Bounds and approximations for sums of dependent log-elliptical random variables.

Dhaene, Denuit, Goovaerts, Kaas and Vyncke [Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke,D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math.Econom. 31 (1), 3-33; Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 (2), 133-161]have studied convex bounds for a sum of dependent random variables and applied these to sums of lognormal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed-form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed-form approximations.We also present numerical examples to show the accuracy of the proposed approximations.

Loss Distribution Approach for Operational Risk Capital Modelling under Basel II: Combining Different Data Sources for Risk Estimation

The management of operational risk in the banking industry has undergone significant changes over the last decade due to substantial changes in operational risk environment. Globalization, deregulation, the use of complex financial products and changes in information technology have resulted in exposure to new risks very different from market and credit risks. In response, Basel Committee for banking Supervision has developed a regulatory framework, referred to as Basel II, that introduced operational risk category and corresponding capital requirements. Over the past five years, major banks in most parts of the world have received accreditation under the Basel II Advanced Measurement Approach (AMA) by adopting the loss distribution approach (LDA) despite there being a number of unresolved methodological challenges in its implementation. Different approaches and methods are still under hot debate. In this paper, we review methods proposed in the literature for combining different data sources (internal data, external data and scenario analysis) which is one of the regulatory requirement for AMA

The total assessment profile, volume 2

Appendices are presented which include discussions of interest formulas, factors in regionalization, parametric modeling of discounted benefit-sacrifice streams, engineering economic calculations, and product innovation. For Volume 1, see

Valuation and Risk Management of Some Longevity and P&C Insurance Products

Numerous insurance products linked to risky assets have emerged rapidly in the last couple of decades. These products have option-embedded features and typically involve at least two risk factors, namely interest and mortality risks. The need for models to capture risk factors\u27 behaviours accurately is enormous and critical for insurance companies. The primary objective of this thesis is to develop pricing and hedging frameworks for option-embedded longevity products addressing correlated risk factors. Various methods are employed to facilitate the computation of prices and risk measures of longevity products including those with maturity benefits. Furthermore, in order to be prepared for the implementation of the new International Financial Reporting Standards (IFRS) 17, the thesis\u27s secondary objective is to provide a methodology for computing risk margins under the impending regulatory requirements. This is demonstrated using a property and casualty (P&C) insurance example and taking advantage of P&C data availability. To accomplish the above-mentioned objectives, five self-contained but related research works are undertaken and described as follows. (i) A pricing framework for annuities is constructed, where interest and mortality rates are both stochastic and dependent. The short-rate process and the force of mortality follow the two-factor Hull-White model and Lee-Carter model, respectively. (ii) The framework in (i) is further developed by adopting the Cox-Ingersoll-Ross model for the short-rate process to price guarantee annuity options (GAOs). The change of measure technique together with the comonotonicity theory is utilised to facilitate the computation of GAO prices. (iii) A further modelling framework extension is attained by considering a two-decrement model for GAO\u27s valuation and risk measurement. Interest rate, mortality and lapse risks are assumed correlated and they are all modelled as affine-diffusion processes. Risk measures are calculated via the moment-based density method. (iv) We introduce a regime-switching set up for the valuation of guaranteed minimum maturity benefits (GMMBs). A hidden Markov model (HMM) modulates the evolution of risk processes and the HMM-based filtering technique is employed to generate the risk-factor models\u27 parameter estimates. An analytical expression for GMMB value is derived with the aid of the change of measure technique in combination with a Fourier-transform approach. (v) Finally, a paid-incurred chain method is customised to model Ontario\u27s automobile claim development triangular data set over a 15-year period, and the moment-based density method is applied to approximate the distributions of outstanding claim liabilities. The risk margins are determined through risk measures as prescribed by the IFRS 17. Sensitivity analysis is performed for risk margins using the bootstrap method