Copyright by Chao Bian 2015

Transcription

1 Copyright by Chao Bian 2015

2 The Dissertation Committee for Chao Bian certifies that this is the approved version of the following dissertation: Two Essays on Asset Pricing Committee: Sheridan Titman, Supervisor Aydoğan Altı Andrés Donangelo Travis Johnson Efstathios (Stathis) Tompaidis

3 Two Essays on Asset Pricing by Chao Bian, B.S.; M.S.; M.S. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN August 2015

4 Dedicated to my parents.

5 Acknowledgments I would like to take this opportunity to thank all those who have made it possible for me to complete this work. My advisor, Sheridan Titman, has provided an enormous amount of guidance and advice throughout my years at UT Austin. He has always been very understanding and supportive, for which I am very grateful. I deeply admire his great knowledge and his work ethic. I owe my special thanks to Andrés Donangelo for his advice, guidance, and critique to make this thesis possible. I am thankful to Aydoğan Altı for his advice and comments. I would like to thank Stathis Tompaidis for his advice about numerical analysis. I thank Travis Johnson and Fernando Anjos for taking the time to read my work and serve on my thesis committee. I thank my classmates and fellow graduate students at UT Austin for their friendship and help. My officemates deserve my special thanks for the countless discussions we had about asset pricing and economics in general. Last but not least, I would like to thank my family. Through the years of my study, my family has been very patient and supportive. They have motivated me through this journey. The unwavering support and love of my family have inspired me to fulfill my goal. v

6 Two Essays on Asset Pricing Publication No. Chao Bian, Ph.D. The University of Texas at Austin, 2015 Supervisor: Sheridan Titman Chapter one of the thesis studies the relationship between corporate cash holdings and expected stock returns. I develop a real options model with external financing costs to relate endogenous firm cash holding policy to expected returns. In the model, firms choose to hold cash optimally to finance investments when productivity is high; and to provide liquidity when productivity is low. The optimal cash holding policy implies a positive relationship between cash-to-assets ratios of firms and expected returns. In addition, I show this positive relation is conditional in nature: it is stronger when cash flow risk dominates the total risk of firms; the positive relation diminishes as firms derive more risk from the investment channel. Using a data set of U.S. pubic companies, I provide empirical support for the predictions of the model. Chapter two investigates the role of intangible capital for firm valuation and risk in the cross section of publicly traded firms. Intangible capital is a durable production factor that is costly to accumulate, hard to evaluate, and vi

7 cannot be easily transferred among firms. I develop a model that firms are endowed with fixed amounts of intangible capital and employ both tangible and intangible capital to produce. Firms incur fixed operating costs that are proportional to intangible capital stock levels. The fixed operating costs introduce operating leverage. In the model, firms with more intangible capital are burdened with more nonproductive capital and face greater operating leverage in bad times. In order to test the model prediction empirically, I construct an empirical measure of intangible capital stock at firm level and show that firms with more intangible capital have average returns that are 0.32% higher than firms with less intangible capital per month. vii

8 Table of Contents Acknowledgments Abstract List of Tables List of Figures v vi x xi Chapter 1. External Financing Costs, Cash Holdings, and Expected Returns Introduction The Model Production Technology and Investment Opportunities Market Imperfections and the Firm s Cash Holdings The Firm s Optimization Problem Model Solution The Benchmark Case Firm Value External Financing Costs and Expected Returns Optimal Cash Holding Policy Firm Value The Value of Mature Firm The Value of Young Firms Quantitative Analysis Optimal Cash Holding and Optimal Exercise Boundary Optimal Cash Holdings and Expected Returns Empirical Investigation Cash Holdings and Firm Risk viii

9 1.4.2 Portfolio Analysis Fama-MacBeth Cross-Sectional Regressions Conclusion Chapter 2. Intangible Capital and Cross-Sectional Stock Returns Introduction A Model of Intangible Capital Model Setup Firm Valuation and Risk Empirical Investigations Measuring Intangible Capital Data and Summary Statistics Intangible Capital and Asset Prices Sorts on intangible capital Sorts on expenses associated with intangible capital Conditional CAPM test Operating Leverage Mechanism Conclusion Appendices 92 Appendix A. Appendix to Chapter 1 93 A.1 Firm Valuation, Risk, and Expected Returns A.2 Firm Value and Risk of the Restricted Case A.2.1 The Mature Firm A.2.2 The Young Firm A.3 Numerical Procedure A.4 Data and Variable Definitions Appendix B. Appendix to Chapter B.1 Proof of Proposition B.2 Sorting Firms on Alternative Measures of Intangible Capital. 108 Bibliography 111 ix

10 List of Tables 1.1 Panel Regressions of Cash Holdings of Firms Summary Statistics of Portfolios Sorted on Cash Holdings Portfolio Returns of Stocks Sorted on Cash Holdings Portfolio Returns of Stocks Sorted on Net Cash Holdings Fama-MacBeth Cross-Sectional Regressions of Monthly Returns Fama-MacBeth Cross-Sectional Regressions of Monthly Returns on Cash Holdings and Interaction Effects Summary Statistics of Portfolios of Firms Sorted by Intangible Capital Intensity Excess Returns to Portfolios Sorted by Intangible Capital Intensity Excess Returns to Portfolios Sorted by Industry Normalized Intangible Capital Intensity Excess Returns and Characteristics of Portfolios Sorted on IE/K Ratio Unconditional and Conditional CAPM Tests Panel Regressions of Degree of Operating Leverage, Conditional Beta, and Productivity B.1 Excess Returns to Portfolios Sorted on Alternative Measure of Intangible Capital B.2 Excess Returns to Portfolios Sorted on Alternative Measure of Intangible Capital x

