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  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Dynamic Portfolio Optimization with a DefaultableSecurity and Regime Switching

    Jose E. Figueroa-Lopez

    Department of StatisticsPurdue University

    figueroa@purdue.edu

    INFORMSCredit and Counterparty Risk

    November 14, 2011(joint work with Agostino Capponi)

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Regime Switching Models and Portfolio Optimization

    1 Why regime switching? Market and credit factors exhibitdifferent behavior depending on the overall state of economy,as measure by, e.g., a macro-economic index Ct .

    2 Portfolio Optimization focused on default-free markets:Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor &Cadenillas (2009);

    3 Defaultable bonds are a significant portion of the market.4 Portfolio optimization problems with defaultable securities has

    focused on Brownian driven risky factors:

    Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang(2008), Jeanblanc & Runggaldier (2010);

    5 Our goal:Develop a framework for solving finite horizon portfoliooptimization problems under regime switching markets with adefaultable bond.

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Regime Switching Models and Portfolio Optimization

    1 Why regime switching? Market and credit factors exhibitdifferent behavior depending on the overall state of economy,as measure by, e.g., a macro-economic index Ct .

    2 Portfolio Optimization focused on default-free markets:Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor &Cadenillas (2009);

    3 Defaultable bonds are a significant portion of the market.4 Portfolio optimization problems with defaultable securities has

    focused on Brownian driven risky factors:

    Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang(2008), Jeanblanc & Runggaldier (2010);

    5 Our goal:Develop a framework for solving finite horizon portfoliooptimization problems under regime switching markets with adefaultable bond.

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Regime Switching Models and Portfolio Optimization

    1 Why regime switching? Market and credit factors exhibitdifferent behavior depending on the overall state of economy,as measure by, e.g., a macro-economic index Ct .

    2 Portfolio Optimization focused on default-free markets:Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor &Cadenillas (2009);

    3 Defaultable bonds are a significant portion of the market.4 Portfolio optimization problems with defaultable securities has

    focused on Brownian driven risky factors:

    Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang(2008), Jeanblanc & Runggaldier (2010);

    5 Our goal:Develop a framework for solving finite horizon portfoliooptimization problems under regime switching markets with adefaultable bond.

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Regime Switching Models and Portfolio Optimization

    1 Why regime switching? Market and credit factors exhibitdifferent behavior depending on the overall state of economy,as measure by, e.g., a macro-economic index Ct .

    2 Portfolio Optimization focused on default-free markets:Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor &Cadenillas (2009);

    3 Defaultable bonds are a significant portion of the market.

    4 Portfolio optimization problems with defaultable securities hasfocused on Brownian driven risky factors:

    Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang(2008), Jeanblanc & Runggaldier (2010);

    5 Our goal:Develop a framework for solving finite horizon portfoliooptimization problems under regime switching markets with adefaultable bond.

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Regime Switching Models and Portfolio Optimization

    1 Why regime switching? Market and credit factors exhibitdifferent behavior depending on the overall state of economy,as measure by, e.g., a macro-economic index Ct .

    2 Portfolio Optimization focused on default-free markets:Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor &Cadenillas (2009);

    3 Defaultable bonds are a significant portion of the market.4 Portfolio optimization problems with defaultable securities has

    focused on Brownian driven risky factors:

    Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang(2008), Jeanblanc & Runggaldier (2010);

    5 Our goal:Develop a framework for solving finite horizon portfoliooptimization problems under regime switching markets with adefaultable bond.

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Regime Switching Models and Portfolio Optimization

    1 Why regime switching? Market and credit factors exhibitdifferent behavior depending on the overall state of economy,as measure by, e.g., a macro-economic index Ct .

    2 Portfolio Optimization focused on default-free markets:Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor &Cadenillas (2009);

    3 Defaultable bonds are a significant portion of the market.4 Portfolio optimization problems with defaultable securities has

    focused on Brownian driven risky factors:

    Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang(2008), Jeanblanc & Runggaldier (2010);

    5 Our goal:Develop a framework for solving finite horizon portfoliooptimization problems under regime switching markets with adefaultable bond.

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Overview of Main Results

    Considered Mertons utility maximization problem from wealth(without consumption) in finite-horizon with a risk-free(default-free) asset, a risky asset, and a defaultable bondunder Markov driven regime switching;

    Established the Hamilton-Jacobi-Bellman (HJB) Equations forthe optimal value function of the problem;

    Proved Verification Theorems for the HJB equations;

    Found closed form representations for the logarithm utilityU(v) = log v in terms of the solution of a system of first orderlinear ODE;

    Determined conditions for the directionality (short or long)of the optimal trading strategies for the defaultable bond;

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Overview of Main Results

    Considered Mertons utility maximization problem from wealth(without consumption) in finite-horizon with a risk-free(default-free) asset, a risky asset, and a defaultable bondunder Markov driven regime switching;

    Established the Hamilton-Jacobi-Bellman (HJB) Equations forthe optimal value function of the problem;

    Proved Verification Theorems for the HJB equations;

    Found closed form representations for the logarithm utilityU(v) = log v in terms of the solution of a system of first orderlinear ODE;

    Determined conditions for the directionality (short or long)of the optimal trading strategies for the defaultable bond;

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Overview of Main Results

    Considered Mertons utility maximization problem from wealth(without consumption) in finite-horizon with a risk-free(default-free) asset, a risky asset, and a defaultable bondunder Markov driven regime switching;

    Established the Hamilton-Jacobi-Bellman (HJB) Equations forthe optimal value function of the problem;

    Proved Verification Theorems for the HJB equations;

    Found closed form representations for the logarithm utilityU(v) = log v in terms of the solution of a system of first orderlinear ODE;

    Determined conditions for the directionality (short or long)of the optimal trading strategies for the defaultable bond;

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Overview of Main Results

    Considered Mertons utility maximization problem from wealth(without consumption) in finite-horizon with a risk-free(default-free) asset, a risky asset, and a defaultable bondunder Markov driven regime switching;

    Established the Hamilton-Jacobi-Bellman (HJB) Equations forthe optimal value function of the problem;

    Proved Verification Theorems for the HJB equations;

    Found closed form representations for the logarithm utilityU(v) = log v in terms of the solution of a system of first orderlinear ODE;

    Determined conditions for the directionality (short or long)of the optimal trading strategies for the defaultable bond;

  • Introduction The Model Portfolio Optimization Verification Theorems Logarithmic Utility Conclusions

    Overview of Main Results

    Considered Mertons utility maximization problem from wealth(without consumption) in finite-horizon with a risk-free(default-free) asset, a risky asset, and a defaultable bondunder Markov driven regime switching;

    Established the Hamilton-Jacobi-Bellman (HJB) Equations forthe optimal value function of the problem;

    Proved Verification Theorems for the HJB equations;

    Found closed form representations for the logarithm utilityU(v) = log v in terms of the solution of a system of first orderlinear ODE;

    Determined conditions for the directionality (short or long)of the optimal trading strategies for the defaultable