model predicts that i) firm-specific be-tas and cross-sectional returns are hump-shaped in sol-vency and liquidity, ii) a portfolio strategy long firmswith high cash and short firms with low cash (HCmLC)has increasing expected returns as firms become lesssolvent, and ii) a portfolio strategy long firms with highsolvency and short firms with low solvency (HSmLS)has increasing expected returns as cash levels decline.

Firm-specific betas

Figure 3 plots firm-specific betas across the solvencyand liquidity measures.

The firm-specific betas are calculated at the monthlyfrequency by regressing daily excess returns on the dailyexcess returns of the CRSP value-weighted “market” in-dex (available in the CRSP file or on prof. Ken French’swebsite). I require a minimum of 10 trading days toestimate a monthly beta. To reduce the influence ofoutliers, I exclude the lowest and highest monthly betafor each firm. Finally, following Vasicek (1973) andFrazzini and Pedersen (2014), I adjust firm i’s estimatedmonthly beta, βit, towards the cross-sectional mean forthe corresponding month, βt, by setting

βit = wit βit + (1 − wit) βt.

Here, wit = 1 − ν2i /(ν

2i + ν2

t ) is a Bayesian adjustmentfactor calculated using the variance of the estimated be-tas for firm i, ν2

i , and the cross-sectional variance of theestimated betas at month t, ν2

t . It places more weight onthe firm’s beta estimates when their variance is small orwhen the cross-sectional variance is high.12

The top panel shows the firm-specific betas across thesolvency measures. In all three plots, and consistentwith the model’s prediction that equity betas decline forlow levels of solvency, the most solvent firms also havethe highest betas, and betas decline almost monotoni-cally as firms become less solvenct. The most solventfirms with a high credit rating, low leverage, and highinterest coverage, have average betas between 0.9 and

12I have also produced a version of Figure 3 where I replace thefirm-specific adjustment factors, wit , with their average across firmsand time, 0.57, and where the cross-sectional mean beta is set asβt = 1 for all months. This mimics the simplified adjustment of firm-specific betas used by Frazzini and Pedersen (2014). The resultingfigure is very similar to Figure 3.

15

Beta and Rating

Rating

Mon

thly

, firm

-spe

cific

CA

PM b

eta

Aaa Aa A Baa Ba B Caa Ca C D

0.65

0.70

0.75

0.80

0.85

0.90

0.95

Beta and Leverage

Market Leverage (deciles)

Mon

thly

, firm

-spe

cific

CA

PM b

eta

Low 2 3 4 5 6 7 8 9 High

0.75

0.80

0.85

0.90

0.95

1.00

Beta and Interest Coverage

Interest Coverage Ratio (deciles)

Mon

thly

, firm

-spe

cific

CA

PM b

eta

High 9 8 7 6 5 4 3 2 Low

0.75

0.80

0.85

0.90

0.95

Beta and Current Ratio

Current Ratio (deciles)

Mon

thly

, firm

-spe

cific

CA

PM b

eta

High 9 8 7 6 5 4 3 2 Low

0.80

0.82

0.84

0.86

0.88

0.90

Beta and Quick Ratio

Quick Ratio (deciles)

Mon

thly

, firm

-spe

cific

CA

PM b

eta

High 9 8 7 6 5 4 3 2 Low

0.78

0.82

0.86

0.90

Beta and Working Capital

Working Capital (deciles)

Mon

thly

, firm

-spe

cific

CA

PM b

eta

High 9 8 7 6 5 4 3 2 Low

0.78

0.80

0.82

0.84

0.86

0.88

0.90

Figure 3. Firm-specific betas across solvency and liquidity measures. This figure shows firm-specific CAPM-betas plotted against the threesolvency measures (top panels: Moody’s credit rating, deciles of market leverage, LT/(LT + ME), and deciles of interest coverage, OI/IX) andthe three liquidity measures (bottom panels: Deciles of current ratio, CA/CL, quick ratio, QA/CL, and working capital, WC/AT ). Firm-specificbetas are estimated at the monthly frequency by regressing daily excess returns on the daily excess returns of the CRSP value-weighted “market”index. Each monthly beta is estimated using a minimum of 10 trading days. To reduce the influence of outliers, the lowest and highest monthlybeta for each firm is excluded, and monthly betas are adjusted towards the cross-sectional mean monthly beta as in Vasicek (1973) and Frazzini andPedersen (2014). Solid lines indicate means within groups while dashed lines indicate medians. In the top (bottom) panels, a horizontal move tothe right corresponds to higher solvency (liquidity) risk.

0.95. At the other end of the solvency spectrum, firmswith a low credit rating, high leverage, and low interestcoverage, have average betas between 0.65 and 0.8.

The bottom panel plots the firm-specific betas againstthe three liquidity measures. The model predicts that asfirms become less solvent, their cash levels decline, andthis was confirmed for the three liquidity measures inFogure 2. Hence, it is not surprising that equity betasalso decline as the liquidity measures decline.

Single-sorted portfolios on solvency and liquidity

The decline in firm-specific equity betas for rising sol-vency and liquidity risk indicate a lower expected returnfor firms in either type of distress. To verify this, Table1 shows excess returns and alphas (abnormal returns)for portfolios formed on the three solvency measures,while Table 2 shows the corresponding results for port-folios formed on the three liquidity measures.

At the beginning of each month, I sort firms intoportfolios according to their solvency or liquidity lev-els. The portfolios are value-weighted, refreshed ev-ery month, and rebalanced every month to maintainthe value-weighting. To ensure that the results are notdriven by outliers, the highest and lowest realized re-turn is excluded for each portfolio. The tables reportthe time-series average of the portfolio returns over the1-month US T-bill as well as alphas, estimated as theintercepts from time-series regressions of excess returnson the “market” index (MKT); the size (SMB) and value(HML) factors of Fama and French (1993); and the mo-mentum factor (UMD) of Carhart (1997).

In Table 1, panel A shows portfolios formed on creditratings, while panels B and C show portfolios formedon deciles of market leverage or interest coverage. Inall three panels, firms in the high end of the solvencyscale (high rating, low leverage, or high interest cov-

16

erage) have positive and significant excess returns andalphas. At the other end of the solvency scale, how-ever, excess returns and alphas turn insignificant andeven significantly negative for the least solvent firms.Furthermore, the decline in excess returns and alphas isalmost monotonic when using leverage and interest cov-erage to proxy for solvency. This is consistent with themodel’s prediction that returns decline in the probabilityof insolvency for low levels of solvency.

Table 2 shows similar results for the portfoliosformed on the deciles of the three liquidity measures:The current ratio in panel A, the quick ratio in panelB, and the working capital ratio in panel C. In all threepanels, excess returns and alphas are significantly posi-tive for high levels of liquidity and generally decline asthe liquidity measures decline. This is consistent withthe model’s prediction that the declining returns on dis-tressed equity is prevalent across both the solvency- andthe liquidity dimension.

High cash minus low cash across credit ratings

The model predicts that a portfolio strategy long firmswith high cash and short firms with low cash will haveincreasing average returns as solvency decreases. Totest this prediction, Table 3 shows results for double-sorted portfolios according to credit ratings and liquid-ity measures. To ensure a sufficient number of firmsin each portfolio, I re-code the original 9 credit ratingsinto three groups: Aaa-A, Baa-B, and Caa-C. I form theportfolios using conditional sorts—first into the threecredit ratings portfolios, and then into three portfoliosbased on terciles of liquidity measures. I then calculate,for each credit rating group, the difference between thereturns on the “high cash” portfolio and the “low cash”portfolio (hereafter, a HCmLC portfolio).

