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This paper extends the standard model of real option by allowing managerial biases originating from the cognition about the economic states in the future to examine the impacts of internal liquid funds on the firm’s investment decisions. The model indicates that the interactions between the cognition biases and firm’s volatility determine the extent to which the manager overestimates the growth rate, hence subjectively overvalues the firm’s value and distorts the investment decisions. The managerial cognition biases enhance the sensitivity of corporate investment to cash flow and increase the convexity of the investment decisions against the firm’s volatility. Relying on the amount of internal liquidity, the manager with cognition biases can both delay and expedite investing.

Many financial economists have exerted great efforts to study the interactions between corporate investment and financing strategies by assuming that the firm’s managers are fully rational, largely ignoring common personality traits of managers in modeling the complex decision-making process of corporate executives. How- ever, at the present time, a lot of studies about human behavior find that managers are inclined to display irrational characteristics, such as time-inconsistent preferences, optimism, overconfidence and over-extrapo- lative biases, and so on^{1}. In fact, all of these characteristics can be generalized as the managerial cognition biases about the economic states. Besides, to study how the firm’s internal capital impacts its investment decisions is another important topic. Thus, in this paper, based on the standard theories of real options, we develop a continuous-time model with managerial biases originating from the manager’s cognition about the economic states in the future to investigate the impacts of internal liquid funds on the firm’s investment decisions.

The cognition biases capturing the manager’s characteristics also can be interpreted as a bias of ability to derive information from the real world. We consequently look at what happens inside the firm when the manager is rational all respects, except for his cognition associated with the economic states. Interestingly, our model shows that it is the interaction between the biased cognition of the manager and the firm’s volatility determines the extend to which the manager who has biased cognition about the economic states perceives the firm’s growth rate. In contrast, in the literature studying optimism and overconfidence, such as [

By assuming the manager has biased cognition on the economic states, our paper finds that the corresponding biases enhance the sensitivity of corporate investment to cash flow. Meanwhile, managerial cognition biases result in distortion of the firm’s investment. Specifically, the manager with biased cognition tends to delay investment if the firm just has few internal funds, but is inclined to accelerate the investment timing if the firm has sufficient internal liquidity. In addition, differing from the studies about overconfidence, the interaction between the managerial biased cognition and the firm’s volatility imposes effects on the growth rate of the firm in the future from the managerial point of view and hence gives rise to that the managerial cognition biases increase the convexity of the investment decision against the firm’s volatility.

This paper relates to several strands of literature. Firstly, by incorporating the managerial cognition biases and indicating the corresponding effects on firms’ investment decisions, it extends the study on real options deve- loped by [

Secondly, our paper continues the line of research about the effects of internal liquidity on firm’s investment decisions. [

Finally, our paper is related to the studies about subjective evolution, such as [

The rest of the paper is organized as follows. Section 2 describes the setup of the model with managerial cognition biases. Section 3 provides solutions for the model. Implication analysis and numerical results are given in Section 4. Finally, Section 5 concludes the whole paper. All Proofs are presented in the Appendix section.

In this section, we develop a continuous-time model with managerial cognition biases associated with the cash flows of the firm in the future. Firstly, we assume that the firm has an investment project and some internal funds at the initial time 0. The investment perpetually generates cash flows

where

In addition to the real probability space

Comparing Equation (1) with (2) indicates that the manager can accurately predict the firm’s growth rate

Conceptionally speaking, probability measures

where ^{2}. Moreover, it is easy to check that

From Equations (1) and (3) , we have the following process

which implies that it is the interaction between the biased cognition of the manager and the firms volatility determines the extend to which the manager overestimates the growth rate of firm in the future. However, this effects are completely different from those arising from overconfidence and/or overoptimism because the volatility has no impacts on the growth rate such as [

Finally, we assume that the manager run the firm to maximize the value of the existing shareholders subject to his knowledge biases on the economic environment. This also implies that there is not agency conflict between shareholders and the manager. In addition, the internal funds

Once the investment is undertaken, the profit is perpetually generated, following the process (1). Thus, the real valuation of the investment conditional on the current cash flow

On the other hand, based on his knowledge about the project, the manager thinks that the generated cash flow by the investment evolves according to Equation (5) and hence his subjective value of the investment con- ditional on ^{3}:

This implies that the managerial subjective evaluation of the investment depends on his subjective biases, which result in an overestimated value of the firm after investment.

