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.
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.
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; u0,(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 ), subjectto u0, 0 1, cu+ g/r, which is an LP. This is ageneralized bang-bang (i.e,extreme point) solution.The Hamiltonian maximization problem can be statedas follows:
we have two cases: Case A: g r and Case B: g > r ,under each of which,we can solve the linearprogramming (5.40) graphically in a closed form. Thisis done in Figure 5.4 and Figure 5.5.
Synthesis of Optimal PathsDefine 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 rFeasible 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.
Interpretation:Control switches to u*=*=0 at . i.e, it requires atleast units of time to retain a dollar of earnings tobe worthwhile. That means, it pays to invest as much earnings as feasible before and it does not payto invest any earnings after .Thus, is thepoint of indifference between retaining earnings orpaying dividends out of earnings.Suppose the firm invests one dollar of earnings at .Since this is the last time any earnings invested willpay off, it is obvious that this dollar will yield dividendsat the rate r from to T. (i.e, under the optimalpolicy)
The value of this dividend stream in terms of dollars is
But this must be equated to one dollar to find theindifference 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 isincreasing.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 theright-hand side of (5.69) is increasing. That means and continues to increase. Therefore, thefirm continues to stay in subcase B3.
Interpretation of Since external equity is more expensive than retainedearnings as a source of financing, investment financedby external equity requires more time to be worthwhile.Thus,
should be the time required to compensate for the floatation cost of external equity.
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 is
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
(5.76) reminds us of the Gordon’s classic formula. represents the marginal worth per additional unit ofearnings. Obviously, a unit increase in earnings willmean 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, wedo not have any which satisfies the limiting condition in (5.75).Note that for g, and infinite horizon, the objectivefunction can be made infinite.
For example, any control policy with earnings growingat rate q[, g] coupled with a partial dividend payout(i.e., 0 1) has an infinite value for the objectivefunction. That is, with , we have
Case B: g > r The limit of the finite horizon optimal solution is to growat the maximum growth rate with
Since disappears in the limit, the stockholders willnever collect dividends. The firm has become aninfinite 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.