Optimal Portfolio of Corporate Investment and Consumption Problem under Market Closure: Inflation Case

We present the model of corporate optimal investment with consideration of the influence of inflation and the difference between the market opening and market closure. In our model, the investor has three market activities of his or her choice: investment in project A, investment in project B, and consumption. The optimal strategy for the investor is obtained using the HamiltonJacobi-Bellman equation which is derived using the dynamic programming principle. Further along, a specific case, the Hyperbolic Absolute Risk Aversion case, is discussed in detail, where the explicit optimal strategy can be obtained using a very simple and direct method. At the very end, we present some simulation results along with a brief analysis of the relationship between the optimal strategy and other factors.


Introduction
Inflation originally referred to the debasement of the currency.When the government increases the total number of the currency, the relative value of the currency decreases.As a result, consumers would need more money to exchange for the same goods and services.The term inflation usually refers to a measured rise in a broad price index that represents the overall level of prices in goods and services in the economy.The Consumer Price Index (CPI), the Personal Consumption Expenditures Price Index (PCEPI), and the GDP deflator are some examples of broad price indices.These factors play an important role in economics, because of which they are studied in the theory.
Portfolio diversification plays an important role in finance.In Merton [1] and Duffie [2], a stochastic model is first given to analyze the security market, while Karatzas [3], based on the stochastic analysis, presented the model considering the split of consumption and investment of the investors.Choi [4], on the other hand, paid attention to the international corporate investment where investments were made into real corporate projects.Afterwards, Bellalah and Wu [5] extended the theory of corporate international investment from Choi into an environment with the presence of incomplete information and taxation from governments.Their model was concerned with the international diversification problem in finance and gave reasonable analysis to the "home bias puzzle." In 2006, a model giving the optimal corporate portfolio and consumption choice was presented by Wu and Zhang [6], while Bellalah and Wu [7] modeled the market closure together with the international securities portfolio management with incomplete information.
In our paper, we present a model of corporate investment and consumption choice problem with market closure in inflation case, so that we can derive the optimal investment strategy.Our model is an extension of the usual corporate investment model from Bellalah and Wu [5,7], Wu and Zhang [6], Huang and Wu [8], and D. T. Zhang and T. Zhang [9] since we take into consideration the influence of inflation and the investment on two projects.
In this model, we suppose that the investor has an option to invest his money into two different projects, A and B. In real life, the production of the factory may be different between the day and the night; therefore, in our model, we have equations describing the input price and the output price of the production in the daytime and night with specified instantaneous expected rates and instantaneous volatilities.Also, we develop two pieces of production equations corresponding to the daytime and nighttime production which is a function of the output price.With the input and output price model and the production quantity model, we can easily derive the expression for the net profit of the corporate investment after considering taxation of the government.
Finally, instead of making investment, the investor has the option to make consumption for satisfaction as well.
During the nighttime, banking activities will not be allowed; therefore, the investment portfolio will be the same throughout the night.However, the investor is still allowed to consume to increase utility.Clearly, the consumption in the daytime and the night will be different.
In general, we consider corporate investment and consumption altogether to maximize the utility of wealth for the investor aiming to find out the optimal strategy for the investment.For the general model, it is difficult to solve for the optimal value function explicitly.Therefore, we considered a specific and important case of utility function, Hyperbolic Absolute Risk Aversion (HARA) case, to get the optimal solution.It turns out that we can write out the explicit optimal solution using a very simple and direct method which will be discussed in Section 3 of the paper.
In the next section, we will present the model and the assumptions of the optimal investment problem.In the HARA cases, besides giving the explicit optimal solution of the corporate investment model using dynamic programming principle, we will give analysis of the model in the economical point of view in Section 3.
At the end of this paper, some simulation results will be given to provide a more numerical interpretation of the model.Relationship between the optimal portfolio and important factors such as the volatility and the interest rate will be given together with computer-drawn graphs.