11 List of Figures 1.1 The marginal costs of external financing and conditional betas of a young firm (low productivity) The marginal costs of external financing and conditional betas of a young firm (high productivity) Optimal cash holdings for young and mature firms as a function of firms productivity Optimal cash holdings for a young firm around optimal growth option exercise boundary Optimal growth option exercise boundary as a function of cash-to-assets ratio Conditional beta as a function of optimal cash-to-assets ratio Conditional beta as a function of optimal cash-to-assets ratio at growth option exercise boundary The average and median intangible-to-tangible capital ratios from 1980 to Decomposition of intangible capital from 1980 to xi

12 Chapter 1 External Financing Costs, Cash Holdings, and Expected Returns 1.1 Introduction The cash holdings of publicly traded U.S. firms have increased dramatically since the mid-1980 s. Large U.S. corporations are now holding recordhigh amounts of cash, which has drawn considerable amount of attention from the media. 1 Cash holdings on firms balance sheets are typically viewed as a poor investment and a drag on stock returns. In addition, conventional wisdom hints firms that hold high amounts of cash should be safer than cash-poor firms. Therefore, one would expect that cash-rich firms generate lower future returns than firms that hold less cash. However, previous empirical studies documented a positive, rather than negative, relationship between corporate cash holdings and future stock returns. 2 Why does high cash holdings predict high stock returns in the future? In this paper, I argue that, in an economy with external financing costs, firms 1 See, for example, Katy Burne, Companies Hold On to Their Cash, The Wall Street Journal, July 29, For example, Simutin (2010) documents that firms excess cash holdings can predict future stock returns. Palazzo (2012) shows firms cash-to-assets ratios positively correlate with their future stock returns. 1

13 cash holdings are positively related to their conditional betas. I propose a real options model that firms choose to hold liquid assets optimally when facing external financing costs. The model ties a firm s optimal cash holding policies to the riskiness of the firm, i.e., the correlation between the firm s cash flows and the aggregate shock. In the model, an equity value maximizing firm produces a stochastic revenue stream and incurs fixed operating expenses. The revenue of the firm is driven by a log-normal diffusion process. The firm may expand its production capacity by making capital investments and can choose to hold a cash inventory. The firm may retain some of its profits on its cash balance, or pay profits out to shareholders as dividends. The firm can finance its capital investments or cover operating losses by drawing down its cash inventory or by issuing additional equities. While equity issuance involves costs, such as bankers fees, the firm s cash holding earns a return that is strictly lower than the risk-free interest rate due to agency costs or tax distortions. The model captures two essential roles of cash holdings of a firm. First, cash holdings can be used to provide liquidity during the time of financial distress, which may lead to expensive external financing or inefficient closedown of the firm. Second, cash holdings may also be used to finance future investments when the firm is productive and wants to expand its operation. In the model, the firm accumulate cash in order to avoid resorting to costly external financing. If the firm has been hit by adverse productivity shocks, it can either raise costly external funds to cover the operating losses or shutdown. 2

14 In this case, the firm benefits from its cash holdings by avoiding undertaking costly equity issuance. Therefore, as its risk of financial distress increases, the firm increases its cash holdings in response. The increase in cash holdings only partially offsets the change in the firm risk. As a result, a higher level of cash holdings reflects increases in the firm s risk and implies high future returns. On the other hand, if the firm has experienced favorable productivity shocks, it may want to invest and to expand its production capacity. Similarly, the firm saves in order to avoid using external funds. The firm then increases its cash holdings as likelihood of making investment increases. Therefore, cash holdings of the firm are informative about its risk and expected returns when there are external financing costs. In addition, I show that the relationship between cash holdings and expected returns is rather complex and conditional in nature: when productivity is low, the firm s expected returns increase with expected external financing costs, therefore, its cash holdings; as firm s productivity increases, the positive relationship between cash holdings and stock returns diminishes and, depending on the relative value the firm derives from its growth options, might become a negative one. The intuition goes as follows. The firm is a portfolio of its cash holdings, assets in place, operating liabilities, and growth options. The expected returns of the firm are value-weighted expected returns of this portfolio. When the firm is doing poorly, cash flow risk dominates the overall risk of the firm. In this case, the external financing costs introduce additional liabilities to the firm, therefore, increase its cash flow risk, hence the overall 3

