Complexity

Volume 2019, Article ID 8479503, 11 pages

https://doi.org/10.1155/2019/8479503

## Optimal Financing and Dividend Strategies with Time Inconsistency in a Regime Switching Economy

^{1}College of Finance, Chongqing Technology and Business University, Chongqing 400067, China^{2}College of Economics and Business Administration, Chongqing University, Chongqing 400030, China

Correspondence should be addressed to Yehong Yang; moc.qq@280154428

Received 25 November 2018; Revised 6 March 2019; Accepted 19 March 2019; Published 4 April 2019

Academic Editor: Ludovico Minati

Copyright © 2019 Yehong Yang and Guohua Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper incorporates a manager’s time-inconsistent preferences into the unified dynamic q-theoretic framework to investigate their impact on the optimal external financing and dividend payout strategies in a regime switching economy. We find that with a higher degree of time inconsistency, either in a favorable market condition or in a financial crisis, dividends are paid out earlier and the equity issues are smaller in size in each occurrence; in a favorable market condition equity financing occurs particularly early. Hence, time inconsistency would result in a decreasing of a firm’s precautionary savings, which may directly cause capital chain rupture and make liquidation more likely. It also implies that corporate external financing and dividend payout are highly dependent on the degree of the manager’s time-inconsistent preferences in a regime switching economy.

#### 1. Introduction

Recent studies show that a firm’s financing and investment behavior changes dramatically during financial crises. For instance, Campello et al. [1] and Campello et al. [2] show that more financially constrained firms planned more cuts in investment spending, expended more cash, drew more credit from banks, and engaged in more asset sales during a crisis. A stochastic model of regime switching which is proposed by Hamilton [3] has been applied previously in the literature (e.g., Guo et al. [4], Sotomayor and Cadenillas [5], and Jang and Kim [6]). These studies illustrate that firms’ policies, including the optimal investment policy, consumption-investment policy, optimal reinsurance, and asset allocation strategies, are affected by regime shifts. On economic grounds, there are indeed reasons to believe that regime shifts contain the possibility of significant impact on firms’ policy choices. For example, business cycle expansion and contraction “regimes” potentially have sizable effects on the profitability or riskiness of investment and, hence, on firms’ willingness to invest in physical or human capital. We still, however, know little about the implications of changes in financial conditions on external financing and payout decisions in a regime switching economy.

There are some studies of external financing and payout strategies, such as Bolton et al. [7, 8], Chen et al. [9], and Chen et al. [10]. Bolton et al. [7] proposes a q-theoretic model of investment and financing for financially constrained firms, and shareholders’ dividend payments are discounted exponentially at a constant discount rate which means that shareholders have time-consistent preferences. Bolton et al. [8] investigate external financing and payout strategies under regime switching and time-consistent preferences. They find that the transition intensity out of favorable market condition affects the firm’s market-timing behavior, including the firm’s investment, external financing, and payout strategies. Loewenstein and Prelec [11] and Rabin [12] use experimental studies of time preferences suggesting that the assumption of time-consistent preferences is unrealistic, and decision-makers often have time-inconsistent preferences (also called present-biased preferences). Hence, Chen et al. [9] and Chen et al. [10] characterize the optimal dividend strategy of an insurance company whose manager has time-inconsistent preferences.

Most economic decisions are intertemporal in nature and involve tradeoffs between current and future rewards. An important component of any intertemporal model is the discount function that discounts delayed rewards to the present for decision-making. Overwhelming evidence has been documented in the psychology and behavioral science literature that suggests time inconsistency is prevalent in human preferences (e.g., Thaler and Shefrin [13], Ainslie and Herrnstein [14], Kirby and Herrnstein [15], Myerson and Green [16], Mcclure et al. [17], and Dellavigna and Malmendier [18]). In pursuing immediate gratification, individuals often exhibit a reversal of preferences when choosing between a smaller, earlier reward and an alternative larger but later reward. That is, when two rewards are both far away in time, a larger-later reward is preferred to a smaller-sooner reward (e.g., subjects prefer 101 dollars in 31 days to 100 dollars in 30 days). When both rewards are brought forward in time, however, the sooner-smaller reward becomes preferred (e.g., subjects prefer 50 dollars today to 51 dollars tomorrow). Such preferences exhibit present biases and highlight a conflict between today’s preferences and future preferences. They also show that an individual’s impatience decreases with time, meaning that a shareholder or manager also has time-inconsistent preferences.

