Johnny Siu-Hang Li

Uncertainty in mortality forecasting: an extension to the classical Lee-Carter approach

Traditionally, actuaries have modeled mortality improvement using deterministic reduction factors, with little consideration of the associated uncertainty. As mortality improvement has become an increasingly significant source of financial risk, it has become important to measure the uncertainty in the forecasts. Probabilistic confidence intervals provided by the widely accepted Lee-Carter model are known to be excessively narrow, due primarily to the rigid structure of the model. In this paper, we relax the model structure by considering individual differences (heterogeneity) in each age-period cell. The proposed extension not only provides a better goodness-of-fit based on standard model selection criteria, but also ensures more conservative interval forecasts of central death rates and hence can better reflect the uncertainty entailed. We illustrate the results using US and Canadian mortality data.

Key Q-Duration: A Framework for Hedging Longevity Risk

When hedging longevity risk with standardized contracts, the hedger needs to calibrate the hedge carefully so that it can effectively reduce the risk. In this article, we present a calibration method that is based on matching mortality rate sensitivities. Specifically, we introduce a measure called key q-duration, which allows us to estimate the price sensitivity of a life-contingent liability to each portion of the underlying mortality curve. Given this measure, one can easily construct a longevity hedge with a small number of q-forward contracts. We further propose an extension for hedging the longevity risk associated with multiple birth cohorts, and another extension for accommodating population basis risk.

On Pricing and Hedging the No-Negative-Equity Guarantee in Equity Release Mechanisms

In a roll-up mortgage, the borrower receives a loan in the form of a lump sum. The loan is rolled up with interest until the borrower dies, sells the house, or moves into long-term care permanently. The house is sold at that time, and the proceeds are used to repay the loan and interest. Most roll-up mortgages are sold with a no-negative-equity guarantee (NNEG), which caps the redemption amount at the lesser of the face amount of the loan and the sale proceeds. The core of this study is to develop a framework for pricing and managing the risks of the NNEG.

Modeling Period Effects in Multi-Population Mortality Models: Applications to Solvency II

Recently Cairns et al. introduced a general framework for modeling the dynamics of mortality rates of two related populations simultaneously. Their method ensures that the resulting forecasts do not diverge over the long run by modeling the difference in the stochastic factors between the two populations with a mean-reverting autoregressive process. In this article, we investigate how the modeling of the stochastic factors may be improved by using a vector error correction model. This extension is highly intuitive, allowing us to visualize the cross-correlations and the long-term equilibrium relation between the two populations. Another key benefit is that this extension does not require the user to assume which one of the two populations is dominant. This benefit is important because, as we demonstrate, it is not always easy to identify the dominant population, even if one population is much larger than the other. We illustrate our proposed extension with data from a pair of populations and apply it to the calculation of Solvency II risk capital.

The CBD Mortality Indexes: Modeling and Applications

Most extrapolative stochastic mortality models are constructed in a similar manner. Specifically, when they are fitted to historical data, one or more series of time-varying parameters are identified. By extrapolating these parameters to the future, we can obtain a forecast of death probabilities and consequently cash flows arising from life contingent liabilities. In this article, we first argue that, among various time-varying model parameters, those encompassed in the Cairns-Blake-Dowd (CBD) model (also known as Model M5) are most suitably used as indexes to indicate levels of longevity risk at different time points. We then investigate how these indexes can be jointly modeled with a more general class of multivariate time-series models, instead of a simple random walk that takes no account of cross-correlations. Finally, we study the joint prediction region for the mortality indexes. Such a region, as we demonstrate, can serve as a graphical longevity risk metric, allowing practitioners to compare the longevity risk exposures of different portfolios readily.

Pricing Standardized Mortality Securitizations: A Two‐Population Model with Transitory Jump Effects

Mortality dynamics are subject to jumps that are due to events such as wars and pandemics. Such jumps can have a significant impact on prices of securities that are designed for hedging catastrophic mortality risk, and therefore should be taken into account in modeling. Although several single‐population mortality models with jump effects have been developed, they are not adequate for modeling trades in which the hedgers population is different from the population associated with the security being traded. In this article, we first develop a two‐population mortality model with transitory jump effects, and then we use the proposed model and an economic‐pricing framework to examine how mortality jumps may affect the supply and demand of mortality‐linked securities.

Canonical Valuation of Mortality-Linked Securities

A fundamental question in the study of mortality�?linked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This article develops a framework for pricing mortality�?linked securities on the basis of canonical valuation. This framework is largely nonparametric, helping us avoid parameter and model risk, which may be significant in other pricing methods. The framework is then applied to a mortality�?linked security, and the results are compared against those derived from other methods.

