To obtain an optimal path, we must synthesize an optimal sequence of subcases. The usual procedure employed is that of the reverse-time construction, first developed by Isaacs (1965). Reverse time can only be defined for finite horizon problems. However, the infinite horizon solution can usually be inferred from the finite horizon solution if sufficient care is exercised. This will be done in Sect.5.2.4.
Our analysis of the finite horizon problem (5.21) proceeds with the assumption that the terminal time T is assumed to be sufficiently large.
We will make this assumption precise during our analysis. Moreover, we will discuss the solution when T is not sufficiently large in Remarks5.2 and 5.4.
Define the reverse-time variableτ as τ =T −t,
5.2. Optimal Financing Model 171 so that
y◦= dy dτ = dy
dt dt
dτ =−y.˙
As a consequence, y◦= −y,˙ and the reverse-time versions of the state and adjoint equations (5.18) and (5.23), respectively, can be obtained by simply replacing ˙y by y◦ and changing the signs of the right-hand sides.
The transversality condition on the adjoint variable
λ(t=T) =λ(τ = 0) = 0 (5.29) becomes the initial condition in the reverse-time sense. Furthermore, let us parameterize the terminal state by assuming that
x(t=T) =x(τ = 0) =αA, (5.30) where αA is a parameter to be determined.
From now on in this section, everything is expressed in the reverse- time sense unless otherwise specified. Using the definitions of x◦ and λ◦ and the conditions (5.30) and (5.29), we can write reverse-time versions of (5.18) and (5.23) as follows:
x=◦ −r(cu+v)x, x(0) =αA, (5.31) λ= (1◦ −u−v)−λ{ρ−r(cu+v)}, λ(0) = 0. (5.32) This is the starting point for our switching point synthesis. First, we consider Case A.
Case A:g≤r.
Note that the constraint v ≤ 1 is superfluous in this case and the only feasible subcases are A1, A2, and A3. Since λ(0) = 0, we have W1(0) =W2(0) =−1 and, therefore, Subcase A1.
Subcase A1: W2=rλ−1<0.
From Row (1) of Table 5.1, we have u∗ = v∗ = 0, which gives the state equation (5.31) and the adjoint equation (5.32) as
x= 0 and◦ λ= 1◦ −ρλ. (5.33)
172 5. Applications to Finance With the initial conditions given in (5.29), the solutions forx and λare
x(τ) =αAand λ(τ) = (1/ρ)[1−e−ρτ]. (5.34) It is easy to see that because of the assumption 0≤c <1,it follows that if W2 = rλ−1 < 0, then W1 = crλ−1 < 0. Therefore, to remain in this subcase as τ increases, W2(τ) must remain negative for some time as τ increases. From (5.34), however, λ(τ) is increasing asymptotically toward the value 1/ρ and therefore, W2(τ) is increasing asymptotically toward the value r/ρ−1. Since, we have assumed r > ρ, there exists a τ1 such thatW2(τ1) = (1−e−ρτ1)r/ρ−1 = 0.It is easy to compute
τ1= (1/ρ) ln[r/(r−ρ)]. (5.35) From this expression, it is clear that the firm leaves Subcase A1 provided τ1 < T. Moreover, this observation also makes precise the notion of a sufficiently largeT in Case A by havingT > τ1.
Remark 5.2 When T is not sufficiently large, i.e., when T ≤ τ1 in Case A, the firm stays in Subcase A1. The optimal solution in this case is u∗ = 0 andv∗ = 0,i.e., a policy of no investment.
Remark 5.3 Note that if we had assumed r < ρ,the firm would never have exited from Subcase A1 regardless of the value of T. Obviously, there is no use investing if the rate of return is less than the discount rate.
At reverse time τ1, we have W2 = 0 and W1 < 0 and the firm, therefore, is in Subcase A2. Also, λ(τ1) = 1/rsinceW2(τ1) = 0.
Subcase A2: W2=rλ−1 = 0.
