Deterministic shock vs. stochastic value-at-risk — an analysis of the Solvency II standard model approach to longevity risk

In general, the capital requirement under Solvency II is determined as the 99.5% Value-at-Risk of the Available Capital. In the standard model’s longevity risk module, this Value-at-Risk is approximated by the change in Net Asset Value due to a pre-specified longevity shock which assumes a 25% reduction of mortality rates for all ages. We analyze the adequacy of this shock by comparing the resulting capital requirement to the Value-at-Risk based on a stochastic mortality model. This comparison reveals structural shortcomings of the 25% shock and therefore, we propose a modified longevity shock for the Solvency II standard model. We also discuss the properties of different Risk Margin approximations and find that they can yield significantly different values. Moreover, we explain how the Risk Margin may relate to market prices for longevity risk and, based on this relation, we comment on the calibration of the cost of capital rate and make inferences on prices for longevity derivatives.     

Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

随机需求下具有不对称信息的闭环供应链的最优差别定价模型

在随机需求条件下建立了由一个制造商和一个零售商所组成的闭环供应链差别定价模型。在制造商成本信息对称和信息不对称情况下,分别讨论了闭环供应链的最优差别定价问题,得到了在不同信息条件下的最优定价组合以及最优的订购量.最后通过数值算例分析了成本信息不对称对两种产品的零售价以及订购量的影响.     

Parametric mortality indexes: From index construction to hedging strategies

In this paper, we investigate the construction of mortality indexes using the time-varying parameters in common stochastic mortality models. We first study how existing models can be adapted to satisfy the new-data-invariant property, a property that is required to ensure the resulting mortality indexes are tractable by market participants. Among the collection of adapted models, we find that the adapted Model M7 (the Cairns–Blake–Dowd model with cohort and quadratic age effects) is the most suitable model for constructing mortality indexes. One basis of this conclusion is that the adapted model M7 gives the best fitting and forecasting performance when applied to data over the age range of 40–90 for various populations. Another basis is that the three time-varying parameters in it are highly interpretable and rich in information content. Based on the three indexes created from this model, one can write a standardized mortality derivative called K-forward, which can be used to hedge longevity risk exposures. Another contribution of this paper is a method called key K-duration that permits one to calibrate a longevity hedge formed by K-forward contracts. Our numerical illustrations indicate that a K-forward hedge has a potential to outperform a q-forward hedge in terms of the number of hedging instruments required.     

Multi-population mortality models: A factor copula approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short-term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA–GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.     

Longevity risk and retirement income tax efficiency: A location spending rate puzzle

In this paper we model and solve a retirement consumption problem with differentially taxed accounts, parameterized by longevity risk aversion. The work is motivated by some observations on how Canadians de-accumulate financial wealth during retirement — which seem rather puzzling. While the Modigliani lifecycle model can justify a variety of (pre-tax) de-accumulation or draw down rates depending on risk preferences, the existence of asymmetric taxes implies that certain financial accounts should be depleted faster than others. Our analysis of data from the Survey of Financial Security indicates that Canadian retirees maintain approximately two-thirds of their financial wealth in tax-sheltered accounts and a third in taxable accounts regardless of age. The ratio of taxable to tax-sheltered wealth increases slightly or remains relatively constant depending on household income which is not what one would expect from the lifecycle model. Indeed, using our model we cannot locate a plausible tax function that justifies a constant “account ratio” regardless of age. For example under flat rates taxable accounts should be depleted well before tax-sheltered accounts are ever touched. The account ratio should go to zero quite rapidly in the absence of government mandated withdrawals. We also demonstrate that under progressive income taxes withdrawals are made from both accounts but at different rates depending on account size, pension income and longevity risk preferences. Again, the “account ratio” should eventually decline. We postulate that this sort of behavior is likely due to irrational considerations linked to mental accounting, etc. It remains to be seen whether this will persist over time and under a more careful analysis of Canadian cohorts or if retirees in other countries exhibit the same behavior.     

A step-by-step guide to building two-population stochastic mortality models

Two-population stochastic mortality models play a crucial role in the securitization of longevity risk. In particular, they allow us to quantify the population basis risk when longevity hedges are built from broad-based mortality indexes. In this paper, we propose and illustrate a systematic process for constructing a two-population mortality model for a pair of populations. The process encompasses four steps, namely (1) determining the conditions for biological reasonableness, (2) identifying an appropriate base model specification, (3) choosing a suitable time-series process and correlation structure for projecting period and/or cohort effects into the future, and (4) model evaluation.For each of the seven single-population models from Cairns et al. (2009), we propose two-population generalizations. We derive criteria required to avoid long-term divergence problems and the likelihood functions for estimating the models. We also explain how the parameter estimates are found, and how the models are systematically simplified to optimize the fit based on the Bayes Information Criterion. Throughout the paper, the results and methodology are illustrated using real data from two pairs of populations.     

Solvency II reporting: How to interpret funds’ aggregate solvency capital requirement figures

Depending on the current risk exposure of an insurance company, the impact of buying an additional unit of a fund on an insurer’s overall Solvency II capital charges, i.e., the Solvency Capital Requirement (SCR), will differ. We call this impact the fund’s SCR contribution and show in which boundaries it lies if only the fund’s aggregate sub-SCR figures are known but not the risk exposures of the insurance company buying the fund. The upper bound of this range, the worst-case SCR contribution, can be used as a conservative measure to assess funds’ Solvency II risk contributions or to assign them to different Solvency II risk categories. We believe that providing funds’ worst-case SCR contributions can be useful information to insurance companies when screening from a broad investment universe.     

Contracting with an urgent supplier under cost information asymmetry

We investigate a contract setting problem faced by a manufacturer who can procure major modules from an overseas supplier, as well as a local supplier. The overseas supplier is prime and offers quality products, whereas the local supplier is viewed only as a backup, and its products are inferior in quality. As the local supplier needs to put in additional effort to fulfill the urgent orders, it is difficult for the manufacturer to estimate this urgent supplier’s production cost. This asymmetric cost information becomes an obstacle for the manufacturer in managing the urgent supplier. In this paper, we study two types of contingent contracts. One is the common price-only contract, and the other is a contract menu consisting of a transfer payment and a lead time quotation. We construct a Stackelberg game model and evaluate how the involvement of an urgent supplier with private cost information affects performances of the prime supplier and the manufacturer in different scenarios (with or without the urgent supplier, under different contingent contracts). We also conduct numerical experiments to show how the parameters of the contracts affect profits of the manufacturer.     

Optimal transfer pricing schemes for work averse division managers with private information

This paper models a decentralized firm under information asymmetry and effort disutility on the part of managers. We assume that managers choose efforts before observing some private information. However, after the effort choice managers receive private information on their cost parameters which they report to the headquarters. There exist many situations in which managers need to take efforts before obtaining private information; for example, the regular maintenance effort on the machine, the effort on R&D for reducing costs and the effort taken to build relationships with the supplier. Two models are considered in this paper based on the timing of acquisition of private information by the managers. We derive optimal coordination mechanisms to facilitate internal transactions for the models. The equilibrium outcome of this paper suggests that: 1) regardless of the timing of managers' information acquisition, the optimal output level under asymmetric information can have overproduction or underproduction when compared with the full information optimal output; 2) under certain demand conditions managers cannot receive any information rent benefit for their private information even if they have the option to renege on the contract after obtaining their private information.