Securitization of Longevity Risk: Pricing Survivor Bonds With Wang Transform in the Lee-Carter Framework

Longevity risk is a major issue for insurers and pension funds, especially in the selling of annuity products. In that respect, securitization of this risk could offer great opportunities for hedging. This article proposes to design survivor bonds which could be issued directly by insurers. In order to guaranty some transparency in the product, the survivor bond is based on a public mortality index. The classical Lee‐Carter model for mortality forecasting is used to price a risky coupon survivor bond based on this index.     

A General Semi-Markov Model for Coupled Lifetimes

Joint-life annuities with a high last survivor benefit play an important role in the optimal annuity portfolio for a retired couple. The dependence between coupled lifetimes is crucial for valuing joint-life annuities. Existing bivariate modeling of coupled lifetimes is based on outdated data with limited observation periods and does not take into account mortality improvement. In this article, we propose a transparent and dynamic framework for modeling coupled lifetime dependence caused by both marital status and common mortality improvement factors. Dependence due to marital status is captured by a semi-Markov joint life model. Dependence due to common mortality improvement, which represents the correlation between mortality improvement patterns of coupled lives, is incorporated by a two-population mortality improvement model. The proposed model is applied to pricing the longevity risk in last survivor annuities sold in the United States and the United Kingdom.     

On systematic mortality risk and risk-minimization with survivor swaps

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time, this new class of financial contract is interesting from an investor's point of view, since it increases the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically, this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk, we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.     

Yes We Can (Price Derivatives on Survivor Indices)

We propose a simulation approach to value derivatives when the underlying dynamics are estimated using the survivor indices directly. Our results show that survivor forward and swap premiums increase with maturity and with the market price of risk. Our results also confirm that taking the optionality into consideration is important from a pricing perspective, for both U.S. women and men. We compare our results to what is obtained using an alternative modeling approach in which a Lee–Carter model is used to indirectly model the survivor index. Compared to this method, our estimated premiums and prices are higher for all longevity products. Moreover, comparing American‐style with European‐style options we find that, although the early exercise option has value when using survivor indices directly, the relative value of the early exercise option is significantly less than when the Lee–Carter model is used to indirectly model the survivor index. It follows that the assumed mortality dynamics have important implications for the term structure of forward and swap premiums and for the effect that changes in the market price of risk has on them.     

Survivor Bonds: A Comment on Blake and Burrows

This article offers a critical assessment of the “survivor bonds” (SBs) proposal recently put forward by Blake and Burrows, which calls for the government to issue bonds whose coupon payments are contingent on the proportions of retirees surviving to particular ages. It suggests that the proposal has considerable merit and discusses the circumstances in which SBs would be useful risk management tools for insurance companies. It also discusses alternatives such as reinsurance, hedging with life contracts, dynamic hedging, and other forms of survivor derivative. Finally, it evaluates and rejects the argument that SBs should be issued by the state.     

Survivor Derivatives: A Consistent Pricing Framework

Survivorship risk is a significant factor in the provision of retirement income. Survivor derivatives are in their early stages and offer potentially significant welfare benefits to society. This article applies the approach developed by Dowd et al. (2006) , Olivier and Jeffery (2004) , Smith (2005) , and Cairns (2007) to derive a consistent framework for pricing a wide range of linear survivor derivatives, such as forwards, basis swaps, forward swaps, and futures. It then shows how a recent option pricing model set out by Dawson et al. (2009) can be used to price nonlinear survivor derivatives, such as survivor swaptions, caps, floors, and combined option products. It concludes by considering applications of these products to a pension fund that wishes to hedge its survivorship risks.     

Securitization of Longevity Risk in Reverse Mortgages

Abstract

The reverse mortgage market has been expanding rapidly in developed economies in recent years. The onset of demographic transition places a rapidly rising number of households in an age window in which reverse mortgages have potential appeal. Increasing prices for residential real estate over the last decade have further stimulated interest.

Reverse mortgages involve various risks from the provider-s perspective that may hinder the further development of these financial products. This paper addresses one method of transferring and financing the risks associated with these products through the form of securitization. Securitization is becoming a popular and attractive alternative form of risk transfer of insurance liabilities. Here we demonstrate how to construct a securitization structure for reverse mortgages similar to the one applied in traditional insurance products.

