LINEBURG


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2002]) is that the algorithms applied to solve these models produce optimal lev-
els 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.
An Optimal Corporate Financing Model 93

In terms of algorithm, there is a scope for improvement in the existing al-
gorithms for computing optimal corporate financial models as well. The limi-
tation of the existing literature on optimal control with switching times is that
computation of such models is performed by algorithms based on discretiza-
tion 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 sub-
divisions 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 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
= market price of a share of stock
= 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 in-
creasing 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)
94 OPTIMAL CONTROL MODELS IN FINANCE

= discount on share price resulting from flotation cost
= a positive constant denoting the responsiveness of the price to changes in
earnings and dividends
T = planning horizon of the optimal financing program


The objective function of this model is to maximize the present value of the
owners™ shares. Expressed by an integral plus an end-point term:



subject to:




where, and the terminal condition is given by the fixed
planning horizon T.
Davis and Elzinga used the reverse-time construction technique which was
given in reference [35, 1965], which will be described below, to systematically
construct a complete solution from solution cases. The solution cases are clas-
sified according to the solution of a mathematical programming problem of
each instant at time arising from the Maximum Principle [69, 1962].

3. Analytical Solution
Davis and Elzinga used the reverse-time construction technique that is par-
ticularly significant in solution synthesis. Starting from the terminal manifold
and moving backwards in time, the entire space of state is filled with optimal
trajectories solving the problem for arbitrary initial states. A linear program
solved by inspection originally determined the solution of this model. The
Maximum principle was successfully used and also modified to allow ease in
handling discounted objective functional. The existence of the optimal control
for this problem was established by corollary 2 of Theorem 4 in reference [50,
1967]:
Theorem 4. Consider the non-linear process in




The data are as follows:
An Optimal Corporate Financing Model 95

1. The initial and target sets and are non-empty compact sets
varying continuously in for all in the basic prescribed compact interval

2. The control restraint set is a non-empty compact set varying
continuously n for
3. The state constraints are (possibly vacuous)
a finite or infinite family of constraints, where are real continuous
functions on
4. The family of admissible controllers consists of all measurable func-
tions on various time intervals in such that each
has a response on steering to
and
5. The cost for each is:



where in and and are continuous in
Assume:
a. The family of admissible controllers is not empty.
b. There exists a uniform bound:




for all responses to controllers
c. The extend velocity set:




is convex in for each fixed
Then there exists an optimal controller on in minimiz-
ing
Corollary 2. Consider the control process in




with cost:
96 OPTIMAL CONTROL MODELS IN FINANCE

where:



where the matrices A, B, are functions on and
are continuous in and is a convex function of for each fixed
Assume that the restraint set is compact and convex for all
Then hypothesis c. is valid. If we assume 1. to 4. and a. b., then the existence
of an optimal control on in is assured.
Due to the Maximum Principle , the necessary conditions for to be
optimal are:



where:




Here, The Hamiltonian can be modified by introducing new “steady
state” variables
An Optimal Corporate Financing Model 97

The adjoint variables are defined as follows:




with boundary conditions
Parameterizing terminal values of the state variables as and

The maximization of H with respect to can be characterized as follows:

(a)
(b)
(c)
(d)
(e)
(f)
(g)

The synthesis of the solution was done by a construction technique in the
reverse time sense. The complete solution is given as follows:

In the case (a)
A. 1.
B. 2.
C. 3.

The solution shows the classical bang-bang control. Although the singular
arc cases appear in the synthesis, none of them are optimal. The computational
methods established in Chapter 2 and Chapter 3 will be used to verify this
solution in Section 4.6.
98 OPTIMAL CONTROL MODELS IN FINANCE

4. Penalty Terms
A substantial class of optimal control problems will deal with the terminal
constraint as well as other constraints that may describe physical limitations on
some process. For computational reasons, some penalty terms are required to
be used to replace these constraints in order to obtain an unconstrained prob-
lem. There are some approaches with good reputations. Since the financial
model discussed in this chapter has an end-point constraint on the market price
of a share of stock at the end of the time planning horizon, the penalty term is
used in the terminal constraint transformation. First an approach is introduced
to deal with a minimization problem subject to both inequality and equality
constraints:




Consider the objective function as a cost to be minimized; then additional
penalty costs are added to when does not satisfy the constraints. Define
vector if
if The problem (1) is replaced by the following uncon-
strained problem:




The terms in and consist of the penalty functions. They are zero when
satisfy all the constraints. is a positive parameter, that can more generally be
replaced by different parameters for each component of and
and are Lagrange multipliers. From the theory of augmented Lagrangian [14,
1995], this unconstrained problem with the penalty terms is minimized at the
same point as the given constrained problem [A], provided that the Lagrange
multipliers and are suitably chosen.
Before the penalty method of terminal constraints is stated, the Delta func-
tion should be introduced since it will be used to include the terminal con-
straints into the integral later. The Dirac delta function is described by:




Now consider an optimal control problem of the form:
An Optimal Corporate Financing Model 99




A terminal constraint can be replaced by a penalty term added to




where is a positive parameter, and approximates to thus
and where is the Lagrange multiplier.
Thus an “end-point term” in a control problem can be included in the inte-
grand as shown:




and so is not an additional case to be treated separately.
A constraint such as which involves controls at different
times, can be treated by adjoining an additional state component:




and imposing the state constraint The latter can be handled by a
penalty term where the parameter is small.