Consistent with the models prediction, the HCmLCportfolios have monotonically increasing returns, al-phas, and Sharpe ratios as credit ratings deteriorate.For firms rated Aaa-A, the HCmLC portfolios have in-significant returns (between −0.05% and 0.08% on amonthly average), insignificant alphas, and low Sharperatios (between −0.06 and 0.10 on an annual basis). Asthe credit ratings deteriorate into Baa-B, returns and al-phas become significantly positive, and the Sharpe ratiorises considerably. In fact, for the riskiest firms rated

Caa-C the HCmLC portfolios have significantly posi-tive returns (between 1.16% and 1.52% on a monthlyaverage), significant alphas, and large Sharpe ratios (be-tween 0.38 and 0.51 on an annual basis).

High solvency minus low solvency across liquiditymeasures

The model also predicts that a portfolio strategy longfirms with high solvency and short firms with low sol-vency will have increasing average returns as cash hold-ings decrease. To test this prediction, Table 4 showsresults for double-sorted portfolios formed first on liq-uidity terciles and then on credit rating groups: Aaa-A,Baa-B, and Caa-C. For each liquidity tercile, I calculatethe difference between the returns on the Aaa-A portfo-lio and the Caa-C portfolio (hereafter, a HSmLS portfo-lio).

Consistent with the models prediction, the HCmLCportfolios have monotonically increasing returns, al-phas, and Sharpe ratios as all three liquidity measuresdecrease.

Concluding remarksThis paper has shown, theoretically and empirically,that cash holdings used to offset liquidity risk can helprationalize the anomalous returns of distressed equity.In my model, levered firms with financing constraintscan default because of liquidity or solvency, but firmsseek to manage their cash to avoid the former. Usingdata on rated US firms between 1970 and 2013, I findempirical evidence consistent with my theoretical pre-dictions: i) the average insolvent firm holds cash thatmeets or exceeds its current liabilities; ii) firm-specificbetas and risk-adjusted returns decline as firms becomeless solvent and as cash levels decline; and iii) a port-folio long firm with high cash and short firms with lowcash has increasing returns for less solvent firms, whilea portfolio long firms with high solvency and short firmswith low solvency has increasing returns for firms withless cash. In sum, my results suggest that there is nodistress anomaly for insolvent but liquid firms.

17

Tabl

e1.

Mon

thly

exce

ssre

turn

sof

port

folio

sfo

rmed

onso

lven

cym

easu

res.

Thi

sta

ble

show

sex

cess

retu

rns,

alph

as,a

ndre

late

dqu

antit

ies

for

port

folio

sfo

rmed

onso

lven

cym

easu

res.

Att

hebe

ginn

ing

ofea

chca

lend

arm

onth

,Ias

sign

firm

sin

topo

rtfo

lios

acco

rdin

gto

thei

rcre

ditr

atin

g(P

anel

A)o

rdec

iles

ofm

arke

tlev

erag

e(P

anel

B)o

rin

tere

stco

vera

gera

tio(P

anel

C)f

orth

epr

evio

usm

onth

.The

port

folio

sar

eva

lue-

wei

ghte

dus

ing

the

prev

ious

mon

th’s

mar

kete

quity

valu

es,r

efre

shed

ever

yca

lend

arm

onth

,an

dre

bala

nced

ever

yca

lend

arm

onth

tom

aint

ain

valu

e-w

eigh

ting.

The

high

esta

ndlo

wes

trea

lized

retu

rnis

excl

uded

for

each

port

folio

.E

xces

sre

turn

isth

etim

e-se

ries

aver

age

ofth

em

onth

lypo

rtfo

liore

turn

s(i

npe

rcen

tage

s)in

exce

ssof

the

1-m

onth

US

Trea

sury

bill

rate

.Vol

atili

tyis

the

annu

aliz

edst

anda

rdde

viat

ion

ofth

em

onth

lyex

cess

retu

rn(i

npe

rcen

tage

s).

Shar

pera

tiois

the

annu

aliz

edav

erag

eex

cess

retu

rndi

vide

dby

the

annu

aliz

edvo

latil

ity.

CA

PMal

pha

and

CA

PMbe

taar

eth

ein

terc

epta

ndsl

ope

estim

ates

from

atim

e-se

ries

regr

essi

onof

mon

thly

exce

ssre

turn

son

the

exce

ssre

turn

sof

the

valu

e-w

eigh

ted

CR

SP“m

arke

t”in

dex

(MK

T).

Thr

ee-

and

four

-fac

tor

alph

asar

eth

ein

terc

epts

from

time-

seri

esre

gres

sion

sof

mon

thly

exce

ssre

turn

son

the

thre

eFa

ma

and

Fren

ch(1

993)

fact

ors

(MK

T,SM

B,a

ndH

ML

)or

thes

eth

ree

fact

ors

asw

ella

sth

eC

arha

rt(1

997)

fact

or(U

MD

).Pa

rent

hese

sin

subs

crip

tgiv

et-

stat

istic

s.Fo

rth

eex

cess

retu

rns

and

the

alph

as,t

henu

llva

lue

isze

ro,w

hile

itis

one

for

the

beta

.St

atis

tical

sign

ifica

nce

atth

e5%

leve

lis

indi

cate

din

bold

.

Pane

lAC

redi

tRat

ing

Aaa

Aa

AB

aaB

aB

Caa

Ca

CE

xces

sre

turn

(avg

.mon

.%)

0.81

(4.3

6)1.

00(5.3

7)0.

92(4.4

2)0.

90(4.0

3)1.

26(5.1

3)0.

83(2.4

7)0.

82(1.8

6)−

0.46

(−0.

59)−

6.09

(−4.

75)

CA

PMal

pha

(ann

.%)

0.59

(4.7

0)0.

76(7.1

5)0.

64(7.8

9)0.

55(6.7

3)0.

90(8.2

6)0.

42(2.0

8)0.

25(0.7

5)−

1.06

(−1.

46)−

6.56

(−5.

31)

Thr

ee-f

acto

ralp

ha(a

nn.%

)0.

57(5.1

7)0.

68(7.1

5)0.

61(7.5

7)0.

51(6.2

7)0.

87(8.2

5)0.

45(2.3

2)0.

27(0.8

4)−

1.05

(−1.

46)−

6.40

(−5.

28)

Four

-fac

tora

lpha

(ann

.%)

0.54

(4.8

6)0.

64(6.6

5)0.

65(8.0

3)0.

56(6.7

6)0.

87(8.1

0)0.

55(2.8

1)0.

53(1.6

5)−

0.67

(−0.

93)−

6.65

(−5.

44)

CA

PMbe

ta(p

ortf

olio

)0.

67(−

12.0

2)0.

76(−

10.3

7)0.

96(−

2.44

)1.

05(2.6

6)1.

09(3.8

6)1.

32(7.2

5)1.

34(4.7

0)1.

10(0.6

5)1.

08(0.3

3)

Vola

tility

(ann

.%)

14.3

314

.47

16.2

417

.32

19.0

626

.27

30.9

949

.52

63.3

0Sh

arpe

Rat

io(a

nn.)

0.68

0.83

0.68

0.62

0.79

0.38

0.32

−0.

