Search Results

This work proposes a backtesting analysis that compares the Lee–Carter and the Cairns–Blake–Dowd mortality models, employing Italian data. The mortality data come from the Italian National Statistics Institute (ISTAT) database and span the period 1975–2014, over which we computed back-projections evaluating the performances of the models compared with real data. We propose three different backtest approaches, evaluating the goodness of short-run forecast versus medium-length ones. We find that neither model was able to capture the improving shock on mortality observed for the male population on the analysed period. Moreover, the results suggest that CBD forecasts are reliable prevalently for ages above 75, and that LC forecasts are basically more accurate for this data. Full article

Much of the debate around a potential British exit (Brexit) from the European Union has centred on the potential macroeconomic impact. In this paper, we instead focus on understanding market expectations for price action around the Brexit referendum date. Extracting implied distributions from the GBPUSD option volatility surface, we originally estimated, based on our visual observation of implied probability densities available up to 13 June 2016, that the market expected that a vote to leave could result in a move in the GBPUSD exchange rate from 1.4390 (spot reference on 10 June 2016) down to a range in 1.10 to 1.30, i.e., a 10–25% decline—very probably with highly volatile price action. To quantify this more objectively, we construct a mixture model corresponding to two scenarios for the GBPUSD exchange rate after the referendum vote, one scenario for “remain” and one for “leave”. Calibrating this model to four months of market data, from 24 February to 22 June 2016, we find that a “leave” vote was associated with a predicted devaluation of the British pound to approximately 1.37 USD per GBP, a 4.5% devaluation, and quite consistent with the observed post-referendum exchange rate move down from 1.4877 to 1.3622. We contrast the behaviour of the GBPUSD option market in the run-up to the Brexit vote with that during the 2014 Scottish Independence referendum, finding the potential impact of Brexit to be considerably higher. Full article

Pareto processes are suitable to model stationary heavy-tailed data. Here, we consider the auto-regressive Gaver–Lewis Pareto Process and address a study of the tail behavior. We characterize its local and long-range dependence. We will see that consecutive observations are asymptotically tail independent, a feature that is often misevaluated by the most common extremal models and with strong relevance to the tail inference. This also reveals clustering at “penultimate” levels. Linear correlation may not exist in a heavy-tailed context and an alternative diagnostic tool will be presented. The derived properties relate to the auto-regressive parameter of the process and will provide estimators. A comparison of the proposals is conducted through simulation and an application to a real dataset illustrates the procedure. Full article

This paper examines the effect of gainsharing provisions on the selection of a discount rate for a defined benefit pension plan. The paper uses a traditional actuarial approach of discounting liabilities using the expected return of the associated pension fund. A stochastic Excel model was developed to simulate the effect of varying investment returns on a pension fund with four asset classes. Lognormal distributions were fitted to historical returns of two of the asset classes; large company stocks and long-term government bonds. A third lognormal distribution was designed to represent the investment returns of alternative investments, such as real estate and private equity. The fourth asset class represented short term cash investments and that return was held constant. The following variables were analyzed to determine their relative impact of gainsharing on the selection of a discount rate: hurdle rate, percentage of gainsharing, actuarial asset method smoothing period, and variations in asset allocation. A 50% gainsharing feature can reduce the discount rate for a defined benefit pension plan from 0.5% to more than 2.5%, depending on the gainsharing design and asset allocation. Full article

The literature on capital allocation is biased towards an asset modeling framework rather than an actuarial framework. The asset modeling framework leads to the proliferation of inappropriate assumptions about the effect of insurance line of business growth on aggregate loss distributions. This paper explains why an actuarial analog of the asset volume/return model should be based on a Lévy process. It discusses the impact of different loss models on marginal capital allocations. It shows that Lévy process-based models provide a better fit to the US statutory accounting data, and identifies how parameter risk scales with volume and increases with time. Finally, it shows the data suggest a surprising result regarding the form of insurance parameter risk. Full article