5. Transformations for the Computer Software Package
for the Finance Model
In this section, in order to develop a computer software package of the opti-
mal control for this financial model, some transformations of the formulas are
introduced here to meet the requirement of the computation. Basically most
transformation techniques with the division of time scales used here were in-
troduced in Chapter 3. Only the different transformations for this particular
computer package are introduced here.
The differential equations in (4.2) and (4.3) are represented as follows:
100 OPTIMAL CONTROL MODELS IN FINANCE




The integral in (4.1) is transformed as:




where

Terminal state is treated separately for nqq package:




Since in this model, time horizon parameter T is assumed to be greater than
1, another transformation of changing time interval [0, T ] to [0,1] is required
for the program which only deals with the time period [0,1].
This transformation was indicated in section 2.3. Let a new time equal
here is the scaled time set for the computational methods.
Then the differential equations and integral of objective function become:




The integral in (4.1) is transformed as:




where
An Optimal Corporate Financing Model 101

The transformations here only deal with the time variable The end-point
term is considered separately in the program.

6. Computational Algorithms for the Non-linear Optimal
Control Problem
From section 4.3, we know that at a particular case (a), the bang-bang con-
trol is the optimal solution of the system. lies on the
vertices of a triangular area:




However, a singular arc is possible, with lying on an edge of the trian-
gle (instead of a vertex), when the parameters etc.) of the functions take
particular values. Since Davis and Elzinga did not have a computer software
package for this optimal control problem, this research is focused on devel-
oping computer software based on the research work which has been done in
Chapter 2 and Chapter 3. In order to get accurate estimates of the switching
times, some computational methods have to divide [0, T] into many subinter-
vals. The time transformation introduced by Goh and Teo [33, 1987] makes
it possible to avoid this difficulty. The technique was also used to construct
the computational algorithms in this section. We establish the control policy
as bang-bang control here. The optimal switching times will be computed.
Several sequences of the control patterns will be experimented with different
initialization of the states. The results will be discussed and analyzed in next
section.

Computational method 4.1 Main Model Program (see model 1_1.m in Ap-
pendix A.3)
Step1. Initialization. First set the a vector of parameters =
which includes all the parameters in this financial model. = is market
capitalization rate, = maximum investment rate, = rate of return to
equity, = discount on share price resulting from flotation costs, = a
positive constant denoting the responsiveness of the price to changes in
earnings and dividends, T = planning horizon of the capital budgeting
program. Then set parameter par = [the number of the state components,
number of control components, nb = the number of big subintervals, ns
= the number of small subintervals, parameters]; and get the total num-
ber of the subintervals by calculating Set the MATLAB
102 OPTIMAL CONTROL MODELS IN FINANCE

“constr” function parameters par (13) = 1 (one equation, constraint in
the minimization problem) , par(14) = the maximum number of function
evaluations, and arbitrary starting lengths of the switching time intervals
Set the vectors of upper bounds uu and
the vector of lower bounds ul of urn, thus Also set the
initial state xinit;
Step2. Call the MATLAB “constr” function. In turn, “constr” calls the “Model2”
to calculate the minimization of the calling program with respect to the op-
timal vector um;
Step3. Input the optimal result um to “Model2”, to obtain the values of the
objective function J(nn) (the last value of the integral) and state vector
xm (xm is a vector of all the values of the state functions take at the grid-
points of the switching time intervals). The result of the objective with
respect to optimal switching times is the solution of this financial system
model.
Computational method 4.2: Model 2 (see model1_2.m in Appendix A.3)
Step1. Initialization. Input um, par and initial state xinit. Set the initial state
xm(1, :) = xinit, and nx = par(1), the number of the state components,
nu = par(2), the number of the control components (in this case, nx and
nu both equal 2, there are two states and two controls,) nb = par(3), the
number of big time intervals, ns = par (4), the number of small time inter-
vals, nn = par(3) * par(4), the number of total time intervals, initial scaled
time subinterval counter it = 1, hs = length of the whole subinter-
vals. Choose the “Model3” as the right side of the differential equations
(4.2) and (4.3), input um;
Step2. Construct the vector sm whose components represent the lengths of
the total time intervals by dividing each big time interval um by ns;
Step3. Call the SCOM package function “nqq” with the stated “Model3” to
solve dynamic equations (4.2) and (4.3). Tabulate the solution for the state
as the vector the result of (4.2),
the result of (4.3);
Step4. Set the initial scaled time subinterval counter it = 1, initial state
zz(: 1) = 0, ma = 1, the number of the input state function. Choose the
“Model3” for SCOM function “nqq”. Input the vector xm(: 1) and um;
Step5. Call SCOM function “nqq” with the stated “Model4” to solve the dif-
ferential equation Tabulate the results in as the
components of the vector
An Optimal Corporate Financing Model 103

Step6. Obtain the last value xm(nn, 1) of the state vector for the “end-point
condition”;
Step7. Call the “End-point condition” to calculate the terminal state;
Step8. Add the result gained from “End-point” condition to the last value of
the vector jm as the result of the objective function in (4.1), and calcu-
late the constraint function of ™model2” which is
Computational method 4.3: Model 3 (see model1_3.m in Appendix A.3)
Step1. Initialization. Input scaled time subinterval counter it, the length of
total subintervals hs, vector um and vector par. Set the value of parameters
T. Set the number of the total subintervals nn = 1/hs;
Step2. Obtain the number of the big time intervals by nb = nn/ns for con-
structing the optimal control policy;
Step3. Set the control policy as vector which jumps between the
vertices of the triangle area (4.18)-(4.19). The control only jumps at the
end points of the big time intervals;
Step4. Construct the right side of the transformation equation for time pt in
(3.10);
Step5. Obtain the right side of the differential equations by using the transfor-
mations in (4.15), (4.16).
Computational method 4.4: Model 4 (see model1_4.m in Appendix A.3)

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