11−

1.15

Mon

ths

496

503

505

505

504

505

415

344

203

Pane

lBM

arke

tLev

erag

eL

ow2

34

56

78

9H

igh

Exc

ess

retu

rn(a

vg.m

on.%

)1.

42(6.2

2)1.

35(7.0

5)1.

06(5.8

5)0.

98(4.9

4)0.

96(4.7

1)0.

89(4.1

5)0.

67(2.9

4)0.

47(1.9

4)0.

02(0.0

6)−

0.88

(−2.

73)

CA

PMal

pha

(ann

.%)

1.07

(10.

67)

1.01

(13.

00)

0.77

(10.

45)

0.66

(8.3

6)0.

63(7.4

3)0.

55(5.6

5)0.

31(2.7

9)0.

05(0.3

9)−

0.39

(−2.

65)−

1.34

(−7.

03)

Thr

ee-f

acto

ralp

ha(a

nn.%

)1.

28(1

5.42

)1.

07(1

4.26

)0.

75(1

0.40

)0.

61(8.0

3)0.

50(6.3

9)0.

37(4.3

8)0.

09(0.9

4)−

0.21

(−2.

19)−

0.68

(−5.

42)−

1.63

(−9.

78)

Four

-fac

tora

lpha

(ann

.%)

1.25

(14.

88)

1.02

(13.

43)

0.75

(10.

24)

0.64

(8.4

2)0.

55(6.9

5)0.

46(5.4

6)0.

25(2.7

3)−

0.03

(−0.

34)−

0.37

(−3.

57)−

1.29

(−8.

72)

CA

PMbe

ta(p

ortf

olio

)1.

04(2.0

3)0.

91(−

5.03

)0.

84(−

9.75

)0.

93(−

4.27

)0.

93(−

3.50

)0.

97(−

1.54

)1.

01(0.5

5)1.

08(3.0

9)1.

16(4.8

7)1.

31(7.2

4)Vo

latil

ity(a

nn.%

)18

.06

15.1

114

.28

15.6

516

.07

16.9

418

.01

19.0

521

.49

25.3

2Sh

arpe

Rat

io(a

nn.)

0.95

1.07

0.89

0.75

0.72

0.63

0.45

0.29

0.01

−0.

41

Mon

ths

520

520

520

520

520

520

520

520

520

520

Pane

lCIn

tere

stC

over

age

Hig

h9

87

65

43

2L

owE

xces

sR

etur

n(a

vg.m

on.%

)1.

20(5.3

6)1.

17(6.1

4)1.

16(6.1

6)1.

04(5.5

5)1.

10(5.7

7)1.

02(4.7

6)1.

01(4.4

0)0.

99(3.8

8)0.

33(1.1

1)−

0.05

(−0.

15)

CA

PMal

pha

(ann

.%)

0.84

(8.6

6)0.

86(1

1.54

)0.

86(1

0.44

)0.

73(9.5

1)0.

80(1

0.22

)0.

68(7.4

5)0.

64(6.0

3)0.

56(4.3

8)−

0.14

(−0.

88)−

0.52

(−2.

86)

Thr

ee-f

acto

ralp

ha(a

nn.%

)1.

08(1

3.64

)0.

92(1

2.69

)0.

86(1

0.84

)0.

65(9.2

6)0.

73(9.5

4)0.

55(6.4

1)0.

49(4.9

0)0.

44(3.6

6)−

0.20

(−1.

33)−

0.46

(−2.

67)

Four

-fac

tora

lpha

(ann

.%)

1.05

(13.

13)

0.89

(12.

12)

0.86

(10.

65)

0.67

(9.4

5)0.

76(9.9

3)0.

60(6.9

2)0.

59(5.9

2)0.

56(4.7

5)0.

04(0.3

2)−

0.24

(−1.

46)

CA

PMbe

ta(p

ortf

olio

)1.

03(1.2

2)0.

89(−

6.40

)0.

86(−

7.63

)0.

87(−

7.47

)0.

89(−

6.44

)0.

98(−

0.80

)1.

03(1.2

2)1.

14(4.9

3)1.

30(8.7

3)1.

33(8.3

2)Vo

latil

ity(a

nn.%

)17

.62

15.0

714

.87

14.7

915

.10

16.9

018

.06

20.0

923

.63

25.2

7Sh

arpe

Rat

io(a

nn.)

0.81

0.93

0.94

0.84

0.88

0.72

0.67

0.59

0.17

−0.

02

Mon

ths

520

520

520

520

520

520

520

520

520

520

Tabl

e2.

Mon

thly

exce

ssre

turn

sof

port

folio

sfo

rmed

onliq

uidi

tym

easu

res.

Thi

sta

ble

show

sex

cess

retu

rns,

alph

as,a

ndre

late

dqu

antit

ies

for

port

folio

sfo

rmed

onliq

uidi

tym

easu

res.

Att

hebe

ginn

ing

ofea

chca

lend

arm

onth

,Ias

sign

firm

sin

topo

rtfo

lios

acco

rdin

gto

deci

les

ofcu

rren

trat

io(P

anel

A),

quic

kra

tio(P

anel

B),

orw

orki

ngca

pita

lrat

io(P

anel

C)

for

the

prev

ious

mon

th.

The

port

folio

sar

eva

lue-

wei

ghte

dus

ing

the

prev

ious

mon

th’s

mar

kete

quity

valu

es,r

efre

shed

ever

yca

lend

arm

onth

,and

reba

lanc

edev

ery

cale

ndar

mon

thto

mai

ntai

nva

lue-

wei

ghtin

g.T

hehi

ghes

tand

low

estr

ealiz

edre

turn

isex

clud

edfo

reac

hpo

rtfo

lio.E

xces

sre

turn

isth

etim

e-se

ries

aver

age

ofth

em

onth

lypo

rtfo

liore

turn

s(i

npe

rcen

tage

s)in

exce

ssof

the

1-m

onth

US

Trea

sury

bill

rate

.Vol

atili

tyis

the

annu

aliz

edst

anda

rdde

viat

ion

ofth

em

onth

lyex

cess

retu

rn(i

npe

rcen

tage

s).S

harp

era

tiois

the

annu

aliz

edav

erag

eex

cess

retu

rndi

vide

dby

the

annu

aliz

edvo

latil

ity.C

APM

alph

aan

dC

APM

beta

are

the

inte

rcep

tand

slop

ees

timat

esfr

oma

time-

seri

esre

gres

sion

ofm

onth

lyex

cess

retu

rns

onth

eex

cess

retu

rns

ofth

eva

lue-

wei

ghte

dC

RSP

“mar

ket”

inde

x(M

KT

).T

hree

-an

dfo

ur-f

acto

ral

phas

are

the

inte

rcep

tsfr

omtim

e-se

ries

regr

essi

ons

ofm

onth

lyex

cess

retu

rns

onth

eth

ree

Fam

aan

dFr

ench

(199

3)fa

ctor

s(M

KT,

SMB

,and

HM

L)o

rthe

seth

ree

fact

ors

asw

ella

sth

eC

arha

rt(1

997)

fact

or(U

MD

).Pa

rent

hese

sin

subs

crip

tgiv

et-

stat

istic

s.Fo

rth

eex

cess

retu

rns

and

the

alph

as,t

henu

llva

lue

isze

ro,w

hile

itis

one

for

the

beta

.St

atis

tical

sign

ifica

nce

atth

e5%

leve

lis

indi

cate

din

bold

.

