Risk vs. profit potential:.

*(English)*Zbl 0875.90045Summary: 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 variance. 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.

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\textit{R. Radner} and \textit{L. Shepp}, J. Econ. Dyn. Control 20, No. 8, 1373--1393 (1996; Zbl 0875.90045)

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