15 riskiness of the firm. When the firm is doing well, the relative value of its growth options increases and the risk of growth options might dominate the firm s total risk. In this case, the external financing costs decrease the value of growth options, and reduces the relative weight of the growth options in the portfolio. Further more, the firm chooses to hold more cash, which further reduces the overall risk of the firm. Therefore, external financing costs make firms safer when they are about to invest. To test the model s predictions, I verify that systematic risk of firms has a significant impact on corporate cash holding policies. Specifically, I estimate firms market betas and test whether the cross-section variation in market betas helps to explain the variation in cash holdings, as predicted by the model. This analysis complements other studies (e.g., Bates, Kahle, and Stulz (2009)) that investigate the determination of firm-level cash holdings. The results show that market beta is positively correlated to cash-to-assets ratio, i.e., firms with higher systematic risk tend to hold more liquid assets on their balance sheets. This result is robust to the inclusion of industry fixed effect, year fixed effect, and lagged cash-to-assets ratios of firms. I also test the model predictions using the portfolio sorts and the standard Fama-MacBeth (Fama and MacBeth (1973)) cross-sectional regressions. I show that firms with higher cash holdings have, on average, higher future stock returns in the cross-section of U.S. publicly traded firms. I first perform a univariate portfolio sort based on cash-to-assets ratio. The univariate sort produces a value-weighted excess return of the high cash-to-assets port- 4

16 folio over the low cash-to-assets portfolio at 0.15% per month. The excess return is not statistically significant. However, after adjusting for risk using the Carhart four-factor model (Carhart (1997)), the risk-adjusted excess return of the high cash-to-assets portfolio over the low cash-to-assets portfolio is 0.40% per month with a test-statistics of I then test the model predictions using the Fama-MacBeth cross-section regressions. I show that the cash predictability holds after controlling for other known cross-section stock return predictors. In addition, I show that this cross-sectional return predictability is weaker for firms that hold cash to finance future investments and for firms that tend to have low marginal costs of issuing equities. I use three dummy variables to proxy for firms investment opportunities or their marginal costs of issuing equities: size, R&D capital, and foreign earnings. Following Chan, Lakonishok, and Sougiannis (2001), I compute R&D capital as the five-year cumulative R&D expenditures, assuming an annual depreciation rate of 20%. Big firms tend to have low marginal costs or agency costs of issuing equities. On the other hand, firms with high R&D capital are likely to have more growth opportunities in the future and, therefore, tend to hold cash to finance future investments. Therefore, one would expect the cash holdings and returns relation is weaker for large firms and firms with high R&D capital. Finally, depending on their foreign earnings, I classify firms to be either domestic or multinational, i.e., foreign earnings dummy. Multinational firms may have incentives to keep foreign earnings abroad due to repatriation taxes, which is not related to the riskiness of these 5

17 firms. Hence, the positive relation between cash holdings and returns should get stronger after controlling the foreign earnings dummy. The Fama-MacBeth regressions confirm the interaction effect between cash holdings and the size or the R&D capital dummy variable. The slope on the interaction term, cash holdings and micro cap, is positive and significant at the 1%. The slope on the cash holdings and low R&D capital interaction term is also positive, but not statistically significant. This evidence suggests that the cash holdings and returns relation strengthens among small firms and firms with low R&D capital, i.e., firms that are more likely to hold cash to avoid financial distress. The slope on the interaction term of cash and earnings dummy is insignificant and close to zero. However, consistent with the model prediction, the slope of the cash-to-assets term increases slightly after controlling the earnings dummy. This paper is related to both the endogenous determination of corporate cash holdings and the stock return predictability. There is a growing theoretical asset pricing literature that attempts to link corporate decisions to asset return predictabilities, such as Berk, Green, and Naik (1999), Gomes, Kogan, and Zhang (2003), and Carlson, Fisher, and Giammarino (2004). My work extends this line of research. Specifically, my paper endogenizes firms liquidity management policies and adds firms cash holdings as a theoretically motivated stock return predictor. My work also extends the real options approach to analyzing firm decisions under uncertainty by introducing external financing costs. The real options approach to analyzing investment decisions 6

18 was introduced by Brennan and Schwartz (1985) and McDonald and Siegel (1986). Dixit and Pindyck (2012) provide a good overview of the real options literature. Recent literature uses the structural model approach of Merton (1974) to evaluate corporate debt, growth options, and choice of capital structure. 3 Early literature often assumes firms can raise external funds at no costs. So the cash holding policy of a firm is trivial in these papers. In my paper, due to the external financing costs, firms cash balances affect their valuations and investment decisions. Palazzo (2012) establishes the link between cash holdings and expected returns using a three-period model of firms investing and financing decisions. In Palazzo s model, firms have risky assets in place, which generate stochastic cash flows, and risk-free investment opportunities. Controlling for the market valuations, riskier firms derive more value from their investment opportunities and have higher expected returns. These firms also choose to hold more cash to finance future investments. Thus, the cash holdings of firms are positively correlated with expected returns. My paper differs from Palazzo s along several dimensions: First, firms only hold cash for future investments in Palazzo s model. My model incorporates the precautionary motive to hold cash, i.e., firms hold cash to better cope with adverse productivity shocks when access to capital markets is costly. Sec- 3 Papers that use structural models to evaluate corporate securities and investment opportunities include Leland (1994), Mauer and Triantis (1994), Abel, Dixit, Eberly, and Pindyck (1996), Abel and Eberly (1996), Leland and Toft (1996), and Hackbarth and Mauer (2011). 7