Specifically, we investigate the merits of developing survivor bonds and survivor swaps for reverse mortgage products. In the case of survivor bonds, for example, we are able to compute premiums, both analytically and numerically through simulations, and to examine how the longevity risk may be transferred to the financial investors. Our numerical calculations provide an indication of the economic benefits derived from developing survivor bonds to securitize the “longevity risk component” of reverse mortgage products. Moreover, some sensitivity analysis of these economic benefits indicates that these survivor bonds provide for a promising tool for investment diversification.     

Competition in British retail banking

This article employs British cross-section time-series data to examine the competitive behavior of retail banks in the period 1985–1989. Product offerings are found to consist of both survivor and dominated products over time. In addition, there is evidence of price discrimination being practiced by banks. Individual banks are found to be influential in the setting of interest rates for retail bank products, though there is wide variation across banks and products.     

A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

In this article, we consider the evolution of the post‐age‐60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so‐called longevity risk. We introduce a two‐factor stochastic model for the development of this curve through time. The first factor affects mortality‐rate dynamics at all ages in the same way, whereas the second factor affects mortality‐rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly. A key component of the article is the proposal and development of a method for calculating the market risk‐adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.     

Longevity/Mortality Risk Modeling and Securities Pricing

Securitizing longevity/mortality risk can transfer longevity/mortality risk to capital markets. Modeling and forecasting mortality rate is key to pricing mortality‐linked securities. Catastrophic mortality and longevity jumps occur in historical data and have an important impact on security pricing. This article introduces a stochastic diffusion model with a double‐exponential jump diffusion process that captures both asymmetric rate jumps up and down and also cohort effect in mortality trends. The model exhibits calibration advantages and mathematical tractability while better fitting the data. The model provides a closed‐form pricing solution for J.P. Morgan’s q‐forward contract usable as a building block for hedging.     

Life expectancy--a commentary on this life table variable

In 1992, I wrote an article on a method of modifying the Decennial US Life Table to accommodate any pattern of excess mortality expressed in terms of excess death rate (EDR), for the specific purpose of calculating the reduced life expectancy, e. I believe this was the first article published in the Journal of Insurance Medicine (JIM) that dealt specifically with life expectancy as an index of survival and risk appraisal, never used in the classification of extra mortality risk in applicants for life insurance. In this commentary, I discuss the 1989-91 US Decennial Life Table in detail. I link the subject matter of the 1992 article with several more recent articles that also focus on the utility of life expectancy in underwriting structured settlement annuities and preparing reports on life expectancy for an attorney in a tort case. A few references are given for further reading on life table methodology and its use in the most accurate estimate of life expectancy, given the inherent limitations of the life table and the limited duration of follow-up studies.     

Mutual Fund Survivorship

This article provides a comprehensive study of survivorshipissues using the mutual fund data of Carhart (1997). We demonstratetheoretically that when survival depends on multiperiod performance,the survivorship bias in average performance typically increaseswith the sample length. This is empirically relevant becauseevidence suggests a multiyear survival rule for U.S. mutualfunds. In the data we find the annual bias increases from 0.07%for 1-year samples to 1% for samples longer than 15 years. Wefind that survivor conditioning weakens evidence of performancepersistence. Finally, we explain how survivor conditioning affectsthe relation between performance and fund characteristics.     

The survivorship bias, share price effect, and small firm effect in Canadian markets

After controlling for survivorship bias, we examine the relation between average returns, firm size, and price levels for Canadian stocks during the 1975-1994 period. Our findings indicate that there is a significant inverse share price level effect in Canadian markets. When we compare the results of the overall sample with the groups of surviving firms and delisted stocks, the latter group shows strong performance for large-size, high-priced stocks. Evidence that supports an independent size effect is less clear for Canadian stocks. A small size effect exists only among the higher share price denominations, which suggests a confounded size-price effect. Although the delisted group returns are statistically different from those of the survivor and the overall groups, which implies some evidence of survivorship bias, the difference between the survivor group and the overall group is weak at best.     

Corporate strategy financialized: Conjuncture,arbitrage and earnings capacity in the S&P500

The conjuncture that ushered in the era of shareholder value served to embed capital market expectations into corporate governance aligning management and shareholder interests. Market arbitrage focussed on modifying contractual relations with stakeholders to extract a (higher) return on invested capital. In this article we focus on cash earnings on capital employed generated by the S&P 500 survivor group of firms covering the period 1990–2008. We use this financial data to construct three complementary perspectives on corporate financial performance: firm, firm-relative and macro. Within this framework the financial numbers and perspectives are analogous to a ‘hall of mirrors’ where ambiguity and contradiction are in play frustrating straightforward performative narratives that connect purpose with financial transformation an era of shareholder value.     