This paper gives a detailed overview of the current state of research in relation to the use of state space models and the Kalman-filter in the field of stochastic claims reserving. Most of these state space representations are matrix-based, which complicates their applications. Therefore, to facilitate the implementation of state space models in practice, we present a scalar state space model for cumulative payments, which is an extension of the well-known chain ladder (CL) method. The presented model is distribution-free, forms a basis for determining the entire unobservable lower and upper run-off triangles and can easily be applied in practice using the Kalman-filter for prediction, filtering and smoothing of cumulative payments. In addition, the model provides an easy way to find outliers in the data and to determine outlier effects. Finally, an empirical comparison of the scalar state space model, promising prior state space models and some popular stochastic claims reserving methods is performed. Full article

Longevity risk constitutes an important risk factor for life insurance companies, and it can be managed through longevity-linked securities. The market of longevity-linked securities is at present far from being complete and does not allow finding a unique pricing measure. We propose a method to estimate the maximum market price of longevity risk depending on the risk margin implicit within the calculation of the technical provisions as defined by Solvency II. The maximum price of longevity risk is determined for a survivor forward (S-forward), an agreement between two counterparties to exchange at maturity a fixed survival-dependent payment for a payment depending on the realized survival of a given cohort of individuals. The maximum prices determined for the S-forwards can be used to price other longevity-linked securities, such as q-forwards. The Cairns–Blake–Dowd model is used to represent the evolution of mortality over time that combined with the information on the risk margin, enables us to calculate upper limits for the risk-adjusted survival probabilities, the market price of longevity risk and the S-forward prices. Numerical results can be extended for the pricing of other longevity-linked securities. Full article

Motivated by the EU Solvency II Directive, we study the one-year ruin probability of an insurer who makes investments and hence faces both insurance and financial risks. Over a time horizon of one year, the insurance risk is quantified as a nonnegative random variable X equal to the aggregate amount of claims, and the financial risk as a d-dimensional random vector $\mathbf{Y}$ consisting of stochastic discount factors of the d financial assets invested. To capture both heavy tails and asymptotic dependence of $\mathbf{Y}$ in an integrated manner, we assume that $\mathbf{Y}$ follows a standard multivariate regular variation (MRV) structure. As main results, we derive exact asymptotic estimates for the one-year ruin probability for the following cases: (i) X and $\mathbf{Y}$ are independent with X of Fréchet type; (ii) X and $\mathbf{Y}$ are independent with X of Gumbel type; (iii) X and $\mathbf{Y}$ jointly possess a standard MRV structure; (iv) X and $\mathbf{Y}$ jointly possess a nonstandard MRV structure. Full article

We prove that the Omega measure, which considers all moments when assessing portfolio performance, is equivalent to the widely used Sharpe ratio under jointly elliptic distributions of returns. Portfolio optimization of the Sharpe ratio is then explored, with an active-set algorithm presented for markets prohibiting short sales. When asymmetric returns are considered, we show that the Omega measure and Sharpe ratio lead to different optimal portfolios. Full article

Building on recent work incorporating recovery risk into structural models by Cohen & Costanzino (2015), we consider the Black-Cox model with an added recovery risk driver. The recovery risk driver arises naturally in the context of imperfect information implicit in the structural framework. This leads to a two-factor structural model we call the Stochastic Recovery Black-Cox model, whereby the asset risk driver A_t defines the default trigger and the recovery risk driver R_t defines the amount recovered in the event of default. We then price zero-coupon bonds and credit default swaps under the Stochastic Recovery Black-Cox model. Finally, we compare our results with the classic Black-Cox model, give explicit expressions for the recovery risk premium in the Stochastic Recovery Black-Cox model, and detail how the introduction of separate but correlated risk drivers leads to a decoupling of the default and recovery risk premiums in the credit spread. We conclude this work by computing the effect of adding coupons that are paid continuously until default, and price perpetual (consol bonds) in our two-factor firm value model, extending calculations in the seminal paper by Leland (1994). Full article

Building a social security system to ensure Singapore residents have peace of mind in funding for retirement has been at the top of Singapore government’s policy agenda over the last decade. Implementation of the Lifelong Income For the Elderly (LIFE) scheme in 2009 clearly shows that the government spares no effort in improving its pension scheme to boost its residents’ income after retirement. Despite the recent modifications to the LIFE scheme, Singapore residents must still choose between two plans: the Standard and Basic plans. To enhance the flexibility of the LIFE scheme with further streamlining of its fund management, we propose some plan modifications such that scheme members do not face a dichotomy of plan choices. Instead, they select two age parameters: the Payout Age and the Life-annuity Age. This paper discusses the actuarial analysis for determining members’ payouts and bequests based on the proposed age parameters. We analyze the net cash receipts and Internal Rate of Return (IRR) for various plan-parameter configurations. This information helps members make their plan choices. To address cost-of-living increases we propose to extend the plan to accommodate an annual step-up of monthly payouts. By deferring the Payout Age from 65 to 68, members can enjoy an annual increase of about 2% of the payouts for the same first-year monthly benefits. Full article

