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Step1. Initialization. Input scaled time subinterval counter it, the length
of subintervals hs, vector xm of values of first state function at switching
times, and also the initial state and vector sm. Set total subintervals
nn = 1/hs;
Step2. Use linear interpolation to get the estimate “xmt” of the state, in a time
between grid points 0h, h, 2h,..., h * nn, where h = 1/nn;
Step3. Add up the time intervals sm to get time in (3.9);
Step4. Construct the right side of the equation (3.9) to obtain time variable


Step5. Calculate the integrand in (4.17) at the scaled time and change the
sign of the integral. (This problem is seeking for a maximum. Since the
computer package only deals with minimization calculation, an opposite
sign of the objective function needs to be changed.)
104 OPTIMAL CONTROL MODELS IN FINANCE

Computational method 4.5 End-point condition (see model1_5.m in Ap-
pendix A.3)
Step1. Initialization. Input the last value of the first state function in vector xm
and the parameters par which are required by the “End-point condition”;
Step2. Construct the terminal term in (4.14).

7. Computing Results and Conclusion
In this section, the algorithms 4.1-4.5 are used in a computer software pack-
age (see in Appendix A.3) which was developed for this financial decision-
making model (4.1)-(4.5). Before we present the computing results, the ana-
lytical solution in the Davis and Elzinga [22, 1970] finance model is described
first for the future comparisons. Figure 6 in Davis and Elzinga [22, 1970]
shows the optimum solution graphically, which means:

Solution case [1]: when the optimal solution has control
in case [C] all the time;

Solution case [2]: when the optimal solution has control
switching from case [B] to [A] at a switching time.

The control regions [A], [B], [C] are described in Section 4.3.

First, set the parameters:




which meets the restriction on in case (a) in Section 4.3.




Initialize the states:
An Optimal Corporate Financing Model 105

Map the control in this order:

[B]
[C]
[A]

in successive time intervals. [B], [C], [A] for nb = 3, or [B], [C], [A], [B],
[C], [A] for nb = 6.
Then run the programs for the algorithms 4.1 -4.5 (details see “model1_1 .m”,
“model1_2.m”, “model1_3.m”, “model1_4.m”, “model1_5.m” in Appendix A.3).
All the cases are put into Table 4.1.




Here, “most [C]” indicates that most of the time is spent in [C].


From the results in Table 4.1:




Then according the analytical solution in Davis™ work, the optimal solution
is supposed to be in solution case [1], which is control [C]. The computing
results in Table 4.1 almost agree with the analytical solution. Since 1.49 is very
close to 1.47, the case [A] will mix with case [C] at some point. This computed
case is close to the theoretical optimum, but it does not exactly agree with i t .
Another solution case is verified next.
Initialize the states:
106 OPTIMAL CONTROL MODELS IN FINANCE

in this order:
Map the optimal control

[C]
[B]
[A]

in successive time intervals. [C], [B], [A] for nb = 3, or [C], [B], [A], [C],
[B], [A] for nb = 6.
Then run the programs. The results are shown in Table 4.2.




From the result in Table 4.2:




The computing results agree with the analytical solution in solution case [2],
which is control case [B] switching to [A] at a certain switching time. In this
example, the computed results agree well with the theory. Another example
with the same initialization but a different given order of control mapping is
given in Table 4.3; the results agree with Table 4.2.
The optimal control are in order:

[B]
[C]
[A]
An Optimal Corporate Financing Model 107

An approximate calculation using the SCOM package, dividing [0,1] into
20 equal subintervals, also confirms the switching patterns for solution case
[2]. The following computation has the same initialization of the parameters
as the above computation. The initial states also take: P(0) = 0.5, E(0) = 1.
The solution of the computation is shown as follows.
The optimal control takes values:




The states are:

P(T) = 1.99
E(T) = 1.96
The objective function is:


the optimal solution
is expected to be in solution case [2] in the analytical solution. The results of
this computation are very close to Table 4.2 and Table 4.3 and also confirm the
analytical solution in Davis and Elzing™s finance model. The optimal control
jumps from [B] to [C] at
The computation in this chapter agrees with the analytical solution in Davis
and Elzinga [22, 1970]. The results might change if the parameters have been
changed. The parameters set in this research are chosen to meet the restriction
on in case (a) (section 4.3). Further research on different parameters and
more subdivisions of the time interval will be very interesting.

8. Optimal Financing Implications
The results of this computation in Tables 4.2 and 4.3 are very close to the
analytical results in Section 3 and also confirm the analytical solution in Davis
and Elzinga [22, 1970]. The model results provide the dynamic structure of
capital of a firm and the optimal switching time from moving from one source
of finance to another. There is one switch in the firm™s financing strategy over
the planning period. The levels of the two sources of fund depend on the
relationships between the rate of return on equity capital and the investor™s
discount rate and the relationship between the equity per share (E) and
the market price of stock(P), thus the optimal control jumps from [B] to [C] at

Since the computational results of the optimal financing model are consis-
tent with the analytical results derived in Section 3 and the results of Davis and
108 OPTIMAL CONTROL MODELS IN FINANCE

Elzinga, they can be applied to understand optimal financing strategies of cor-
porations to determine the optimal mix of structure of long-term funds to use in
actual management of the capital structure of companies. Although theoretical
controversies continue, model results suggest that the optimal structure which
minimizes the firms™ composite cost of capital changes over time. The model
results provide the timing of switching from one fund to another fund. In real
life, these switches depend on the cost of these funds, the rate of return, share
prices, debt capacity, business cycles, business risks, etc. Various results are
generated by different sets of parameter values of the model. Sometimes sub-
jective judgments need to be made to choose the appropriate optimal capital
mix path of the firm.

