Radner & Shepp 1996 - Risk vs Profit Potentials a Model for Corporate Strategies (JEDC)

ELSEVIER Journal of Economic Dynamics and Control

20 (1996) 137331393

Risk vs. profit potential: A model for corporate strategy

Roy Radner and Larry Shepp*

AT&T Bell Laboratories, Murra_v Hill. NJ 07974, USA

(Received July 1995; final version received November 1995)

Abstract

A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and vari- ance. We model the firm’s cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A. The second is an arbitrary nondecreasing process, chosen by the firm. The firm’s strategy must be nonclairvoyant.

The firm is bankrupt at the first time, T, at which the cash reserve falls to zero (T may be infinite), and the firm’s objective is to maximize the expected total discounted dividends from 0 to T, given an initial reserve, x; denote this maximum by V(x). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: (1) pay no dividends if the reserve is less than some critical level, a, and pay out all of the excess above a; (2) choose the drift/volatility pairs from the upper extreme points of the convex hull of A, between the pair that minimizes the ratio of volatility to drift and the pair that maximizes the drift; furthermore, the firm switches to successively higher volatility/drift ratios as the reserve increases to a. Finally, for the optimal policy, the firm is bankrupt in finite time, with probability one.

Key words: Corporate policy; Wiener processes; Economics modelling JEL classijication: D7

1. A theory of corporate decision making

Consider a firm with fixed plant and equipment that produces an uncertain stream of net revenues. (Net revenue equals gross revenue less all costs except

*Corresponding author.

0165-1889/96/$15.00 (3 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1889(95)00904-A

1314 R. Radner, L. Shepp /Journal of Economic Dynamics and Control 20 (1996) 1373-1393

the cost of capital.) Part of this revenue may be paid out in dividends, with the remainder accumulated in a cash reserue. Since the net revenue can be negative as well as positive, the cash reserve can fluctuate up and down, We suppose that the firm starts with a positive cash reserve, say x, and that it becomes bankrupt the first time, if ever, that the cash reserve falls to zero. When the firm becomes bankrupt, it ceases to exist.

The firm’s manager controls the dividends, which must be nonnegative, and can also influence the stochastic process of net revenues. In other words, the manager’s policy has two parts, a dividend policy and an operating policy. (We shall give a more precise description of this in a moment.) We define the pro@ of the firm to be the expected total dividends paid out during the life of the firm, discounted at some fixed rate, say Y > 0. We shall say that the manager’s policy is optimal if it maximizes the firm’s profit. For a Bachelier model of the firm’s net revenue process, we characterize the firm’s optimal policy and the resulting maximum profit, V(x), that the firm can attain starting with a cash reserve x.

In this (simplified) model of the firm, the assets are in two parts: (1) fixed, or illiquid, assets, which produce the stream of net revenues, and (2) the cash reserve, which produces no net revenues but is completely liquid and provides insurance against bankruptcy. There is no reinvestment of net revenues in the illiquid assets. Furthermore, the cash reserve can be increased only from retained earnings, but not by an infusion of outside capital. We have adopted the convention that the firm becomes bankrupt when the cash reserve falls to zero, but any other critical level (including negative ones) could be postulated. For example, if the firm could not obtain a line of credit that exceeds h, then the critical level for the cash reserve would be - b. However, once the cash reserve fell to this critical level, the firm could not be revived with more investment. We have also implicitly assumed that the illiquid assets of the firm have no salvage at the time of bankruptcy, but this assumption can easily be relaxed (see Section 3).

Roughly speaking, we may interpret our model as describing a firm that has reached its optimal scale of operation. A more general model would allow for investment of retained earnings in both new productive (illiquid) capital assets, and the cash reserve. However, such investment would eventually encounter diminishing returns to scale, until the optimal scale is reached. On the other hand, firms can sometimes sell parts of their productive assets with little if any loss in the value of the sold assets, i.e., the productive assets may be more or less liquid, rather than completely illiquid as in our present model. In particular, bankruptcy may result in only a partial loss to creditors and investors, followed by a ‘reorganization’ with the addition of new capital. The analysis of a more general model embodying these features would be desirable, but is beyond the scope of the present paper.

We turn now to a formal statement of the problem. Let X, denote the cash reserve at time t, and let R, denote the accumulated net revenues up through time t (here time is continuous and nonnegative). In the light of the preceding

R. Radner, L. Shepp 1 Journal of Economic Dynamics and Control 20 (1996) 1373-1393 1315

discussion it is natural to assume that R, evolves according to the Bachelier additive model,

dR, = pdt + adW,, 0 <t < zo,

0, t > To,

where z0 is the first t for which X, I 0, i.e., when the firm becomes bankrupt. Here p is the expected (instantaneous) rate of net revenue, and ad W, represents the contribution to net revenue due to random fluctuations beyond the firm’s control, with W, a standard Brownian motion or Wiener process. The effect of the manager’s operating policy is to control p and a as a function of time, as described below.