19 ond, Palazzo s model is a static trade-off model whereas the model presented in this paper is a dynamic model. In my model, firms optimal cash holdings change as their productivity varies. In Palazzo s model, a firm s optimal cash holdings depends on the relative value of the firm s investment opportunities to its assets in place. Lastly, due to the dynamic nature of my model, I show that the positive relationship between cash holdings and expected returns is only conditional. This paper is also related to the literature in corporate finance that studies the endogenous determination of corporate cash holdings. Opler, Pinkowitz, Stulz, and Williamson (1999) examine the determinants and implications of corporate cash holdings empirically. They show that firms hold liquid assets when cash flows from operation are low relative to planned investments and when outside funds are expensive. Almeida, Campello, and Weisbach (2004) provide evidence that firms with greater frictions in raising external funds save a higher portion of their cash flows as cash than do those with fewer frictions. Similarly, Faulkender and Wang (2006) also conclude that cash holdings are more valuable for constrained firms than for unconstrained firms. There has been substantial progress on the modeling of cash retention decisions in recent years, such as Gamba and Triantis (2008), Riddick and Whited (2009),and Bolton, Chen, and Wang (2011). Bolton et al. (2011) propose a dynamic model of investment and corporate risk management for a financially constrained firm. They use the model to study the endogenous 8

20 response of corporate hedging, cash accumulation, and payout policies to financing constraints. Bolton et al. (2011) model a firm s operating revenue as an arithmetic Brownian motion process. Therefore, the productivity is not a state variable in their model, and the riskiness of the firm s assets does not change over time. Unlike Bolton et al. (2011), I model the production process as a geometric Brownian motion and focus on the asset pricing implications of cash holdings. Hence, both the productivity and the cash holdings are state variables in my model, and cash holdings are related to the conditional beta of the firm. The paper proceeds as follows. Section 1.2 presents a real options based asset pricing model to understand the empirical evidence. Section 1.3 solves the model numerically and discusses its properties. Section 1.4 presents the empirical facts in the data regarding the relationship between cash holdings, conditional beta, and stock return predictability. Finally, Section 1.5 concludes. 1.2 The Model I develop a model of a firm s optimal cash holding policy in the presence of growth option, shutdown option, and costly access to external financing. The main goal of the model is to show that cash holdings can be positively correlated with the systematic risk of a firm, measured by the equity beta of the firm, and to discuss the economic mechanisms behind this counter-intuitive relationship. I first describe the firm s production and investment opportuni- 9

21 ties. Then, I introduce the external financing costs and the opportunity cost of holding cash inside the firm. Finally, I state the firm s optimization problem. I explain the valuation technique used in the paper in the appendix Production Technology and Investment Opportunities The firm employs physical capital, K, for production. For simplicity, I assume the physical capital of the firm does not depreciate. The revenue process of the firm is assumed to be X t K, where X t is an exogenous state variable. I specify that X t evolves according to a diffusion process as dx t X t = θdt + σdz t, t 0, where Z t is a standard Wiener process. Both parameters θ > 0 and σ > 0 are constant and are the mean and the volatility of the productivity shocks, respectively. Therefore, the productivity of the firm follows a geometric Brownian motion process. This production specification is commonly used in the real options literature. In order to simplify the model, I allow only two capital levels, K l < K h. Firms with these capital levels are called young and mature firms, respectively. I assume that a typical firm is endowed with an initial level of capital K l and one option to expand its capital stock to K h by investing additional capital in the amount I = K h K l > 0. 4 The price of capital is 4 I assume the firm s investment is lumpy. The lumpiness of investments can be justified by fixed adjustment costs of capital in a neoclassical investment model. As a partial equilibrium 10

22 normalized to one, and there are no adjustment costs to make investments. When exercising its growth option, if the firm has accumulated enough cash, i.e., W t I, it can use its cash holdings to finance the investment. Otherwise, the firm has to raise money by issuing additional equity, F t, to pay for the investment. Issuing equity is costly, and I will provide more details about external financing costs in the next section. Capital investments are completely irreversible, i.e., the value of capital may not be recovered once it is installed. In addition, I assume the firm also incurs a fixed operating cost f(k) > 0 per unit of time. The fixed operating cost f( ) is strictly increasing in capital level, K. For convenience, I denote f l f(k l ) and f h f(k h ), where f l < f h. I further assume there is no variable cost associated with production. Therefore, the firm s operating profit over time interval dt is given by dπ t = (X t K i f i ) dt, where i = l, h is the capital level of the firm. The firm can distribute its profits as dividends to shareholders. Any remaining profit is accumulated in an internal cash account. Let W t 0 denote the firm s cash balance. 5 I will discuss the firm s cash account in the next section. model, I choose not to model the capital adjustment costs but simply assume a lumpy investment process. I also assume the firm has only limited investment opportunities. The primary goal of this paper is to demonstrate the impact of growth opportunities on the firm s cash holdings and risk. Therefore, restricting the firm s capital levels simplifies the model while preserves economic intuitions. 5 I do not model the line of credit by specifically requiring the cash balance of the firm has to be greater than zero. 11

