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Risks, Volume 5, Issue 2 (June 2017)

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Research

Open AccessArticle State Space Models and the Kalman-Filter in Stochastic Claims Reserving: Forecasting, Filtering and Smoothing
Risks 2017, 5(2), 30; doi:10.3390/risks5020030
Received: 1 April 2017 / Revised: 7 May 2017 / Accepted: 15 May 2017 / Published: 27 May 2017
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Abstract
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
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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
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Open AccessFeature PaperArticle Actuarial Geometry
Risks 2017, 5(2), 31; doi:10.3390/risks5020031
Received: 12 April 2017 / Revised: 16 May 2017 / Accepted: 2 June 2017 / Published: 16 June 2017
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Abstract
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
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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
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Open AccessArticle Multivariate Functional Time Series Forecasting: Application to Age-Specific Mortality Rates
Risks 2017, 5(2), 21; doi:10.3390/risks5020021
Received: 26 October 2016 / Revised: 15 March 2017 / Accepted: 21 March 2017 / Published: 25 March 2017
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Abstract
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.
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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
(This article belongs to the Special Issue Ageing Population Risks)
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Open AccessFeature PaperArticle Effects of Gainsharing Provisions on the Selection of a Discount Rate for a Defined Benefit Pension Plan
Risks 2017, 5(2), 32; doi:10.3390/risks5020032
Received: 21 March 2017 / Revised: 9 June 2017 / Accepted: 12 June 2017 / Published: 20 June 2017
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Abstract
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
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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
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Open AccessArticle Asymmetric Return and Volatility Transmission in Conventional and Islamic Equities
Risks 2017, 5(2), 22; doi:10.3390/risks5020022
Received: 13 December 2016 / Revised: 27 March 2017 / Accepted: 27 March 2017 / Published: 29 March 2017
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Abstract
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
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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
Open AccessFeature PaperArticle Actuarial Applications and Estimation of Extended CreditRisk+
Risks 2017, 5(2), 23; doi:10.3390/risks5020023
Received: 20 January 2017 / Revised: 22 March 2017 / Accepted: 26 March 2017 / Published: 31 March 2017
Cited by 1 | PDF Full-text (654 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
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
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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
(This article belongs to the Special Issue Ageing Population Risks)
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Open AccessArticle Applying spectral biclustering to mortality data
Risks 2017, 5(2), 24; doi:10.3390/risks5020024
Received: 6 October 2016 / Revised: 22 March 2017 / Accepted: 29 March 2017 / Published: 4 April 2017
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Abstract
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
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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
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Open AccessArticle Enhancing Singapore’s Pension Scheme: A Blueprint for Further Flexibility
Risks 2017, 5(2), 25; doi:10.3390/risks5020025
Received: 6 December 2016 / Revised: 4 April 2017 / Accepted: 7 April 2017 / Published: 13 April 2017
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Abstract
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
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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
(This article belongs to the Special Issue Designing Post-Retirement Benefits in a Demanding Scenario)
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Open AccessFeature PaperArticle Bond and CDS Pricing via the Stochastic Recovery Black-Cox Model
Risks 2017, 5(2), 26; doi:10.3390/risks5020026
Received: 18 January 2017 / Revised: 4 April 2017 / Accepted: 10 April 2017 / Published: 19 April 2017
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Abstract
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.
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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 At defines the default trigger and the recovery risk driver Rt 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
Open AccessFeature PaperArticle Risk Management under Omega Measure
Risks 2017, 5(2), 27; doi:10.3390/risks5020027
Received: 10 January 2017 / Revised: 7 April 2017 / Accepted: 27 April 2017 / Published: 6 May 2017
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Abstract
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
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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
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Open AccessFeature PaperArticle Asymptotic Estimates for the One-Year Ruin Probability under Risky Investments
Risks 2017, 5(2), 28; doi:10.3390/risks5020028
Received: 28 March 2017 / Revised: 22 April 2017 / Accepted: 22 April 2017 / Published: 6 May 2017
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Abstract
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
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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 Y consisting of stochastic discount factors of the d financial assets invested. To capture both heavy tails and asymptotic dependence of Y in an integrated manner, we assume that 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 Y are independent with X of Fréchet type; (ii) X and Y are independent with X of Gumbel type; (iii) X and Y jointly possess a standard MRV structure; (iv) X and Y jointly possess a nonstandard MRV structure. Full article
Open AccessArticle Maximum Market Price of Longevity Risk under Solvency Regimes: The Case of Solvency II
Risks 2017, 5(2), 29; doi:10.3390/risks5020029
Received: 25 February 2017 / Revised: 30 April 2017 / Accepted: 4 May 2017 / Published: 10 May 2017
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Abstract
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
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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
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