Stochastic Inter-Temporal Optimization in Discrete Time

The standard model of inter-temporal optimization is based upon certainty equivalence and ignores risk and uncertainty. We solve a modification of the standard model of inter-temporal optimization in an environment where the return to capital is stochastic, and we impose the constraint that there be no default on the short-term debt. We derive benchmarks for optimal foreign debt in a world of uncertainty. Insofar as the actual debt exceeds the benchmark, the expected utility of consumption is reduced. Default occurs with probability (1-p) when the debt exceeds the maximum debt f(2)max. The main reasons for a deviation between the actual debt and the optimal debt are that the borrower is overly optimistic about the distribution function of the return to investment, does not optimize with the "no default" constraint, and/or there is a moral hazard problem. We also consider an inter-temporal optimization model involving extreme prudence. The lender, who may be an institutional investor, has infinite risk aversion and will only lend for projects where the profitability of the investment is almost sure. In this case also, we derive the optimal debt, which is our benchmark for risk management