Chapter 5 Finance Application
(i) Simple Cash Balance Problem
(ii) Optimal Equity Financing of a corporation
(iii) Stochastic Application
Cash Balance Model
To determine optimal cash levels to meet the demand
for cash at minimum total discounted cost.
Too much cash  opportunity loss of not being able to
earn higher returns by buying securities.
Too little cash  will incur higher transaction costs
when securities are sold to meet the cash demand.
NOTATION
T = the time horizon,
x(t) = the cash balance in $.
y(t) = the security balance in $.
d(t) = instantaneous cash demand (d(t)<0  accounts
receivable.)
u(t) = sale of securities, -U2 u  U1 (u<0  purchase),
U2 0,U1 0.
r1(t) = interest on Cash (demand deposits).
r2(t) = interest on Securities (e.g. bonds).
 = Broker’s Commission in dollars per dollar’s worth
of Securities bought or sold.
State Equations
Control constraints
Objective
Solution. By Maximum Principle
Interpretation: 1(t) is the future value (at timeT ) of
one dollar held in the cash account from time t to T.
2(t) is the future value (at timeT ) of one dollar in
invested securities from time t to T. Thus, the adjoint
variables have natural interpretations as the actuarial
evaluations of competitive investments at each point
of time.
Optimal Policy
In order to deal with the absolute value function we
write the control variables u as the difference of two
nonnegative variables,i.e.,
we impose the quadratic constraint
Given (5.10) and (5.11) we can write
We can now substitute (5.10) and (5.12) into the
Hamiltonian (5.5) and reproduce below the part which
depends on control variables u1 and u2 , and denote it
by W. Thus,
W is linear in u1 and u2 so that the optimal strategy is
bang-bang and is as follows:
where
Interpretation:
Sell at the max allowable rate if the future value of
a dollar less than broker’s commission (i.e. the future
value of (1-) dollars ) is greater than the future value
of a dollar’s worth of securities, and not to sell if the
future values are in reverse order.
Interpretation:
 u2* =U2
i.e.,purchase securities at the maximum rate if the
future value of a dollar plus broker’s commission is less
than the future value of a dollar’s worth of securities.
Figure 5.1: Optimal Policy Shown in ( 1,2 ) Space
Figure 5.2: Optimal Policy Shown in ( t,2 /1) Space
Optimal Financing Model
y(t) = the value of the firm’s assets or invested capital at time t,
x(t) = the current earnings rate in dollars per unit time at time t,
u(t) = the external or new equity financing expressed as a
multiple of current earnings; u0,
(t) = the fraction of current earnings retained,i.e.,1- (t)
represents the rate of dividend payout; 0(t) 1,
1-c = the proportional floatation (i.e.,transaction) cost for
external equity; c a constant, 0 c 1,
 = the continuous discount rate (assumed constant); known
commonly as the stockholder’s required rate of return,
r = the actual rate of return (assumed constant) on the firm’s
invested capital; r > ,
g = the upper bound on the growth rate of the firm’s assets,
T = the planning horizon; T< ( T=  in section 5.2.4)
Objective: Maximize the net present value of future
dividends that accrue to the initial shares.That is,
Assume no salvage value for the time being.
For convenience, we restate this problem as
Solution. By Maximum Principle
the current-value Hamiltonian
the current-value adjoint variable  satisfies
with the transversality condition
Rewrite the Hamiltonian as
where
So given , u* is defined by max ( W1u+W2 ), subject
to u0, 0 1, cu+  g/r, which is an LP. This is a
generalized bang-bang (i.e,extreme point) solution.
The Hamiltonian maximization problem can be stated
as follows:
we have two cases:
Case A: g r and Case B: g > r ,
under each of which,we can solve the linear
programming (5.40) graphically in a closed form. This
is done in Figure 5.4 and Figure 5.5.
Figure 5.4 Case A: g  r
Figure 5.5 Case B: g > r
Table 5.1 Characterization of Optimal Controls
Synthesis of Optimal Paths
Define the reverse-time variable
as
so that
The transversality condition on the adjoint variable
Let
Using the definitions of and and the conditions
(5.42) and (5.41), we can write reverse-time versions
of (5.30) and (5.35) as follows:
Case A: g  r
Feasible subcases are A1, A3, and A6.
(0)=0  W1(0)=W2(0)=-1  A1
Subcase A1:
we have
Since 0 c <1,

Thus, to stay in this subcase requires that
must
remain negative for some time as
increases.
 is increasing asymptotically toward the value 1/
W2 is increasing asymptotically toward the value r/ - 1
Since r >  , there exists
, such that
So, the firm exits subcase A1 provided
Remarks:
i) We assume T sufficiently large so that the firm will
exit A1 at
.
ii) Note that for r < , the firm never exits from Subcase
A1. Obviously, no use investing if the rate of return
is less than the discount rate.
iii) At
, W2 =0, W1<0  Subcase A6.
Subcase A6:
The optimal controls
The optimal controls are obtained by conditions
required to sustain W2=0 for a finite time interval.
substitute =1/r (since W2=0), and equating the
right-hand to zero we obtain
We have assumed r>,so we cannot maintain singular
control. Since r > ,
Therefore, W2 is
increasing from zero and becomes positive after
.