The most common way to model time inconsistency is the quasi-hyperbolic discount function, in which the discount rate decreases in the horizon (see Harris and Laibson [19]). By this means, many researchers show that time inconsistency leads to the tendency to choose actions inducing short-run benefits over those inducing long-run benefits, such as Grenadier and Wang [20], Lien and Yu [21], and Li et al. [22]. Time inconsistency can be interpreted as an intrinsic property of preferences in research on the market-timing motive for external financing/payout decisions. Therefore, an important question is the correlation of time inconsistency and external financing/payout decisions in a regime switching economy.

In this paper, following Bolton et al. [8] (hereafter, BCW), we extend the unified dynamic q-theoretic framework to construct a time-inconsistent model in which the manager has time-inconsistent preferences. Suppose that the manager has time-varying impatience modeled with a quasi-hyperbolic discount function, and her objective is to choose the firm’s optimal investment and payout strategies to maximize firm value. Intuitively, since the manager is working on behalf of shareholders and makes decisions that appeal to shareholders, when shareholders have time-inconsistent preferences, the manager should also exhibit time-inconsistent preferences. This is also assumed in Grendier and Wang [20] and Chen et al. [10]. Shareholders are aware that the manager’s time preferences will change in the future and they would write contracts with the manager to mitigate the problem of inconsistency and maximize shareholder value. However, before writing the contract, shareholders need to know how corporate decisions are influenced by the manager’s time-inconsistent preferences.

In a regime switching economy, the firm can be in one of two states: a favorable market condition or a financial crisis, that is, state or state . Based on the classical BCW model, we treat changes in financing conditions as exogenous. In this paper, we assume that the favorable market condition and financial crisis have a low financing cost and a high financing cost, respectively. There are several reasons for the expensive financing cost in a financial crisis, for example, changes in financial intermediation costs, changes in investors’ risk attitudes, changes in market sentiment, or changes in aggregate uncertainty and information asymmetry. We make predictions about how the degree of time-inconsistent preferences influences optimal external financing and dividend payout strategies and how much difference the manager’s time-inconsistent preferences make in a regime switching economy.

The main results of our analysis are as follows. First, firm value, the firm’s investment, and external financing and payout strategies are affected by two major factors: time inconsistency and regime switching. For instance, the impact of time inconsistency on firm value and the firm’s investment is much higher in a favorable market condition than in a financial crisis. In a favorable market condition, the manager with time-inconsistent preferences pays out dividends and issues equity earlier even without immediate financing needs but raises less equity capital than her time-consistent counterpart. Second, with a higher degree of time inconsistency, dividends are paid out earlier and less equity is issued each time. In particular, the firm chooses to issue equity early in a favorable market condition.

The remainder of this paper proceeds as follows. Section 2 sets up the model. Section 3 presents the model solution. Section 4 provides quantitative analysis. Section 5 concludes.

#### 2. The Model

Our model incorporates a manager’s time-inconsistent preferences into the BCW model of dynamic framework. In this section, we specify the manager’s time preferences, regime switching, production technology, and stochastic financing opportunities for a financially constrained company.

##### 2.1. Time Preferences of the Manager

Based on Grenadier and Wang [20] and Harris and Laibson [19], we use a continuous-time version of the quasi-hyperbolic discount function to reflect the empirically documented declining discount rate. We model the manager as a sequence of successive decision-makers (called self 1,2,...). denotes self ’s discount function for any time , so shareholders value $1 paid in the future as , where is as follows:for all . Here, the planning horizon of self is divided into the present period (from to ()) and the future period (from to ). When the current self dies at time , the next self is born. The preferences of self are again divided into a present and a future , and so on. This discount function means that the manager is more impatient in making short-term decisions than long-term decisions.

