| The price of products may undergo abrupt change in a short time with the development of technology or unexpected events, however there are many products whose price processes are jump-diffusion processes. Under the assumption that the product price processes are jump diffusion processes, this paper mainly makes a research on how an company enter a new industry or project, how to expand its operation, and how to operate the company in view of the preparatory period. Considering the agency costs, the capital structure and the investment decisions involve shareholders, bondholders and managers, but we only consider the conflicts of interest between bond and equity holders, because this thesis bases on the assumption that managers have the same interests with shareholders.Let us consider the company's decision if it enter a new industry. No plant and capital are available, and the investment is fully funded debt or equity. Here we omit constructing time and purchasing well-established plant. Managers operate the company in order to max-imize shareholder equity valve. In view of entering a new plant, the following conclusions are obtained.Theorem 1 If r>μ+y0λ, the new plant is fully funded equity, and products price is governed by dPl=μPt-dt+σPt-dWt+y0Pt-dNt, then the value function of the equity is and the bankruptcy leverTheorem 2 If r>μ+y0λand (c(1-θ))/r>γ, the new plant is fully funded debt, and products price is governed by dPt=μPt-dt+σPt-dWt+y0Pt-dNt, then the value function of the equity is the value function of the debt isTheorem 3 If r>μ-γ, the new plant is fully funded equity, and products price is followed dPt=μPt-dt+σPt-dWt-Pt-dNt, then the value function of the equity is and the bankruptcy leverTheorem 4 the new plant is fully funded debt, and the product price is followed dPt=μPt-dt+σPt-dWt-Pt-dNt, then the values function of the equity and debt are andNow let us go back to the original firm that owns one plant with an option to invest an identical plant cost I, which is funded debt or equity. Let us consider the company how to expand a new plant. The following conclusions are obtained.Theorem 5 , the new plant is fully funded debt, and products price is governed by dPt=μPt-dt+σPt-dWt+y0Pt-dNt, then the value functi of the equity V1(p,c)isτm=inf{t; Pt≤Pm(c)},τd=inf{t; Pt≥Pd(c)}, where A1(c),B1(c),Pm(c),Pd(c)are to meet the formulas(3.13)-(3.16)of§3Theorem 3.4.Theorem 6 If r>μ+y0λ,the new plant is fully funded equity, Iis the investment cost,and products price is governed by dPt=μPt-dt+σPt-dWt+y0Pt-dNt,there exists Pe,Pl such thatτe=inf{t; Pt≤Pe},τl=inf{t; Pt≥Pl}.The value function of the equity is where Pe, Pl and A2,B2 are desribed by§3Theorem3.6's formulas(3.21)-(3.23).Theorem 7 If r>μ-λand (c(1-θ))/r>2γ,the new plant is fully funded equity,and the product price is followed dPt=μPt-dt+σPt-dWt-Pt-dNt,whereτm=inf{t; Pt≤Pm(c)},τd=inf{t; Pt≥Pd(c)},the value function of the equity is where A1(c),B1(c),Pm(c),Pd(c) are described by§4Theorem 4.4's formulas(4.14).Theorem 8 Under the assumption that the product price processes are dPt=μPt-dt+σPt-dWt-y0Pt-dNt, if r>μ-λy0 and (c(1-θ))/r>2γ,there exists Pm(c),Pd(c) such that τm=inf{t; Pt≤Pm(c)},τd=inf{t; Pt≥Pd(c)}, where l1, l2, l3, l4, Pm(c), Pd(c) are described by§4Theorem 4.9's formula(4.31).We can get the same conclusions in the assumption that the another plant is fully funded equity. We further consider a more realistic case when the construction of another plant requires certain period.Consider the company's strategy,we can get following conclusions.Theorem 9 ,the second plant is fully funded debt, and products price is governed by dPt=μPt-dt+σPt-dWt+y0Pt-dNt, there exists Pm,Pd such thatτm=inf{t; Pt≤Pm},τd=inf{t; Pt≥Pd}.Then the value function of the equity is where A1Δ(c),B1Δ(c),Pm(c),Pd(c) are constant,described by§3Theorem 3.7's formulas(3.29).Theorem 10 If r>μ-λand (c(1-θ))/r>2γ,under the assumption that the product price is followed dPt=μPt-dt+σPt-dWt-Pt-dNt, such thatτm=inf{t; Pt≤Pm(c)},τd=inf{t; Pt≥Pd(c)},the value of the equity is where A1Δ(c),B1Δ(c),Pm(c),Pd(c) are constant,such that§4Theorem 4.7's formulas(4.23).Theorem 11 If r>μ-λy0 and (c(1-θ))/r>2γ,under the assumption that the product price processes are dPt=μPt-dt+σPt-dWt-y0Pt-dNt, such thatτm=inf{t; Pt≤Pm(c)}, τd=inf{t; Pt≥Pd(c)}, and the value of the equity is where the constant coefficients and Pm(c), Pd(c) are described by§4Theorem 4.10's formulas (4.34)-(4.39).The result of numerical examples shows that, based on different operating principles, investment lines considered by the business managers are not the same, when prices rise, companies tend to over investment, thereby causing the loss of company value; on the other hand, some terms in claims agreement are forced to tend to under investment, reduce the size of business, resulting in the loss of business investment opportunities. Comparing investment strategy between the geometric Brownian motion model and the jump diffusion model, we find reasonable operating rules to price speculation and testing of the business decisions are of great significance, the use of inappropriate strategies will lead to unnec-essary losses incurred by an company, or the loss of good investment opportunities. The sensitivity analysis of the jump diffusion model parameters shows that the different parame-ter selection will produce different investment strategies. The selection of the parameter will have the opportunity to have a significant business impact on the new projects or expansion of conpanies, so before the decision-making the company must estimate the prices running model exactly, and then make a reasonable investment. When the business face higher risks, the agency cost between shareholders and creditors will gradually become larger, because of the capital substitution effect, the risk faced by the creditors will become larger, so when the creditors sign the contract terms, they'11 propose conditions to bind business investment management activities, to some extent, it will reduce the agency cost. |
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