LINEBURG

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equationвЂќ as the right side of the differential equation (3.1), input um (in
this chapter, the second order equation in the optimal control problem is
transformed into two equivalent first order differential equations; so there
are two state functions).
Step 2. Construct the vector sm whose components represent the lengths of
the all time intervals by dividing each big time interval um by ns.
Step 3. Call the SCOM package function вЂњnqqвЂќ with the stated вЂњInput function
for second order differential equationвЂќ to solve the two equivalent first order
differential equations (3.3), (3.4) for the state function Tabulate the
solution for the state as the vector
is the result of (3.3),
is the result of (3.4).
Step 4. 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 вЂњinput function for the oscillator problemвЂќ for SCOM function вЂњnqqвЂќ
46 OPTIMAL CONTROL MODELS IN FINANCE

to solve the integration of the objective function (3.2). Input the vectors
xm(: 1) and sm.
Step 5. Call SCOM function вЂњnqqвЂќ with the stated вЂњInput function for the os-
cillator problemвЂќ to solve the differential equation
Tabulate results in as the components of vector
Step 6. Take the last result of the vector jm as the value of the objective
function, and calculate the constraint function of вЂњMinimization programвЂќ
which is
Algorithm 3.3: Input function for second order differential equation (see
project2_3.m in Appendix A.2)
Step 1. Initialization. Input scaled time subinterval counter it, the length of
total subintervals hs and vector um. Set the value to parameters T, (set
B for(3.18) and (3.19) in the next section). Set the number of the total
subintervals nn = 1/hs.
Step 2. Obtain the number of the big time intervals by nb = nn/ns for estab-
lishing the control policy.
Step 3. Set the control policy as vector with al-
ternating value 1 and вЂ“1, see (3.8); the control only jumps at the end points
of the big time intervals.
Step 4. Construct the right side of the transformation equation for time pt in
(3.10): for

Step 5. Obtain the right sides of the differential equations by using the trans-
formations in (3.13) and (3.14):

Algorithm 3.4: Input function for the oscillator problem (see project2_4.m
in Appendix A.2)
Step 1. Initialization. Input scaled time subinterval counter it, the length of
total subintervals hs, vector xm of values of first state function at switching
times, and also the initial state and vector sm. Set the number of the total
subintervals nn = 1/hs.
Step 2. Use linear interpolation to get the estimate вЂњxmtвЂќ of the state, in a
time between grid points where
Step 3. Add up the time intervals sm to get time in (3.9):
for
A Financial Oscillator Model 47

Step 4. Construct the right side of the equation (3.9) to obtain time variable

Step 5. Calculate the integrand in (3.15) at scaled time t:

5. Financial Control Pattern
The financial dynamic system introduced in Section 3.2 is a second-order
differential equation, and the forcing function /indexforcing function takes
values such as вЂ“2, 2, вЂ“ 2 , 2 , . . . , switching between them at optimal switching
times determined by an optimization calculation. Note if is restricted
to choose two values вЂ“2 and 2, then both the patterns вЂ“2,2, вЂ“ 2 , 2 , . . . and
2, вЂ“2,2, вЂ“ 2 , . . . have to be computed. The comparison of the solutions of
these two patterns will be discussed in next section. This research studies
a set of bang-bang controls that jump on the extreme points of the feasible
area. However, since the sequences of the control switching are not defined,
all the sequences of the control patterns need to be computed and analyzed. A
numerical example in financial modeling is shown next.

6. Computing the Financial Model: Results and Analysis
In this section, the control is set to be вЂ“2 and 2. The solutions of two
different initial starts of control are discussed and compared.
We can formulate the financial optimal control problem as follows:

subject to:

takes value вЂ“ 2 , 2 , . . . or 2, вЂ“ 2 , . . . in successive time intervals (3.19)

The target value of the target = вЂ“ 5.
48 OPTIMAL CONTROL MODELS IN FINANCE

Computational results are indicated in Table 3.1 above.