23 1.2.2 Market Imperfections and the Firm s Cash Holdings I assume there are two imperfections. First, there is a marginal cost, φ > 0, to issuing additional equity. The external financing costs may vary across firms. The second is that it is costly for the firm to hold liquid assets internally. Specifically, the firm s cash account generates interests at a rate ˆr strictly less than the risk-free rate r. This assumption reflects an agency cost associated with the free cash inside the firm or losses raised from corporate taxes. 6 Since it can always make a lump-sum dividend payment to its shareholders, the firm will not choose to close down if its cash holdings are positive W t > 0 at any time t. When the firm runs out of cash, it has to either raise external funds or shut down if its revenue X t K is lower than the instantaneous operating expense f(k). If the firm chooses to raise external funds to continue operating, it must pay the financing costs specified above. The firm may also raise external funds to invest if it chooses to exercise its growth option but does not have enough cash to finance the investment. In some situations, for example when its productivity X t is sufficiently low, the firm may find optimal to shut down. Let τ denote the firm s shutdown time, where τ [0, + ) is endogenous and stochastic. If τ = +, then the firm will never choose to 6 This assumption is standard in models with cash holdings. For example, see Cooley and Quadrini (2001), Kim, Mauer, and Sherman (1998), and Riddick and Whited (2009). If ˆr r, the firm would strictly prefer to keep cash inside the firm no matter what the value of its cash holdings is. This is because holding cash incurs no costs to its shareholders but has the benefits of reducing the chance of raising equities in the future. 12

24 shut down. Let dd t denote the firm s incremental payout to its shareholders over time interval dt. Payout to shareholders may take the form of a cash dividend, a share repurchase, or both. Let df t denote the firm s incremental external financing (after fees) over time interval dt. Putting everything together, the firm s cash holdings W t evolves according to the following equation, dw t = ˆrW t dt + dπ t I t dt dd t + df t, where I t is the cash outflow caused by investment. I t = K h K l if a (young) firm decides to exercise its growth option at time t. Otherwise, I t = 0. The df t dd t is the net cash flow from financing. This equation is an accounting identity and holds from period to period The Firm s Optimization Problem I assume that the firm is risk-neutral and acts in the best interests of its shareholders. The firm chooses its investment policy I t (if young), payout policy D t, external financing policy F t, and shutdown time τ to maximize its shareholder value (equity value) defined as: [ τ ] Vt i (X, W ) = max E t e rs [dd t+s (1 + φ)df t+s ] + e rτ W τ, (1.1) {I t,d t,f t,τ} t where the expectation is taken under the risk-neutral measure and i = l, h for young and mature firms, respectively. The first two terms of equation (1.1) are the discounted value of net payouts to shareholders and the last term is 13

25 the discounted value at the time of shutting down. Notice that a mature firm may still want to hold some cash to avoid raising external funds. The firm value depends on two state variables, the productivity X t and its stock of cash W t. The conditional expected return on equity is given by β i = d log V i d log X. I provide the details of firm valuation and a derivation of risk and expected returns of the firm in Appendix A Model Solution In this section, I characterize the solution to the firm s problem (1.1) and solve the model numerically. To solve the firm s optimization problem (1.1) is rather complicated. Therefore, I first solve the special case where the firm is not allowed hold cash. I use the solution to this special case as a benchmark and deliver the economic intuition that riskier firms have the incentive to hold more cash than safer firms. Then, I solve the model numerically. I characterize the solution to the problem where the firm chooses to hold cash optimally. I investigate the effect of cash holdings on the firm s optimal growth option exercise policy and its risk. I analyze the dynamics of optimal cash holding policy, compute the marginal value of internal funds, and analyze cash holdings with respect to the conditional beta of the firm. Finally, I propose testable implications of the model. 14

26 1.3.1 The Benchmark Case First, I provide analytic solutions for firm value and risk when the firm is restricted from holding cash, i.e., W t = 0 all the time. The solutions to this special case provide insights that link the firm s cash holdings and expected returns. Since the firm is not allowed to hold a cash balance, it must payout all its profits to shareholders as dividends. When the firm s profit rate is greater than zero, i.e., XK i f i 0, its shareholders receive dividends at the rate XK i f i. Otherwise the firm raises equities to cover its operating losses, which is equivalent to paying negative dividends to shareholders at a rate of (1 + φ)(xk i f i ) per unit of time. Due to the presence of the fixed operating costs, the firm may choose to shut down when its productivity is sufficiently low. Therefore, there exists a productivity boundary XB i, below which the firm will choose to close down. On the other hand, if its productivity is sufficiently high, a young firm may find it optimal to exercise its growth option to become a mature firm. Let X G denote the productivity boundary, above which a young firm will choose to grow endogenously. 15