We note in passing that, in models of individual investors in securities markets, it has become fashionable to describe fluctuations in asset values with

the multiplicative model Black and &holes (1972, 1973) and Samuelson (1965)

dR, = R,(pdt + adW,).

Such a model embodies an assumption of constant retums to scale, which, as we have noted above, is inappropriate for an individual firm, although it might provide a good approximation in the case of a single investor in the securities market whose assets are small relative to the market as a whole.

Returning to our model, we assume that the firm has available n alternate business plans - plan i is identified by the parameters (pi, a;), i = 1, , n, as in (1) - and that the firm is able to switch continuously between plans.’ If the firm uses plan (Pi, a:) E .02 = ((pi, a’), i = 1, . . . , n} at time t. then the accumulated net revenue, R,, of the firm evolves according to the model

dR, = p*dt + a,dW,, 0 I t I To. (3)

Unless the firm distributes some dividends at some time, there is not much point to being in business. We denote the total dividends distributed up to time r by Z,, where dZ, 2 0. Hence cash reserve X, of the firm evolves according to

dX, = p,dt + a,dW, - dZ,, 0 I t 5 To, (4)

’ This is a serious assumption which is not appropriate in reality since management cannot appear

to be too capricious, but if n is large and if (pi, a’) varies slowly with i, then it does not seem

unreasonable when the actual switching takes place only between adjacent plans, i and i + 1. which

will be proven to always take place under optimal corporate management.


where z. (as before) is the first t 2 0 for which X, = 0, and

dX,=dZ,=O for t>zo. (5)

Note that X0 = x 2 0, given. The firm wants to choose its corporate policy [&, a:), Z,] in such a way that the total projt, i.e., the expected total discounted dividends,

s cc

ES e-“dZ,, 0

is maximized, where Y > 0 is a given rate of discount. Accordingly, define

s 02

I’(x) = sup E, e-“dZ,. (6) 0

Here the sup or max is taken over all policies [(Pi, a:), Z,] which do not see into the future, i.e. (pt, a:) and Z, are independent of the future increments d W,, s > t, of the noise process, W. More precisely, pL,, a:, and Z, should be ‘nonclairvoyant’ in the sense of adaptedness to the standard filtration; of course it is necessary to assume that pLr, of, Z, should be jointly measurable in w and t.

We will find V(x) in Section 2 as well as the optimal policy for the firm, including both the optimal choice of (v*, a:) and the optimal dividend stream, Z,, t 2 0. In each case the solution will be stationary in time and will only be - a function of the current cash reserve, X,. We will see that there is a threshold, a, such that dividends are paid only when X, 2 a, and then are paid at an infinite rate so that, in fact, 0 I X, I a for all t > 0, under optimal management.’ The optimal choice of plan at each time t, (pL,, c$), is also only a function of X,. This is a simple consequence of the formulation of the problem; if we change the problem, even slightly, say using a finite horizon instead of exponential discounting, the optimal solution no longer is of stationary form and appears impossible to find in closed, explicit, form. This is familiar in problems of this type (e.g., Benes et al., 1980; Shepp and Shiryaev, 1993,1994; Dubins et al., 1993). We will show in Section 2 that the firm will go bankrupt at time 20 < GO with probability one under optimal management. This seems somewhat paradoxical, since one believes optimally managed firms prefer to remain solvent, but in fact it is a theorem, and might explain why profit-oriented firms actually operate on the edge of bankruptcy and often wind up on the wrong side of it. If instead, one redefines the criterion (6) of optimality to include a term involving P{zo = co}

‘Policies of this type, called barrier policies by M. Harrison, arise in a variety of problems of ‘regulated Brownian motion’, see Harrison (1985).

R. Radner, L. Shepp /Journal of Economic Dynamics and Control 20 (1996) 1373-1393 1311

in the maximization, then the solution exhibits nonstationarity and apparently

cannot be found explicitly, as discussed in Section 3. We find (in Section 2) the exact condition on d = {(pi, G?), i = 1, . . . , n)

under which each of the n (distinct) plans, (pi, a?) , has a chance to actually

become part of the corporate policy (only if X, ever enters the set where (pi, 0:) is the optimal choice). If p1 < p2 5 ... I 11, is the ordering of the drift parameters, without loss of generality, then one also needs that the plans satisfy:

(7)

or else one of the plans will completely dominate another one and any time the one plan, say (pi, G:), would be used, (p2, 0:) would be preferable - if (7) failed and say &p2 < ~:/pi -because plan 2 would be more profitable (p2 2 pi) and its risk would be sufficiently small. There is however one more condition needed for each (pi, of) E .r4 to be needed. We must have that each (pi, a;) is actually an extreme point of -c9*, the convex hull of d. This is because, through rapid

switching between the plans in &‘, the firm can actually obtain any plan in .d*, and if a plan in &* dominates one of (pi, CJ~) in the above sense, then (Iii, a!) can be eliminated from it. One can restate this last condition as:

oi’ - of_ 1 2 2

Pi-Pi-1 ’

Oi+l -Oi

3 i=2,3, . . . . n-l, Pi+1 -Pi

(8)

so that (7) and (8) will be shown to be necessary and sufficient for each plan in .d to be needed, i.e., so that V(&, x) > V(J#, x) for d’ c d. Note that the essential condition here is (7), which for n = 2 indicates the way the two different

plans, (pi, a:) and (p2, ot), should be compared, and thus allows an intrinsic comparison of drift and volatility, two parameters which appeared difficult to quantitatively compare until now. It is particularly reassuring that the conditions (7) and (8) do not involve the (interest or discount) rate r, i.e., (7) gives a universal way to compare drift and volatility, and is even independent of other plans in _~4 for n > 2.

In Section 3 we consider the somewhat more general problem of determining

is a,

V,(x) = supE, e-” dZ, + ce- rTo ,

0 1

where c is a given parameter. This problem formulation arises in two situations: (1) there is a positive salvage value of the firm’s assets at the time of bankruptcy (c > 0), and (2) the manager is paid a constant salary throughout the life of the firm, and the goal is to maximize some weighted average of the expected


discounted totals of dividends and the manager’s salary (c < O).3 Because of the form of the criterion, the optimal solution is again stationary, and also z. < co w.p.1. Furthermore, z. is stochastically decreasing in c.

Finally in Section 4 we consider a more technical variant of the theory where &’ consists of a continuum of plans (i.e., d* has infinitely many extreme points). Here we give a technique based on the smooth-fit principle (e.g., Gigelions and Shiryaev, 1966; Shepp and Shiryaev, 1993; Benes et al., 1980) for guessing and proving the optimal strategy and obtaining V(&, x). In particular we consider the case where for each 0 I u I p < co the company has a plan with P(U) = u and a(u) = &A where 0 is fixed. We give in Section 4 the exact solution in this case by way of illustrating the general method in the continuous case. It is interesting that, as p + cc, V(x;p, 0) ---f 00, so if the firm has a choice of unboundedly aggressive policies (with a/p fixed, but c2/p unbounded), then it can obtain an infinite profit.4

2. Finitely many plans; which one to choose?

In order to solve (6) for V we must first guess it -for which we use the heuristic ‘principle of smooth fit’ as in (e.g., Benes et al., 1980; Shepp and Shiryaev, 1993). Suppose we can find a function V = Y(x), x 2 0, which satisfies:

(a) B(0) = 0,

(b) 0 I V(x) I const. x, 0 I x < cc, (10)

(c) B E c2 [O, co), V’(x) 2 1, x 2 0, (11)

and

- rB(X)+~iP’(X)+~~~p”(X)jO, VX20, Vi=l, . . ..Iz. (12)

Then it is easy to see that I’(&, x) < V(x). To see this let (,u~, 02) be any choice in JZZ for each t, and 2, an arbitrary dividend scheme. Define the process Y,, t 2: 0,

3 For a related class of problems, see Dutta and Radner (1994)

4Dutta and Radner (1994) consider the situation in which the set ~8’ is compact and strictly convex. They first study the case in which the ‘dividend rate’ is bounded, and then let the bound increase

without limit. Of course, without further specification of the set d they obtain only qualitative

properties of the optimal policy and the value function. The methods of this paper can be used for

arbitrary J&‘.


by

Y, = V(XJe-*' + s ‘eersdZ,, 0

(13)

where X, is the fortune stream of the firm employing the plan (p*, a:) and Z,, i.e., X0 = x, and, as in (4),

dX, = ptdt + o,dW, - dZ,, 0 i t < 50. (14)

We see that Y has the Ito differential

dY, = em”*{ - rV(X,)dt + V’(XJ(pLtdt + a,dW, - dZ,)

+ + V”(X,) CT: + dZ,} . (15)

Thus the expected increment of dY, is (since EV’(X,)a,dW, = 0 because X, is independent of dW, under Ito calculus, which applies because of (10.~))

EdY, = e-*‘dt { - rV(X,) + v’(X,)rpt + i V”(X,)o:

+ (1 - V’(X,)) dZ,}. (16)

The latter gives E dY, I 0 because dZ, 2 0 and (12) and (11) hold. Thus Y, is an expectation decreasing (submartingale) process, and so EY, I EYo, which gives from (13) and an (easy) argument based on (10a) and (lob)

E,Y,,, = E, s

me-r’ dZ, < E,Yo = V(x). 0

(17)

Since (pt, a:) and Z, are arbitrary, we obtain I’(&‘, x) I V(x). We now need to construct V satisfying (lo)-(12) in such a way that equality will hold in the above argument.