We apply spectral biclustering to mortality datasets in order to capture three relevant aspects: the period, the age and the cohort effects, as their knowledge is a key factor in understanding actuarial liabilities of private life insurance companies, pension funds as well as national pension systems. While standard techniques generally fail to capture the cohort effect, on the contrary, biclustering methods seem particularly suitable for this aim. We run an exploratory analysis on the mortality data of Italy, with ages representing genes, and years as conditions: by comparison between conventional hierarchical clustering and spectral biclustering, we observe that the latter offers more meaningful results. Full article

We introduce an additive stochastic mortality model which allows joint modelling and forecasting of underlying death causes. Parameter families for mortality trends can be chosen freely. As model settings become high dimensional, Markov chain Monte Carlo (MCMC) is used for parameter estimation. We then link our proposed model to an extended version of the credit risk model CreditRisk+. This allows exact risk aggregation via an efficient numerically stable Panjer recursion algorithm and provides numerous applications in credit, life insurance and annuity portfolios to derive P&L distributions. Furthermore, the model allows exact (without Monte Carlo simulation error) calculation of risk measures and their sensitivities with respect to model parameters for P&L distributions such as value-at-risk and expected shortfall. Numerous examples, including an application to partial internal models under Solvency II, using Austrian and Australian data are shown. Full article

Abstract: This paper analyses the interdependence between Islamic and conventional equities by taking into consideration the asymmetric effect of return and volatility transmission. We empirically investigate the decoupling hypothesis of Islamic and conventional equities and the potential contagion effect. We analyse the intra-market and inter-market spillover among Islamic and conventional equities across three major markets: the USA, the United Kingdom and Japan. Our sample period ranges from 1996 to 2015. In addition, we segregate our sample period into three sub-periods covering prior to the 2007 financial crisis, the crisis period and the post-crisis period. We find weak support for the decoupling hypothesis during the post-crisis period. Full article

This study considers the forecasting of mortality rates in multiple populations. We propose a model that combines mortality forecasting and functional data analysis (FDA). Under the FDA framework, the mortality curve of each year is assumed to be a smooth function of age. As with most of the functional time series forecasting models, we rely on functional principal component analysis (FPCA) for dimension reduction and further choose a vector error correction model (VECM) to jointly forecast mortality rates in multiple populations. This model incorporates the merits of existing models in that it excludes some of the inherent randomness with the nonparametric smoothing from FDA, and also utilizes the correlation structures between the populations with the use of VECM in mortality models. A nonparametric bootstrap method is also introduced to construct interval forecasts. The usefulness of this model is demonstrated through a series of simulation studies and applications to the age-and sex-specific mortality rates in Switzerland and the Czech Republic. The point forecast errors of several forecasting methods are compared and interval scores are used to evaluate and compare the interval forecasts. Our model provides improved forecast accuracy in most cases. Full article

We consider the financial planning problem of a retiree wishing to enter a retirement village at a future uncertain date. The date of entry is determined by the retiree’s utility and bequest maximisation problem within the context of uncertain future health states. In addition, the retiree must choose optimal consumption, investment, bequest and purchase of insurance products prior to their full annuitisation on entry to the retirement village. A hyperbolic absolute risk-aversion (HARA) utility function is used to allow necessary consumption for basic living and medical costs. The retirement village will typically require an initial deposit upon entry. This threshold wealth requirement leads to exercising the replication of an American put option at the uncertain stopping time. From our numerical results, active insurance and annuity markets are shown to be a critical aspect in retirement planning. Full article

