Abstract
Bond immunization is an important topic in portfolio management. This paper presents a scenario based optimization framework for solving a cash flow matching problem. In this problem, the time horizon of the cash flow generated by the liability is longer than the maturities of the available bonds, and the interest rates are uncertain. Bond purchase decisions are made each period to generate cash flows to cover the obligations due in the future. We use buffered probability of exceedance (bPOE) and conditional value-at-risk (CVaR) to control for the risk of shortfalls. The initial cost of the hedging portfolio of bonds is minimized and optimal positions in bonds are calculated at all time periods. We also study the methodology when solving the optimization problem to minimize bPOE instead of CVaR, which has important practical relevance. The methodology we present in this paper is quite general and can be extended to other financial optimization problems. We use portfolio safeguard optimization package to solve the optimization problems.
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Notes
The essential supremum of the random value L is the smallest number a such that probability of the set \(\{L>a\}\) equals zero.
See examples of such problems at this case study: http://www.ise.ufl.edu/uryasev/research/testproblems/financial_engineering/case-study-portfolio-replication-with-cardinality-and-buyin-constraints/.
The modeling of interest rate in real world may involve a more sophisticated mathematical design and concern other variables, such as credit default risk in a long run. The purpose of this case study is to demonstrate the approach, which can be extended to other models.
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We thank Prof. Ken Kortanek for providing the simulation data. We are also grateful to the referees for their useful comments. All errors remain our responsibility.
Appendix: Portfolio safeguard (PSG) codes
Appendix: Portfolio safeguard (PSG) codes
PSG code for minimizing CVaR with problem (7):
minimize
CVaR(0.9,lmax(matrix_1L,...,matrix_120L))
Constraint:<= 1172.368
linear(matrix_0)
Box:>= 0
PSG code for minimizing bPOE with problem (8):
minimize
bPOE(0,lmax(matrix 1,...,matrix 120))
Constraint:<= 1172.368
linear(matrix_0)
Box:>= 0
PSG code for minimizing bPOE with problem (13):
minimize
pm_pen(-1,lmax(matrix_1L0,...,matrix_120L0))
Constraint:<= 0
linear(matrix_0)
-1172.368*variable(lambda)
Box:>= 0
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Shang, D., Kuzmenko, V. & Uryasev, S. Cash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-risk. Ann Oper Res 260, 501–514 (2018). https://doi.org/10.1007/s10479-016-2354-6
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DOI: https://doi.org/10.1007/s10479-016-2354-6