A firm's optimizing behaviour under a Value-at-Risk constraint

In this paper we employ the concept of Value-at-Risk to model a kind of risk-averse behaviour of a firm which seeks to maximize profit à la Greenwald-Stiglitz [Greenwald, B.C., Stiglitz, J.E., 1993, Financial Market Imperfections and Business Cycles, Quarterly Journal of Economics 108 (1), 77-114]. It is shown that there exists a unique well-defined solution function which relates output to the firm's net worth, but that this function is not monotone. The latter is due to the fact that, whenever the VaR-constraint is not binding, the firm behaves in a risk-neutral fashion. It is also shown that in this context the Modigliani-Miller theorem applies only in the special case where there is no risk of bankruptcy.