Pane

lAC

urre

ntra

tioH

igh

98

76

54

32

Low

Exc

ess

Ret

urn

(avg

.mon

.%)

1.17

(4.2

6)1.

24(4.9

6)1.

15(4.7

6)1.

11(5.2

2)1.

12(5.2

6)1.

11(5.2

2)0.

96(4.5

6)0.

99(5.3

0)0.

98(5.0

3)1.

09(6.1

5)

CA

PMal

pha

(ann

.%)

0.74

(5.4

6)0.

82(7.0

1)0.

76(7.0

7)0.

73(8.5

8)0.

75(8.6

7)0.

73(9.3

9)0.

61(8.0

7)0.

69(8.7

4)0.

66(7.5

2)0.

82(9.5

4)T

hree

-fac

tora

lpha

(ann

.%)

1.00

(9.2

3)1.

00(9.8

0)0.

87(8.4

2)0.

77(8.9

6)0.

75(8.7

1)0.

75(9.5

1)0.

58(7.8

5)0.

62(8.4

5)0.

57(6.7

2)0.

70(8.9

3)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

98(8.9

3)1.

00(9.6

4)0.

93(8.9

2)0.

79(9.0

9)0.

78(8.8

2)0.

79(9.8

2)0.

59(7.8

9)0.

61(8.2

1)0.

60(6.9

4)0.

72(9.0

7)

CA

PMbe

ta(p

ortf

olio

)1.

20(6.7

1)1.

15(5.7

7)1.

08(3.5

6)1.

01(0.6

3)1.

02(0.8

9)1.

03(1.9

1)0.

99(−

0.72

)0.

86(−

8.10

)0.

88(−

6.40

)0.

79(−

10.8

7)Vo

latil

ity(a

nn.%

)21

.66

19.7

319

.00

16.8

416

.86

16.8

416

.57

14.6

915

.37

14.0

4Sh

arpe

Rat

io(a

nn.)

0.65

0.75

0.72

0.79

0.80

0.79

0.69

0.80

0.76

0.93

Mon

ths

520

520

520

520

520

520

520

520

520

520

Pane

lBQ

uick

ratio

Hig

h9

87

65

43

2L

owE

xces

sR

etur

n(a

vg.m

on.%

)1.

35(4.9

4)1.

23(4.9

1)1.

17(5.1

6)1.

10(5.2

7)1.

03(4.9

9)1.

02(5.0

4)1.

01(5.1

7)0.

94(4.8

3)1.

06(5.5

0)1.

07(5.4

8)

CA

PMal

pha

(ann

.%)

0.93

(6.7

2)0.

85(7.0

0)0.

77(8.0

7)0.

73(9.6

4)0.

67(8.3

4)0.

65(7.9

7)0.

69(9.2

6)0.

63(7.6

8)0.

75(7.7

6)0.

71(7.7

1)T

hree

-fac

tora

lpha

(ann

.%)

1.22

(11.

17)

1.06

(9.8

4)0.

90(9.8

7)0.

76(1

0.08

)0.

66(8.2

0)0.

62(7.7

9)0.

65(8.8

9)0.

52(6.8

2)0.

63(6.8

3)0.

65(7.0

8)Fo

ur-f

acto

ralp

ha(a

nn.%

)1.

21(1

0.86

)1.

05(9.6

0)0.

94(1

0.15

)0.

77(1

0.02

)0.

64(7.7

8)0.

60(7.3

1)0.

70(9.4

5)0.

56(7.3

1)0.

66(7.1

3)0.

69(7.3

5)

CA

PMbe

ta(p

ortf

olio

)1.

19(6.1

5)1.

11(4.0

1)1.

07(3.0

8)1.

01(0.8

3)1.

00(−

0.26

)0.

95(−

2.59

)0.

92(−

4.96

)0.

90(−

5.47

)0.

84(−

7.73

)0.

88(−

5.72

)

Vola

tility

(ann

.%)

21.5

619

.87

17.8

616

.48

16.3

515

.94

15.4

415

.40

15.1

615

.37

Shar

peR

atio

(ann

.)0.

750.

750.

780.

800.

760.

770.

790.

730.

840.

83

Mon

ths

520

520

520

520

520

520

520

520

520

520

Pane

lCW

orki

ngca

pita

lrat

ioH

igh

98

76

54

32

Low

Exc

ess

Ret

urn

(avg

.mon

.%)

1.36

(4.5

6)1.

32(5.2

5)1.

26(5.5

3)1.

19(5.6

3)1.

09(5.0

8)1.

00(4.6

5)0.

94(4.6

9)1.

04(5.2

8)0.

94(4.8

1)1.

07(6.2

3)

CA

PMal

pha

(ann

.%)

0.84

(5.4

3)0.

93(7.7

7)0.

86(9.1

2)0.

82(9.7

4)0.

71(8.5

9)0.

61(7.8

2)0.

58(6.8

4)0.

73(8.6

8)0.

62(6.7

8)0.

81(8.9

3)T

hree

-fac

tora

lpha

(ann

.%)

1.17

(9.2

1)1.

14(1

0.94

)0.

94(1

0.07

)0.

88(1

0.50

)0.

77(9.5

7)0.

67(8.6

6)0.

53(6.4

0)0.

64(7.8

5)0.

52(6.0

2)0.

70(8.2

6)Fo

ur-f

acto

ralp

ha(a

nn.%

)1.

15(8.9

1)1.

17(1

0.96

)0.

96(1

0.17

)0.

90(1

0.59

)0.

84(1

0.31

)0.

72(9.1

5)0.

52(6.1

2)0.

67(8.1

1)0.

53(5.9

9)0.

72(8.4

1)

CA

PMbe

ta(p

ortf

olio

)1.

31(8.9

5)1.

14(5.0

8)1.

08(3.5

6)1.

01(0.4

8)1.

03(1.5

4)1.

04(2.1

2)0.

94(−

3.24

)0.

91(−

4.92

)0.

87(−

6.49

)0.

75(−

12.3

4)Vo

latil

ity(a

nn.%

)23

.56

19.9

317

.93

16.6

616

.90

16.9

115

.80

15.5

715

.43

13.6

3Sh

arpe

Rat

io(a

nn.)

0.69

0.80

0.84

0.86

0.77

0.71

0.71

0.80

0.73

0.95

Mon

ths

520

520

520

520

520

520

520

520

520

520

Tabl

e3.

Hig

hca

shm

inus

low

cash

retu

rnsa

cros

scre

ditr

atin

gsT

his

tabl

esh

ows

retu

rns,

alph

as,a

ndre

late

dqu

antit

ies

forc

ondi

tiona

l-so

rted

port

folio

sfo

rmed

oncr

edit

ratin

gsan

dliq

uidi

tym

easu

res.

Att

hebe

ginn

ing

ofea

chca

lend

arm

onth

,Ias

sign

firm

sin

to3

port

folio

sac

cord

ing

toth

eirc

redi

trat

ing

(Aaa

-A,B

aa-B

,and

Caa

-C)a

ndth

enin

toth

ree

port

folio

sac

cord

ing

tote

rcile

sof

curr

entr

atio

(Pan

elA

),qu

ick

ratio

(Pan

elB

),or

wor

king

capi

talr

atio

(Pan

elC

).T

hepo

rtfo

lios

are

valu

e-w

eigh

ted

usin

gth

epr

evio

usm

onth

’sm

arke

tequ

ityva

lues

,ref

resh

edev

ery

cale

ndar

mon

th,a

ndre

bala

nced

ever

yca

lend

arm

onth

tom

aint

ain

valu

e-w

eigh

ting.

