** Case study background and problem formulations**

**PROBLEM 1: problem_cash_flow_matching_1**

Minimize linear (minimizing the cost of portfolio)

subject to

Cvar_risk ≤ const (constraint on downside risk)

Box constraints (variables are not negative)

——————————————————————–

Cvar_risk = CVaR Risk for Loss

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 1331 | 200 | 1172.368 | 11.13 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 2: problem_cash_flow_matching_2**

Minimize Cvar_risk (minimizing the downside risk)

subject to

linear ≤ const (constraint on cost of portfolio)

Box constraints (variables are not negative)

——————————————————————–

Cvar_risk = CVaR Risk for Loss

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 1331 | 200 | 0.000000000001 | 5.41 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 3: problem_cash_flow_matching_3**

Minimize bPOE (minimizing the buffered probability of downside risk)

subject to

linear ≤ const (constraint on cost of portfolio)

Box constraints (variables are not negative)

——————————————————————–

bPOE = buffered probability of exceeding

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 1331 | 200 | 0.100000000006 | 8.16 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 4: problem_cash_flow_matching_4**

Minimize PM_pen (minimizing the Partial Moment)

subject to

linear ≤ 0 (constraint on cost of portfolio)

Box constraints (variables are not negative)

——————————————————————–

bPOE = Partial Moment

Box constraints = constraints on individual decision variables (br /)

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 1332 | 200 | 0.10000000008 | 7.83 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 5: problem_cash_flow_matching_5**

For 12 problems

Minimize bPOE (minimizing the buffered probability of downside risk)

subject to

linear ≤ const (constraint on budget of portfolio)

Box constraints (variables are not negative)

End for

——————————————————————–

bPOE = buffered probability of exceeding

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 1331 | 200 | 1.000000000000 | 15.65 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**CASE STUDY SUMMARY**

Bond immunization is an important topic in portfolio management. This case study demonstrates a scenario based optimization framework for solving a cash flow matching problem where the time horizon of the liabilities is longer than the maturities of available bonds and the interest rates are uncertain. Bond purchase decisions are made each period to generate cash flow for covering the obligations in future. Since cash flows depend upon future prices of bonds, which are not addressed precisely, some risk management approach needs to be used to handle uncertainties in cash flows.

This case study finds optimal portfolio providing the necessary cash flow with high probability and controlling the total initial portfolio cost.