The Investment And Consumption Of Entrepreneurs Under Incomplete Markets

In this dissertation, we set up formal model of mathematical finance, in which we solve the optimization problems for entrepreneurs to maximize their total discounted consumption utility in infinite horizon by real investment, market portfolio allocation, precautionary saving, consumption and business entry/exit under incomplete markets resulted from the non-diversifiable business risk and liquidity constraints.The contributions of the dissertation are summed up as follows:(1) Based on the proportional investment friction and the assumption of Epstein-Zin recursive utility, we obtain semi-closed-form solutions of the entrepreneurial certainty equivalent wealth and optimal decisions by using the technique of reducing dimension according to the homogeneous property of the value function. The numerical results show that the business risk and liquidity constraints have the following significant economic effects: In contrast to the complete-market benchmark, these frictions and the risk aversion make the entrepreneur invest substantially less in the business and financial assets; The non-diversifiable idiosyncratic risk and liquidity constraints increase the precautionary saving and the"effective"coefficient of relative risk aversion, and also make the marginal value of the liquid wealth greater than one; Entrepreneurial firm's entry and exit both depend on outside option, start-up cost and the marginal value of the entrepreneurial wealth under incomplete markets, and the entrepreneur will reduce the initial size of the firm significantly to decrease the idiosyncratic risk exposure; The flexibility of the exit option controls the downside risk effectively; High risk aversion appetite makes the entrepreneur entry later and exit earlier; The liquid wealth could mitigate the negative effect from non-diversifiable business risk and liquidity constraints; Especially, the incompleteness of the markets have no effects on entrepreneurial intertemporal decisions and the valuation of asset/capital when his liquid wealth gets sufficiently high.(2) Non-diversifiable risk and liquidity constraints invalidate the risk-return rule to fix the cost of the capital. For this reason, we measure the expected return via the firm private value, and find that the entrepreneurs obtain not only the system risk premium for taking the aggregate system risk, but also the idiosyncratic risk premium for exposuring to non-diversifiable business risk. The results enrich the classical CAPM theory under incomplete markets. In addition, the numerical results show that the incompleteness of the markets has significant impacts on the entrepreneurial expected return, beta coefficient, system risk premium and idiosyncratic risk premium.(3) The dissertation studies impacts of the fixed investment cost and pricing wedge on entrepreneurial decisions and asset/capital pricing. Theoretic and numerical results show that: The fixed investment cost and pricing wedge have significant effects on enterprises'value, marginal value of liquid wealth, real investment strategy, marginal propensity to consume, beta coefficient, optimal initial investment size, the value of investment option and the threshold level for liquid wealth; However, the impacts on consumption and market portfolio allocation, liquidation and idiosyncratic risk premium are not obvious; Especially, the price wedge decreases the motivation of the reverse real investment and initial investment size, and on the contrary, the fixed investment cost does not affect on the reverse real investment and increases the initial investment size.(4) We explore the optimal investment, consumption and pricing based on risk financing under the assumption of CARA utility. Thanks to analytical derivation, we find that enterprises'value, real investment, consumption, business exit (default or cash-out) depend only on the capital stock. Theoretic analysis and numerical results show that: The incompleteness of the markets has negative effects on the pricing of entrepreneurial equity and real investment decision, but no effects on default decision; The entrepreneur will cash out on time to manage the upside risk if the idiosyncratic risk is high enough; And entrepreneurs obtain idiosyncratic risk premium for taking non-diversifiable business risk; Moreover, the incompleteness of the markets also has substantial impacts on optimal debt financing strategy, initial capital structure, asset/capital pricing and real investment decision under different equity financing strategies.(5) Based on the assumption of CARA utility, we explore the optimal investment and consumption with an infinite horizon and obtain the consumption-utility based indifference price of real option under partial information. Thanks to Kalman filtering, dynamic programming and Hamilton-Jacobi-Bellman theory, an implied option value is given by the semi-closed-form solution to the free-boundary PDE under the separation principle of control system. By Monte Carlo simulation, the difference of strategies between partial information and full information is discussed. Finally, we explore the relation between the two value functions under partial and full information, and the economic value of information is derived by comparing the welfare values under the two different information levels.