Advances in Consumption-Investment Problems with Applications to Pension

We present the optimal consumption and investment strategy for an investor, endowed with labor income, searching to maximize utility from consumption and terminal wealth when he is restricted to fulfill a binding capital constraint of a European (constraint on terminal wealth) or an American (constraint on the wealth process) type. In both cases the optimal strategy is proven to be of the option based portfolio insurance type. The optimal strategy combines a long position in the optimal unrestricted allocation with a put option. In the American case, where the investor is restricted to fulfill a capital guarantee at every intermediate time point over the interval of optimization, we prove that the investor optimally changes his budget constraint for the unrestricted allocation whenever the constraint is active. The strategy is explained in a step by step manner and graphically illustrations are presented in order to support the intuition and to compare the restricted optimal strategy with the unrestricted optimal counterpart.