Enrico Biffis

Stochastic mortality under measure changes

We provide a self-contained analysis of a class of continuous-time stochastic mortality models that have gained popularity in the last few years. We describe some of their advantages and limitations, examining whether their features survive equivalent changes of measures. This is important when using the same model for both market-consistent valuation and risk management of life insurance liabilities. We provide a numerical example based on the calibration to the French annuity market of a risk-neutral version of the model proposed by Lee & Carter (1992).

Securitizing and tranching longevity exposures

We consider the problem of optimally designing longevity risk transfers under asymmetric information. We focus on holders of longevity exposures that have superior knowledge of the underlying demographic risks, but are willing to take them off their balance sheets because of capital requirements. In equilibrium, they transfer longevity risk to uninformed agents at a cost, where the cost is represented by retention of part of the exposure and/or by a risk premium. We use a signalling model to quantify the effects of asymmetric information and emphasize how they compound with parameter uncertainty. We show how the cost of private information can be minimized by suitably tranching securitized cashflows, or, equivalently, by securitizing the exposure in exchange for an option on mortality rates. We also investigate the benefits of pooling several longevity exposures and the impact on tranching levels.

Pricing life insurance contracts with early exercise features

In this paper we describe an algorithm based on the Least Squares Monte Carlo method to price life insurance contracts embedding American options. We focus on equity-linked contracts with surrender options and terminal guarantees on benefits payable upon death, survival and surrender. The framework allows for randomness in mortality as well as stochastic volatility and jumps in financial risk factors. We provide numerical experiments demonstrating the performance of the algorithm in the context of multiple risk factors and exercise dates.

Regression-Based Algorithms for Life Insurance Contracts with Surrender Guarantees

We present a general framework for pricing life insurance contracts embedding a surrender option. The model allows for several sources of risk, such as uncertainty in mortality, interest rates and other financial factors. We describe and compare two numerical schemes based on the Least Squares Monte Carlo method, emphasizing underlying modeling assumptions and computational issues.

Mortality-Linked Securities and Derivatives

In the last few years, the risk of mortality improvements has become increasingly capital intensive for pension funds and annuity providers to manage. The reason is that longevity risk has been systematically underestimated, making balance sheets vulnerable to unexpected increases in liabilities. The traditional way of transferring longevity risk is through insurance and reinsurance markets. However, these lack the capacity and liquidity to support an estimated global exposure in excess of

The Cost of Counterparty Risk and Collateralization in Longevity Swaps

20tr (e.g., Loeys et al., 2007). Capital markets, on the other hand, could play a very important role, offering additional capacity and liquidity to the market, leading in turn to more transparent and competitive pricing of longevity risk. Blake and Burrows (2001) were the first to advocate the use of mortality-linked securities to transfer longevity risk to the capital markets. Their proposal has generated considerable attention in the last few years, and major investment banks and reinsurers are now actively innovating in this space (see Blake et al., 2008, for an overview). Nevertheless, despite growing enthusiasm, longevity risk transfers have been materializing only slowly. One of the reasons is the huge imbalance in scale between existing exposures and willing hedge suppliers. Another reason is that a traded mortality-linked security has to meet the different needs of hedgers (concerned with hedge effectiveness) and investors (concerned with liquidity and with receiving adequate compensation for assuming the risk), needs that are difficult to reconcile when longevity risk, a long-term trend risk that is difficult to quantify, is involved. A third reason is the absence of an established market price for longevity risk. We provide an overview of the recent developments in capital markets aimed at overcoming such difficulties and at creating a liquid market in mortality-linked securities and derivatives.

Lee-Carter Goes Risk-Neutral

Derivative longevity risk solutions, such as bespoke and indexed longevity swaps, allow pension schemes and annuity providers to swap out longevity risk, but introduce counterparty credit risk, which can be mitigated if not fully eliminated by collateralization. We examine the impact of bilateral default risk and collateral rules on the marking to market of longevity swaps, and show how longevity swap rates must be determined endogenously from the collateral flows associated with the marking-to-market procedure. For typical interest rate and mortality parameters, we find that the impact of collateralization is modest in the presence of symmetric default risk, but more pronounced when default risk and/or collateral rules are asymmetric. Our results suggest that the overall cost of collateralization is comparable with, and often much smaller than, that found in the interest-rate swaps market, which may then provide the appropriate reference framework for the credit enhancement of both indemnity-based and indexed longevity risk solutions.

Informed Intermediation of Longevity Exposures

We consider a class of stochastic intensities of mortality that generalizes the model proposed by Lee and Carter (1992), allowing general diffusions to drive the mortality time-trend. We analyze the stability of such class of intensities under measure changes and show how a risk-neutral version of the generalized Lee-Carter model can be employed for fair valuation purposes. We provide an example of model calibration based on the Italian annuity market.

Optimal Insurance with Counterparty Default Risk

We examine pension buyout transactions and longevity risk securitization in a common framework, emphasizing the role played by asymmetries in capital requirements and mortality forecasting technology. The results are used to develop a coherent model of intermediation of longevity exposures, between defined benefit pension plans and capital market investors, through insurers operating in the pension buyout market. We derive several predictions consistent with the recent empirical evidence on pension buyouts and offer insights on the role of buyout firms and regulation in the emerging market for longevity-linked securities. A multiperiod version of the model is used to explore the effects of longevity risk securitization on the capacity of the pension buyout market.

Satellite data and machine learning for weather risk management and food security

We study the design of optimal insurance contracts when the insurer can default on its obligations. In our model default arises endogenously from the interaction of the insurance premium, the indemnity schedule and the insurers assets. This allows us to understand the joint effect of insolvency risk and background risk on optimal contracts. The results may shed light on the aggregate risk retention schedules observed in catastrophe reinsurance markets, and can assist in the design of (re)insurance programs and guarantee funds.