Optimal risk-exposure management with costly refinancing opportunities

Andrea Barth, Santiago Moreno
In this paper the decisions of a firm's manager, in terms of exposure to a profitable but risky technology, distribution of dividends and (costly) re-injection of cash to ward off bankruptcy are studied. The analysis of the manager's optimal choices is done via a value function whose state variable is the firm's current level of reserves. Contingent on whether proportional or fixed costs of reinvestment are considered, singular stochastic control or stochastic impulse control techniques are
more » ... techniques are used. 2000 Mathematics Subject Classification. 91G10; 91G50; 91G80. Key words and phrases. Portfolio choice, risk exposure, singular stochastic control, stochastic impulse control. We would like to thank Jean-Charles Rochet for his comments and advice. The research leading to these results has received funding from the ERC (grant agreement 249415-RMAC) and from NCCR FinRisk (project Banking and Regulation). 1 2 ANDREA BARTH AND SANTIAGO MORENO-BROMBERG the level of cash reserves follows a diffusion process, this results in a localized optimal strategy. The solution of the manager's problem may be, therefore, identified with a Skohorod problem, where the cash-reserves process is reflected at level x * . The mathematical methodologies for this kind of problems have been studied, for example, in [5] and [3] . The two main departures from [13] and [8] found in the literature are: • Introducing the possibility of varying the size of the risky project. Højgaard and Taksar study in [7] a model of an insurance firm whose manager controls the firm's risk exposure via proportional reinsurance. This is translated into a stochastic control problem where reinsurance is represented by a factor 1 − α ∈ [0, 1]. The second control variable, the cumulative distribution of dividends, keeps the firm's cash reserves below the optimal dividend-distribution barrier x * . The possibility of continuously tweaking the reinsurance level results in two additional (relative to [8]) boundary points 0 < x * * * < x * * < x * . Below x * * * there is partial reinsurance and on (x * * * , x * * ) there is none. In the partial-reinsurance region, the optimal choice α * results in a reserves level that follows a geometric Brownian motion: the proportion of reinsurance tends to one as the reserves approach zero, thus preventing the firm from going bankrupt. The model could also be interpreted as a Merton-style problem of optimal portfolio design, where α represents the proportion of wealth invested in the risky asset, and where the savings account pays zero interest. Rochet-Villeneuve [15] address a similar optimal portfolio problem, but from the point of view of a corporation. They assume the firm's debt level is fixed and its reserves may not become negative. As a consequence the amount of cash that may be invested in the risky asset is bounded above by the firm's current liabilities. Notice the departure from a proportional scenario in [7] to one in nominal terms. Here debt and reserves accrue interest at the same (fixed) rate. This feature, together with the non-proportional setting, results in a value function whose corresponding HJB variational inequality cannot be solved in closed form. This issue notwithstanding, the authors are able to show that the manager's optimal strategy is quite similar to that in [7]: invest a multiple of the firm's equity into the risky asset, keep the rest as cash reserves, and distribute dividends when the value of the firm exceeds some threshold. The resulting value function is rational with exponent smaller than one for low levels of equity, which the authors interpret as corporate risk aversion stemming from the risk of bankruptcy. Interestingly, even though the HJB variational inequality in [15] is more complicated than that in [7] and cannot be solved in closed form, the structures of the solutions to both problems are quite similar. • Introducing reinvestment possibilities. Natural extensions to [8] were considered by Lokka and Zervos in [12] and by Déscamps, Mariotti, Rochet and Villeneuve in [4]. These authors allow for additional equity issuance under proportional or fixed transaction costs, respectively. In both cases closed form solutions for the firm's optimal balance sheet are given, and conditions for the viability of costly equity issuance are found. Furthermore, in [4] the impact of financial frictions on firm governance in terms of the model's predictions is studied, as well as how these governance issues affect the volatility of stock returns. The fixed-transaction-costs setting is further extended by Akyildirim, Güney, Rochet and Soner in [1], where interest rates and issuance costs are governed by an exogenous Markov chain. This provides insight, in a stylized model, on how the business cycle affects managerial decisions and, mapping back to [4], stock price
doi:10.5167/uzh-93703 fatcat:wetaxio5gbeflnlqzk3ngi637i