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Risks, Volume 1, Issue 3 (December 2013), Pages 81-212

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Research

Open AccessArticle Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates
Risks 2013, 1(3), 81-100; doi:10.3390/risks1030081
Received: 8 August 2013 / Revised: 15 October 2013 / Accepted: 17 October 2013 / Published: 25 October 2013
Cited by 1 | PDF Full-text (430 KB) | HTML Full-text | XML Full-text
Abstract
In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use
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In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Optimal Deterministic Investment Strategies for Insurers
Risks 2013, 1(3), 101-118; doi:10.3390/risks1030101
Received: 30 September 2013 / Revised: 28 October 2013 / Accepted: 2 November 2013 / Published: 7 November 2013
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Abstract
We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility
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We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility and the probability of ruin. Following recent research, we assume that investment strategies have to be deterministic. This leads to deterministic control problems, which are quite easy to solve. Moreover, it turns out that there are some interesting links between the optimal investment strategies of these problems. Finally, we also show that this approach works in the Lévy process framework. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Optimal Dynamic Portfolio with Mean-CVaR Criterion
Risks 2013, 1(3), 119-147; doi:10.3390/risks1030119
Received: 6 August 2013 / Revised: 19 October 2013 / Accepted: 4 November 2013 / Published: 11 November 2013
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Abstract
Value-at-risk (VaR) and conditional value-at-risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of a Neyman–Pearson type binary solution. We add a constraint on expected return to investigate the mean-CVaR
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Value-at-risk (VaR) and conditional value-at-risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of a Neyman–Pearson type binary solution. We add a constraint on expected return to investigate the mean-CVaR portfolio selection problem in a dynamic setting: the investor is faced with a Markowitz type of risk reward problem at the final horizon, where variance as a measure of risk is replaced by CVaR. Based on the complete market assumption, we give an analytical solution in general. The novelty of our solution is that it is no longer the Neyman–Pearson type, in which the final optimal portfolio takes only two values. Instead, in the case in which the portfolio value is required to be bounded from above, the optimal solution takes three values; while in the case in which there is no upper bound, the optimal investment portfolio does not exist, though a three-level portfolio still provides a sub-optimal solution. Full article
(This article belongs to the collection Systemic Risk and Reinsurance)
Open AccessArticle A Risk Model with an Observer in a Markov Environment
Risks 2013, 1(3), 148-161; doi:10.3390/risks1030148
Received: 11 October 2013 / Revised: 5 November 2013 / Accepted: 5 November 2013 / Published: 11 November 2013
Cited by 7 | PDF Full-text (311 KB) | HTML Full-text | XML Full-text
Abstract
We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation
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We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle U.S. Equity Mean-Reversion Examined
Risks 2013, 1(3), 162-175; doi:10.3390/risks1030162
Received: 16 October 2013 / Revised: 13 November 2013 / Accepted: 18 November 2013 / Published: 4 December 2013
Cited by 1 | PDF Full-text (656 KB) | HTML Full-text | XML Full-text
Abstract
In this paper we introduce an intra-sector dynamic trading strategy that captures mean-reversion opportunities across liquid U.S. stocks. Our strategy combines the Avellaneda and Lee methodology (AL; Quant. Financ. 2010, 10, 761–782) within the Black and Litterman framework (BL; J. Fixed
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In this paper we introduce an intra-sector dynamic trading strategy that captures mean-reversion opportunities across liquid U.S. stocks. Our strategy combines the Avellaneda and Lee methodology (AL; Quant. Financ. 2010, 10, 761–782) within the Black and Litterman framework (BL; J. Fixed Income, 1991, 1, 7–18; Financ. Anal. J. 1992, 48, 28–43). In particular, we incorporate the s-scores and the conditional mean returns from the Orstein and Ulhembeck (Phys. Rev. 1930, 36, 823–841) process into BL. We find that our combined strategy ALBL has generated a 45% increase in Sharpe Ratio when compared to the uncombined AL strategy over the period from January 2, 2001 to May 27, 2010. These new indices, built to capture dynamic trading strategies, will definitely be an interesting addition to the growing hedge fund index offerings. This paper introduces our first “focused-core” strategy, namely, U.S. Equity Mean-Reversion. Full article
Open AccessArticle Impact of Climate Change on Heat Wave Risk
Risks 2013, 1(3), 176-191; doi:10.3390/risks1030176
Received: 1 November 2013 / Revised: 3 December 2013 / Accepted: 5 December 2013 / Published: 12 December 2013
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Abstract
We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring
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We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring the potential impact of climate change on heat wave risk. Numerical illustrations are given. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
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Open AccessArticle Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach
Risks 2013, 1(3), 192-212; doi:10.3390/risks1030192
Received: 6 November 2013 / Revised: 2 December 2013 / Accepted: 5 December 2013 / Published: 13 December 2013
Cited by 1 | PDF Full-text (340 KB) | HTML Full-text | XML Full-text
Abstract
This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator,
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This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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