Mortality Modelling With Renewal Process and Optimal Hedging Strategy Under Basis Risk

Abstract

In this thesis, we address the risks that are related to the random residual lifetime of insureds. These risks could be classified as catastrophic mortality risk and longevity risk. Catastrophic mortality risk represents the sudden increases in mortality rates which means that the insurance companies or pension plans would have to make sudden payments to many policyholders. While catastrophic mortality risk describes the shorter lifetime than anticipated of an individual or a group, its counterpart longevity risk represents the uncertain evolution in mortality rates. When a group or an individual live longer than anticipated, insurance companies or pension plans would make annuity payments longer than expected. Since the catastrophic mortality risk and longevity risk could cause serious financial consequences, management of these risks is important for insurance companies and pension plans.

Catastrophic mortality risk often causes transitory jumps on the mortality curve. Several stochastic mortality models have been developed to capture these jump effects. To the best of author’ knowledge, all these jump models in the actuarial literature assume that the mortality jumps occur once a year, or they used a Poisson process for their jump frequencies. Due to their low probability and high-impact nature, the timing and the frequency of future catastrophic events and hence mortality jumps are unpredictable, however, the history of events could give information about their future occurrences. In this thesis, a new approach for the modelling of the frequency of catastrophic mortality risk is introduced and a specification of the Lee-Carter model using a renewal process is proposed. The history of events can be included in jump modelling by using this process.

We perform several statistical tests on the inter-arrival times data of the catastrophic events to show that the renewal process could be used for jump frequencies. For this purpose, first, we detect outliers on the mortality time index. The statistical tests are applied to the inter-arrival times of these detected outliers. According to the test results, we can use the lognormal renewal process to model jump frequencies for all selected countries.

Longevity risk is another risk factor that we examined in this thesis. We use index-based longevity swaps to hedge this risk. Index-based securities have many advantages. In such capital market solutions, it is possible to transfer the longevity risk to capital markets at lower costs. However, the potential differences between hedging instruments and pension or annuity portfolio cause longevity basis risk. Furthermore, we extended the proposed mortality model to incorporate longevity basis risk. We modelled reference population’s mortality by using the proposed mortality model and then the portfolio’s mortality is modelled by using the information of the reference population. According to our analysis, the common age effect is important for both populations.

Since the longevity-linked derivatives are traded in the over-the-counter markets, an insurer or a pension plan can be exposed to counterparty default risk. In this thesis, we provide a hedging framework for longevity basis risk in the context of collateralization. We assume that both parties are posting the collateral and they re-hypothecate it to increase the benefits of this transaction.

We build hypothetical pension plan and index-based longevity swap transaction to show the effects of collateralization and risk reduction level. Our analysis present that bilateral collateral posting increases longevity basis risk reduction level and hedge effectiveness.