In this subcase, the optimal controls
u∗ = 0, 0≤v∗≤g/r (5.36)
from Row (3) of Table 5.1 are singular with respect to v. This case is termed singular because the Hamiltonian maximizing condition does not yield a unique value for the controlv.In such cases, the optimal controls are obtained by conditions required to sustain W2 = 0 for a finite time interval. This means we must have W◦= 0,which in turn implies λ= 0.◦ To compute λ,◦ we substitute (5.36) into (5.32) and obtain
λ= (1◦ −v∗)−λ[ρ−rv∗]. (5.37)
5.2. Optimal Financing Model 173 Substituting λ= 1/r, its value at τ1, in (5.37) and equating the right- hand side to zero we obtain
r=ρ (5.38)
as a necessary condition required to maintain singularity over a finite time interval following τ1. Condition (5.38) is fortuitous and will not generally hold. In fact we have assumed r > ρ. Thus, the firm will not stay in Subcase A2 for a nonzero time interval. Furthermore, sincer > ρ, we have λ◦ (τ1) = (1−ρ/r) > 0. Therefore, W2 is increasing from zero and becomes positive afterτ1.Thus, atτ+1 the firm switches to Subcase A3.
Subcase A3: W2=rλ−1>0.
The optimal controls in this subcase from Row (2) of Table5.1 are
u∗ = 0, v∗ =g/r. (5.39)
The state and the adjoint equations are
x=◦ −gx, x(τ1) =αA, (5.40) λ= (1◦ −g/r)−λ(ρ−g), λ(τ1) = 1/r, (5.41) with values atτ =τ1 deduced from (5.34) and (5.35).
Sinceλ◦ (τ1)>0, λis increasing atτ1from its value of 1/r.A further examination of the behavior of λ(τ) as τ increases will be carried out under two different possible conditions: (i) ρ > g and (ii)ρ≤g.
(i) ρ > g: Under this condition, as λ increases, λ◦ decreases and becomes zero at a value obtained by equating the right-hand side of (5.41) to zero, i.e., at
¯λ= 1−g/r
ρ−g . (5.42)
This value ¯λis, therefore, an asymptote to the solution of (5.41) starting atλ(τ1) = 1/r.Since r > ρ > gin this case,
W2=rλ¯−1 = r(1−g/r)
ρ−g −1 = r−ρ
ρ−g >0, (5.43) which implies that the firm continues to stay in Subcase A3.
(ii) ρ ≤ g: Under this condition, as λ(τ) increases, λ◦ (τ) increases.
So W2(τ) = rλ(τ)−1 continues to be greater than zero and the firm continues to remain in Subcase A3.
174 5. Applications to Finance Remark 5.4 With ρ ≤g, note that λ(τ) increases to infinity as τ in- creases to infinity. This has important implications later when we deal with the solution of the infinite horizon problem.
Since the optimal decisions for τ ≥τ1 have been found to be inde- pendent ofαAforT sufficiently large, we can sketch the solution for Case A in Fig.5.5starting withx0.This also gives the value of
αA=x0eg(T−τ1)=x0egT[1−ρ/r]g/ρ, as shown in Fig.5.5.
[ ]
ln A
A
Figure 5.5: Optimal path for case A:g≤r
Mathematically, we can now express the optimal controls and the optimal state, now in forward time, as
u∗(t) = 0, v∗(t) =g/r, x∗(t) =x0egt, t∈[0, T−τ1], (5.44)
5.2. Optimal Financing Model 175 u∗(t) = 0, v∗(t) = 0, x∗(t) =x0eg(T−τ1), t∈(T−τ1, T], (5.45) As for λ(t),from (5.34) we have
λ(t) = 1
ρ[1−e−ρ(T−t)], t∈(T−τ1, T]. (5.46) For t∈[0, T −τ1],we have from (5.41),
λ(t) =˙ λ(ρ−g)−(1−g/r), λ(T−τ1) = 1/r. (5.47) Following Sect.A.1, we can solve this equation as
λ(t) = 1
re−(ρ−g)(T−τ1−t)+ 1−g/r
ρ−g [1−e−(ρ−g)(T−τ1)], t∈[0, T −τ1].
(5.48) In this solution for Case A, there is only one switching point provided thatT is sufficiently large (i.e.,T > τ1 in this case). The switching time t=T−τ1has an interesting economic interpretation. Namely, it requires at least τ1 units of time to retain a dollar of earnings to be worthwhile for investment. That means, it pays to invest as much of the earnings as feasible before T −τ1, and it does not pay to invest any earnings after T−τ1.Thus,T−τ1 is the point of indifference between retaining earnings or paying dividends out of earnings. To see this directly, let us suppose the firm retains one dollar of earnings at T −τ1. Since this is the last time that any of the earnings invested will be worthwhile, it is obvious (because all earnings are paid out) that the dollar just invested atT −τ1 yields dividends at the rate r from T−τ1 toT. The value of this dividend stream in terms of (T −τ1)-dollars is
τ1
0 re−ρsds= r
ρ[1−e−ρτ1], (5.49) which must be equated to one dollar to find the indifference point. Equat- ing (5.49) to 1 yields precisely the value of τ1 given in (5.35).
