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 the 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 the existence and uniqueness of a global regular solution. Our explicit characterization of the strategy has direct financial implications: it indicates that the investor purchases a high number of bond shares when his borrowing costs are low and the bond sufficiently safe, and reduces the size of his long position or even sells short when his financing costs are high or the bond very risky.