Designing post retirement benefits requires access to appropriate investment instruments to manage the interest rate and longevity risks. Post retirement benefits are increasingly taken as a form of income benefit, either as a pension or an annuity. Pension funds and life insurers offer annuities generating long term liabilities linked to longevity. Risk management of life annuity portfolios for interest rate risks is well developed but the incorporation of longevity risk has received limited attention. We develop an immunization approach and a delta-gamma based hedging approach to manage the risks of adverse portfolio surplus using stochastic models for mortality and interest rates. We compare and assess the immunization and hedge effectiveness of fixed-income coupon bonds, annuity bonds, as well as longevity bonds, using simulations of the portfolio surplus for an annuity portfolio and a range of risk measures including value-at-risk. We show how fixed-income annuity bonds can more effectively match cash flows and provide additional hedge effectiveness over coupon bonds. Longevity bonds, including deferred longevity bonds, reduce risk significantly compared to coupon and annuity bonds, reflecting the long duration of the typical life annuity and the exposure to longevity risk. Longevity bonds are shown to be effective in immunizing surplus over short and long horizons. Delta gamma hedging is generally only effective over short horizons. The results of the paper have implications for how providers of post retirement income benefit streams can manage risks in demanding conditions where innovation in investment markets can support new products and increase the product range. Full article

We explore the extent to which claim incidence in Disability Income Insurance (DII) is affected by changes in the unemployment rate in Australia. Using data from 1986 to 2001, we fit a hurdle model to explore the presence and magnitude of the effect of changes in unemployment rate on the incidence of DII claims, controlling for policy holder characteristics and seasonality. We find a clear positive association between unemployment and claim incidence, and we explore this further by gender, age, deferment period, and occupation. A multinomial logistic regression model is fitted to cause of claim data in order to explore the relationship further, and it is shown that the proportion of claims due to accident increases markedly with rising unemployment. The results suggest that during periods of rising unemployment, insurers may face increased claims from policy holders with shorter deferment periods for white-collar workers and for medium and heavy manual workers. Our findings indicate that moral hazard may have a material impact on DII claim incidence and insurer business in periods of declining economic conditions. Full article

Previous research has shown that risk preferences are sensitive to the financial domain in which they are framed. In the present paper, we explore whether the effect of negative priming on risk taking is moderated by financial context. A total of 120 participants completed questionnaires, where risky choices were framed in six different financial scenarios. Half of the participants were allocated to a negative priming condition. Negative priming reduced risk-seeking behaviour compared to a neutral condition. However, this effect was confined to non-experiential scenarios (i.e., gamble to win, possibility to lose), and not to ‘real world’ financial products (e.g., pension provision). The results call into question the generalisability of priming effects on different financial contexts. Full article

Coherent models were developed recently to forecast the mortality of two or more sub-populations simultaneously and to ensure long-term non-divergent mortality forecasts of sub-populations. This paper evaluates the forecast accuracy of two recently-published coherent mortality models, the Poisson common factor and the product-ratio functional models. These models are compared to each other and the corresponding independent models, as well as the original Lee–Carter model. All models are applied to age-gender-specific mortality data for Australia and Malaysia and age-gender-ethnicity-specific data for Malaysia. The out-of-sample forecast error of log death rates, male-to-female death rate ratios and life expectancy at birth from each model are compared and examined across groups. The results show that, in terms of overall accuracy, the forecasts of both coherent models are consistently more accurate than those of the independent models for Australia and for Malaysia, but the relative performance differs by forecast horizon. Although the product-ratio functional model outperforms the Poisson common factor model for Australia, the Poisson common factor is more accurate for Malaysia. For the ethnic groups application, ethnic-coherence gives better results than gender-coherence. The results provide evidence that coherent models are preferable to independent models for forecasting sub-populations’ mortality. Full article

This paper considered a dependent discrete-time risk model, in which the insurance risks are represented by a sequence of independent and identically distributed real-valued random variables with a common Gamma-like tailed distribution; the financial risks are denoted by another sequence of independent and identically distributed positive random variables with a finite upper endpoint, but a general dependence structure exists between each pair of the insurance risks and the financial risks. Following the works of Yang and Yuen in 2016, we derive some asymptotic relations for the finite-time and infinite-time ruin probabilities. As a complement, we demonstrate our obtained result through a Crude Monte Carlo (CMC) simulation with asymptotics. Full article