After a little experimentation, it is easy to guess that V and V must be of the following form. If we assume that

0 < 111 < j.42 < “. < /A”, (18)

then we must try to find thresholds

a, = 0 < a, < ‘.. < a,, (19)


for which, for i = 1, . . , n,

2 - ~P(X)+~jY'(X)+O'P"(X)=O for C2i-r <xl@,

2 (20)

V(x) = t + x for a, S x, (21)

V(0) = 0, T/EC2.

Now equality will hold throughout (17) provided we set

(22)

(A d) = (Pi, 0;) when X, E (ai- 1, ai), (23)

and

dZ, = 0 for X, < a,, dZ, = cc for X, > a,,. (24)

We need only check the details. In order to do this we need to simultaneously determine when each of the plans (pi, 0’) are actually needed and correspond to a nonzero interval (Ui- r, Ui). We will show for ‘neededness’ the plans must also satisfy

r&l 0s -L-<-L Pi-1 Pi’

i = 1, . ,n,

as well as

a; - ai”_ 2 gi+l - 0:

Pi-Pi-1 < Pi+l-Pi’

i=l,2, . . . . n-l.

(25)

(26)

Equivalently, (26) says that each plan (pi, a;) must be an extreme point of d*, the convex hull of d.

So suppose (IS), (25), and (26) hold for & = {(pi, a:); i = 1, . . . , n} and define Eli and pi as the roots of iv’o” + ypi - r = 0, i.e.,

cli = -_i+Jm

Pi = -

p-Jp-cq CT”

> o? . (27) 1

Nowseta,=O,andfori=1,2, . . ..n-ldefineaiby

1 ai-ui-1 =--

4 - Pi

log pi+l(Bi)

[

Pi-l(ai)

Pi+lt”li) pi-l(Bi) 1 ’ (28)

R. Radner. L. Shepp /Journal ofEconomic Dynamics and Control 21) (1996) 1373-1393 1381

where the quadratic polynomials P; are defined for i = 0. 1, . . . , n by

Pi(X) = f X'Oi' + X~i - Y, (2%

where B 0 = cl0 = 0, and so P,(x) z - I, a degenerate quadratic. We will check that (19) holds and then we set

Y(x) = i

Ai$V - &&4X for a;_r<xla;, i=l. . . . . n,

.‘c + 5 for a, < x. (30)

We will now show that AiF I$, i = 1, . . . , n, and 4 can be chosen so that p satisfies all of (lo)-( 12) and that equality holds in (17).

Lemma 1. The roots pi < 0 c ai cf Pi satisfy

p1 < “. < /?” < 0 <a, < ... < a,. (31)

Proof: If i < j, then since r > 0 and 0 < p; < pj by (18) and (25)

L-c’ or 2 2

Pj Pi

z_af+aj<f’al+a;<fO’ir:+cc;. 14

Pj Pi pj

(32)

It follows that {4(Oj’/pj) (gj + a;) + l}(Orj - LY;) < 0, SO Clj < Cl; since Clj and LX; are positive. Similarly, ($(e;/pj) (bj + fi;) + 1}(/3j - ,!I;) < 0, but (ej/pj)flj =

(l/PI) ( - pj - J’m) < - 1 and (ef/pj)fl; < (ef/p;) P; < 1 by the fact that p; < 0 and the same argument, SO j?i < /lj. Q.E.D.

Because of (31) and the fact that P; is a quadratic which is large at + cc and has zeros at Or;, fi;, it follows that Pi(ai+ 1) < 0, Pi(Ni- 1) > 0, Pi(pi+ 1) < 0, Pi(fii-,) > 0. To prove (19) holds we must show that the argument of the logarithm in (28) is > 1, or in terms of positive quantities, we must show

Writing this last term out completely, using (29), reduces to (26) and (33) follows. Thus (19) is proved.


