Optimal Credit Investment with Borrowing Costs

We consider the portfolio decision problem of a risky investor. The investor borrows at a rate higher than his lending rate, and invests in a risky bond whose market price is correlated with the credit quality of the investor. By viewing the concave drift of the wealth process as a continuous function of the admissible control, we characterize the optimal strategy in terms of a relation between a critical borrowing threshold and solutions of a system of first order conditions. We analyze the nonlinear dynamic programming equation and prove singular growth of its coefficients. Using a truncation technique relying on the locally Lipschitz-continuity of the optimal strategy, we remove the singularity and show existence and uniqueness of a global regular solution. Our explicit characterization of the strategy has direct financial implications: it indicates that the investor switches from overinvesting in the bond when his borrowing costs are low and the bond sufficiently safe to underinvesting or short-selling it when his financing costs are high or the bond very risky