The

high

esta

ndlo

wes

trea

lized

retu

rnis

excl

uded

fore

ach

port

folio

.Ret

urn

isth

etim

e-se

ries

aver

age

ofth

em

onth

lyre

turn

sfo

rthe

port

folio

that

islo

ngth

e“h

igh

cash

”po

rtfo

lioan

dsh

ortt

he“l

owca

sh”

port

folio

.Vo

latil

ityis

the

annu

aliz

edst

anda

rdde

viat

ion

ofth

em

onth

lyre

turn

(in

perc

enta

ges)

.Sh

arpe

ratio

isth

ean

nual

ized

aver

age

retu

rndi

vide

dby

the

annu

aliz

edvo

latil

ity.

CA

PMal

pha

and

CA

PMbe

taar

eth

ein

terc

epta

ndsl

ope

estim

ates

from

atim

e-se

ries

regr

essi

onof

mon

thly

retu

rns

onth

eex

cess

retu

rns

ofth

eva

lue-

wei

ghte

dC

RSP

“mar

ket”

inde

x(M

KT

).T

hree

-and

four

-fac

tora

lpha

sar

eth

ein

terc

epts

from

time-

seri

esre

gres

sion

sof

mon

thly

retu

rns

onth

eth

ree

Fam

aan

dFr

ench

(199

3)fa

ctor

s(M

KT,

SMB

,and

HM

L)o

rthe

seth

ree

fact

ors

asw

ella

sth

eC

arha

rt(1

997)

fact

or(U

MD

).Pa

rent

hese

sin

subs

crip

tgiv

et-

stat

istic

s.Fo

rthe

retu

rns

and

the

alph

as,t

henu

llva

lue

isze

ro,w

hile

itis

one

fort

hebe

ta.S

tatis

tical

sign

ifica

nce

atth

e5%

leve

lis

indi

cate

din

bold

.

Pane

lA:

Cur

rent

Rat

ioA

aa-A

Baa

-BC

aa-C

Hig

hca

shm

inus

low

cash

Hig

hca

shm

inus

low

cash

Hig

hca

shm

inus

low

cash

Ret

urn

(avg

.mon

.%)

0.01

(0.0

5)0.

37(2.3

7)1.

16(2.2

1)

CA

PMal

pha

(ann

.%)

−0.

05(−

0.43

)0.

29(1.9

2)1.

05(2.0

0)T

hree

-fac

tora

lpha

(ann

.%)

0.07

(0.5

9)0.

40(2.9

9)1.

14(2.1

7)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

09(0.8

4)0.

37(2.7

4)0.

97(1.8

3)

CA

PMbe

ta(p

ortf

olio

)0.

19(−

30.9

4)0.

26(−

22.2

8)0.

25(−

6.53

)

Vola

tility

(ann

.%)

9.58

12.1

936

.39

Shar

peR

atio

(ann

.)0.

010.

370.

38M

onth

s50

250

540

4

Pane

lB:

Qui

ckR

atio

Aaa

-AB

aa-B

Caa

-CH

igh

cash

min

uslo

wca

shH

igh

cash

min

uslo

wca

shH

igh

cash

min

uslo

wca

shR

etur

n(a

vg.m

on.%

)−

0.05

(−0.

41)

0.34

(2.3

2)1.

24(2.3

3)

CA

PMal

pha

(ann

.%)

−0.

09(−

0.81

)0.

26(1.8

7)1.

08(2.0

4)T

hree

-fac

tora

lpha

(ann

.%)

0.08

(0.7

5)0.

48(3.9

4)1.

39(2.7

0)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

04(0.3

6)0.

43(3.4

8)1.

41(2.6

9)

CA

PMbe

ta(p

ortf

olio

)0.

12(−

32.9

1)0.

24(−

24.5

9)0.

36(−

5.43

)

Vola

tility

(ann

.%)

9.17

11.3

237

.08

Shar

peR

atio

(ann

.)−

0.06

0.36

0.40

Mon

ths

503

505

404

Pane

lC:

Wor

king

Cap

italR

atio

Aaa

-AB

aa-B

Caa

-CH

igh

cash

min

uslo

wca

shH

igh

cash

min

uslo

wca

shH

igh

cash

min

uslo

wca

shR

etur

n(a

vg.m

on.%

)0.

08(0.6

4)0.

43(2.8

1)1.

52(2.9

6)

CA

PMal

pha

(ann

.%)

0.02

(0.1

3)0.

35(2.4

1)1.

46(2.8

4)T

hree

-fac

tora

lpha

(ann

.%)

0.17

(1.4

9)0.

53(4.3

1)1.

52(2.9

4)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

22(1.9

2)0.

51(4.1

0)1.

35(2.5

8)

CA

PMbe

ta(p

ortf

olio

)0.

18(−

30.5

6)0.

27(−

22.1

7)0.

14(−

7.59

)

Vola

tility

(ann

.%)

9.67

11.9

835

.77

Shar

peR

atio

(ann

.)0.

100.

430.

51M

onth

s50

350

440

5

Tabl

e4.

Hig

hso

lven

cym

inus

low

solv

ency

retu

rnsu

sing

cred

itra

tings

acro

ssliq

uidi

tym

easu

resT

his

tabl

esh

ows

retu

rns,

alph

as,a

ndre

late

dqu

antit

iesf

orco

nditi

onal

-so

rted

port

folio

sfo

rmed

onliq

uidi

tym

easu

res

and

cred

itra

tings

.A

tthe

begi

nnin

gof

each

cale

ndar

mon

th,I

assi

gnfir

ms

into

3po

rtfo

lios

acco

rdin

gto

terc

iles

ofcu

rren

tra

tio(P

anel

A),

quic

kra

tio(P

anel

B),

orw

orki

ngca

pita

lrat

io(P

anel

C),

and

then

into

thre

egr

oups

acco

rdin

gto

cred

itra

ting

(Aaa

-A,B

aa-B

,and

Caa

-C).

The

port

folio

sar

eva

lue-

wei

ghte

dus

ing

the

prev

ious

mon

th’s

mar

kete

quity

valu

es,r

efre

shed

ever

yca

lend

arm

onth

,and

reba

lanc

edev

ery

cale

ndar

mon

thto

mai

ntai

nva

lue-

wei

ghtin

g.T

hehi

ghes

tand

low

estr

ealiz

edre

turn

isex

clud

edfo

rea

chpo

rtfo

lio.

Ret

urn

isth

etim

e-se

ries

aver

age

ofth

em

onth

lyre

turn

sfo

rth

epo

rtfo

lioth

atis

long

the

“hig

hso

lven

cy”

(Aaa

-A)

port

folio

and

shor

tthe

“low

solv

ency

”(C

aa-C

)po

rtfo

lio.

Vola

tility

isth

ean

nual

ized

stan

dard

devi

atio

nof

the

mon

thly

retu

rn(i

npe

rcen

tage

s).