Next we turn to determining Ai, Bi, i = 1, . . . , n, and t in (30). For 1 I i 2 n - 1 we have from (30) and the smooth-fit heuristic

V(u;) = V(a+) or

P’(u;) = V’(a’) or

Jo” = P’(a’) or

We also have

V(O)=0 or A, =Bt, (37)

and for i = n we have

V(a,) = V(a,‘) or Aneanan - B,ePnnn = a, + t, (38)

V’(u,) = V’(u,‘) or A,a,eanan - B,/3,esnan = 1, (39)

Y”(u,) = Y”(a,‘) or A,a~ennon - B,&ePnan = 0. (40)

Now multiply (34) (35), (36) by - I, ,Ui, fa?, respectively, and add to obtain, using the fact that Cli and pi are roots of (29) for 1 5 i < n,

0 = Ai+lPi(~i+l)e”l+‘“l - Bi+lPi(Pi+I)ea’+“‘, (41)

noting that (41) also holds for i = 0, consistently with (37). Similarly multiplying

(34) (35), (36) by - r, pi+ 1, $J,“+ 1, and adding, we get for 1 < i < n,

0 = AiPi+ 1 (ai)F’ - BiPi+ l(pJeaiat, (42)

which is consistent with (28) and shows where (28) was obtained. From (39) and (40) we obtain A,, and B,, both nonnegative,

R. Radner, L. Shepp 1 Journal of Economic Dynamics and Control 20 (1996) 1373-1393 1381

Now the rest of Ai, Bi, and [ are fixed as follows: 4 is given by (38). We can still use (34) since we only used two combinations of (34)-(36) above so that, since (41) holds for i = 0, . . . , n - 1, we have, replacing Bi and Bi+ 1 in (34) by the values from (41) and (42)

(44)

which determines A,, . . . , A,_ 1. Finally B1, , B,_ 1 are given by (42). Now that P is consistently defined we must show that (lo)-(12) hold and that

equality holds in (17). We see that Y’(a,) = 1 and so P’(x) 2 1 will hold if we show that I/“(X) < 0, 0 I x I a,. From (30),

so we need to show that

and it is enough to check this for x = ai since C(i - pi > 0. Using (42), we need to show that CrFPi+ 1(8i) I flfPi+ I(ai) , or from (29)

Cancelling the LX?/?: terms we need that ((ai + pi)r - C(iBi~i+ l)(Xi - /Ii) 2 0. But Xi + /?i = - 2/Li/Of, ai/?i = - 2r/af, and pi - pi > 0, SO this is the same as pi+ I 2 pi, which is true by (18). SO (10) and (11) hold. TO prove (12), suppose xE(aj-r,aj),lIj<n.Wemustshowforalli,jE{l, . . ..n}.

- r~(X) + ~i8’(X) + 4 ~~ v"(X) < 0. (410

Since this is a combination of two exponentials by (30), it suffices to check that (48) holds at the endpoints x = aj_ 1, x = Uj, i.e., for all i and j,

0 2 _ y(AjeainJ _ Bjeoj”J) + pi(Ajajeniaj _ Bj/jjepPJ)

since aj_ I is just another endpoint, because a sum of two exponentials negative at the end points of an interval is negative throughout the interval. We may rewrite (49) as

0 2 AjeaJaJ( - r + ~jpi + 4 CX~~O?) - Bjel’J”J( - r + Bj~j + f /?;a’). (50)


Using (29) and (42) we may rewrite this as (since Aj, Bj are positive)

0 2 Pj+l(bj)Pi(Crj) - Pj+l(CCj)P@j) for all i,je {I, ..‘> H>. (51)

Putting in the definition of Pi, Pj+ 1 in (29) and cancelling terms, we may rewrite (51) as

Dividing out by Clj - pj > 0 and noting OljBj = - 2r/ojz, aj + flj = - 2pjLj/Of, and cancelling a factor of r, we see that (52) is the same as

O2tPj+l -&)Of +(~jFLj-_j+l)O? +(Pi-Pj)“j2+1, (53)

which is the same as the convexity condition (26). Indeed, if i = j or i = j + 1, this vanishes, while otherwise if, say, i < j < j + 1, then

Pj - Pi a?+Pj+l-Pj 2 1

pj+l -Pi Pj+l -Pi

Cj+l IG$ (54)

is the same as (53) and also convexity of (pi, c$). Similarly for j < j + 1 < i. Thus under (25) and (26), (lo)-(12) hold for r as defined. Now equality holds in the argument in (17) because, when d =j or i = j + 1, we have seen that (53) vanishes.

If instead (25) or (26) fails, it is easy to see that some points may be omitted from &, and a function Y may be constructed for which V(x, &‘) = V(x) where d’ c d. Indeed the largest subset &’ of d for which (25) and (26) hold will provide B(&‘, x), which will give an upper bound on I’(&‘, x), i.e., I’(&, X) I V(a’, X) so constructed and so VI&Q’, x) = V(J&“, x). In particular we must have /Li > 0 for all plans in d.

We see that the optimal plan then pays dividends only when x 2 a, and that V(x) = x + < for x 2 a,, where 5 is given by (30)

a” + Bn A =--- -a,=- -a,,

4lBll r (55)

R. Radner, L. Shepp J Journal of Economic Dynamics and Control 20 (1996) 1373-1393 1385

where a, is given as the sum over i = 1, . , n in (28). It is a consequence of the theory that 5 > 0 of course since V(x) 2 x, because one strategy is to take all profits at once.