All the graphs shown below and on the following few pages are the com-
puting results of the financial optimal control problem (3.16)-(3.22). There are
three sets of the graphs, which represent the solutions at nb = 2, nb = 4,
and nb = 6, where nb is the number of the big time intervals. In each set,
the big time interval is subdivided by ns = 1,2,4,6,8,10 respectively, where
ns is the number of the small time intervals. Each set has six pairs of graphs
representing six different subdivisions. All the graphs are shown in the same
order. A pair of graphs represent вЂњthe state function and the given fitting
function against time and the comparison of two state functions, and
A Financial Oscillator Model 49
50 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 51
52 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 53
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Set 1 for nb = 2

The first set of the graphs is the solution at nb (the number of big time
subintervals) equals 2. The time period [0,1] is divided into and
ns (the number of small time subintervals) takes 1,2,4,6,8,10 respectively.
As mentioned in Section 3.3, the control policy allows the control only to jump
at the end points of the big time intervals (so the control does not jump at the
end of the small time intervals). The control takes value вЂ“2,2 in two big time
intervals and switches once at time The first pair of graphs
are the results of nb = 2 and ns = 1, that is, the time period is divided by
2 big intervals and each subinterval has no further subdivision. It is obvious
that the approximation between state and the given fitting function is not very
close because of the small nb and ns. As ns (the number of small intervals)
increases, a better approximation is obtained. But since control only jumps
once during the whole time horizon, it is hard to reach the global minimum. It
is understandable that more jumps are helpful for searching a better fit вЂ“ to find
a more stable financial system. Although the ns still increases, the decreasing
of the objective function slows down, thus nb needs to be increased. The results
are also reported in Table 3.1. These results demonstrate the complexities of
the dynamics of the financial system with damped oscillator.
A Financial Oscillator Model 55
56 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 57
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A Financial Oscillator Model 59
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Set 2 for
A Financial Oscillator Model 61

This set of the graphs is the graphical solution at nb = 4 and ns = 1,2,4,6,
8,10. The control policy is in successive time intervals
and control jumps three times at A better
approximation than the results at nb = 2 is shown. Most of the results confirm
the computational algorithms 3.1-3.4 except a typical case nb = 4, ns = 8.
In this case, the value of the objective function is bigger than the value of the
objective function at nb = 4, ns = 6. That means the decrease of the objective
function stopped at this point and increased instead. We also discovered that
the value of the objective function at this point is same as the value of the other
control pattern 2, вЂ“2, 2, вЂ“2 at the same point. They are shown in Table 3.1 and
Table 3.2. From this test, it may be conjectured that the optimal search stops at
a certain point for some reason. A small test was also made in the experiment,
another control policy, which was created as 2, вЂ“2.05,2, was
put into the program to replace the old control policy. A better search was
obtained and the value of the objective function of the financial model became
J = 0.2034. The new control policy enables the optimal search to continue
until the optimum is reached.
62 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 63
64 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 65
66 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 67

Set 3 for nb = 6

This is the set of the graphical results at nb = 6 and ns = 1, 2, 4, 6, 8, 10.
A better approximation than nb = 4 is obtained as expected. A similar com-
putation as in case nb = 4, ns = 8 happened at case nb = 6, ns = 8 and case
nb = 6, ns = 64. In case nb = 6, ns = 8, the optimal switching times are
the control stays
at вЂ“2 from the third subinterval until the end of the time period. Although the
control policy was set to switch five times in the program, the real computation
did not show that the control jumped so often. As in the experiment we did for
case nb = 4, ns = 8, this problem also can be solved by perturbing the control
by a small amount. From the results shown in the Table 3.1, a conclusion can
be made that when the number of big intervals and small intervals increases,
the result of the objective function decreases. Although there are some unex-
pected results, a small disturbance of the control can easily obtain the correct
answers. The experiment gave some promising results which confirm the ac-
curacy of the computational algorithms in this chapter.
Another experiment is also presented to indicate the effects of the different
patterns of the control.
68 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 69
70 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 71
72 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 73

Set 1 for nb = 2
74 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 75
76 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 77
78 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 79

Set 2 for
80 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 81
82 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 83
84 OPTIMAL CONTROL MODELS IN FINANCE
A Financial Oscillator Model 85

