Optimal Investment and Consumption When Allowing Terminal Debt

We analyze an optimal consumption and investment problem for a representative agent who may have different preferences for consumption and for terminal wealth. The utility for consumption is characterized by constant relative risk aversion so consumption is always positive. In contrast, the agent's risk aversion with regard to final wealth does not necessarily imply he must end up with positive terminal wealth. Indeed, we define the risk aversion for both positive and negative wealth levels and require it to be positive but not monotone. There is a point of maximal risk aversion at zero wealth and the agent may continue to consume when his wealth is negative.Using dual methods we can derive explicit solutions for this problem in a multi-asset economy which takes survival probabilities of the agent into account. This allows us to study the optimal patterns for consumption and investment and compare them to the case where terminal debt is not allowed