Thus, at
, we switch to subcase A3.
Subcase A3:
The optimal controls:
The state and the adjoint equations are:
Since
,  is increasing at from its value of 1/r.
(i) >g :
As  increases, decreases and becomes zero at a
value obtained by equating the right-hand side of
(5.53) to zero,i.e., at
Since r >  > g ,
The firm will continue to stay in A3.
(ii)   g: As
increases,
increases. So
.
The firm continues to stay in Subcase A3.
Figure 5.6: Optimal Path for Case A: gr
Interpretation:
Control switches to u*=*=0 at
. i.e, it requires at
least
units of time to retain a dollar of earnings to
be worthwhile. That means, it pays to invest as much
earnings as feasible before
and it does not pay
to invest any earnings after
.Thus,
is the
point of indifference between retaining earnings or
paying dividends out of earnings.
Suppose the firm invests one dollar of earnings at
.
Since this is the last time any earnings invested will
pay off, it is obvious that this dollar will yield dividends
at the rate r from
to T. (i.e, under the optimal
policy)
The value of this dividend stream in terms of
dollars is
But this must be equated to one dollar to find the
indifference point,i.e.,
So under the optimal policy:
The firm grows at rate g until
,Since g  r, the
entire growth can be financed out of retained earnings.
Thus, there is no need to resort to external financing
which is more expensive (0 c 1). Then from
,
the firm issues dividends since there is no salvage
value at T.
Case B: g > r
Since g/r >1, the constraint   1 is relevant.
Subcase B1:
The analysis is the same as subcase A1. The firm
swithches out at time
to subcase B8.
Subcase B8:
The optimal controls are:
As in subcase A6, the singular case cannot be
sustained since r >  .
W2 is increasing at
from zero and becomes positive
after
.Thus, at
,the firm find itself in subcase B4.
Subcase B4:
The optimal controls are:
The state and the adjoint equations are:
Since
,we have
As  increases, W1 increases and become zero at time
defined by
which gives
At
, the firms switches to subcase B7.
Subcase B7:
The optimal controls are;
To maintain this singular control over a finite time
period, we must keep W1= 0 in the interval. This
means we must have
which implies
To compute , we substitute (5.64) into (5.44) and
obtain:
Substituting
from (5.62) and equating the
right-hand side to zero, we obtain
Since r >  , the firm cannot stay in B7.
From (5.65), we have
which implies that  is increasing and therefore, W1 is
increasing.Thus at ,the firm switches to subcase B3.
Subcase B3:
The optimal controls are
The reverse-time state and the adjoint equations are:
Since
,
is increasing. In case B, we assume
g > r, But r >  has been assumed throughout the
chapter. Therefore,  < g and the second term in the
right-hand side of (5.69) is increasing. That means
and
continues to increase. Therefore, the
firm continues to stay in subcase B3.
Interpretation of
Since external equity is more expensive than retained
earnings as a source of financing, investment financed
by external equity requires more time to be worthwhile.
Thus,
should be the time required to compensate for the
floatation cost of external equity.
Figure 5.7: Optimal Path for Case B: g > r
5.2.4 Solution for the Infinite Horizon Problem
For the infinite horizon case the transversality
condition must be changed to
This condition is only a sufficient condition (not a
necessary one).
Case A: g  r
The limiting solution in this case is given as subcase
A3, i.e.,
and
For  > g, the analysis of subcase A3 shows that
increasing asymptotically toward the value
Thus
clearly satisfies (5.75). Furthermore,
which implies that the firm stays in subcase A3, i.e,
the maximum principle holds. Thus,
The corresponding state trajectory
.
The adjoint trajectory
is
(5.76) reminds us of the Gordon’s classic formula.
represents the marginal worth per additional unit of
earnings. Obviously, a unit increase in earnings will
mean an increase of 1- * or 1-g/r units in dividends.
This should be capitalized at the rate equal to the
discount rate less the growth rate (i.e.,  - g ).
For   g, the reverse-time construction implies that
increase without bound as increase. Thus, we
do not have any  which satisfies the limiting condition
in (5.75).
Note that for   g, and infinite horizon, the objective
function can be made infinite.
For example, any control policy with earnings growing
at rate q[, g] coupled with a partial dividend payout
(i.e., 0  1) has an infinite value for the objective
function. That is, with
, we have
Case B: g > r
The limit of the finite horizon optimal solution is to grow
at the maximum growth rate with
Since
disappears in the limit, the stockholders will
never collect dividends. The firm has become an
infinite sink for investment.
Remarks:
Let
denote the optimal control for the finite
horizon problem in Case B. Let
denote any
optimal control for the infinite horizon problem in Case
B. We already know that
. Define an infinite
horizon control
by extending
as follows:
We now note that for our model in Case B, we have
Obviously,
is not an optimal control for the
infinite horizon problem.
If we introduce a salvage value Bx(T), B > 0, for the
finite horizon problem, the new objective function,
is a closed mapping in the sense that
for the modified model.
Example 5.2:
Solution:
Since g  r  Case A ,
The optimal controls are
The optimal state trajectory is;
The value of the objective function is;
Infinite horizon solution
Since g <  , and g < r, the problem is not ill-posed.
The optimal controls are:
and