Set 3 for nb = 6
86 OPTIMAL CONTROL MODELS IN FINANCE

In this group of graphs, the pattern of control is 2, вЂ“2,2, вЂ“2,.... Like the
previous experiment, this experiment starts from nb = 2, ns = 1 and increases
nb and ns gradually. As expected, better approximation is obtained by increas-
ing either nb or ns. Unexpected answers were also found at certain points. A
small perturbation is helpful to gain the accurate results. Here, we only put
the attention on the comparison of these two different patterns of the control.
From Table 3.1 and Table 3.2, it is found that the values of the objective func-
tion of two different control patterns at starting points (nb = 2, ns = 1) have
big differences. Pattern вЂ“ 2 , 2 , вЂ“ 2 , 2 , . . . gives much better result than pattern
2, вЂ“2, 2, вЂ“ 2 , . . . . It is an interesting phenomenon that when nb and ns be-
come very big (more jumps and better gradient), the results of the objective
function with different control patterns are very close. We can conclude that
when optimal control jumps infinitely and the integration calculation has more
subdivisions of the time period, the better fit for the problem can be reached
whatever pattern of the optimal control is used.
Figure 3.1 and Figure 3.2 show the results of the objective function against
1/ns at two different control patterns. The value of the objective function tends
to zero when the number nb of subdivisions increases. It is also confirmed
that a greater number of subdivisions of the time interval will lead to a better
integral calculation. The three lines in each figure also show that more jumps
of the control will give better approximation. In the next figure, a cost is
added to the objective function. A complete description of the cost of switching
control has been discussed in Chapter 2. In this chapter, a cost is only attached
to the number of large subintervals concerned with the control jumping, shown
as follows:

K is the cost of changing control, nb is the number of large subintervals.
In this computation, K is set to be 0.01, and ns = 64 for the accuracy of the
A Financial Oscillator Model 87

calculation. The same constraints (3.17)-(3.22) are included for this modified
optimal control problem (3.23).

Figure 3.1. Plot of integral F against 1/ns at ut=-2,2

Figure 3.2. Plot of integral F against 1/ns at ut=2,-2
88 OPTIMAL CONTROL MODELS IN FINANCE

Figure 3.3. Plot of cost function F against the number of large time intervals nb

As discussed earlier, two different levels of the subdivision of the time in-
tervals are used in this example. But the control policy is only constructed so
that the control switches at the end of the large subintervals. In two experi-
ments shown in Figure 3.3, a cost attached to nb (the number of the
large subintervals) is added to the objective function. It is known that a greater
number of subintervals lead the system to a more accurate minimization. A
more accurate value of the objective function can be obtained when
The cost is attached to the results of the objective function at ns = 64. Two
different lines in Figure 3.3 represent the results of the two different patterns
of the control which are shown in Figure 3.1 and Figure 3.2. At point nb = 6,
both of them reach the optimum. It is confirmed that the financial system with
two different control patterns will reach the same minimum with the same cost
when the number nb of subdivisions increases.
Since there are some non-switching times obtained from the computation, it
is conjectured that whether an optimal control with a zero value in the middle to
match the non-switching times can bring a better approximation. An example
with this matching control will be discussed in Chapter 5.
A Financial Oscillator Model 89

7. Financial Investment Implications and Conclusion
Since the financial sector is volatile, a financial oscillator model is neces-
sary to study the non-linear complex dynamic behavior of the sector. Incorpo-
ration of a damping function to stabilize the oscillatory dynamics of the finan-
cial sector can facilitate an understanding of the control mechanism useful for
smooth functioning of the financial market. The modeling and computational
experiments in this chapter show similar results as those obtained in Chapter 2.
Switching times and costs of switching control are significant factors in the de-
termination of optimal investment planning strategy for the economy. Higher
costs of switching control reduce the optimal number of switching times es-
sential in evolving an optimal investment strategy overtime. The dampening
factor is also significant in designing stable optimal investment planning.
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 вЂњsmartвЂќ 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 вЂњpenalty termвЂќ is introduced for
solving optimal control problems with constraints and end-term condition. By
using the вЂњpenalty termвЂќ, all the constraints become easy to be included into
the cost function. The transformations of this model are described next in Sec-
tion 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-
92 OPTIMAL CONTROL MODELS IN FINANCE

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:

where = the value of stock, = the current value of the firm - the value of
all outstanding claims against the firmвЂ™s assets, = the cost of equity, = the
value of retained earnings, = the opportunity cost of the retained earnings.
It is academically interesting and practically useful to determine the opti-
mal 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 op-
timal 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 develop-
ment or application of a computer program) are not well known (except Islam
and Craven [38, 2002]). One limitation of the existing literature on the compu-
tation of optimal corporate financing models (including Islam and Craven [38,
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