No abstract

The dynamic portfolio selection problem with bankruptcy and nonlinear transaction costs is studied. The portfolio consists of a risk-free asset, and a risky asset whose price dynamics is governed by geometric Brownian motion. The investor pays transaction costs as a (piecewise linear) function of the traded volume of the risky asset. The objective is to find the stochastic controls (amounts invested in the risky and risk-free assets) that maximize the expected value of the discounted utility of terminal wealth. The problem is formulated as a non-singular stochastic optimal control problem in the sense that the necessary condition for optimality leads to explicit relations between the controls and the value function. The formulation follows along the lines of Merton (Merton 1969 Rev. Econ. Stat. 51 , 247–257; Merton 1971 J. Econ. Theory 3 , 373–413) and Bensoussan & Julien (Bensoussan & Julien 2000 Math. Finance 10 , 89–108) in the sense that the controls are the amounts of the risky asset bought and sold, and they are bounded. It differs from the works of Davis & Norman (Davis & Norman 1990 Math. Oper. Res. 15 , 676–713), who use, in the presence of proportional transaction costs, a singular-control formulation in which the controls are rates of buying and selling of the risky asset, and they are unbounded. Numerical results are presented for buy/no transaction and sell/no transaction interfaces, which characterize the optimal policies of a constant relative risk aversion investor. The no transaction region, in the presence of nonlinear transaction costs, is not a cone. The Merton line, on which no transaction takes place in the limiting case of zero transaction costs, need not lie inside the no transaction region for all values of wealth.

The multi-period portfolio-selection problem is formulated as a Markowitz meanvariance optimization problem. It is shown that the single-period Markowitz quadratic programming algorithm can be used to solve the multi-period assetallocation problem with suitable modifications in the covariance and linear constraint matrices. It is assumed that the number of shares invested in risky assets is deterministic and the amount of money invested in the risk-free asset is random at future trading dates when short sales on assets are allowed. The general covariance matrix in the multi-period setup contains intertemporal correlations between assets, in addition to correlations between assets at all trading dates. Analytical solutions for the optimal trading strategy, which is linear in the risk-aversion parameter, are obtained. The efficient frontier is a straight line in the expected return/standard deviation of the portfolio space. When the dynamics of the risky assets follow geometric Brownian motion, it is shown that the time-zero allocations to the risky assets coincide with those obtained by Merton in the continuous-time framework. When short sales are not allowed on assets, the values of the portfolio at future trading dates may not be conserved. In the modified Markowitz mean-variance formulation, the value of the portfolio at future trading dates is conserved in the expected sense, and the optimal trading strategy is selected so that the deviation is minimized in the least-square sense. The efficient frontier is a parabola in the expected return/total variance space when short-sales are allowed and when the portfolio consists of risky assets only. When short sales are not allowed, it is shown that finding the optimal trading strategy is equivalent to solving the single-period Markowitz quadratic programming problem, with suitable modifications in the covariance and linear constraint matrices. By solving the Karush-Kuhn-Tucker conditions, analytical solutions are obtained for a two-period two-assets case, and the Markowitz mean-variance principle is illustrated by solving a test problem numerically using the revised simplex method.

This paper derives the theoretical underpinnings behind the following observed empirical facts in credit risk modeling: The probability of default, the seniority, the thickness of the tranche, the debt cushion, and macroeconomic factors are the important determinants of the conditional probability density function of the recovery rate given default (RGD) of a firm’s debt and its tranches. In a portfolio of debt securities, the conditional probability density functions of the recovery rate given default of tranches have point probability masses near zero and one, and the expected value of the recovery rate given default increases as the seniority or debt cushion increases. The paper derives other results as well, such as the fact that the conditional probability distribution function associated with any senior tranche dominates that of any junior tranche by first-order. The standard deviation of the recovery rate given default of a senior security need not be greater than that of a junior security. It is proved that the expected value of the recovery rate given default need not increase as the proportional thickness of the tranche increases.

The guidelines of various Accounting Standards require every financial institution to measure lifetime expected credit losses (LECLs) on every instrument, and to determine at each reporting date if there has been a significant increase in credit risk since its inception. This paper models LECLs on bank loans given to a firm that has promised to repay debt at multiple points over the lifetime of the contract. The LECL can be written as a sum of ECLs (estimated at reporting date) incurred at debt repayment times. The ECL at any debt repayment time can be written as a product of the probability of default (PD), the expected value of loss given default and the exposure at default. We derive a stochastic dynamical equation for the value of the firm’s asset by incorporating the dynamics of the factors. Also, we show how the LECL and the term structure of the PD can be estimated by solving a Black–Scholes–Merton like partial differential equation. As an illustration, we present the numerical results for the various credit loss indicators of a fictitious firm when the dynamics of the short-term interest rate is characterized by a Cox–Ingersoll–Ross mean-reverting process.