Eliminating proxy errors from capital estimates by targeted exact computation

Abstract

Determining accurate capital requirements is a central activity across the life insurance industry. This is computationally challenging and often involves the acceptance of proxy errors that directly impact capital requirements. Within simulation-based capital models, where proxies are being used, capital estimates are approximations that contain both statistical and proxy errors. Here, we show how basic error analysis combined with targeted exact computation can entirely eliminate proxy errors from the capital estimate. Consideration of the possible ordering of losses, combined with knowledge of their error bounds, identifies an important subset of scenarios. When these scenarios are calculated exactly, the resulting capital estimate can be made devoid of proxy errors. Advances in the handling of proxy errors improve the accuracy of capital requirements.

Error propagation and attribution in simulation-based capital models

“…The loss error bound can be attributed to individual proxy functions, leading to a mechanism for improvements to the approximations, and when combined with the proxy error elimination technique of Crispin & Kinsley (2022), gives the possibility of completely removing proxy errors from capital estimates. Christiansen (2008) describes examples of uses of first order sensitivity analysis within life insurance applications, referred to as the sensitivity concept.…”

“…This approach does not form part of this study, since we are interested in establishing analytic bounds on approximated loss scenarios. Crispin & Kinsley (2022) show that once analytic bounds on approximated loss scenarios are known, the potential error in the corresponding capital estimates may be calculated. The calculation of error bounds on capital estimates allows the practitioner to pursue error reduction and, possibly, error elimination.…”

“…The loss error bound can be attributed to individual proxy functions, leading to a mechanism for improvements to the approximations, and when combined with the proxy error elimination technique of Crispin & Kinsley (2022), gives the possibility of completely removing proxy errors from capital estimates.…”

“…Following the approach of Lemma 2 of Crispin & Kinsley (2022), we pose the ordered approximate bounds as having potential practical application in estimating bounds on the unknown ordered losses: for . In particular, consider the basic percentile L-estimator , where is the index associated with the 99.5% loss, due to our convention that positive values of are gains in own funds.…”

“…If these prudent bounds are suitably validated, exact methods of error analysis can be performed. Lemma 1 and Theorem 1 of Crispin & Kinsley (2022) imply that the proxy error in the capital estimate can be bounded, and the method of targeted exact computation may apply resulting in the elimination of proxy error from the estimate. Approximate bounds may be attributed across the individual data proxies to identify where improvements to data proxies may be most useful, and then to use improved proxies to further refine the approximate capital bounds (Section 4.5). The proxy loss model and approximate error bounds could be used to cross-validate separately developed proxies, including those aimed at directly approximating the loss function whose development and validation may have encountered the curse-of-dimensionality. …”