9. Conclusion
The determination of the optimal structure of corporate capital and the switch-
ing times for different methods of financing are essential for the actual manage-
ment of capital structure of corporations. The computation of optimal switch-
ing time in this paper agrees with the analytical solution in Davis and Elzinga
[22, 1970]. The results might change if the parameters are changed. The pa-
rameters set in this research is chosen to meet the restriction on in case (a)
(Section 4.3). Further research on different parameters and more subdivisions
of the time interval will be very interesting. Development of an algorithm to
coincide the time subdivisions with switching times are also another important
area of further research.
Chapter 5

FURTHER COMPUTATIONAL EXPERIMENTS AND
RESULTS




1. Introduction
This chapter will give some experiments that have been done for examin-
ing the computational algorithms developed in Chapter 2 and Chapter 3 for
computing dynamic optimization financial models. The computation tests for
the algorithms in Chapter 2 are presented in Section 5.2. Three computing
examples are included. The problems met during computation are discussed
and the solutions for those problems are indicated. Section 5.3 contains a dif-
ferent control policy with one possible pattern of the financial optimal control
problem (3.2)-(3.6). The computing results of this new control policy are also
included. The objectives of the modeling experiments are to provide examples
of additional modeling structures which can be adopted in financial modeling
including some new forms of the objective functions and control patterns, and
to test the computational algorithms and the computer program by applying
them to some further models.

2. Different Fitting Functions
In this section, three different given fitting functions of OCPWCS (Optimal
Control Problems When Control are Step functions) are indicated. The diffi-
culty and non-accuracy in the computation of financial models are analyzed
and the possible solutions of those problems are given.

2.1 Calculation with square criterion in the objective
function
Consider a financial optimal control model as follows:
110 OPTIMAL CONTROL MODELS IN FINANCE




subject to:




The target function is: 1/2*t.
Now apply the algorithms 2.1 - 2.4 to solve this problem. Note the objective
function is no longer an absolute value. The algorithm 2.4 needs to be modified
to meet the square value. The time division technique will be used here. The
following graphs and tables are the computing results of this control problem.
In the figures, “*-” represents the state function and “.” represents the
given fitting function
This model is the same investment planning model as in Section 2.7. How-
ever this model has a different fitting functions and the objective function is
in a squared form which is a preferred form of decision criteria in economies
(Islam [36, 2001]).
Figure 5.1 is the result of (numbers of the time intervals).




Figure 5.1. Plot of n=4, forcing function ut=1,0,1,0
Further Computational Experiments and Results 111

The outputs of this computation are shown as follows:
The real optimal switching times are:




Time period [0,1] is divided into four subintervals:




The control is mapped as:




As results, the state variables of the financial system takes values as:




The minimization of the objective function at is:




Figure 5.2 is the graph result of
112 OPTIMAL CONTROL MODELS IN FINANCE




Figure 5.2. Plot of n=10, forcing function ut=1,0,1,0,1,0,1,0,1,0


The outputs of are:
The real optimal switching times are:




Time period [0,1] is divided into 10 subintervals:



The control is mapped into:




As results, the state takes values as:
Further Computational Experiments and Results 113

The minimization of the objective function at is:




When the jump of the control increases, a better approximation between state
and given fitting function is obtained.
In this experiment, several other computations have also been done in the
case of and, 8. The results of the objective function of all these cases
are shown in Figure 5.3 and in Table 5.1. The value of the objective function
decreases when increases. It is confirmed that the value of the objective
function tends to zero when the control jumps infinitely often.




Figure 5.3. Results of objective function at n=2,4,6,8,10


The above results imply that for the development of a stable investment plan
aimed at certain target value for the stock price, flexibility in switching among
investment strategies is essential.
114 OPTIMAL CONTROL MODELS IN FINANCE




2.2 Calculation with absolute value criterion in the
objective function
In this section, a similar financial model similar experiments as in Section
5.2.1 are introduced. The difference is that the objective function in this section
is the absolute value of the state approximating the given fitting function.
The financial optimal control problem is shown as follows:




subject to:




The definition of the variables and parameters are same as in Section 3.2.
However the present model has a different forcing function.
This is an optimal financing model for a firm where the decision problem
involves whether to buy back (“1) or issue (1) some stocks or to maintain the
present situation (0) of the amount of stocks used in the market so that the
firm™s stock price remains stable.
Apply the algorithms 2.1-2.4 on the problem (5.5)-(5.9). Only one modifi-
cation is made in algorithm 2.4 for this given fitting function There are
Further Computational Experiments and Results 115

two graphs shown as follows which represent and respectively.
As in the last section, the lines represent “state function” by “*-”, and “given
fitting function” by “.”.
The results gained from are shown as follows:
The real optimal switching times are:




Time period [0,1] is divided into four subintervals:




The control is mapped into:




As results, the state variable of the financial system variable of the financial
system takes values as:




The minimization of the objective function at is:




Figure 5.4 also shows the approximation between the state function and
given fitting function Three jumps of the optimal control are shown.
116 OPTIMAL CONTROL MODELS IN FINANCE




Figure 5.4. Plot of n=4, forcing function ut=1,0,1,0




An interesting situation was met during the computation at the case
Clearly the wrong result was gained in which the control only jumped once at
and the minimum of the objective function was greater
than the value of the minimum at the case It is possible that the com-
putation was finding some local minimum, instead of the global minimum.
A check for this mistake was also made. It was found that error was in the
initialization of the lengths of time intervals um0. Originally um0 was set as
[1/2,1/2,1/2,1/2,1/2,1/2,1/2]. Since the constraint on time is
then the original set of um0 did not meet this restriction. It also misled the
computation to a local minimum. The correct initialization of um is very crit-
ical for the whole computation. The mistake also occurred in the computation
of the case in Section 5.2.3. Now set um0 as Accurate results of

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