In the first part of the paper, the positive and negative jumps of NASDAQ daily (log-) returns and three of its stocks are estimated based on the methodology presented by Theodosiadou et al. 2016, where jumps are assumed to be hidden random variables. For that reason, the use of stochastic state space models in discrete time is adopted. The daily return is expressed as the difference between the two-sided jumps under noise inclusion, and the recursive Kalman filter algorithm is used in order to estimate them. Since the estimated jumps have to be non-negative, the associated pdf truncation method, according to the non-negativity constraints, is applied. In order to overcome the resulting underestimation of the empirical time series, a scaling procedure follows the stage of truncation. In the second part of the paper, a nonparametric change point analysis concerning the (variance–) covariance is applied to the NASDAQ return time series, as well as to the estimated bivariate jump time series derived after the scaling procedure and to each jump component separately. A similar change point analysis is applied to the three other stocks of the NASDAQ index. Full article

The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note,weelaborateononespecificformulationofareplicatingportfolioproblem. Incontrasttothetwo most popular replication approaches, it does not yield an analytic solution (if, at all, a solution exists andisunique). Further,althoughconvex,theobjectivefunctionseemstobenon-smooth,andhencea numericalsolutionmightthusbemuchmoredemandingthanforthetwomostpopularformulations. Especially for the second reason, this formulation did not (yet) receive much attention in practical applications, in contrast to the other two formulations. In the following, we will demonstrate that the (potential) non-smoothness can be avoided due to an equivalent reformulation as a linear second order cone program (SOCP). This allows for a numerical solution by efficient second order methods like interior point methods or similar. We also show that—under weak assumptions—existence and uniqueness of the optimal solution can be guaranteed. We additionally prove that—under a further similarly weak condition—the fair value of the replicating portfolio equals the fair value of liabilities. Based on these insights, we argue that this unloved stepmother child within the replication problem family indeed represents an equally good formulation for practical purposes. Full article

I show that risk-sharing pension plans can reduce some of the shortcomings of defined benefit and defined contributions plans. The risk-sharing pension plan presented aims to improve the stability of benefits paid to generations of members, while allowing them to enjoy the expected advantages of a risky investment strategy. The plan does this by adjusting the investment strategy and benefits in response to a changing funding level, motivated by the with-profits contract proposed by Goecke (2013). He suggests a mean-reverting log reserve (or funding) ratio, where mean reversion occurs through adjustments to the investment strategy and declared bonuses. To measure the robustness of the plan to human factors, I introduce a measurement of disappointment, where disappointment is high when there are many consecutive years over which benefit payments are declining. Another measure introduced is devastation, where devastation occurs when benefit payments are zero. The motivation is that members of a pension plan who are easily disappointed or likely to get no benefit, are more likely to exit the plan. I find that the risk-sharing plan offers more disappointment than a defined contribution plan, but it eliminates the devastation possible in a plan that tries to accumulate contributions at a steadily increasing rate. The proposed risk-sharing plan can give a narrower range of benefits than in a defined contribution plan. Thus it can offer a stable benefit to members without the risk of running out of money. Full article

Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum $X_1+X_2$ occur as result of a single big jump with heavy tails whereas $X_1,X_2$ are of equal order of magnitude in the light-tailed case. The method is based on the ratio $|X_1-X_2|/(X_1+X_2)$ , for which sharp asymptotic results are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot. Full article

Optimal forms of reinsurance policies have been studied for a long time in the actuarial literature. Most existing results are from the insurer’s point of view, aiming at maximizing the expected utility or minimizing the risk of the insurer. However, as pointed out by Borch (1969), it is understandable that a reinsurance arrangement that might be very attractive to one party (e.g., the insurer) can be quite unacceptable to the other party (e.g., the reinsurer). In this paper, we follow this point of view and study forms of Pareto-optimal reinsurance policies whereby one party’s risk, measured by its value-at-risk (VaR), cannot be reduced without increasing the $\mathrm{VaR}$ of the counter-party in the reinsurance transaction. We show that the Pareto-optimal policies can be determined by minimizing linear combinations of the $\mathrm{VaR}$ s of the two parties in the reinsurance transaction. Consequently, we succeed in deriving user-friendly, closed-form, optimal reinsurance policies and their parameter values. Full article

