We develop a stochastic control model for an investor's optimal investment and consumption over an uncertain planning horizon when the investor is endowed with a defaultable income stream. The distributions of the random time of default and the random terminal time are prescribed by deterministic hazard rates, and the investor makes investments in a standard financial market with a bond and a stock, modeled by geometric Brownian motion. In addition, the investor purchases insurance against both default and the terminal date, the default insurance serving as a proxy for the investor's disutility for default. We approximate the original continuous-time problem with a sequence of discrete-time Markov chain control problems by applying dynamic programming and the Markov chain approximation. We demonstrate how the problem can be solved numerically through a logarithmic transformation of the investor's wealth variable, even when the utilities are CRRA with large risk aversion parameter. The model and computational approach are applied to a retiree's optimal annuity decision in the presence of default risk, and we demonstrate that default risk can lead a retiree to annuitize significantly smaller proportions of savings, even when a portion of the defaulted annuity can be recovered, than is traditionally considered optimal by the retirement-finance community. Hence, we show that credit risk may play an important role in resolving the annuity puzzle.

This paper solves a long-standing open problem in mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dual problem. We show that this is indeed possible if the dual problem and its domain are carefully defined. More precisely, we show that the optimal terminal wealth is equal to the inverse of marginal utility evaluated at the solution to the dual problem, which is in the form of the regular part of an element of(L∞)* (the dual space of L∞). (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

We introduce a new insurance instrument allowing retirees to hedge against risk of mortality and risk of default. At retirement, the retiree is allowed to purchase an annuity that provides a defaultable income stream over his lifetime. The time of mortality and time of default are both uncertain, but are accompanied by determined hazard rates. The retiree will make consumption and investment choices throughout his lifetime, which have certain restrictions: the retiree can never enter a bankruptcy state (negative total wealth), and the investment choices are made in a risk-free financial instrument (such as a treasury bill or bond) and a risky instrument (such as commodities or stock). The retiree also makes insurance premium payments which hedge against mortality and default risks simultaneously. This new form of insurance is one which can be implemented by financial institutions as a means for retirees to protect their illiquid assets. In doing so, we calculate the optimal annuity rate a retiree should purchase to maximize his utility of consumption and bequest.Throughout the paper, we develop stochastic control models for a retiree's optimal investment and consumption policies over an uncertain planning horizon in several models which may or may not allow for insurance purchases. We find exact solutions to several models, and apply dynamic programming and the logarithmic transformation to other models to find numerical solutions when constraints are needed. We also analyze the effects of loading on insurance, analyzing the effects of more expensive insurance on the retiree's control policies and value functions. In particular, we will consider the model in which the retiree can purchase life insurance and credit default insurance (in the form of a credit default swap, or CDS) separately to hedge against life events. CDS's do not exist for annuities, but we extend this model by incorporating life insurance and the CDS into a single entity, which can be a viable, and realistic, option to hedge against risk. This model is beneficial in providing a solution to the annuity problem by showing that minimal annuity purchase is optimal.