27 Firm Value The value of equity of a mature firm satisfies the expression: ( Kh (1 + φ) V h δ X f ) h + A h r 11X λ 1 + A h 12X λ 2 X [XB, h f h /K h ) (X, 0) = K h δ X f h r + Ah 21X λ 1 X [f h /K h, + ) (1.2) where λ 1 < 0 and λ 2 > 1. The expression for the equity value and its derivation can be found in Appendix A.2. The value of the mature firm given in equation (1.2) is similar to the mature firm valuation derived in Gomes and Schmid (2010). 7 The XK h /δ term in equation (1.2) is the present value of the future cash flows generated by the assets in place of the firm, K h. The f h /r term captures the present value of all future operating obligations. The X λ 1 terms show the impact of the option to shut down on the value of the firm. Due to the existence of external financing costs, expression (1.2) differs from the firm valuations derived in previous literature in two ways. First, the present value of future cash flows and future operating expenses is reduced by a factor of 1 + φ when the firm is distressed X t K h < f h. In this case, the firm raise external funds to cover its operating losses by paying a proportional fee. Second, the A h 12X λ 2 term also captures the effect of external financing and is entirely missing from the expression for the value of a mature firm derived in Gomes and Schmid (2010). When the firm is not distressed, X t K h f h, this term disappears. 7 See equation (9) in Gomes and Schmid (2010). 16

28 The value of equity of a young firm satisfies the expression: ( Kl (1 + φ) V l δ X f ) l + A l r 11X λ 1 + A l 12X λ 2 X [XB, l f l /K l ) (X, 0) = K l δ X f l r + Al 21X λ 1 + A l 22X λ 2 X [f l /K l, X G ) (1.3) After exercising its growth option, the young firm becomes a mature firm. The value of the firm is, then, given by V h (X, 0) less the cost of investing (1 + φ)(k h K l ) for X X G. The first three terms, XK l /δ, f l /r, and X λ 1, in equation (1.3) are similar to those of a mature firm except for different levels of assets and operating costs. They reflect the present value of the future cash flows, the present value of future obligations, and the value of the option to shut down to avoid these obligations, respectively. The last term, X λ 2, however, reflects both the costs of external financing, when productivity is low, and the value of future growth option, when productivity is high, of the young firm. Furthermore, the endogenous shutdown boundary of the young firm is lower than that of a mature firm, i.e., XB l Xh B.Due to the existence of growth opportunities, young firms choose to shut down at a lower productivity level than mature firms. 17

29 External Financing Costs and Expected Returns I express the conditional equity betas β i, for any types of firms i = l, h, in a general form as follows 1 + ( V i) ( 1 (1 + φ) f ) i r + (λ 1 1)A i 11X λ 1 + (λ 2 1)A i 12X λ 2 X [XB, i f β i i /K i ) = 1 + ( V i) ( ) 1 fi r + (λ 1 1)A i 21X λ 1 + (λ 2 1)A i 22X λ 2 X [f i /K i, + ) (1.4) where A h 22 = 0. For a young firm (i = l), its conditional betas converge to the conditional betas of a mature firm for X t X G. The first term in equation (1.4) is common to both young and mature firms, and is simply the firm s revenue beta. Since operating profits are linear in productivity, this term is effectively normalized to one. It captures the riskiness of a firm s assets in place. The f i /r term reflects the effects of operating leverage on risk and expected returns. A i 11X λ 1 and A i 21X λ 1 terms capture the impact of shutdown on firm risk. The possibility of shutting down reduces risk of both types of firms. The A l 22X λ 2 term captures the leverage effect from growth options. One can interpret A l 22X λ 2 as the value of the growth option. So the ratio A l 22X λ 2 /V l gives the percentage of (young) firm value in growth option. The growth options are riskier than assets in place as they are call options on the underlying assets of firms. 8 8 Mathematically, one can see that growth options are riskier than assets in place because λ 2 > 1. In fact, λ 2 1 > 0 is the excess riskiness of growth options relative to assets in 18

30 There are two novel features in equation (1.4) due to the existence of external financing costs. First, when a firm is distressed, the leverage effect of the f i /r term is further amplified by a factor of 1 + φ. This is very intuitive as shareholders have to contribute more to keep the firm afloat when external financing is costly. Second, the A h 12X λ 2 term reflects the fact that the firm chooses to shut down at a higher productivity level when external financing is not free. The endogenous shutdown boundary X h B increases as the marginal external financing cost φ increases. The A h 12X λ 2 in equation (1.4) reduces the conditional betas of the firm. This is true for young firms as well. 9 Although the firm is not allowed to hold cash in this benchmark case, to analyze equation (1.4) will help us develop intuition why cash holdings may provide information about expected returns in addition to book-to-market and size. First, the second term of equation (1.4) is proportional to f i /V i, which represents the book-to-market ratio of the firm. Hence, book-to-market ratio describes the operating leverage component of risk up to a first-order approximation. When productivity is low, the f i /r term dominates and the firm s risk increases as external financing costs increase. Therefore, controlling for book-to-market ratio, firms that have higher external financing costs are place. 9 For mature firms, A h 12 = φf ( h 1 λ1 λ 2 λ 1 δ + λ 1 r ) ( ) λ2 Kh < 0. We do not have closed-form expression, A l 12, for young firms. We can show A l 12 satisfies A l 12 < A l 22 instead. So the growth option of a young firm is not as valuable when the firm is financially distressed. f h 19