This article considers an economy where risk is insurable, but selection determines the pool of individuals who take it up. First, we demonstrate that the comparative statics of these economies do not necessarily depend on its marginal selection (adverse versus favorable), but rather other characteristics. We then use repeated cross-sections of medical expenditures in the U.S. to understand the role of changes in the medical risk distribution on the fraction of Americans without medical insurance. We find that both the level and the shape of the distribution of risk are important in determining the equilibrium quantity of insurance. Symmetric changes in risk (e.g., shifts in the price of medical care) better explain the shifting insurance rate over time. Asymmetric changes (e.g., associated with a shifting age distribution) are not as important. Full article

We consider an insurer who faces an external jump-diffusion risk that is negatively correlated with the capital returns in a multidimensional regime switching model. The insurer selects investment and liability ratio policies continuously to maximize her/his expected utility of terminal wealth. We obtain explicit solutions of optimal policies for logarithmic and power utility functions. We study the impact of the insurer’s risk aversion, the negative correlation between the external risk and the capital returns, and the regime of the economy on the optimal policy. We find, among other things, that the regime of the economy and the negative correlation between the external risk and the capital returns have a dramatic effect on the optimal policy. Full article

Panel data of our interest consist of a moderate number of panels, while the panels contain a small number of observations. An estimator of common breaks in panel means without a boundary issue for this kind of scenario is proposed. In particular, the novel estimator is able to detect a common break point even when the change happens immediately after the first time point or just before the last observation period. Another advantage of the elaborated change point estimator is that it results in the last observation in situations with no structural breaks. The consistency of the change point estimator in panel data is established. The results are illustrated through a simulation study. As a by-product of the developed estimation technique, a theoretical utilization for correlation structure estimation, hypothesis testing and bootstrapping in panel data is demonstrated. A practical application to non-life insurance is presented, as well. Full article

The regulation on the Belgian occupational pension schemes has been recently changed. The new law allows for employers to choose between two different types of guarantees to offer to their affiliates. In this paper, we address the question arising naturally: which of the two guarantees is the best one? In order to answer that question, we set up a stochastic model and use financial pricing tools to compare the methods. More specifically, we link the pension liabilities to a portfolio of financial assets and compute the price of exchange options through the Margrabe formula. Full article

We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this. Full article

We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping framework, we discuss the choice of residuals, namely the Pearson residuals, the deviance residuals and the Anscombe residuals. In addition, several possible residual adjustments are discussed and compared in a case study. We carry out a practical implementation and comparison of methods using real-life insurance data to estimate reserves and their prediction errors. We propose to consider proper scoring rules for model validation, and the assessments will be drawn from an extensive case study. Full article

In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum benefit criteria: released capital, expected profit and exponential-fractional-logarithmic utility from the insurer’s point of view. Numerical examples are presented to explain how the optimal initial surplus and retention level are changed according to the individual claim amounts, loading factors and weights of the criteria. In the decision making process, we use The Analytical Hierarchy Process (AHP) and The Technique for Order of Preference by Similarity to ideal Solution (TOPSIS) methods as the Multi-Attribute Decision Making methods (MADM) and compare our results considering different combinations of loading factors for both exponential and Pareto individual claims. Full article

Reinsurance is often empirically hailed as a value-adding risk management strategy which an insurer can utilize to achieve various business objectives. In the context of a distortion-risk-measure-based three-party model incorporating a policyholder, insurer and reinsurer, this article formulates explicitly the optimal insurance–reinsurance strategies from the perspective of the insurer. Our analytic solutions are complemented by intuitive but scientifically rigorous explanations on the marginal cost and benefit considerations underlying the optimal insurance–reinsurance decisions. These cost-benefit discussions not only cast light on the economic motivations for an insurer to engage in insurance with the policyholder and in reinsurance with the reinsurer, but also mathematically formalize the value created by reinsurance with respect to stabilizing the loss portfolio and enlarging the underwriting capacity of an insurer. Our model also allows for the reinsurer’s failure to deliver on its promised indemnity when the regulatory capital of the reinsurer is depleted by the reinsured loss. The reduction in the benefits of reinsurance to the insurer as a result of the reinsurer’s default is quantified, and its influence on the optimal insurance–reinsurance policies analyzed. Full article

