×

zbMATH — the first resource for mathematics

Risk vs. profit potential:. (English) Zbl 0875.90045
Summary: 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.

MSC:
91B38 Production theory, theory of the firm
91B62 Economic growth models
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benes, V.E.; Shepp, L.A.; Witsenhausen, H.S., Some solvable stochastic control problems, Stochastics, 4, 39-83, (1980) · Zbl 0451.93068
[2] Black, F.; Scholes, M., The valuation of option contracts and a test of market efficiency, Journal of finance, 27, 399-418, (1972)
[3] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-659, (1973) · Zbl 1092.91524
[4] Dubins, L.B.; Shepp, L.A.; Shiryaev, A.N., Optimal stopping rules and maximal inequalities for Bessel processes, Teoriya veroyatnostey i primenenü, 38, 288-330, (1993), in Russian · Zbl 0807.60040
[5] Dutta, P.K.; Radner, R., Optimal principal-agent contracts for a class of incentive schemes, Economic theory, (1994), forthcoming · Zbl 0815.90036
[6] Dutta, P.K.; Radner, R., Profit maximization and the market selection hypothesis, (1994), AT&T Bell Laboratories Murray Hill, NJ, Unpublished paper · Zbl 0937.91072
[7] Gigelions, B.I.; Shiryaev, A.N., On Stefan’s problem and optimal stopping rules for Markov processes, Teoriya veroyatnostey i primenenü, 11, 611-631, (1966), in Russian · Zbl 0178.53303
[8] Harrison, J.M., Brownian motion and stochastic flow systems, (1985), Wiley New York, NY · Zbl 0659.60112
[9] Samuelson, P.A., Rational theory of warrant pricing, Industrial management review, 6, 13-39, (1965)
[10] Shepp, L.A.; Shiryaev, A.N., The Russian option: reduced regret, Annals of applied probability, 3, 631-640, (1993) · Zbl 0783.90011
[11] Shepp, L.A.; Shiryaev, A.N., A dual Russian option for selling short, (), forthcoming · Zbl 0867.90014
[12] Shepp, L.A.; Shiryaev, A.N., A new look at the pricing of Russian options, Teoriya veroyatnostey i primenenü, 39, 130-149, (1994), in Russian · Zbl 0834.60072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.