An optimal portfolio problem in a defaultable market

Abstract

We consider a portfolio optimization problem in a defaultable market. The investor can dynamically choose a consumption rate and allocate his/her wealth among three financial securities: a defaultable perpetual bond, a default-free risky asset, and a money market account. Both the default risk premium and the default intensity of the defaultable bond are assumed to rely on some stochastic factor which is described by a diffusion process. The goal is to maximize the infinite-horizon expected discounted log utility of consumption. We apply the dynamic programming principle to deduce a HamiltonJacobi-Bellman equation. Then an optimal Markov control policy and the optimal value function is explicitly presented in a verification theorem. Finally, a numerical analysis is presented for illustration.

Markov decision process algorithms for wealth allocation problems with defaultable bonds

“…In addition, the results allocated a constant fraction of wealth in the Brownian asset, in a similar fashion to Merton [16]. Bo et al [8] approached a perpetual allocation problem for an investor with logarithmic utility, considering a defaultable perpetual bond along with a traditional stock and a risk-free account in a similar manner to Bielecki and Jang [6]. Their work modelled stochastically the intensities and premium process and made use of heuristic arguments in order to postulate the price dynamics of the defaultable bond.…”

“…The numerical analysis has been focused on the dependence of optimal portfolio selections on the risk premium, recovery of market value and several other parameters defining the model, and it has extended the scope of the results in [6], [8], [9], and [15] to broader families of utility functions, highlighting relevant divergences on optimal strategies with respect to variations and generalizations in choices of utilities. In addition, in this paper we have examined the impact of a short-selling restriction within the market, identifying a dependency on optimal stock allocations with respect to default event on a corporate bond.…”

“…In the last decade, however, it is the optimal investment linked to defaultable claims that has attracted major attention. High-yield corporate bonds offer attractive risk-return profiles and have become popular in comparison to stocks or default-free bonds; recent work in this area includes those of Bielecki and Jang [6], Bo et al [8], Lakner and Liang [15], and Capponi and Figueroa-Lopez [9].…”

“…In this paper we make use of the results on credit risk presented in [7] along with the theory for MDPs reviewed in [4] and [18]; and extend the work in [2] and [3] to the context of defaultable markets explored in [6], [8], [9], and [15] and references therein. By means of a conversion of the optimization problem into a MDP, its value function is characterized as the unique fixed point to a dynamic programming operator and optimal wealth allocations are numerically approximated through value iteration.…”

“…This allows us to undertake a numerical analysis exploring the dependence of portfolio selections on the risk premium and different parameters describing the system. In doing so, we are able to examine extensions of the work in [6], [8], [9], and [15] to more general families of logarithmic and exponential utility functions. The results highlight the nature of the significantly different allocation procedures under an exponential family of utilities, and the existence of a dependency on optimal stock allocation to default event, in a model with short-selling restrictions.…”