31 riskier than otherwise similar firms. I illustrate this insight with a simple numerical example. Figure 1.1 shows the conditional betas of a young firm as a function of productivity X t for three different levels of external financing costs. Not surprisingly, expected returns rise as X t decreases. This result is also true for a mature firm. Figure 1.1 also confirms our intuition that the external financing costs amplify the cash flow risk of the firm. So firms that face higher external financing costs tend to have higher betas. These firms have a stronger incentive to hold cash, if allowed, to avoid using costly external financing than firms that have lower external financing costs. The second observation is that the positive relationship between external financing costs and expected returns diminishes as the firm s productivity increases. For a young firm, expected returns might actually decrease with marginal external financing costs when its productivity is high. This observation seems counter-intuitive. However, one can think of a young firm as a portfolio of assets in place and a growth option. The firm risk is a valueweighted average of the risk of assets in place, which is normalized to one, and the risk of the growth option, which is higher than one. The external financing costs decrease the value of the growth option, therefore, decrease expected returns of the young firm. Figure 1.2 confirms this observation. The figure shows the conditional betas of a young firm as a function of productivity X t for three different levels of external financing costs. The figure shows that expected returns start to rise with X t for sufficiently high X because the relative weight of the growth option in total firm value increases. More importantly, 20

32 3.60 φ = 0 φ = 0.15 φ = Conditional β X t Figure 1.1: The marginal costs of external financing and conditional betas of a young firm (low productivity). This figure shows the conditional betas of a young firm as a function of productivity X t for three different levels of external financing costs, φ = 0, 0.15, 0.3. When the productivity X t is low, the financial leverage dominates firm s total risk. Firms that face higher external financing costs have higher conditional betas than otherwise similar firms. Other parameter values used to generate this plot are r = 0.05, δ = 0.02, σ = 0.2, K l = 2, and f l = 0.1. the risk of the young firm decrease with φ as external financing costs reduce the value of its growth option, although the differences in conditional betas tend to be small. 21

33 1.176 φ= 0 φ= 0.15 φ= 0.3 Conditional β X G (φ= 0) X t Figure 1.2: The marginal costs of external financing and conditional betas of a young firm (high productivity). This figure shows the conditional betas of a young firm as a function of productivity X t for three different levels of external financing costs, φ = 0, 0.15, 0.3. When the productivity X t is high, the growth option dominates the firm s total risk. Other parameter values used to generate this plot are r = 0.05, δ = 0.02, σ = 0.2, K l = 2, and f l = 0.1. These parameter choices generate a growth option exercise boundary X G = 0.61 when φ = 0. Although firms that have higher external financing costs have stronger incentive to hold cash to avoid raising external funds, figure 1.1 and figure 1.2 suggests that the relationship between cash holdings and expected returns is conditional in nature. When productivity is low, cash flow risk dominates the 22

34 total risk of both types of firms. In this case, expected returns increase with external financing costs. When productivity is sufficiently high, the risk of growth option dominates the total risk of a young firm. Expected returns, on the other hand, may decrease with external financing costs Optimal Cash Holding Policy In this section and next section, I relax the restrictive assumption made in Section and allow the firm to hold cash. I use the value function iteration method to solve for the optimal policies and the value of the firm. The detailed description and the convergence of the method can be found in Appendix A.3. The results presented in the remaining of this paper are obtained by the value function iteration procedure with 12 time steps per year. I first show that the firm, young (i = l) or matured (i = h), will choose an all-or-nothing type of profit retention policy when it is allowed to hold cash. When raising external funds is costly, the firm faces the tradeoff between hoarding cash at a strictly lower rate of return ˆr < r and paying external financing costs in the future. Intuitively, so long as the marginal value of cash exceeds one, the firm would want to retain all its earnings. When the marginal value of its cash is one or less, the firm would instead pay out all proceeds as dividends. Specifically, I show that there exists three regions: (i) a cash accumulating region, in which the firm does not pay dividends but accumulates cash; (ii) a payout region, in which the firm pays all profits to its 23

35 shareholders; (iii) an external financing region, in which the firm either raises external funds or closes down depending its productivity. The firm s value and expected returns depend on which region the firm is in. The three regions are separated by two cash boundaries: a lower boundary W i (X) below which the firm raises external funds or closes down; an upper boundary W (X) i above which the firm pays all its profits as dividends. The upper bound, W (X), i is indeed the optimal level of cash holdings for the firm. For those (X, W ) region where it is optimal to accumulate cash, the firm value V i (X, W ) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation: (ˆrW + xk i f i ) V i w + (r δ)xv i x σ2 x 2 V i xx rv i = 0. (1.5) The first term, V i w, represents the marginal effect of the firm s cash holdings on firm value V i (X, W ). With a frictionless capital market, the marginal value of cash is always V i w = 1. With costly external financing, on the other hand, the marginal value of cash, which is the expected marginal cost of financing, will be greater than one, V i w > 1, in the cash accumulating region. The second and third terms, (r δ)xv i x σ2 x 2 V i xx, capture the expected change in firm value caused by a fluctuation in the firm s productivity X t. The last term is the required equity return under risk-neutral measure. This term should be equal to the expected return from holding the equity, which is the sum of the first three terms. Since firms make no dividend payment to shareholders, we do not have a source term in equation (1.5). 24

