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Replicating Portfolios—Optimization with a Penalty Term

Professor: 
Type: 
Master's Thesis
Corporate Partner: 
Zurich Insurance Company Ltd
Date Published: 
March 20, 2013

The usage of replicating portfolios in life insurance has been a topic of great interest to (re)insurance companies during the last few years. The timely valuation of life insurance liabilities is particularly challenging due to the complexity and the long-term nature of these liabilities. Replicating portfolios are an approach to approximating the value of such liabilities by a set of financial assets which tries to correctly mimic the market risk behavior inherited in the life liability cash flows. As such, replicating portfolios are a powerful tool to quickly re-valuate life insurance portfolios for internal and regulatory risk management purposes.

In the first part of this thesis a brief description of how to determine the Solvency Capital Requirement within the framework of Solvency II is given. The replicating portfolio contributes to its determination as a tool to quickly re-evaluate life insurance liabilities of the economic balance sheet for a large number of real-world scenarios. Furthermore, an introduction to the theoretical framework of replicating portfolios is presented and a brief insight into the process steps needed to create a replicating portfolio in practice is given. The expertise and information needed for a successful replication are immense, even if we only considerthe necessary understanding of the liability-cash-flow generation with an actuarial model including knowledge of the life-insurance products in order to understand which market risks the whole life insurance portfolio is exposed to, and thus to be able to select appropriate classes of financial assets for the replication. It is also necessary to have in-depth knowledge of the generation of different types of scenarios including market-consistent and real-world varieties generated by the so-called Economic Scenario Generation model.

This thesis only scratches the surface of these topics, but gives the reader a sufficiently sound knowledge-base that they will be able to understand the main aim of the thesis: the calibration of a penalized minimization problem and the investigation of the quality of the impact of the resulting replicating portfolios on real-life insurance portfolios provided by the Zurich Insurance Company Ltd. As the minimization of the squared mismatch of the cash flows of the replicating assets and the benchmark liabilities over all scenarios have the tendency to result in a portfolio with an excessive number of non-zero and offsetting asset positions which can lead to deterioration of the approximation of the liability cash flows, trading constraints are an effective way of regularizing the optimization problem so that it produces sparse and low offsetting replicating portfolios. However, the introduction of the constraint in the form of an added penalty term to the objective function gives rise to more degrees of freedom in the calibration process and the resulting optimization problem becomes more challenging for the solver which has to be dealt with. The results of the different calibrations show that adding the trading constraint leads to RPs with quite different asset compositions. They have in common an effective way of mitigating offsetting positions and, with the L1-penalty approach, also have a method which leads to more sparse portfolios. However, the resulting analysis of the quality criteria to decide which approach of the different calibrations performed the best is ambiguous.

Further, it is investigated whether the different calibrations affect the market risk charge of the Solvency Capital Requirements on a real life insurance portfolio. The analysis shows that the total market risk charge of the Solvency Capital Requirements is quite stable across the different penalty calibrations. But the contributions of the considered single risk market risk factors show larger variations for the Value-at-Risk scenario, but are quite similar when the identical scenario is investigated. Overall, one can thus state that although the replicating portfolios on the investigated real life insurance portfolio have quite different asset compositions, it seems that the risk inherited in the life insurance portfolios can not only be mirrored in a unique replication but that there are several candidate replicating portfolios available.The usage of replicating portfolios in life insurance has been a topic of great interest to (re)insurance companies during the last few years. The timely valuation of life insurance liabilities is particularly challenging due to the complexity and the long-term nature of these liabilities. Replicating portfolios are an approach to approximating the value of such liabilities by a set of financial assets which tries to correctly mimic the market risk behavior inherited in the life liability cash flows. As such, replicating portfolios are a powerful tool to quickly re-valuate life insurance portfolios for internal and regulatory risk management purposes.