In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to characterize the optimal reinsurance strategy which minimizes the insurer’s risk measure of its total loss. Our calculations show that the optimal reinsurance strategy is of the multi-layer form, i.e., $f^{*}\left(x\right)=x\wedge c^{*}+{(x-d^{*})}_{+}$ with $c^{*}$ and $d^{*}$ being constants such that $0\le c^{*}\le d^{*}$ . Full article

The application of stochastic volatility (SV) models in the option pricing literature usually assumes that the market has sufficient option data to calibrate the model’s risk-neutral parameters. When option data are insufficient or unavailable, market practitioners must estimate the model from the historical returns of the underlying asset and then transform the resulting model into its risk-neutral equivalent. However, the likelihood function of an SV model can only be expressed in a high-dimensional integration, which makes the estimation a highly challenging task. The Bayesian approach has been the classical way to estimate SV models under the data-generating (physical) probability measure, but the transformation from the estimated physical dynamic into its risk-neutral counterpart has not been addressed. Inspired by the generalized autoregressive conditional heteroskedasticity (GARCH) option pricing approach by Duan in 1995, we propose an SV model that enables us to simultaneously and conveniently perform Bayesian inference and transformation into risk-neutral dynamics. Our model relaxes the normality assumption on innovations of both return and volatility processes, and our empirical study shows that the estimated option prices generate realistic implied volatility smile shapes. In addition, the volatility premium is almost flat across strike prices, so adding a few option data to the historical time series of the underlying asset can greatly improve the estimation of option prices. Full article

In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall the notion of conditional, dynamic and time-consistent risk measures. We link the latter with its iterated property, which gives us a way to construct time-consistent dynamic risk measures from a backward iteration scheme with the composition of conditional risk measures. We then consider particular cases with the conditional version of the value at risk, tail value at risk and conditional expectation measures. We finally give an application of these measures with the determination of the solvency capital of a pension liability, which offers a fixed guaranteed rate without any intermediate cash-flow. We assume that the company is fully hedged against the mortality and underwriting risks. Full article

This article provides a case study that analyzes national macroprudential insurance regulation in Switzerland. We consider an insurance market that is based on data from the Swiss private insurance industry. We stress this market with several scenarios related to financial and insurance risks, and we analyze the resulting risk capitals of the insurance companies. This stress-test analysis provides insights into the vulnerability of the Swiss private insurance sector to different risks and shocks. Full article

Life insurers are exposed to deflation risk: falling prices could lead to insufficient investment returns, and inflation-indexed protections could make insurers vulnerable to deflation. In this spirit, this paper proposes a market-based methodology for measuring deflation risk based on a discrete framework: the latter accounts for the real interest rate, the inflation index level, its conditional variance, and the expected inflation rate. US inflation data are then used to estimate the model and show the importance of deflation risk. Specifically, the distribution of a fictitious life insurer’s future payments is investigated. We find that the proposed inflation model yields higher risk measures than the ones obtained using competing models, stressing the need for dynamic and market-consistent inflation modelling in the life insurance industry. Full article

In this paper, we quantitatively compare the forecasts from four different mortality models. We consider one discrete-time model proposed by Lee and Carter (1992) and three continuous-time models: the Wills and Sherris (2011) model, the Feller process and the Ornstein-Uhlenbeck (OU) process. The first two models estimate the whole surface of mortality simultaneously, while in the latter two, each generation is modelled and calibrated separately. We calibrate the models to UK and Australian population data. We find that all the models show relatively similar absolute total error for a given dataset, except the Lee-Carter model, whose performance differs significantly. To evaluate the forecasting performance we therefore look at two alternative measures: the relative error between the forecasted and the actual mortality rates and the percentage of actual mortality rates which fall within a prediction interval. In terms of the prediction intervals, the results are more divergent since each model implies a different structure for the variance of mortality rates. According to our experiments, the Wills and Sherris model produces superior results in terms of the prediction intervals. However, in terms of the mean absolute error, the OU and the Feller processes perform better. The forecasting performance of the Lee Carter model is mostly dependent on the choice of the dataset. Full article

Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the flexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and financial data sets. Full article