With this interpretation of τ1, we conclude that enough earnings must be retained so as to make the firm grow exponentially at the max- imum rate of g until t=T−τ1.After this time, all of the earnings are paid out and the firm stops growing. Sinceg≤r (assumed for Case A), the growth in the first part of the solution can be financed entirely from retained earnings. Thus, there is no need to resort to more expensive external equity financing. The latter will not be true, however, in Case B when g > r,which we now discuss.
176 5. Applications to Finance Case B:g > r.
Sinceg/r >1,the constraintv≤1 in Case B is relevant. The feasible subcases are B1, B2, B3, B4, and B5 shown adjacent to the darkened lines in Fig.5.4. As in Case A, it is obvious that the firm starts (in the reverse-time sense) in Subcase B1. Recall that T is assumed to be sufficiently large here as well. This statement in Case B will be made precise in the course of our analysis. Furthermore, the solution whenT is not sufficiently large in Case B will be discussed in Remark 5.4.
Subcase B1: W2=rλ−1<0.
The analysis of this subcase is the same as Subcase A1. As in that subcase, the firm switches out at time τ =τ1 to Subcase B2.
Subcase B2: W2=rλ−1 = 0.
In this subcase, the optimal controls
u∗ = 0, 0≤v∗ ≤1 (5.50)
from Row (3) of Table 5.1 are singular with respect to v. As before in Subcase A2, the singular case cannot be sustained for a finite time because of our assumption r > ρ.As in Subcase A2,W2 is increasing at τ1 from zero and becomes positive after τ1. Thus, atτ+1,the firm finds itself in Subcase B3.
Subcase B3: W1=crλ−1<0, W2 =rλ−1>0.
The optimal controls in this subcase are
u∗= 0, v∗= 1, (5.51)
as shown in Row (5) of Table 5.1. The state and the adjoint equations
are ◦
x=−rx, x(τ1) =αB (5.52)
with αB,a parameter to be determined, and
λ=◦ λ(r−ρ), λ(τ1) = 1/r. (5.53) Obviously, earnings are growing exponentially at rate r and λ(τ) is in- creasing at rate (r−ρ) asτ increases fromτ1.Since λ(τ1) = 1/r,
5.2. Optimal Financing Model 177 we have
λ(τ) = (1/r)e(r−ρ)(τ−τ1) forτ ≥τ1. (5.54) Asλincreases, W1 increases and becomes zero at a time τ2 defined by
W1(τ2) =crλ(τ2)−1 =ce(r−ρ)(τ−τ1)−1 = 0, (5.55) which, in turn, gives
τ2=τ1+ 1 r−ρln
1 c
. (5.56)
At τ+2,the firm switches to Subcase B4.
Before proceeding to Subcase B4, let us observe that in Case B, we can now define T to be sufficiently large whenT > τ2. See Remark5.4 when T ≤τ2.
Subcase B4: W1=crλ−1 = 0.
In Subcase B4, the optimal controls are
0≤u∗≤(g−r)/rc, v∗= 1. (5.57) From Row (6) in Table 5.1, these controls are singular with respect to u. To maintain this singular control over a finite time period, we must keep W1 = 0 in the interval. This means we must have W◦1 (τ2) = 0, which, in turn, implies λ◦ (τ2) = 0.To compute λ,◦ we substitute (5.57) into (5.32) and obtain
λ=◦ −u∗−λ{ρ−r(cu∗+ 1)}. (5.58) Atτ2, W1(τ2) = 0 givesλ(τ2) = 1/rc.With this in (5.58), its right-hand side equals zero only whenr =ρ.But we have assumedr > ρthroughout Sect.5.2, and therefore a singular path cannot be sustained for τ2 > 0, and the firm will not stay in Subcase B4 for a finite amount of time.
Furthermore, from (5.58), we have λ◦ (τ2) = r−ρ
rc >0, (5.59)
which implies that λis increasing and therefore,W1 is increasing. Thus atτ+2,the firm switches to Subcase B5.
178 5. Applications to Finance Subcase B5: W1=crλ−1>0.