All periods consisting of present and future are discounted exponentially with discount factor . The future period is further discounted with uniform weight . The duration of the present period is exponentially distributed with parameter . The parameters and determine the degree of time-inconsistent preferences. determines how much the future period is valued relative to the present period. determines the arrival rate of the future selves and thus how often preferences change. When is lower, the current self is more impatient. When , the current self is only concerned about her actions in the present and completely disregards her future selves. With a greater , the degree of time-inconsistent preferences changes more quickly and the arrival rate of the future period increases. Moreover, when or , discount function (1) degenerates into the traditional exponential discount function which represents time-consistent preferences.

It is important to note the following: first, discount function (1) is also called the “present-biased” or “quasi-geometric” discount function in the literature (see Krusell and Smith Jr. [23] and Hsiaw [24]) and, second, self can also be viewed as the manager of the firm. In this case, and can be seen as the beginning and the end of manager ’s term of office. The manager cares more about the decisions made during her term in office and pays out dividends to the shareholders. The manager will overvalue her career achievements and undervalue the achievements of her successors by discounting them with an additional factor . Her career achievements are measured by the dividends that the shareholders receive.

##### 2.2. Regime Switching

We model regime switching by switching the firm’s external financing conditions. Namely, suppose that the firm has two aggregate states, denoted by =, where state () is the state with favorable external financing condition (financial crisis). A risk-averse manager requires a risk premium to compensate for the risk of the economy switching states. We characterize this risk premium through the wedge between the transition intensity under the physical probability measure and that under the risk-neutral probability measure. Over time increment , the state switches from to (or from to ) with a probability (or ). Let and denote the risk-neutral transition intensities from to and from to , respectively. Thenwhere and capture the risk adjustment for the change of states and , respectively. In other words, () can be interpreted as the price of risk with respect to financing shocks in state (). Let and , so we have and . Then () which implies that the transition intensity out of state () is higher (lower) under the risk-neutral probability measure than under the physical measure. For simplicity, we set . Intuitively, it reflects the fact that a manager’s risk aversion towards a bad state is captured by making it more likely to switch to the bad state and less likely to leave it. In short, it is as if a risk-averse investor were uniformly more “pessimistic” than a risk-neutral investor; she thinks “good times” are likely to be shorter and “bad times” longer.

##### 2.3. Production Technology

The firm employs capital and cash as the only factors of production. We normalize the price of capital to one and denote by and the firm’s capital stock and gross investment, respectively. The firm’s capital stock evolves according towhere is the rate of depreciation in state .

The firm’s operating revenue is proportional to its capital stock and is given by , where is the firm’s revenue shock over time increment . We assume thatwhere is a standard Brownian motion under the risk-neutral measure. Meanwhile, and denote the drift and volatility of the risk-neutral productivity shock in state , respectively. The firm’s operating profit over time increment is then given bywhere is the firm’s operating revenue, is the investment cost over time, and is the additional adjustment cost that the firm incurs in the investment process.

Based on the neoclassical investment literature (Hayashi [25]), we assume that the firm’s adjustment cost is homogeneous of degree one in and . In other words, the adjustment cost takes the form , where is the firm’s investment-capital rate () and is increasing and convex in . We assume that is quadratic:where is the adjustment cost parameter and is a constant in state . In the literature, is usually considered as the rate of deprecation (or zero), which implies a zero adjustment cost when net investment is zero (or zero adjustment cost for zero gross investment).

Finally, the firm can liquidate capital at any moment and obtain a liquidation value , where denotes the recovery value per unit of capital in state . Let denote the optimal liquidation time.