Shar

pera

tiois

the

annu

aliz

edav

erag

ere

turn

divi

ded

byth

ean

nual

ized

vola

tility

.C

APM

alph

aan

dC

APM

beta

are

the

inte

rcep

tand

slop

ees

timat

esfr

oma

time-

seri

esre

gres

sion

ofm

onth

lyre

turn

son

the

exce

ssre

turn

sof

the

valu

e-w

eigh

ted

CR

SP“m

arke

t”in

dex

(MK

T).

Thr

ee-

and

four

-fac

tor

alph

asar

eth

ein

terc

epts

from

time-

seri

esre

gres

sion

sof

mon

thly

retu

rns

onth

eth

ree

Fam

aan

dFr

ench

(199

3)fa

ctor

s(M

KT,

SMB

,and

HM

L)o

rthe

seth

ree

fact

ors

asw

ella

sth

eC

arha

rt(1

997)

fact

or(U

MD

).Pa

rent

hese

sin

subs

crip

tgiv

et-

stat

istic

s.Fo

rthe

retu

rns

and

the

alph

as,t

henu

llva

lue

isze

ro,w

hile

itis

one

fort

hebe

ta.S

tatis

tical

sign

ifica

nce

atth

e5%

leve

lis

indi

cate

din

bold

.

Pane

lA:

Cur

rent

Rat

ioH

igh

cash

Med

ium

cash

Low

cash

Aaa

-Am

inus

Caa

-CA

aa-A

min

usC

aa-C

Aaa

-Am

inus

Caa

-CR

etur

n(a

vg.m

on.%

)0.

33(0.7

0)0.

67(1.5

0)0.

98(2.0

7)

CA

PMal

pha

(ann

.%)

0.55

(1.2

0)0.

88(2.0

2)1.

20(2.5

9)T

hree

-fac

tora

lpha

(ann

.%)

0.63

(1.3

8)0.

81(1.8

8)1.

18(2.7

4)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

46(1.0

0)0.

51(1.2

0)0.

81(1.9

1)

CA

PMbe

ta(p

ortf

olio

)−

0.51

(−14.9

6)−

0.49

(−15.7

3)−

0.51

(−14.7

5)Vo

latil

ity(a

nn.%

)32

.63

30.6

333

.55

Shar

peR

atio

(ann

.)0.

120.

260.

35M

onth

s39

739

341

6

Pane

lB:

Qui

ckR

atio

Hig

hca

shM

ediu

mca

shL

owca

shA

aa-A

min

usC

aa-C

Aaa

-Am

inus

Caa

-CA

aa-A

min

usC

aa-C

Ret

urn

(avg

.mon

.%)

0.61

(1.3

9)1.

20(2.4

7)1.

28(2.9

8)

CA

PMal

pha

(ann

.%)

0.92

(2.1

7)1.

38(2.8

7)1.

47(3.4

8)T

hree

-fac

tora

lpha

(ann

.%)

0.97

(2.3

2)1.

30(2.8

1)1.

69(4.4

0)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

82(1.9

4)0.

82(1.8

0)1.

43(3.7

2)

CA

PMbe

ta(p

ortf

olio

)−

0.60

(−16.3

7)−

0.41

(−13.1

4)−

0.40

(−14.8

2)Vo

latil

ity(a

nn.%

)30

.40

32.8

730

.09

Shar

peR

atio

(ann

.)0.

240.

440.

51M

onth

s40

037

841

3

Pane

lC:

Wor

king

Cap

italR

atio

Hig

hca

shM

ediu

mca

shL

owca

shA

aa-A

min

usC

aa-C

Aaa

-Am

inus

Caa

-CA

aa-A

min

usC

aa-C

Ret

urn

(avg

.mon

.%)

0.38

(0.8

9)0.

42(0.9

6)1.

24(2.6

2)

CA

PMal

pha

(ann

.%)

0.54

(1.2

9)0.

73(1.7

4)1.

43(3.0

3)T

hree

-fac

tora

lpha

(ann

.%)

0.58

(1.4

3)0.

85(2.0

5)1.

37(3.1

6)Fo

ur-f

acto

ralp

ha(a

nn.%

)0.

50(1.1

9)0.

52(1.2

5)0.

91(2.1

5)

CA

PMbe

ta(p

ortf

olio

)−

0.36

(−14.6

9)−

0.65

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23

AppendicesA Additional details and proofsThis appendix gives additional details of the model’s de-velopment that were omitted from the main text, as wellas the proofs of the paper’s propositions and corollaries.

A.1 Change of measureThe firm’s cumulated earnings and the economy’sstochastic discount factor were given by the P-dynamicsin (1) and (3), i.e.

dXt = µPXtdt + σXtdWPt

and

dΛt = −rΛtdt − ηΛtdZPt ,

where WPt and ZPt are standard P-Brownian motionswith correlation ρ. There thus exists a standard P-Brownian motion, ZPt , independent of ZPt , such that

WPt = ρZPt +√

1 − ρ2 ZPt .

Now, suppose Λ0 = 1, let Lt = ertΛt, and note that Lt

may be written as

Lt = exp(− 1

2η′η t − η′ ZPt

),

where η = (η, 0)′ and ZPt = (ZPt , ZPt )′. Fix T and define a

risk-neutral pricing measure Q by dQdP = LT . Girsanov’s

Theorem then gives that the process ZQt = (ZQt , ZQt ), de-

fined by

ZQt = ZPt + ηt,

is a standard Q-Brownian motion.Finally, let WQt = ρZQt +

√1 − ρ2 ZQt , and note that

WQt is standard Q-Brownian motion which may be writ-ten as

WQt = ρ(ZPt + ηt

)+

√1 − ρ2 ZPt = WPt + ηρt.

It thus follows that the Q-dynamics of the firm’s cumu-lated earnings process may be written as

dXt = µQXtdt + σXtdWQt ,

where µQ = µP − ηρσ, exactly as in (4).

A.2 Proof of Proposition 1Suppose cash holdings are of the form Ct = C(Xt) forsome twice continuously differentiable function C(X).It follows by Ito’s Lemma along with (2) and (4) that

dDt =[rC(Xt) + (1 − τ)(µQXt − k)

−CX(Xt)µQXt −12CXX(Xt)σ2X2

t

]dt

+[(1 − τ)σXt −CX(Xt)σXt

]dWQt , (17)

where subscripts denote partial deviates. Since the firmhas no access to external financing beyond time 0, itcan, in particular, not issue additional equity. This im-plies that the cumulated dividend process has to be non-decreasing for all t > 0, which is satisfied if and onlyif i) the drift-term in (17) is nonnegative and ii) thevolatility-term in (17) is zero.

The requirement that the volatility-term of dDt has tobe zero implies the simple differential equation CX(X) =

(1 − τ), which has the general solution

C(X) = (1 − τ)X + L

for some constant L. Plugging this solution into thedrift-term of dDt, and imposing the requirement that thedrift has to be nonnegative, it follows that the solutionhas to satisfy

C(X) ≥ (1 − τ) kr

for all X. To solve for the constant L, note that sinceCX(X) = (1 − τ) > 0, the target cash level is increasingin X. As the insolvency trigger X? is a lower boundfor X, it follows that C(X) ≥ C(X?) ≥ (1 − τ) k

r for allX. Combining this inequality with the general solutionthen gives that the constant L must satisfy

L ≥ −(1 − τ)X? + (1 − τ) kr = (1 − τ)

[−X? + k

r

].

By choosing L as low as possible (i.e. the value wherethe inequality is binding), the expression for the solutiongiven in Proposition 1 follows.