36 When the firm s cash holding is high enough, the firm is better off distributing some cash to its shareholders to avoid the cash-carrying cost, r ˆr. Let W (X) i denote this endogenous payout boundaries for both types of firms. Notice this boundary depends on the productivity of the firm. For a mature firm, W h (X) is a decreasing function of productivity. When it is very productive (high X), the firm is less likely to incur a net operating loss. Therefore, as X increases, the marginal value of a mature firm s cash holdings decreases. The firm chooses to hold less cash. W h (X) goes to zero when X is sufficiently high. For a young firm, on the other hand, W (X) l decreases first as X increases for the same reason as in the case of a mature firm. As X approaches the endogenous growth option exercise boundary X G (W ), the young firm is more likely to exercise its growth option and, possibly, incur external financing costs if it is financially constrained. Therefore, the young firm retains its profits and increases its cash reserve as a response. The W (X) l increases again. Notice, when the firm is allowed to hold cash, the growth option exercise boundary X G (W ) is affected by the firm s cash level, W, as well. Intuitively, if the firm starts with a large amount of cash, W > W (X), i then it is optimal for the firm to distribute the excess cash as a lump sum dividend payment and bring down its cash reserve to the optimal level, W (X). i Moreover, the firm value must be continuous before and after cash distribution. Therefore, we have the following equation for V i (X, W ) V i (X, W ) = V i ( X, W i (X) ) + W W i (X), W > W i (X). (1.6) 25

37 Further exploiting the continuity of V i (X, W ), we take the limit of equation (1.6) as cash holdings approach the optimal level from above, W W i (X) +, V i w ( X, W i (X) ) = 1. (1.7) Equation (1.7) simply shows the marginal value of cash equals one at optimal cash holding boundary and the firm is indifferent between distributing and retaining its cash. Equation (1.7) is also known as the smooth pasting condition. Since the payout boundary W i (X) is optimally chosen, we also have the following super contact condition: 10 V i ww ( X, W i (X) ) = 0. (1.8) Both mature and young firms hold cash to cover operating losses whereas young firms may also hold cash to finance growth option exercises. When the firm s cash holding is less than some pre-specified boundary W i (X), the firm either incurs costs to raise external funds or closes down. Depending on its productivity X, the firm may prefer either refinancing or shutdown. Although the firm can choose to refinance at any time, it can be shown that it is not optimal for the firm to do so before it runs out of cash, i.e., W i (X) = 0 for both types of firms regardless of productivity X. The intuition is as follows. 10 For the optimal control of diffusion processes, smoothness requires twice differentiability of the value function, and is sometimes known as the super contact condition. In pure stopping problems, smoothness requires that the value function is once differentiable, and is known as the smooth pasting condition. The super contact condition has been extensively used to characterize optimal solutions. See, e.g., DeMarzo and Sannikov (2006) and DeMarzo, Fishman, He, and Wang (2012). 26

38 First, cash within the firm earns a below-market interest rate ˆr < r. Second, it is optimal to postpone refinancing due to the time value of money for the external financing costs. Therefore, it is always better to postpone the external financing as long as possible. Since the external financing costs are proportional to the amount of funds raised, the firm will raise the exact amount either to cover its instantaneous operating loss or to pay the cost of investment Firm Value The firm will prefer to shut down if its productivity X is below some endogenous cutoff value XB i (W ). The endogenous shutdown boundary decreases in firm s cash holdings W when external financing is costly. Intuitively, this is true for two reasons. First, the firm may use its cash reserve to pay for the operating costs when its productivity is low. This essentially reduces external financing fees that the firm would otherwise have to pay. So the firm may shut down at a lower productivity level. Second, the firm value increases in the volatility of productivity, i.e., the V i xx term in equation (1.5). The firm strictly prefers to stay longer in order to capture the upside of productivity. When it closes down, the firm pays out all its cash holdings to shareholders as a final dividend and then ceases operating. Therefore, the firm value upon shutting down is V i (X i B(W ), W ) = W X X i B(W ). (1.9) 27

39 Further exploiting the continuity of V i (X, W ) at XB i, we have the smooth pasting condition at X i B, ( Vx i X i B, W ) = 0. (1.10) The Value of Mature Firm A mature firm holds cash to avoid paying external financing costs when distressed. The higher the value of X, the further away the firm is from financial distress. When the productivity is sufficiently high X +, the marginal value of cash holding decreases to one, lim X + V h w (X, W ) = 1. Therefore, we have the following boundary condition: V h (X, W ) = V h (X, 0) + W X +, (1.11) where V h (X, 0) is the value of the mature firm when the firm is restricted from holding cash. V h (X, 0) is given by equation (1.2). To summarize, the complete solutions for the mature firm s value V h (X, W ), its optimal shutdown boundary XB h (W ), and its optimal dynamic cash holding policy W h (X) are given by: (i) the HJB equation (1.5), (ii) the shutdown boundary conditions (1.9) and (1.10), (iii) the infinite boundary condition (1.11), and (iv) the payout boundary conditions (1.6), (1.7), and (1.8) The Value of Young Firms A young firm may choose to exercise their growth options endogenously when productivity reaches some optimal exercise boundary, X G (W ). This 28