In the first part of this thesis a brief description of how to determine the Solvency Capital Requirement within the framework of Solvency II is given. The replicating portfolio contributes to its determination as a tool to quickly re-evaluate life insurance liabilities of the economic balance sheet for a large number of real-world scenarios. Furthermore, an introduction to the theoretical framework of replicating portfolios is presented and a brief insight into the process steps needed to create a replicating portfolio in practice is given. The expertise and information needed for a successful replication are immense, even if we only considerthe necessary understanding of the liability-cash-flow generation with an actuarial model including knowledge of the life-insurance products in order to understand which market risks the whole life insurance portfolio is exposed to, and thus to be able to select appropriate classes of financial assets for the replication. It is also necessary to have in-depth knowledge of the generation of different types of scenarios including market-consistent and real-world varieties generated by the so-called Economic Scenario Generation model.

This thesis only scratches the surface of these topics, but gives the reader a sufficiently sound knowledge-base that they will be able to understand the main aim of the thesis: the calibration of a penalized minimization problem and the investigation of the quality of the impact of the resulting replicating portfolios on real-life insurance portfolios provided by the Zurich Insurance Company Ltd. As the minimization of the squared mismatch of the cash flows of the replicating assets and the benchmark liabilities over all scenarios have the tendency to result in a portfolio with an excessive number of non-zero and offsetting asset positions which can lead to deterioration of the approximation of the liability cash flows, trading constraints are an effective way of regularizing the optimization problem so that it produces sparse and low offsetting replicating portfolios. However, the introduction of the constraint in the form of an added penalty term to the objective function gives rise to more degrees of freedom in the calibration process and the resulting optimization problem becomes more challenging for the solver which has to be dealt with. The results of the different calibrations show that adding the trading constraint leads to RPs with quite different asset compositions. They have in common an effective way of mitigating offsetting positions and, with the L1-penalty approach, also have a method which leads to more sparse portfolios. However, the resulting analysis of the quality criteria to decide which approach of the different calibrations performed the best is ambiguous.

Further, it is investigated whether the different calibrations affect the market risk charge of the Solvency Capital Requirements on a real life insurance portfolio. The analysis shows that the total market risk charge of the Solvency Capital Requirements is quite stable across the different penalty calibrations. But the contributions of the considered single risk market risk factors show larger variations for the Value-at-Risk scenario, but are quite similar when the identical scenario is investigated. Overall, one can thus state that although the replicating portfolios on the investigated real life insurance portfolio have quite different asset compositions, it seems that the risk inherited in the life insurance portfolios can not only be mirrored in a unique replication but that there are several candidate replicating portfolios available.The usage of replicating portfolios in life insurance has been a topic of great interest to (re)insurance companies during the last few years. The timely valuation of life insurance liabilities is particularly challenging due to the complexity and the long-term nature of these liabilities. Replicating portfolios are an approach to approximating the value of such liabilities by a set of financial assets which tries to correctly mimic the market risk behavior inherited in the life liability cash flows. As such, replicating portfolios are a powerful tool to quickly re-valuate life insurance portfolios for internal and regulatory risk management purposes.

In the first part of this thesis a brief description of how to determine the Solvency Capital Requirement within the framework of Solvency II is given. The replicating portfolio contributes to its determination as a tool to quickly re-evaluate life insurance liabilities of the economic balance sheet for a large number of real-world scenarios. Furthermore, an introduction to the theoretical framework of replicating portfolios is presented and a brief insight into the process steps needed to create a replicating portfolio in practice is given. The expertise and information needed for a successful replication are immense, even if we only considerthe necessary understanding of the liability-cash-flow generation with an actuarial model including knowledge of the life-insurance products in order to understand which market risks the whole life insurance portfolio is exposed to, and thus to be able to select appropriate classes of financial assets for the replication. It is also necessary to have in-depth knowledge of the generation of different types of scenarios including market-consistent and real-world varieties generated by the so-called Economic Scenario Generation model.

This thesis only scratches the surface of these topics, but gives the reader a sufficiently sound knowledge-base that they will be able to understand the main aim of the thesis: the calibration of a penalized minimization problem and the investigation of the quality of the impact of the resulting replicating portfolios on real-life insurance portfolios provided by the Zurich Insurance Company Ltd. As the minimization of the squared mismatch of the cash flows of the replicating assets and the benchmark liabilities over all scenarios have the tendency to result in a portfolio with an excessive number of non-zero and offsetting asset positions which can lead to deterioration of the approximation of the liability cash flows, trading constraints are an effective way of regularizing the optimization problem so that it produces sparse and low offsetting replicating portfolios. However, the introduction of the constraint in the form of an added penalty term to the objective function gives rise to more degrees of freedom in the calibration process and the resulting optimization problem becomes more challenging for the solver which has to be dealt with. The results of the different calibrations show that adding the trading constraint leads to RPs with quite different asset compositions. They have in common an effective way of mitigating offsetting positions and, with the L1-penalty approach, also have a method which leads to more sparse portfolios. However, the resulting analysis of the quality criteria to decide which approach of the different calibrations performed the best is ambiguous.