Unisex-Tafeln in der betrieblichen Altersversorgung

Due to the so-called “Test-Achats”-decision of the European Court the insurance business is stipulated to use unisex-calculations from December 21th, 2012 onwards. According to prevailing view this also impacts occupational pension schemes. The present article deals with an exact as possible implementation of the unisex-approach in the mortality table RT05G of Klaus Heubeck as this mortality table is relevant for the pension schemes in Germany. For pension schemes—in contrast to the private insurance industry—one has to deal with the difficulties that on the one hand the different kinds of benefit like old age pension, disability benefits and widow’s pension are evaluated at the same time, and on the other hand the mortality and disability rates contain no (or little) safety margins. Therefore one has to establish an exact as possible approach for all biometric probabilities simultaneously. Apart from the natural demixing due to mortality a closer look has to be taken at the probabilities concerning the surviving dependants—probabilty of being married at death, age of the surviving dependants and mortality of surving dependants.     

Industry Structure and Ripple Effects of Bankruptcy Announcements

The market structure of an industry plays an important role in determining the stock market performance of surviving firms during intra-industry bankruptcy announcements. On evaluating the announcement effects of a survivor sample from each of two industries with very different market structures, namely the airline industry and the railroad industry, we find that the airline sample received significant abnormal returns (positive ripple) while the railroad sample experienced significant abnormal losses (negative ripple). Furthermore, the differences of the abnormal returns from the two samples also are statistically significant. These findings demonstrate support for the market structure hypothesis (MSH), but cast doubt on the contagion effect hypothesis (CEH).     

Modeling Mortality With Jumps: Applications to Mortality Securitization

In this article, we incorporate a jump process into the original Lee–Carter model, and use it to forecast mortality rates and analyze mortality securitization. We explore alternative models with transitory versus permanent jump effects and find that modeling mortality via transitory jump effects may be more appropriate in mortality securitization. We use the Swiss Re mortality bond in 2003 as an example to show how to apply our model together with the distortion measure approach to value mortality-linked securities. Pricing the Swiss Re mortality bond is challenging because the mortality index is correlated across countries and over time. Cox, Lin, and Wang (2006) employ the normalized multivariate exponential tilting to take into account correlations across countries, but the problem of correlation over time remains unsolved. We show in this article how to account for the correlations of the mortality index over time by simulating the mortality index and changing the measure on paths.     

A Bühlmann Credibility Approach to Modeling Mortality Rates

In this article, we first propose a Bühlmann nonparametric credibility approach to forecasting mortality rates, and we then compare forecasting performances between the proposed Bühlmann approach and the Lee-Carter/Cairns-Blake-Dowd (CBD) models. Empirical results based on mortality data for both genders of Japan, the United Kingdom, and the United States with two age spans, a wide range of fitting year spans, and three forecasting periods show that the credibility approach contributes to much better forecasting performances measured by the mean absolute percentage error. Moreover, we give an informative credibility interpretation regarding the average decrements of an individual time trend for age x and a group time trend for all ages, and we discuss the effects of the slope and intercept of the linear functions for the forecasted mortality rates under the proposed Bühlmann nonparametric credibility approach and the Lee-Carter/CBD models. Finally, we also estimate the parameters of the Bühlmann credibility approach in a semiparametric framework, and we provide a stochastic version of forecasting mortality rates for the Bühlmann credibility approach.     

The political economy of derived pension rights

Derived pension rights (including survivor benefits and spousal compensations for one-earner couples) exist in most Social Security systems but with variable generosity. They are mainly viewed as a means to alleviate poverty among older women living alone. The purpose of this paper is to explain how they can emerge from a political economy process when Social Security is a combination of Bismarckian and Beveridgean pillars. We find that the pension system should be contributive but with a positive level of derived rights. We also show that such a system encourages stay-at-home wives, thus revealing an unpleasant trade-off between female labor participation and poverty alleviation.     

The CAPM is Wanted,Dead or Alive

Kothari, Shanken, and Sloan (1995) claim that βs from annual returns produce a stronger positive relation between β and average return than βs from monthly returns. They also contend that the relation between average return and book-to-market equity (BE/ME) is seriously exaggerated by survivor bias. We argue that survivor bias does not explain the relation between BE/ME and average return. We also show that annual and monthly βs produce the same inferences about the β premium. Our main point on the β premium is, however, more basic. It cannot save the Capital asset pricing model (CAPM), given the evidence that β alone cannot explain expected return.