Model and Formulation of the Problem
Let (Ω, F, ) be a complete probability space endowed with a filtration {F  : 0 ≤  ≤ }.{  } 0≤≤ , { 1  } 0≤≤ , and { 2  } 0≤≤ are three 1-dimensional Brownian motions defined in this space, which represent the external sources of uncertainty in the market, and the correlation coefficients between them are Denote the rate of inflation by   , which is stochastic and depends on the situation of the economic.Like some pricing processes of the stochastic assets, we use a geometric Brownian motion to describe the inflation rate level at time , that is, The drift of the process  is the expected rate of inflation per unit of time, and it is defined by where () 2 is the variance of the process per unit of time defined by Without loss of generality, we assume that  0 = 1; that is, there is no inflation at initial time  = 0.
In this model, we suppose that the investor can choose two real projects, A and B, with some production to get a higher return but with certain risk.The input price and output price are not the same during the day and the night.The duration of the production in the daytime is , and the duration at the nighttime is .Then the daytime is [0, ], [ + , 2 + ], . .., [( − 1)( + ),  + ( − 1)], denoted by , while the nighttime is [,  + ], [2 + , 2 + 2], . .., [( + ( − 1), ( + )], denoted by   .
For project A, we denote () and () as the the input prices in the daytime and nighttime, respectively.() and () are the output prices, while () and () are the quantity of the production and (), and () are the cash flow.
In economic theory, the quantity of the production is the function of the input price, () = [()]  1 , where  1 < 0 in general.The cash flow of project A in daytime and nighttime are as follows: Similarly, we can define the corresponding variables of project B using the letters which have the symbol * .And the cash flow of project B in daytime and nighttime are as follows: We suppose the input price and output price satisfy the following equation: where () and () have initial values  0 ,  0 , respectively.
Here   ,   represent the instantaneous expected input rates in the day and the night, respectively, while   ,   represent the instantaneous expected output rates.,  are instantaneous volatilities in the day and the night, respectively.According to Ito's formula, we know that the cash flow for project A and B satisfy the following equations: where where with Similarly, we can get  * () and  * () just substitute  * ( * ) for  () and substitute  * ( * ) for  ().
Since the existence of the inflation, the actual value of the input price should be   =   /  .Notice that the proportional rate of change of the real return of input price and output price should be described by Finally the real cash flow for project A and B, Λ() = ()/(), Λ(), . .., satisfy Let  , () denote the total wealth at time , and () represents the proportion of the wealth invested in project B; then (1 − ()) , () is the amount invested in project A. Taking inflation into consideration, both of them are calculated by its actual value.So we have Substituting Λ  , Λ * (), Λ  , and Λ *  parts by ( 12) and ( 13), respectively, we get that, for an investor whose initial wealth is  > 0, the total wealth at time  in inflation case satisfies In the daytime, the investor can choose investment with proportion  and consumption rate  1 to maximize his wealth, but at the nighttime, he cannot change his portfolio and only can choose a different consumption rate  2 ; the portfolio at the nighttime stays the same as the optimal portfolio in the daytime.So the investor wants to maximize the following utility of wealth by choosing his investment strategy  and consumption rate  1 and  2 .Let  1 () the value of wealth  starting the daytime and  2 () be the value of the wealth  starting the nighttime.The whole investment duration can be divided into  days, that is,  daytime duration and  the nighttime duration.Consider The optimal problem (15)-( 19) can be solved by the classical methods, such as Pontryagin's maximum principle (cf., e.g., Yong and Zhou [10], Wang et al. [11], Wang and Wu [12], etc.) which gives the necessary conditions for the optimal strategies and dynamic programming principle (cf., e.g., Yong and Zhou [10]).In the following section, we will use a simple method, called construction method, to give an explicit solution to this problem in a special HARA case and give the economic analysis.

HARA Case: Optimal Solutions and Economic Analysis
In this section, we want to use the construction method to get the explicit optimal strategy in the special Hyperbolic Absolute Risk Aversion (HARA) case.This method can also be found when solving the linear quadratic (LQ) optimal control problem.It is often along with the dynamic programming principle; see Yong and Zhou [10], for example.For simplicity, we only consider the case where the entire time interval is +,  is the duration of the daytime, and  is the duration of the nighttime.For general case, the whole interval is ( + ), and we can get the explicit result by repeating the procedure with the same method.Let Here  and  are constants, where  > 0,  ∈ (0, 1).We try to get explicit optimal decision , consumption rates  1 ,  2 , and the optimal value function in this case.
Proof.We first consider the optimal problem (21) at nighttime.During the night, the investor cannot change his portfolio.His consumption  2 is the only activity that varies.So, the wealth equation in the duration [,  + ] is, Here  * is the optimal portfolio in the daytime, and we will solve it afterward.
We let Θ  be one nonnegative deterministic continuous function whose dynamic will be given later.Applying Ito's formula to ( −(−) /(1 − ))( , ()) 1− Θ  from  to  + , we have So we can write where Now we let Θ  be nonnegative and the solution of the following ordinary differential equation: Here So We can easily get  = 0,  = 0.
If we take then we can check that  attains its maximum at point ( 2 ) * and  = 0.So And then applying Ito's formula to( − /(1−))( ,  ) 1− Θ  from 0 to  and taking expectation on both sides, we have So we can write where Here (39) If we take it is easy to check that   ( * ()) = 0 and   ( * ) < 0.
One can check that  attains its maximum at point ( 1  ) * ,  = 0. Then the optimal value function is optimal investment and consumption strategy and then some simulation results illustrating the influence of market's volatility parameters to the optimal choice.In addition, there are several interesting problems about the underlying topic, such as the corresponding convex restriction problem and the partial information problem which needs to use the Pontryagin's maximum principle under partial information (Wang and Wu [12,13], etc.).Since these problems are more consistent with the practical situation in real market, we will furthermore investigate them in further work and then desire to solve some practical optimization investment problems in financial market.