Chapter 4

AN OPTIMAL CORPORATE FINANCING MODEL

1. Introduction

In this chapter, an optimal corporate financing model (Davis and Elzinga [22, 1970]) has been used since it is one of the well known and pioneering models on optimal control in finance. Some other recent work in this area include [21, 1998] and [61, 2001]. The approaches that were constructed in Chapter 2 and Chapter 3 for the optimal control problems are applied. The model discusses investment allocation in order to decide what proportion of its earnings should be retained for internal investment and what proportion should be distributed for shareholders and dividends in a public utility. The aim is to choose the smart investment program that the owners can get most benefits from. In the real world, these kinds of problems are common.

Section 4.2 defines the problem of this financial model. Then the analytical solution, which was created by Davis and Elzinga, is discussed in Section 4.3. In Section 4.4, an important technique called penalty term is introduced for solving optimal control problems with constraints and end-term condition. By using the penalty term , all the constraints become easy to be included into the cost function. The transformations of this model are described next in Section 4.5 for the computation. The computational algorithms for this model are constructed in Section 4.6. A computer software package for the algorithms is shown in Appendix A.3. The analysis and discussion of the computing results are presented in Section 4.7.

2. Problem Description

A firm decides how it should generate its finance to maximize the value of the firm, the stock value, or to achieve any other specified objectives. Two well known papers by Modigliani and Miller [59, 1958] [60, 1963] were instru-

mental in developing the literature on the modern theory of optimal corporate finance. According to this theory, the optimal financial structure of the firm is determined by the optimal financing level, the cost of capital or the weighted average cost of capital is equal to the weighted average costs of alternative sources of financing. For a firm, if funds can be obtained from debt or equity and retaining earnings, the optimal financial structure of the firm is given at the marginal investment, where:

P = {q/v)c+ (e/v)r

where g = the value of stock, и = the current value of the firm - the value of all outstanding claims against the firms assets, с = the cost of equity, e = the value of retained earnings, = the opportunity cost of the retained earnings.

It is academically interesting and practically useful to determine the optimal proportion between different funds that minimize the cost of capital and to maximize the value of the firm. In a dynamic framework, there may be switches among funds over time depending on the developments in the firm and in the financial market and the economy. It is, therefore, important to know the optimal timing for switching from one source of funds to another, which is optimal for the firm in minimizing the cost of capital and maximizing the value of firm. Unlike Modigdiani and Miller, it is assumed in this paper that the structure of capital has impact on the values of shares and the firm.

Modeling optimal corporate financial structure is a useful area for research in corporate finance since such models can provide information about the optimal proportion of sources of finances of the corporation, its investment and dividend strategies over a period of time. Dynamic optimization models in the form of optimal control in addressing optimal corporate financial structure have been developed initially by Davis and Elzinga [22, 1970], Krouse and Lee [49, 1973], Elton and Gruber [26, 1975] and Sethi [78, 1978] (see also Craven [14, 1995], Sethi and Thompson [79, 2000]).

A general limitation of the existing literature on optimal corporate financing is that although there are some analytical studies in this area, computational exercises (numerical model building, application of an algorithm and development or application of a computer program) are not well known (except Islam and Craven [38, 2002]). One limitation of the existing literature on the computation of optimal corporate financing models (including Islam and Craven [38, 2002]) is that the algorithms applied to solve these models produce optimal levels of various funds, but do not generate optimal timing for switching from one fund to another fund. In this paper, a computational approach (algorithm and program) which can generate optimal switching time among different funds is presented.

In terms of algorithm, there is a scope for improvement in the existing algorithms for computing optimal corporate financial models as well. The limitation of the existing literature on optimal control with switching times is that computation of such models is performed by algorithms based on discretization of the switching time. After switching time discretion, the resulting model is a discrete time continuous variable optimal control model with switching times. If the switching times are sub-division times of time discretization, this method can generate a unique solution. In models where switching time is made a parameter or a variable, the determination of optimal switching time may also be difficult due to the difficulty of the switching times and time subdivisions not coinciding. If the sub-division times and the switching times are different, then the computation involves two steps. First, to find the optimal solution for the time sub-division and later to find optimal solution in terms of optimal switching time. Computing an optimal financial model with too many time steps may be difficult in some cases due to computer memory and time required to compute.

A computational approach, which can overcome the above computational problems by suitable transformation of the original time is developed in this chapter.

A simplified non-linear optimal control model which can address the above optimal financing problems for corporations is described below, involving two state variables and two control variables. Price and equity per share are the state variables, and the earnings retention rate and stock financing rate us are the control variables. The objective of the utility is defined as the discounted sum of dividends and capital gains, thus, the present value of share ownership is to be maximized. Two differential equations describe the change in stock price and equity per share.

The state, control variables , and parameters are expressed as follows: State variables

P(t) = market price of a share of stock

E(t) = equity per share of outstanding common stock (net worth of utility divided by the number of shares outstanding)

Control variables

= retention rate which describes the fraction of earnings retained for increasing the capital assets

= stock financing rate concerning new money invested in the company

Parameters

= market capitalization rate (or investor discount rate) = maximum investment rate

= rate of return to equity (maximum return allowed by government)