Further, it is investigated whether the different calibrations affect the market risk charge of the Solvency Capital Requirements on a real life insurance portfolio. The analysis shows that the total market risk charge of the Solvency Capital Requirements is quite stable across the different penalty calibrations. But the contributions of the considered single risk market risk factors show larger variations for the Value-at-Risk scenario, but are quite similar when the identical scenario is investigated. Overall, one can thus state that although the replicating portfolios on the investigated real life insurance portfolio have quite different asset compositions, it seems that the risk inherited in the life insurance portfolios can not only be mirrored in a unique replication but that there are several candidate replicating portfolios available.The usage of replicating portfolios in life insurance has been a topic of great interest to (re)insurance companies during the last few years. The timely valuation of life insurance liabilities is particularly challenging due to the complexity and the long-term nature of these liabilities. Replicating portfolios are an approach to approximating the value of such liabilities by a set of financial assets which tries to correctly mimic the market risk behavior inherited in the life liability cash flows. As such, replicating portfolios are a powerful tool to quickly re-valuate life insurance portfolios for internal and regulatory risk management purposes.

In the first part of this thesis a brief description of how to determine the Solvency Capital Requirement within the framework of Solvency II is given. The replicating portfolio contributes to its determination as a tool to quickly re-evaluate life insurance liabilities of the economic balance sheet for a large number of real-world scenarios. Furthermore, an introduction to the theoretical framework of replicating portfolios is presented and a brief insight into the process steps needed to create a replicating portfolio in practice is given. The expertise and information needed for a successful replication are immense, even if we only considerthe necessary understanding of the liability-cash-flow generation with an actuarial model including knowledge of the life-insurance products in order to understand which market risks the whole life insurance portfolio is exposed to, and thus to be able to select appropriate classes of financial assets for the replication. It is also necessary to have in-depth knowledge of the generation of different types of scenarios including market-consistent and real-world varieties generated by the so-called Economic Scenario Generation model.

This thesis only scratches the surface of these topics, but gives the reader a sufficiently sound knowledge-base that they will be able to understand the main aim of the thesis: the calibration of a penalized minimization problem and the investigation of the quality of the impact of the resulting replicating portfolios on real-life insurance portfolios provided by the Zurich Insurance Company Ltd. As the minimization of the squared mismatch of the cash flows of the replicating assets and the benchmark liabilities over all scenarios have the tendency to result in a portfolio with an excessive number of non-zero and offsetting asset positions which can lead to deterioration of the approximation of the liability cash flows, trading constraints are an effective way of regularizing the optimization problem so that it produces sparse and low offsetting replicating portfolios. However, the introduction of the constraint in the form of an added penalty term to the objective function gives rise to more degrees of freedom in the calibration process and the resulting optimization problem becomes more challenging for the solver which has to be dealt with. The results of the different calibrations show that adding the trading constraint leads to RPs with quite different asset compositions. They have in common an effective way of mitigating offsetting positions and, with the L1-penalty approach, also have a method which leads to more sparse portfolios. However, the resulting analysis of the quality criteria to decide which approach of the different calibrations performed the best is ambiguous.

Further, it is investigated whether the different calibrations affect the market risk charge of the Solvency Capital Requirements on a real life insurance portfolio. The analysis shows that the total market risk charge of the Solvency Capital Requirements is quite stable across the different penalty calibrations. But the contributions of the considered single risk market risk factors show larger variations for the Value-at-Risk scenario, but are quite similar when the identical scenario is investigated. Overall, one can thus state that although the replicating portfolios on the investigated real life insurance portfolio have quite different asset compositions, it seems that the risk inherited in the life insurance portfolios can not only be mirrored in a unique replication but that there are several candidate replicating portfolios available.