The optimal controls in this subcase from Row (4) of Table5.1 are u∗ = g−r
rc , v∗ = 1. (5.60)
Then from (5.31) and (5.32), the reverse-time state and the adjoint equa- tions are
x=◦ −gx, (5.61)
λ=◦ −(g−r
rc ) +λ(g−ρ). (5.62)
Since λ◦ (τ2) > 0 from (5.59), λ(τ) is increasing at τ2 from its value λ(τ2) = 1/rc >0.Furthermore, we haveg > rin Case B, which together withr > ρ,assumed throughout Sect.5.2, makesg > ρ.This implies that the second term on the right-hand side of (5.62) is increasing. Moreover, the second term dominates the first term for τ > τ2, since λ(τ2) = 1/(rc) > 0, and r > ρ and g > r imply g−ρ > g −r > 0. Thus, λ◦ (τ) > 0 for τ > τ2, and λ(τ) increases with τ . Therefore, the firm continues to stay in Subcase B5.
Remark 5.5 Note that λ(τ) in Case B increases without bound as τ becomes large. This will have important implications when dealing with the infinite horizon problem in Sect.5.2.4.
As in Case A, we can obtain this optimal solution explicitly in forward time, and we ask you to do this in Exercise 5.9. We now can sketch the complete solution for Case B in Fig.5.6. In this solution, there are two switching points instead of just one as in Case A. The reason for two switching points becomes quite clear when we interpret the significance of τ1 andτ2.It is obvious thatτ1 has the same meaning as before. Namely, ifτ1 is the remaining time to the horizon, the firm is indifferent between retaining a dollar of earnings or paying it out as dividends. Intuitively, it seems that 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. That is,
τ2−τ1 = 1 r−ρln
1 c
>0 (5.63)
as obtained in (5.56), should be the time required to compensate for the floatation cost of external equity. Let us see why.
5.2. Optimal Financing Model 179
=
B5
B3
B1
Earnings
Figure 5.6: Optimal path for case B:g > r
When the firm issues a dollar’s worth of stock at time t = T −τ2, it incurs a future dividend obligation in the amount of one (T −τ2)- dollar, even though the capital acquired is only c dollars because of the floatation cost (1−c). Since we are attempting to find the breakeven time for external equity, it is obvious that retaining all of the earnings for investment is still profitable. Thus, there is no dividend from (T−τ2) to (T−τ1),and the firm grows at the rater. Therefore, this investment ofcdollars at time (T−τ2) grows intocer(τ2−τ1)dollars at time (T−τ1).
From the point of view of a buyer of the stock at time (T−τ2),since no dividend is paid until time (T−τ1) and since the stockholder’s required rate of return isρ,the firm’s future dividend obligation at time (T−τ1) is eρ(τ2−τ1) in terms of (T−τ1)-dollars. But then we must have
eρ(τ2−τ1)=cer(τ2−τ1), (5.64)
180 5. Applications to Finance which can be rewritten precisely as (5.63). Moreover, the firm is marginally indifferent between investing any costless retained earnings at time (T−τ1) or paying it all out as dividends. This also means that the firm will be indifferent between having the new available capital of cer(τ2−τ1) dollars at time (T−τ1) as a result of issuing a dollar’s worth of stock at time (T −τ2),or not having it. Thus, we can conclude that the firm is indifferent between issuing a dollar’s worth of stock at time (T−τ2) or not issuing it. This means that before time (T−τ2),it pays to issue stocks at as large a rate as feasible, and after time (T −τ2),it does not pay to issue any external equity at all.
We have now provided an intuitive justification of (5.63) and con- cluded that all earnings must be retained from time (T−τ2) to (T−τ1).
Because r > ρ, it follows that the excess return on the proceeds c from the new stock issue is cer(τ2−τ1)−ceρ(τ2−τ1)at time (T−τ1).When dis- counted this amount back to time (T −τ2),we can use (5.63) or (5.64) to see that
%
cer(τ2−τ1)−ceρ(τ2−τ1)
&
e−ρ(τ2−τ1) =celn(1/c)−c= 1−c.
Thus, the excess return from time (T−τ2) to (T−τ1) recovers precisely the floatation cost.
Remark 5.6 WhenT is not sufficiently large, i.e., whenT < τ2in Case B, the optimal solution is the same as in Remark 5.1 when T ≤τ1. If τ1 < T ≤ τ2, then the optimal solution is u∗ = 0 and v∗ = 1 until t=T −τ1.Fort > T −τ1,the optimal solution is u∗ = 0 andv∗= 0.
Having completely solved the finite horizon case, we now turn to the infinite horizon case.