The remaining part of the proof is to assert that thetarget cash level, C(X), is twice continuously differen-tiable in current cumulated earnings, X. If X′ ≥ X?

is a discontinuity point for C(X), then C(X) must jumpdownwards at X′, or else it would not have been the

24

target cash level an (infinitesimal) instance before X′.However, even a downwards jump at X′ would implythat C(X) is not the target cash level an instance beforeX′. Hence, C(X) must be continuous for all X.

On the other hand, if C(X) is continuous but non-differentiable at some X′ ≥ X?, then C(X) would for allX , X′ satisfy the same differential equation as before.However, this differential equation and the associatedinequalities imply a solution that is twice continuouslydifferentiable for all X. �

A.3 Proof of Proposition 2

To prove that the conjectured dividend policy in (7),dD?

t , is optimal, it must first be shown that it attains themaximal equity value in (5), E(X,C), and, second, thatno other dividend policy implies a higher equity value—that is, it must be shown that

i) E(X,C) = EQt,X,C

[∫ τ

t e−r(u−t) dD?u + e−r(τ−t)Cτ

], and

that

ii) E(X,C) ≥ EQt,X,C[∫ τ

t e−r(u−t) dDu + e−r(τ−t)Cτ

]for

all nonnegative, non-decreasing, adapted dividendprocesses, Dt.

In both parts, the proof will make use of the followingtwo general results.

First, note that a general dividend process, Dt, is notnecessarily continuous, meaning that it, and the corre-sponding cash process, Ct, may have jumps. Still, a gen-eral Dt may be decomposed into the sum of its purelycontinuous part and its jumps: Dt = Dc

t +∑

s≤t(Ds−Ds−).Using this along with the generalized Ito’s Lemma andthe dynamics of Xt and Ct in (2) and (4), it follows thatthe discounted gains process of equity, as defined byGt = e−rtE(Xt,Ct) +

∫ t0 e−rsdDs, has the dynamics

dGt = e−rt [−rE(Xt,Ct−) +AE(Xt,Ct−)] dt

+ e−rt [1 − EC(Xt,Ct−)] dDt

+ e−rtσXt [EX(Xt,Ct−) + (1 − τ)EC(Xt,Ct−)] dWQt+ e−rt [E(Xt,Ct) − E(Xt,Ct−)

−EC(Xt,Ct−) (Ct −Ct−)] . (18)

Here, the infinitesimal operator,AE(X,C), is given by

AE(X,C) = µQXEX(X,C)+

[rC + (1 − τ)(µQX − k)

]EC(X,C)

+ 12σ

2X2EXX(X,C)

+ 12 (1 − τ)2σ2X2ECC(X,C)

+ (1 − τ)σ2X2EXC(X,C),

while the jumps of the cash process, by (2), are given byCt −Ct− = −(Dt −Dt−) for all t. Since equity is a tradedasset, Gt is a (possibly non-continuous) Q-martingale,which will impose restrictions on the first two terms in(18) depending on the specific form of Dt.

Second, it follows by definition of the equity valuefunction in (5) that

E(Xt,Ct)→ Cτ for t → τ. (19)

Proof of Part i)

Assume that dDt = dD?t for all t, and consider, in turn,

the three regions implied by the conjectured dividendpolicy in (7).

First, when 0 < Ct < C, dividend payouts are givenby dD?

t = 0 for all t, which is continuous and impliesthat Ct is also continuous. By (18), this implies that Gt isaQ-martingale if and only if −rE+AE = 0. Combiningthis with the implied form of (18) then gives

d(e−rtE(Xt,Ct))= e−rtσXt [EX(Xt,Ct) + (1 − τ)EC(Xt,Ct)] dWQt .

Assuming that the derivatives of the equity value func-tion are sufficiently integrable, it follows by integratingthis expression between t and v ∧ τ for any v ≥ t andtaking expectations that

e−rtE(X,C) = EQt,X,C

[e−r(v∧τ)E(Xv∧τ,Cv∧τ)

].

Since v was arbitrary, it follows by letting v → ∞ andapplying (19) that

E(X,C) = EQt,X,C

[e−r(τ−t)Cτ

],

which proves part i) in the first region.Next, when Ct = C(Xt), dividend payouts are given

by dD?t = rC(Xt) − (1 − τ)k dt = r(1 − τ)(Xt − X?) dt,

25

which again is continuous. In this region, equity valueis given by E(X,C(X)) = E(X) and satisfies the ODE in(9)—hence, −rE +AE = 0. Furthermore, Proposition3 and its proof (see Section A.4 of this appendix) givethat the solution to (9) is separable in X and C and lin-ear in C, which implies EC(X,C(X)) = 1. Combiningthese results with the implied form of (18), integratingbetween t and v ∧ τ, and taking expectations, one thenarrives at

E(X,C) = EQt,X,C

[∫ v∧τ

te−r(u−t)dD?

u

+e−r((v∧τ)−t)E(Xv∧τ,Cv∧τ)].

Letting v→ ∞ and applying (19) thus proves part i) forthe second region.

Finally, when Ct > C(Xt), dividend payouts aredD?

t = Ct − C(Xt) and thus not continuous. However,since this by (7) occurs if and only if Ct− = C(Xt),it holds that E(Xt,Ct−) = E(Xt,C(Xt)). Therefore, byProposition 3 and its proof, equity value at time t−is separable in X and C and linear in C, which givesEC(Xt,Ct−) = EC(Xt,C(Xt)) = 1. Combining these re-sults with (18), Gt is a martingale if and only if

rE(Xt ,Ct−) +AE(Xt,Ct−)= −rE(Xt,C(Xt)) +AE(Xt,C(Xt)) = 0,

while the jump-term of (18) becomes

E(Xt ,Ct) − E(Xt,Ct−) − EC(Xt,Ct−) (Ct −Ct−)= Ct −C(Xt) − (Ct −C(Xt)) = 0.

Using the implied form of (18) and repeating the stepsof the proof in the second region thus also proves part i)in the third region.

Proof of Part ii)

From part 1), the dividend policy in (7) attains themaximal equity value in (5), E(X,C). The task isnow to show that no other nonnegative, non-decreasing,adapted dividend process implies a higher equity value.Let Dt = Dc

t +∑

s≤t(Ds − Ds−) be such a dividend pro-cess and let Ct be its corresponding cash process withjumps Ct − Ct− = −(Dt − Dt−) for all t. Plugging this

form of Dt into (18), the discounted gains process underthe dividend policy Dt has the dynamics

dGt = e−rt [−rE(Xt,Ct−) +AE(Xt,Ct−)] dt

+ e−rtdDt − e−rtEC(Xt,Ct−) dDct

+ e−rtσXt [EX(Xt,Ct−) + (1 − τ)EC(Xt,Ct−)] dWQt+ e−rt [E(Xt,Ct) − E(Xt,Ct−)] .

