Optimal Consumption and Investment with Labor Income and European/American Capital Guarantee

We present the optimal consumption and investment strategy for an investor,endowed with labor income, searching to maximize utility from consumption and terminalwealth when facing 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 strategyis proven to be of the option-based portfolio insurance type. The optimal strategy combinesa long position in the optimal unrestricted allocation with a put option. In the Americancase, where the investor is restricted to fulfill a capital guarantee at every intermediatetime point over the interval of optimization, we prove that the investor optimally changeshis budget constraint for the unrestricted allocation whenever the constraint is active. Thestrategy is explained in a step-by-step manner, and numerical illustrations are presented inorder to support intuition and to compare the restricted optimal strategy with the unrestrictedoptimal counterpart