##### 2.4. Stochastic Financing Opportunities

For simplicity, we only research external equity financing as the source of external funds for the firm. When the firm increases external equity financing, it incurs a fixed cost , where is the fixed cost parameter in state . Besides the fixed cost , it also causes a variable cost for each incremental dollar it raises. Let , , and denote the process for the firm’s cumulative external financing, the firm’s cumulative issuance costs, and the firm’s cumulative nondecreasing payout process to shareholders up to time , respectively. Hence, denotes the net proceeds from external financing over time interval , denotes the financing costs to raise net proceeds from external financing, and denotes the payout over time interval .

The manager distributes cash to shareholders in order to avoid paying a carry cost on the firm’s retained cash holdings. We assume that cash inside the firm earns a below-market riskless return and incurs the carry cost denoted by in state . Then the dynamics for the firm’s cash evolves as follows:where is the risk-free interest rate in state . Observe that and are endogenously determined by the firm.

#### 3. The Solution

This section investigates the firm optimality in a regime switching economy when the manager has time-consistent or time-inconsistent preferences. Let and denote firm value with time consistency and time inconsistency in state , respectively. The manager would choose to pay out cash once its stock grows sufficiently large in order to avoid the cash-carrying cost. Hence, the payout boundary with time consistency (time inconsistency) is denoted by (). Similarly, if the firm’s cash holdings are low, it could choose to issue equity. Let and denote the issuance boundary with time consistency and time inconsistency, respectively. We first characterize the solution for the case in which a manager has time-consistent preferences. In this case, the firm’s decision-making and firm value rely on the following three regions: an external financing/liquidation region (), an internal financing region (), and a payout region ().

##### 3.1. Time-Consistent Model

Suppose that the manager with time-consistent preferences chooses firm’s investment , cumulative payout policy , cumulative external financing , and liquidation time to maximize firm value defined at time (under the risk-neutral measure):The first term is the discounted value of net payouts to shareholders; the second term is the discounted value upon liquidation. Optimality could imply that the firm never liquidates, so we have .

When the firm’s cash flow is in the internal financing region, i.e., , satisfies the following system of Hamilton-Jacobi-Bellman (HJB) equations (under the risk-neutral measure):where denotes the state that is different from . The first term on the right side of (9) represents the impact of capital stock changes on firm value. The second term represents the effects of the volatility of on firm value. And the third term represents the effects of the expected change in the firm’s cash holdings on firm value. The last term captures the expected change of firm value when the state changes from to .

We use the scale invariance of the firm’s technology to write firm value as and investment as in the case of time consistency, where is the cash-capital ratio. Substituting these terms into (9), we obtain the following ordinary differential equations (ODEs) for :

The first-order condition for the investment-capital ratio is given byTaking the derivative of investment-capital ratio in (11) with respect to , we getEquation (12) shows that the firm’s investment increases with if and only if firm value is concave.

Firstly, we specify the payout boundary. When the marginal value of cash flow held by the firm is less than the marginal value of cash flow held by shareholders, the manager starts paying out the excess cash. Thus the endogenous payout boundary satisfies the following value matching and super contact conditions (Dumas [26]):On the other hand, if , the firm is better off distributing the excess cash as a lump sum and reducing its cash holdings to . So we haveIntuitively, this condition reflects the fact that the firm holds too much cash or the firm’s cash holdings in state are such that and the state of the economy could suddenly switch from to .