Suppose that the manager undertakes the project at time

In fact, the first term in the second line is the firm value based on the managerial cognition biases, and the second and third terms capture the costs of external equity funding in the manager’s point of view. If the manager has not any knowledge biases, that is

Note that before investment, internal capital is reinvested in a risk-free asset and hence generates interest

As stated above, the manager undertakes the investment to maximize the firm value for the existing shareholders based on his biased knowledge. It is equivalent to choose the timing of the investment. Ac- cordingly, the investment decisions determined by the manager can be represented as the following optimal problem before the investment

subject to

Intuitively, Equation (9) shows that the manager exercises the option to realize the value of the project by costing

Note that once the accruing internal funds exceed the investment costs I before undertaking the investment, the manager distributes the interests

with boundary conditions:

and

These are very intuitive. Condition (13) represents that the firm value valuated by the manager equals the value of internal capital

where

and

Equation (16) implies that it is optimal for the manager to put off investing if the current profit

Particularly, the solutions in Equations (16)-(18) are consistent with those to maximize the physical value of the firm if no managerial biases, i.e.

In this section, we focus on the effects of the managerial cognition biases on investment decisions by comparing the solutions in our model with those for the standard model. In the numerical analysis, according to [

Similarly, we define the first-order derivative

in the perspective of the timing of undertaking investment because a reduction in

Proposition 1. The increase in the magnitude of cognition biases

Proposition (1) shows that increments in internal funds induce the manager to expedite investing sooner when his cognition biases are greater, providing useful empirical implications about the relationship between the managerial cognition biases and the sensitivity of corporate investment to cash flow.

Proposition 2. If

The first part of Proportion (2) indicates that if the internal capital of the firm is enough to cover the whole investment costs I, the manager with biased cognition prefers to expedite investing relative to the manager without biased cognition. This is due to that the manager overestimates the economic states of the firm, that is he thinks that the firm has a higher growth rate so it is optimal to invest earlier. This result is consistent with the empirical studies about behavioral corporate finance, such as [

In addition, because the manager with biased cognition trades-off the subjective benefits from undertaking the investment and costs arising from external equity issuance, the effects of cognition biases on the investment threshold intuitively depend on the level of internal funds. When the internal funds are less, the more the cognition biases of the manager, the later he undertakes the investment. Otherwise, the manager expedites investing.

Now, we focus on that how the firm’s volatility influences the firm’s investment decisions and that how the managerial cognition biases distort the investment decisions along with the firm’s volatility. For convenience, we rewrite Equation (17) as follows:

Thus, the corresponding threshold for the standard model can be represented as

Comparing Equation (22) with Equation (23) shows that how the volatility influences the investment threshold through

extreme low or high level of the firm’s volatility induces the manager who has biased cognition to delay undertaking the investment. In contrast, the moderate volatility induces the manager with cognition biases to accelerate the investment timing.

In this paper, based on the standard theories of real options, we develop a continuous-time model with man- agerial cognition biases associated with the firm’s cash flow in the future to investigate the impacts of internal funds on the firm’s investment decisions and provide closed-form solutions for the relationship among the managerial cognition biases, the internal funds and investment strategies. Our model shows that the interactions between the biased cognition of the manager and the firm’s volatility determine the extend to which the manager with cognition biases overestimates the growth rate and accordingly the consequent valuation of the firm.

In addition, we find that the biased cognition of the manager enhances the sensitivity of corporate investment to cash flow. Moreover, relative to the manager without any cognition biases as in the standard models of real options, the limited internal funds induce the manager who has biased cognition to put off investing, but the sufficient internal capital in the firm induces the manager to accelerate the investment timing. This implies that the managerial cognition biases greatly distort the effects of the internal financing on the firm’s investment decisions. Finally, our paper concludes that the managerial cognition biases increase the convexity of the investment decisions against the firm’s volatility, which implies that the impacts resulting from managerial cognition biases are completely different from those arising from the manager’s overconfidence, which only translates the curve of the investment threshold against the firm’s volatility.

The paper just uses a parsimonious model to study that how managerial cognition biases influence the firm’s investment strategies by using internal liquidity. There is some limitations in our research. But our model can serve as a benchmark for future work. For example, if we allow the firm to issue risky corporate debt, what are firm’s investment decisions and how managerial cognition bases impact the firm’s capital structure. In fact, to study the role played by human behavior in finance has very extensive applications and significant economic implications.

We thank the Editor and referees for their constructive comments and suggestions, which help us substantially improve the paper. Chunhui Wen acknowledgments support by National Social Science Foundation of China (No. 13CJY008) and Social Science Foundation of Hunan Province (No. 15YBA206).

Zhigang Liu,Congming Mu,Chunhui Wen, (2016) Impacts of Internal Financing on Investment Decisions by Managers with Cognition Biases. Journal of Mathematical Finance,06,431-442. doi: 10.4236/jmf.2016.63034

Proof. Note that

Meanwhile, note that

Differentiating it with respect to

For convenience, we denote

Proof. The proof for the first part, please refer to the proof of proposition 2 in [

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