Now that we see that the firm remains always in the interval 0 I X, I u,, we see that w.p.1, z0 < cc, i.e., the firm will go bankrupt in finite time, almost surely. This is a consequence of the fact that every process with bounded drift and reflecting barrier at a, will eventually hit 0, w.p.1.

We have shown that the integral with respect to W implicit in the above proof is a martingale rather than a local martingale because V has a bounded derivative and this makes our proof complete. The fact that P grows at most linearly as stated in (lob) was of course also used to obtain that I/(XJe-” tends to zero as t + ac..

3. Staying alive; maximizing the time until bankruptcy

Is the criterion (6) cogent, or are firms rather trying to maximize not profits but perhaps P,(zO = cc > or E,ro? It is possible to work out ExzO for the optimal policy (for (6)) of Section 2, but of course it will not be the policy that maximizes this (indeed this can be made infinite if /li > 0 for some policy). But perhaps real firms are attempting to maximize profits under a fixed level, Px{zO = CC } 2 E, of survival, where E > 0 is given. Such problems are inherently very difficult because of a technical reason akin to the reason that made the treatment of Russian option Shepp and Shiryaev (1993) much more amenable to analysis then the American option in the Black-Scholes classic papers; see Black and Scholes (1972, 1973). Namely, the model formulated in (6) is inherently one- dimensional in that the optimal strategy for the firm is stationary and depends on the value of Xt, and not also upon t. Suppose instead of (6) we want to find

e-“dZ, + cJ’, (tO = (K } , (56)

where c > 0 is given. This (using Lagrange multipliers) is seen to be equivalent to the E problem stated above. But (56) is impossible to solve, it seems, because one cannot easily guess the form of the optimal policy which now depends on the solution of a partial differential equation instead of a simple ordinary differential equation as in Section 2. Indeed, imagine letting t run for a short interval, h. Then (56) will involve a new problem where c changes to c exp(rh) . This problem appears to be much more difficult to solve for the reason that the value of c keeps changing as time runs. It is completely analogous to the American option analysis where the time until maturity must be tracked. The advantage of the discounting in (6) was to keep the problem invariant in time.


There is one special way to formulate a problem of the form of (56), qualita- tively, which has some of the same features as (56) but allows for explicit solution. This is to find

e-“dZ, + cE,eCrTO 1 , (57)

for fixed c. In fact, this optimization problem arises naturally in two situations of interest. First, suppose that the firm has a ‘salvage value’ when it becomes bankrupt, which is distributed to the investors. Then the total expected discounted return to the investors is given by (57), where c is the salvage value.

Second, suppose that the manager is paid a fixed salary, say s dollars per unit time, during the life of the firm. Then his expected total discounted salary is

s TO

5% eP”sdt = (S/T) (1 - E,e-*‘O). 0

(58)

Suppose further that the firm wants to maximize its profit, subject to some given lower bound on the manager’s expected total discounted salary. For some value of the Lagrangian multiplier m, this is equivalent to (57), with c = - ms/r. Note that in this case c is negative which ‘encourages’ z0 to be large.

Now the problem (57) remains invariant as time moves ahead because each of the terms in (57) changes by exp( - rh) if t changes from 0 to h, and the problem remains stationary. The same methods will work to find U(x) as used in Section 2 to find V(x) except that U(0) # O! Instead, the boundary condition U’(0) = 0 must be used. We leave it to the interested reader to pursue this direction.

4. A continuum of corporate policies

We suppose here that the set d of available corporate policies is a continuum, i.e., for each drift p there is a least volatility 02(p) in the convex set ~2 and p fills out an interval, which for simplicity we take to be 0 < p I p. We again assume that the optimality criterion is (6). We will show how to guess the optimal corporate policy using the (heuristic) principle of smooth fit to help guess the reward function, V(x) . The least volatility in & with drift ,U is given, we call it a’(,~). As we have seen in Section 1 we may as well assume that a’(p) is a concave

function of/L, 0 I p I /?. In distinction to Section 2 where C’(U) had only a finite number of extreme points and was linear in-between them, here we will assume for simplicity that o’(p) is strictly concave for 0 I p I /? so that every (,u, a”(p)) is an extreme point of &. Of course we will also assume a2(p)/p increases in p as in (7), otherwise some policies in C& are dominated and not needed.