By the proof of part i), the maximal equity value satis-fies −rE(X,C) +AE(X,C) = 0 for all X and C. Hence,once more assuming that the deviates of the maximalequity value function are sufficiently integrable, it fol-lows by integrating the last expression between t andv ∧ τ, taking expectations, and rearranging, that

E(Xt,Ct) = EQt,X,C

[e−r((v∧τ)−t)E(Xv∧τ,Cv∧τ)

]+ EQt,X,C

[∫ v∧τ

te−r(u−t)EC(Xu,Cu−) dDc

u

]

− EQt,X,C

∑t≤u≤v∧τ

e−r(u−t) [E(Xt,Ct) − E(Xt,Ct−)]

.Since it in general holds that EC(X,C) ≥ 1 (see foot-note 11), the middle-term in the above expression canbe bounded from below. Furthermore, the same prop-erty and the relation Ct − Ct− = −(Dt − Dt−) ≤ 0 (sincethe dividend process is non-decreasing) imply

− [E(Xt,Ct) − E(Xt,Ct−)] ≥ Dt − Dt−,

which thus bounds the last term. Therefore,

E(Xt,Ct) ≥ EQt,X,C

[e−r((v∧τ)−t)E(Xv∧τ,Cv∧τ)

]+ EQt,X,C

[∫ v∧τ

te−r(u−t)dDc

u

]

+ EQt,X,C

∑t≤u≤v∧τ

e−r(u−t)(Du − Du−)

= EQt,X,C

[e−r((v∧τ)−t)E(Xv∧τ,Cv∧τ)

]+ EQt,X,C

[∫ v∧τ

te−r(u−t)dDu

],

where the last equality uses Dt = Dct +

∑s≤t(Ds − Ds−).

Letting v → ∞ and applying (19) thus proves part ii),which completes the proof. �

26

A.4 Proof of Proposition 3

The ODE (9) has the general solution

E(X) = (1 − τ)r[

Xr−µQ −

X?

r

]+ M1Xφ+

+ M2Xφ−

= C(X) + (1 − τ)[µQXr−µQ −

kr

]+ M1Xφ+

+ M2Xφ− ,

where the second equality follows by the form of thetarget cash holdings, C(X), in (6). Here, M1 and M2are real-valued constants to be determined by boundaryconditions, while the exponents φ+ and φ− are given by

φ± =σ2 − 2µ ±

√(σ2 − 2µ)2 + 8rσ2

2σ2 .

Since φ+ > 1, the value matching condition (10) impliesM1 = 0. Given this, the limited liability condition (11)implies

M2 = (X?)−φ−

(1 − τ)[

kr −

µQX?

r−µQ

].

By plugging the constants into the general solution, anddefining πQ(X) = (X/X?)φ

−

, the expression for E(X) inthe proposition follows. �

A.5 Proof Corollary ??

By the chain rule and the expressions in (12) and (6) forthe equilibrium equity value and the target cash level,the parametric derivative of E(X) with respect to C(X)is given by

dE(X)dC(X)

=∂E(X)∂X

(∂C(X)∂X

)−1

= 1 +µQ

r − µQ+

[kr−µQX?

r − µQ

]φ−πQ(X)

X.

Since πQ(X)→ 0 if and only if X → ∞, it follows that

limπQ(X)→0

dE(X)dC(X)

= 1 +µQ

r − µQ> 1,

because r > µQ > 0 by assumption, thus proving part i).

On the other hand, since πQ(X) → 1 if and only ifX → X?, it follows that

limπQ(X)→1

dE(X)dC(X)

= 1 +µQ

r − µQ+

[k

rX? −µQ

r − µQ

]φ−

= 1 +µQ

r − µQ−

µQ

r − µQ= 1,

using the expression for X? in (13), proving part ii). �

A.6 Proof of Proposition 4

Using the expression in (3) for the equilibrium equityvalue, it follows by differentiating that

ψEt = Xt

EX(Xt)E(Xt)

=Xt

E(Xt)

[(1 − τ) + (1 − τ) µQ

r−µQ

+(1 − τ)[

kr −

µQX?

r−µQ

]φ−πQ(Xt)

Xt

]=

1E(Xt)

[E(Xt) −C(Xt) + (1 − τ)Xt + (1 − τ) k

r

−(1 − τ)(1 − φ−)[

kr −

µQX?

r−µQ

]πQ(Xt)

],

where the last equality follows by the expressions forE(X) and C(X) in (12) and (6). The form of ψE

t given inthe proposition then follows by dividing E(Xt) into eachterm in the parentheses. �

A.7 Proof of Corollary 4.1

The proofs of parts i) and ii) of the corollary will makeuse of the following results about the earnings sensitiv-ity, ψE

t .First, using the expression for C(X) in (6), the sensi-

tivity can be compactly written as

ψEt = 1 +

(1 − τ)X? − (1 − φ−) M πQ(Xt)E(Xt)

, (20)

where M = (1 − τ)[

kr −

µQX?

r−µQ

].

27

Second, by differentiating the form of ψEt in (20) with

respect to Xt, it follows that

∂ψEt

∂Xt= −

(1 − φ−) M φ−πQ(Xt)Xt

E(Xt)

−EX(Xt)

[(1 − τ)X? − (1 − φ−) M πQ(Xt)

]E2(Xt)

= −(1 − φ−)M φ−πQ(Xt)

Xt

E(Xt)−ψE

t (ψEt − 1)Xt

,

where the last equality uses the definition ψEt = Xt

EX (Xt)E(Xt)

as well as (20). Hence, by the chain rule, the parametricderivative of ψE

t with respect to πQ(Xt) is given by

dψEt

dπQ(Xt)=

∂ψEt

∂Xt

(∂πQ(Xt)∂Xt

)−1

= −(1 − φ−)M

E(Xt)−ψE

t (ψEt − 1)

φ−πQ(Xt). (21)

Proof of Part i)

Suppose πQ(Xt) → 0. Then Xt → ∞, so the expressionin (6) for C(Xt) gives that

limπQ(Xt)→0

C(Xt) = (1 − τ)Xt < (1 − τ) µQXt

r−µQ ,

since r > µQ > 12 r by assumption.

Next, since φ− < 0, Proposition 3 gives that as X →∞, πQ(X) approaches zero faster than E(X) approachesinfinity. Combined with the form of ψE

t in (20), it thusfollows that

limπQ(Xt)→0

ψEt > 1. (22)

Finally, consider the expression in (21). As πQ(Xt)→0, E(Xt) → ∞, implying that the first term vanishes.By (22), the numerator of the second term is asymptoti-cally positive, while φ− < 0 implies that the denomina-tor is negative. The second term is therefore as a wholeasymptotically positive, implying in total that

limπQ(Xt)→0

dψEt

dπQ(Xt)> 0,

which proves part i).

Proof of Part ii)

Suppose now that πQ(Xt)→ 1. Then Xt → X?, so

limπQ(Xt)→1

C(Xt) = (1 − τ) kr > (1 − τ) µ

QX?

r−µQ ,

by the expression in (13) for X?.Next, the limited liability condition (11) and the ex-

pression for C(X) in (6) give that

limπQ(Xt)→1

ψEt = 1 +

(1 − τ)X? − (1 − φ−) MC(X?)

=r − µQ

µQφ−

φ− − 1< 1, (23)

because µQ > 12 r by assumption and φ−

φ−−1 ∈ (0, 1). Notethat µQ > 1

2 r is a sufficient but not necessary conditionfor (23) to hold.

Finally, consider again the expression in (21). AsπQ(Xt) → 1, E(Xt) → C(X?) by the limited liabil-ity condition (11), which implies that the first term isasymptotically negative. By (23), the numerator ofthe second term will be asymptotically negative, whileφ− < 0 implies that the denominator is negative. Thesecond term is therefore as a whole asymptotically neg-ative, implying in total that

limπQ(Xt)→1

dψEt

dπQ(Xt)< 0,

thus proving part ii) and completing the proof. �

28