This paper is a report from the Extreme Events Working Party. The paper considers some of the difficulties in calculating capital buffers to cover potential losses. This paper considers the reasons why a purely mechanical approach to calculating capital buffers may bot be possible or justified. A range of tools and techniques is presented to help address some of the difficulties identified.

of the London Discussion Dr D. J. P. Hare, F.I.A. (Chairman):I am going to invite Parit Jakhria, who is one of the authors of tonight's paper, to introduce it. He is an experienced actuary who works for the Prudential in the capital modelling team and has worked on a number of working parties for the profession. He has helped to produce what I think is an excellent paper.Mr P. C. Jakhria, F.I.A., CFA: Welcome to the discussion of our professional paper entitled 'Difficult Risks and Capital Models'. We talk about modelling in general and capital models in particular. Before I proceed any further, I should like to acknowledge and thank the contributions of all the members of the Extreme Events Working Party. It is on behalf of the working party that I will be summarising the paper this evening.My intention is to provide a very high level summary of the themes of the paper. The paper itself has quite a number of ideas, so I will try to capture the key ideas and also use three examples to highlight some of the ideas that we think are important and/or new.As background, it is worth talking a little about modelling in general. Models permeate every corner of the actuarial world. Although our paper discusses capital modelling in particular, many of the insights may apply to actuarial modelling in general.Right at the start of the paper there are some real, as well as hypothetical, examples of where capital models have gone wrong.Before discussing the paper, I will give a brief summary of our understanding of capital models. We gather information from the past, plus some insight into present conditions. We combine that with the knowledge of a particular problem and model the future, with the ultimate aim of making decisions about the future, for example how much capital to put aside.We would like to highlight that a model is a simplified representation of the real world. Thinking about it philosophically, if one wanted a perfectly accurate model of the universe it would need to be as big as the universe. In fact, we want much more than an accurate model of the real world in capital modelling. We want to able to project faster than real-time, so as to extrapolate to 'tail' scenarios. You have to make some choices in terms of how to simplify the model to match the real world. It is really this process of simplifying which requires a huge amount of choice and judgement. Later we will go into one of the examples in a little more detail. 617We try to list the different areas in which you can encounter choices when modelling. These are not exhaustive, but we try and cover a lot of different things. The single biggest decision, for example, is simply what risk factors to model or, more precisely, what risk factors to model stochastically.Other than that, you can choose your overall framework (e.g. whether you want to use a building block approach, a risk factor approach, etc.). For each model component you could choose which model to use. You could choose: how to calibrate the model; what data to use; whether there is any judge...

Under the European Union’s Solvency II regulations, insurance firms are required to use a one-year VaR (Value at Risk) approach. This involves a one-year projection of the balance sheet and requires sufficient capital to be solvent in 99.5% of outcomes. The Solvency II Internal Model risk calibrations require annual changes in market indices/term structure for the estimation of risk distribution for each of the Internal Model risk drivers. This presents a significant challenge for calibrators in terms of: Robustness of the calibration that is relevant to the current market regimes and at the same time able to represent the historically observed worst crisis; Stability of the calibration model year on year with arrival of new information. The above points need careful consideration to avoid credibility issues with the Solvency Capital Requirement (SCR) calculation, in that the results are subject to high levels of uncertainty. For market risks, common industry practice to compensate for the limited number of historic annual data points is to use overlapping annual changes. Overlapping changes are dependent on each other, and this dependence can cause issues in estimation, statistical testing, and communication of uncertainty levels around risk calibrations. This paper discusses the issues with the use of overlapping data when producing risk calibrations for an Internal Model. A comparison of the overlapping data approach with the alternative non-overlapping data approach is presented. A comparison is made of the bias and mean squared error of the first four cumulants under four different statistical models. For some statistical models it is found that overlapping data can be used with bias corrections to obtain similarly unbiased results as non-overlapping data, but with significantly lower mean squared errors. For more complex statistical models (e.g. GARCH) it is found that published bias corrections for non-overlapping and overlapping datasets do not result in unbiased cumulant estimates and/or lead to increased variance of the process. In order to test the goodness of fit of probability distributions to the datasets, it is common to use statistical tests. Most of these tests do not function when using overlapping data, as overlapping data breach the independence assumption underlying most statistical tests. We present and test an adjustment to one of the statistical tests (the Kolmogorov Smirnov goodness-of-fit test) to allow for overlapping data. Finally, we explore the methods of converting “high”-frequency (e.g. monthly data) to “low”-frequency data (e.g. annual data). This is an alternative methodology to using overlapping data, and the approach of fitting a statistical model to monthly data and then using the monthly model aggregated over 12 time steps to model annual returns is explored. There are a number of methods available for this approach. We explore two of the widely used approaches for aggregating the time series.

This paper overviews a practical approach to the assessment of operational risk in life insurance companies. It considers how actuaries, working in conjunction with risk management professionals and senior management, can develop a framework to assess the capital requirements relating to operational risk, taking into account the capital requirements of other risks and their interaction.This paper recognises that we do not live in an ideal world, and that a lot of the data which one might want for operational risk assessment are not, and in some cases never will be, available. Consequently, the approach outlined in this paper takes into account the fact that management and assessment of operational risk is at an early stage of development in the life industry. In addition, it outlines some of the areas where development is necessary or desirable in the coming years.There is a section on the operational risks against which it is appropriate to hold capital. As this is a new area for insurance companies, and given the governance requirements for Individual Capital Assessments, it is important to explain the results effectively to senior management. Therefore, a brief review of techniques for reporting the results of the assessment is provided.The paper concludes with some thoughts on how operational risk management can be embedded more in the business, and then considers what future work will help develop the framework. To echo the thoughts of the authors of the general insurance paper on this topic (Tripp et al., 2004), we hope that the paper will sow seeds for the development of best practice in dealing with operational risk, and will raise the awareness and increase the interest of actuaries in this emerging topic.

Some UK insurers have been using real-world economic scenarios for more than 30 years. Popular approaches have included random walks, time series models, arbitrage-free models with added risk premiums or 1-year Value at Risk distribution fits. Based on interviews with experienced practitioners as well as historical documents and meeting minutes, this paper traces historical model evolution in the United Kingdom and abroad. We examine the possible catalysts for changes in modelling practice with a particular emphasis on regulatory and socio-cultural influences. We apply past lessons to provide some guidance to the direction of capital market modelling in future, which has been key for business and strategy decisions.