Markovian Approaches to Joint-Life Mortality

Abstract Many insurance products provide benefits that are contingent on the combined survival status of two lives. To value such benefits accurately, we require a statistical model for the impact of the survivorship of one life on another. In this paper we first set up two models, one Markov and one semi-Markov, to model the dependence between the lifetimes of a husband and wife. From the models we can measure the extent of three types of dependence: (1) the instantaneous dependence due to a catastrophic event that affect both lives, (2) the short-term impact of spousal death, and (3) the long-term association between lifetimes. Then we apply the models to a set of jointlife and last-survivor annuity data from a large Canadian insurance company. Given the fitted models, we study the impact of dependence on annuity values and examine the potential inaccuracy in pricing if we assume lifetimes are independent. Finally, we compare our Markovian models with two copula models considered in previous research on modeling joint-life mortality.

Longevity Risk and Capital Markets: The 2012–2013 Update

This Special Issue of the North American Actuarial Journal contains 15 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the articles were presented at Longevity Eight: The Eighth International Longevity Risk and Capital Markets Solutions Conference, which was held in Waterloo, Ontario, Canada, on September 7–8, 2012. It was hosted by the Department of Statistics and Actuarial Science at the University of Waterloo, Waterloo Research Institute in Insurance, Securities and Quantitative Finance (WatRISQ), and the Pensions Institute. It was sponsored by Prudential Financial, Inc., Sun Life Financial of Canada, Société Générale Corporate and Investment Banking, the Society of Actuaries (SOA), and the Canadian Institute of Actuaries (CIA). Longevity risk and related capital market solutions have grown increasingly important in recent years, both in academic research and in the markets we refer to as the new Life Market, that is, the capital market that trades longevity-linked assets and liabilities.1 Mortality improvements around the world are putting more and more pressure on governments, pension funds, and life insurance companies as well as individuals to deal with the longevity risk they face. At the same time, capital markets can, in principle, provide vehicles to hedge longevity risk effectively and transfer the risk from those unwilling or unable to handle it to those willing to invest in such risk in exchange for appropriate risk-adjusted returns or who have a counterpoising risk that longevity risk can hedge, such as life offices with mortality risk on their books. Many new investment products have been created both by the insurance/reinsurance industry and by the capital markets. Mortality catastrophe bonds are an example of a successful insurancelinked security. Some new innovative capital market solutions for transferring longevity risk include longevity (or survivor) bonds, longevity (or survivor) swaps, and mortality (or q-) forward contracts. The aim of the International Longevity Risk and Capital Markets Solutions Conferences is to bring together academics and practitioners from all over the world to discuss and analyze these exciting new developments. The conferences have followed closely the developments in the market. The first conference (Longevity One) was held at Cass Business School in London in February 2005. This conference was prompted by the announcement of the Swiss Re mortality catastrophe bond in December 2003 and the European Investment Bank/BNP Paribas/PartnerRe longevity bond in November 2004. The second conference was held in April 2006 in Chicago and hosted by the Katie School at Illinois State University.2 Since Longevity One, there have been further issues of mortality catastrophe bonds, as well as the release of the Credit Suisse Longevity Index. In the United Kingdom, new life companies backed by global investment banks and private equity firms were setting up for the express purpose of buying out the defined benefit pension liabilities of U.K. corporations. Goldman Sachs announced it was setting up such a buyout company itself (Rothesay Life) because the issue of pension liabilities was beginning to impede its mergers and acquisitions activities. It decided that the best way of dealing with pension liabilities was to remove them altogether from the balance sheets of takeover targets. So there was now firm evidence that a new global market in longevity risk transference had been established. However, as with many other economic activities, not all progress follows a smooth path. The EIB/BNP/PartnerRe longevity bond did not attract sufficient investor interest and was withdrawn in late 2005. A great deal, however, was learned from this failed issue about the conditions and requirements needed to launch a successful capital market instrument.

A Semi-Markov Multiple State Model for Reverse Mortgage Terminations

Abstract Reverse mortgages provide a mechanism for seniors to release the equity that has been built up in their home. At termination, the mortgagors are usually guaranteed to owe no more than the value of their property. The value of the reverse mortgage guarantee is heavily dependent on the maturity or termination date, which is uncertain. In this paper, we model reverse mortgage terminations using a semi-Markov multiple state model which incorporates three different modes of exit: death, entrance into a long-term care facility, and voluntary prepayment. We apply the proposed model specifically to develop the valuation formulas for roll-up mortgages in the UK and Home Equity Conversion Mortgages (HECMs) in the USA. We examine the significance of each mode of termination by valuing the contracts allowing progressively for each mode. On the basis of our model and assumptions, we find that both health related terminations and voluntary (non-health related) terminations significantly impact the contract value. In addition we analyze the premium structure for US reverse mortgage insurance, and demonstrate that premiums appear to be too high for some borrowers, and substantial cross-subsidies may result.