R. Radner, L. Shepp J Journal of Economic Dynamics and Control 20 (1996) 1373-3393 1387

Again we seek a function V(x) with the following properties analogous to

(10))(12):

(a) V(0) = 0,

(b) 0 I V(x) I const. x, 0 I x < cc, (59)

(4 v E cz 10, co), V’(x) 2 1, 0 5 x < ‘CC ) (60)

- r-V(x) + pV’(x) + fcr’(p)V”(X) IO, 0 < p I /L 0 2 X < cc. (61)

We also seek a function U: [0, co) + [0, 81; (p, c?) = (u(x), a2(u(x))) will be the policy to use when X, = x. How do we find V (which will be I/ as in (6)) and u?

Because of a fortuitous technique which we now describe it is not so hard.

Indeed we should expect that for (11, 02) = (u(x), a2(x)) the inequality (61) is an equality:

- TV(X) + u(x) V’(x) + ~o”(u(x)) V”(X) = 0. (62)

But because of (61) the derivative on p of (61) must be zero when p = u(x) since the maximum of the left side of (61) over p is attained at p = u(x), at least for

0 < x < p(p). Thus we expect

V’(x) + a(p)a’(p)P”(x) = 0 at 1~ = u(x), 0 < p < fl, (63)

where CT = U(U) is the given function. We may eliminate 8” from (62) and (63) to

get

- rV(x) + ( u(x) - + g$) Y’(?c) = 0.

Now we may formally differentiate on x in (64) to get

- t-l/‘(x) + f u’(x) V’(x) + ( fJ(4x)) - u(x) - + ____

(f(W) > V”(X) = 0.

Again eliminating p” from (63) noting p = u(x) in (63) gives

(64)

(65)

- rV’(x) + f u’(x) V’(x) - ( 444) u(x) - 4 ___

o’(u(x)) 1 V’(x)

a(u(x))o’(u(x)) = O. (66)


We may now divide by V’(x) which is 2 1 by (60), and we obtain a differential equation for u(x) not involving I/!

u’(x) = 2r + 2uw 1 2

4u(x))+44) - +4x)) ’ ( > which we can write as

du

2r + 2u/a(u)o’(u) - (l/a’(u))2 = dx.

(67)

(68)

Now (68) is separable in u and x, so we may integrate both sides, taking into account that u(0) = 0 because all points of _& are extreme points (again this is just intuition and heuristic) and obtain x = x(u) and, by inversion, u = u(x). We will have x(p) = b for some b, and u(b) = 8. We will also expect that u(x) increases in x from x = 0 to x = b. For b 5 x 5 a, where a is some point > b to be found, we will use u(x) = 8, the most aggressive policy in A!, and finally for x > a we will expect

V(x) = x + 5, a I x, (69)

by analogy with (21) since we will set dZ = co for x > a. It remains to find only a and 5. We expect Y(0) = 0, and V E C2 by (59), and these must determine them. Of course, everything in our plan is heuristic, but once V is guessed we can use the methods of Section 2 to show that V z P.

To illustrate the above we choose the simplest special case as an example, o2 = 02(p) given by

cT2(p) = e2p2, 05p5pp, (70)

where 0 and fl are given parameters. We seek

s co

V(x) = sup E, e-“dZ,, (71) 0

where the sup is taken over all pt E [0, /3], t 2 0, and Z, with Z, 2 0, with both p and Z nonclairvoyant. We assume X0 = x and

dX, = /.~L,dt + Bptd W, - dZ,, O<t<z,, (72)

where TV is the first t < cc for which X, = 0 (if there is such) and dX, = dZ, = 0 for t 2 TV.

R. Radner, L. Shepp / Journal of Economic Dynamics and Control 20 (1996) 1373- 1393 1389

From (68) we obtain simply for 0 5 x < /I

du

2r + l/P = dx or U(X) = kx with k = 2r + l/0”. (73)

Thus b = p/k and U(X) = /I for x 2 b, heuristically. We will use pt = u(X,), t > 0; dZ, = cc for X, 2 b so that X, will remain in [IO, b]. It is simplest to use (64) to guess P and we have, easily, since V(0) = 0,

- ?-p(x) + 4 kxP'(x) = 0 or p(x) = Alx2'lk, 0 5 x 5 b = p/k, (74)

B(x) = A2eai2x - B2eSzX, b<x<a; V(x) = x + (, a < x, (75)

where a2 and j2 are the roots of - r + fly + ij?‘t12y2 = 0, i.e.,

- cI2 =

l+&ziP l-JCT%

o’p ’ P2 = - e2p . (76)

It remains to find Al, AZ, B2, 5, a. There are three conditions at each of b and a for B E C2, but one of these is redundant because b = p/k is already guessed.

Since V(b + ) = V(b - ) and V’(b+) = B’(b-), we have from (74) and (75)

Alb2’lk = A2eazb - B2epzb, (77)

A,;b 2’lk-1 = A2M2eazb _ B2P2ePzb. (78)

That P”(b + ) = V(b - ) follows automatically as noted above since b = p/k.

