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; 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.

Figure 5.4 Case A: g r

Figure 5.5 Case B: g > r

Table 5.1 Characterization of Optimal Controls

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.

Figure 5.6: Optimal Path for Case A: gr

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.

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 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.

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