An insurance company offers an insurance contract $(p,K)$ , consisting of a premium p and a deductible K. In this paper, we consider the problem of choosing the premium optimally as a function of the deductible. The insurance company is facing a market of N customers, each characterized by their personal claim frequency, α, and risk aversion, β. When a customer is offered an insurance contract, she/he will, based on these characteristics, choose whether or not to insure. The decision process of the customer is analyzed in detail. Since the customer characteristics are unknown to the company, it models them as i.i.d. random variables; $A_1,\dots ,A_N$ for the claim frequencies and $B_1,\dots ,B_N$ for the risk aversions. Depending on the distributions of $A_i$ and $B_i$ , expressions for the portfolio size $n(p;K)\in [0,N]$ and average claim frequency $\alpha (p;K)$ in the portfolio are obtained. Knowing these, the company can choose the premium optimally, mainly by minimizing the ruin probability. Full article

Variable annuities represent certain unit-linked life insurance products offering different types of protection commonly referred to as guaranteed minimum benefits (GMXBs). They are designed for the increasing demand of the customers for private pension provision. In this paper we analytically price variable annuities with guaranteed minimum repayments at maturity and in case of the insured’s death. If the contract is prematurely surrendered, the policyholder is entitled to the current value of the fund account reduced by the prevailing surrender fee. The financial market and the mortality model are affine linear. For the surrender model, a Cox process is deployed whose intensity is given by a deterministic function (s-curve) with stochastic inputs from the financial market. So, the policyholders’ surrender behavior depends on the performance of the financial market and is stochastic. The presented pricing scheme incorporates the stochastic surrender behavior of the policyholders and is only based on suitable closed-form approximations. Full article

The family of Liouville copulas is defined as the survival copulas of multivariate Liouville distributions, and it covers the Archimedean copulas constructed by Williamson’s d-transform. Liouville copulas provide a very wide range of dependence ranging from positive to negative dependence in the upper tails, and they can be useful in modeling tail risks. In this article, we study the upper tail behavior of Liouville copulas through their upper tail orders. Tail orders of a more general scale mixture model that covers Liouville distributions is first derived, and then tail order functions and tail order density functions of Liouville copulas are derived. Concrete examples are given after the main results. Full article

Life annuities are attractive mainly for healthy people. In order to expand their business, in recent years, some insurers have started offering higher annuity rates to those whose health conditions are critical. Life annuity portfolios are then supposed to become larger and more heterogeneous. With respect to the insurer’s risk profile, there is a trade-off between portfolio size and heterogeneity that we intend to investigate. In performing this, there is a second and possibly more important issue that we address. In actuarial practice, the different mortality levels of the several risk classes are obtained by applying adjustment coefficients to population mortality rates. Such a choice is not supported by a rigorous model. On the other hand, the heterogeneity of a population with respect to mortality can formally be described with a frailty model. We suggest adopting a frailty model for risk classification. We identify risk groups (or classes) within the population by assigning specific ranges of values to the frailty within each group. The different levels of mortality of the various groups are based on the conditional probability distributions of the frailty. Annuity rates for each class then can be easily justified, and a comprehensive investigation of insurer’s liabilities can be performed. Full article

In the linear coinsurance problem, examined first by Mossin (1968), a higher absolute risk aversion with respect to wealth in the sense of Arrow–Pratt implies a higher optimal coinsurance rate. We show that this property does not hold for health insurance under ex post moral hazard; i.e., when illness severity cannot be observed by insurers, and policyholders decide on their health expenditures. The optimal coinsurance rate trades off a risk-sharing effect and an incentive effect, both related to risk aversion. Full article

Because of the profitable nature of risk businesses in the long term, de Finetti suggested that surplus models should allow for cash leakages, as otherwise the surplus would unrealistically grow (on average) to infinity. These leakages were interpreted as ‘dividends’. Subsequent literature on actuarial surplus models with dividend distribution has mainly focussed on dividend strategies that either maximise the expected present value of dividends until ruin or lead to a probability of ruin that is less than one (see Albrecher and Thonhauser, Avanzi for reviews). An increasing number of papers are directly interested in modelling dividend policies that are consistent with actual practice in financial markets. In this short note, we review the corporate finance literature with the specific aim of fleshing out properties that dividend strategies should ideally satisfy, if one wants to model behaviour that is consistent with practice. Full article