Since V(a + ) = Y(a - ), V’(a+) = V’(a-), and P”(a+) = V”(K), we have from

(75)

A2eaza - B2eP2a = a + t, (79)

A2cx2eaza - B2f12eBZa = 1, (80)

A2a;ea2a - B2/3:eP2a = 0. (81)


We find

2pe a=h+---log

Jz + l/0

Ji Ji - l/S’ b=/i’/k, k=2r+1/6=,

A2 = JZ + l/e ea2a eg

JC - i:eT$ ’

B = & - l/O ew8za ep

- 2 &+1/02$ ’

P-4

(83)

where L = log[(Jrk + l/Q/($ - l/8)]. From this we see that as /J + co, P(x, fi) -+ cc for each x, so that if the company has the chance to make arbitrarily large risk policies (p, 02p2) with unbounded ,u, then it can make infinite profits in the sense of (6).

It is now easy to verify that V satisfies (59)-(61), and so the method of proof of Section 2 shows easily that V of (71) satisfies V(x) = V(x). It is sufficient to check that r”(x) < 0 for all x.

5. An alternative model to Bachelier; BlackScholes-!Samuelson

In Section 1 we argued that the multiplicative model (2) would be inappropriate for a model of the firm because it embodied the assumption of constant returns to scale at all scales, however large. In addition, the usual model of finance theory assumes that all assets are liquid. With this in mind, recall that, with the multiplicative model, if there were no dividends, then the assets of the firm would evolve according to

dX, = X,(,udt + adW,). (85)

In this section we show that with this model the criterion (6) leads to an empty problem - either V(x) = co, or V(x) = x, and all the profits of the company are drawn at once. Note that in the case of (85), X, > 0 for all t > 0 w.p.1, i.e., r0 = co, and the firm never goes bankrupt (not such a good model, since real firms often do!). But suppose we do use (85) in place of (3).

Let us consider for simplicity the case when ATZ is a singleton; the general case is similar. Suppose that, if dividends are paid, the assets of the firm evolve

R. Radner. L. Shepp /Journal of Economic Dynamics and Control 20 (1996) 1373-1393 1391

according to

dX, = X,(pdt + adW,) - d.Z,, O<tlzo, (86)

where Z, is the accumulated dividends, dZ, 2 0, and of course now bankruptcy

at ~~ < CC is possible. The firm wants to determine

s ;x,

V(x) = sup E, e-“dZ,, 0

(87)

where the sup is over all choice of nonclairvoyant Z. We will now show that if p i r, then V(x) = x, which may be obtained by

paying the dividend x at t = 0 so that r. = 0. Then we will show that if p > r,

then V(x) = cc. For any choice of Z,, we set V(x) E x, x > 0, and define

s t Y, = I/(XJe-” + em*“dZ,, 0 < t. 638)

0

We find using (86) that Y has the differential,

dY, = e-” - rV(X,)dt + V’(X - t)[X&dt + adW,) - d&l

+ v”(X,)X$ dt + dZ, I

=e --** f(p - r)X,dt + oX,dW,}, (89)

since V’(x) E 1 and V”(x) = 0. Now, if p < r, we see that Y, is a submartingale,

EdY, I 0, so that

E,Y,, I ExYo. (90)

This gives from (88)

s ‘X

J% eprsdZ, I V(xo) = Ytx) = x. 0

(91)

Since this holds for every choice of Z,, we conclude V(x) I x. Suppose now that p > r. Then one simple strategy for the firm is to let the

assets accumulate (dZ, = 0) until time T. Since X, > 0 for 0 5 t I T, the fortune

1392 R. Radner, L. Shepp /Journal ofEconomic Dynamics and Control 20 (1996) 1373-1393

of the firm will be XT, which is given by the well-known exponential form

_yT = xeaWr + (II - o”P)T (92)

At time T we liquidated all the assets. Since this is an allowable strategy for the firm, we have that I/ is certainly bigger than the profit X,exp( - rT) obtained in this (trivial) way, i.e.,

But since W is a standard Wiener process,

and so from (93)

V(x) 2 e(p-r)T.

(94)

(95)

But ,U > r and T is arbitrary so V(x) = co. Q.E.D.

Acknowledgement of referee’s comments

A referee pointed out that our formulation is not so novel, and indeed several authors have considered quite closely related models although, as he states, they were less explicit and concrete in their analyses. The referee kindly supplied the following rough list of authors and dates of papers which deal with similar problems for the reader’s comparison and reference: Hotelling, Merton [1974], Cox, Ingersoll, and Ross [1985], Taksar [1985], Holmstrom and Milgrom [1987], and Sung [1995].

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Documents

Radner & Shepp 1996 - Risk vs Profit Potentials a Model for Corporate Strategies (JEDC)