Secondly, we give the issuance boundary. The firm chooses to raise external funds by issuing shares so as to bring its cash stock back into the interior region when it is sufficiently valuable. In this case, the firm could suddenly change from the state with the financing boundary into the other state with a higher financing boundary . And its cash holdings could be midway between the two lower financing boundaries (). The firm optimally chooses to issue equity financing when its going-concern value is higher than its liquidation value or the cost of external financing is low. Let denote the firm’s cash target level after equity issuance. We define and . Hence, when , firm value satisfiesWe have the following value matching and smooth pasting conditions:Intuitively, if the firm chooses to issue equity before it runs out of cash, it must be the case that the marginal value of cash at the issuance boundary is equal to the marginal issuance cost . So we have

Finally, if (17) fails to hold, the firm does not raise funding until it runs out of cash. Then the optimal liquidation boundary with time consistency is and firm value is given by

##### 3.2. Time-Inconsistent Model

In this section, we assume that the time-inconsistent manager chooses the firm’s investment , cumulative payout policy , cumulative external financing , and liquidation time to maximize firm value defined as follows (under the risk-neutral measure) at any time :If , it could display that the firm’s optimality is to never choose liquidation. The firm’s decision-making and firm value also rely on the following three regions: an external financing/liquidation region (), an internal financing region (), and a payout region ().

Let and denote firm value and investment in the case of time inconsistency, respectively. Hence, in the internal financing region, using Ito’s generalized lemma and the dynamic programming method, firm value with time inconsistency satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:where the function is defined byNote that the last term on the right side of (20) captures the effect of the manager’s time-inconsistent preferences on firm value. It is intuitive. Once the current self (who decides on during the interval ) loses control at rate , firm value has a sudden reduction . The function means that the current self expects her successor (who makes decisions during the interval ) will also apply nonexponential discounting. Hence, the succeeding self will use the investment strategy . In a word, the last term represents the average changes in firm value due to the random changes in the manager’s time preferences modeled by a Poisson process with .

We also can write firm value as , and investment as , where is the cash-capital ratio. Hence, we obtain the following ordinary differential equations (ODEs) for :where the function is defined by

The first-order condition for the investment-capital ratio is given by

Firstly, we specify the payout boundary. As with time inconsistency, the endogenous payout boundary satisfies the following value matching and super contact conditions:On the other hand, if , we haveAdditionally, the upper boundary condition of is defined bywhere depends on (25). When , it also has .

Secondly, we give the issuance boundary. Similar to the case of time consistency, if the firm is sufficiently valuable it then chooses to raise external funds through an equity issuance so as to bring its cash stock back into the interior region. Let denote the firm’s cash target level after equity issuance. We can write and . Hence, when , firm value satisfiesWe have the following value matching and smooth pasting conditions:Intuitively, if the firm chooses to issue equity before it runs out of cash, it must be the case that the marginal value of cash at the issuance boundary is equal to the marginal issuance cost . So we haveOf course, the lower boundary condition of is defined bywhere and depend on (29) and (30), respectively.

Finally, if condition (30) fails to hold, the firm does not raise funding until it runs out of cash. Then liquidation could be preferred; we have

#### 4. Quantitative Analysis

This section provides quantitative analysis of the time-inconsistent model and time-consistent model in a regime switching economy. We firstly exhibit our choice of parameters and then illustrate the model’s solutions in the case of time inconsistency and time consistency. In order to make a reliable comparison, we select some parameter values following the BCW model.

The parameters remain the same in both states: , , , , , and (under the risk-neutral probability measure). We rely on the technology parameters estimated by Eberly et al. [27] for these parameter choices. The cash-carrying cost is set to . We do not take a firm stand on the precise interpretation of the cash-carrying cost; it can be due to a tax disadvantage of cash or to agency frictions.

We set the marginal cost of issuance in both states to be . We keep this parameter constant across the two states for simplicity and focus only on changes in the fixed cost of equity issuance to capture changes in the firm’s financing opportunities. The fixed cost of equity issuance in state is set at . As for the issuance costs in state , we choose . Although in reality these parameter values can also change with the aggregate state, we keep them fixed in this paper to isolate the effects of changes in external financing conditions.

We assume that the transition intensity out of state is set to to reflect an average duration of eight years for state . Our paper chooses the price of risk with respect to financing shocks in state to be . So, the risk-adjusted transition intensity out of states and is and , respectively. In addition, we assume that the discount factor equals the risk-free rate . Table 1 summarizes the parameter values in the model.