Statistics and Actuarial Science

Permanent URI for this collectionhttps://uwspace.uwaterloo.ca/handle/10012/9934

This is the collection for the University of Waterloo's Department of Statistics and Actuarial Science.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

Waterloo faculty, students, and staff can contact us or visit the UWSpace guide to learn more about depositing their research.

Browse

Now showing 1 - 5 of 5

Multivariate Risk Measures for Portfolio Risk Management
(University of Waterloo, 2021-01-29) Jia, Huameng; Cai, Jun
In portfolio risk management, the main foci are to control the aggregate risk of the entire portfolio and to understand the contribution of each individual risk unit in the portfolio to the aggregate risk. When univariate risk measures are used to quantify the risks associated with a portfolio, there is usually a lack of consideration of correlations between individual risk units and the aggregate risk and of dependence among these risks. For this reason, multivariate risk measures defined by considering the joint distribution of risk units in the portfolio are more desirable. In this thesis, we define new multivariate risk measures by minimizing multivariate loss functions subject to various. constraints. With the proposed multivariate risk measures, we obtain risk measures for the entire portfolio and each individual risk unit in the portfolio at the same time. In Chapter 2, we introduce a multivariate extension of Conditional Value-at-Risk (CVaR) based on a multivariate loss function associated with different risks related to portfolio risk management. We prove that the defined multivariate risk measure satisfies many desirable properties such as positive homogeneity, translation invariance and subadditivity. Then, we provide numerical illustrations with multivariate normal distribution to show the effects of the parameters in the model. After that, we also perform a comparison between our multivariate CVaR and other traditional univariate risk measures such as VaR and CVaR. In Chapter 3, we define a multivariate risk measure for capital allocation purposes. Unlike most of the existing allocation principles that assume the total capital is exogenously given, we obtain the optimal total capital for the entire portfolio and the optimal capital allocation to all the individual risk units in the portfolio at the same time. In this chapter, we first discuss our model with a two-level organization/portfolio structure. Then, we move to a more complex three-level organization/portfolio structure. We find that many of the existing allocation principles can be seen as special or limiting cases of our model. In addition, our model can explain those allocation principles as solutions to optimization problems. Finally, we provide a numerical example for the two-level organization/portfolio structure model with two different error functions. In Chapter 4, we introduce a multivariate shortfall risk measure induced by cumulative prospect theory (CPT) and give the corresponding risk allocations under the multivariate shortfall risk measure. To obtain this risk measure, we make an extension of previously studied univariate generalized shortfalls induced by CPT and incorporate the idea of systemic risk. In this study, we discuss the properties of the risk measure and conditions for its existence and uniqueness. Also, we perform a simulation study and a comparison to a previously studied multivariate shortfall to show that our model can provide a more reasonable risk measure and allocation result.
Optimization, model uncertainty, and testing in risk and insurance
(University of Waterloo, 2024-07-11) Jiao, Zhanyi; Cai, Jun; Wang, Ruodu
This thesis focuses on three important topics in quantitative risk management and actuarial science: risk optimization, risk sharing, and statistical hypothesis testing in risk. For the risk optimization, we concentrate on risk optimization under model uncertainty where only partial information about the underlying distribution is available. One key highlight, detailed in Chapter 2, is the development of a novel formula named the reverse Expected Shortfall (ES) optimization formula. This formula is derived to better facilitate the calculation of the worst-case mean excess loss under two commonly used model uncertainty sets – moment-based and distance-based (Wasserstein) uncertainty sets. Further exploration reveals that the reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. Chapter 3 considers a different approach to derive the closed-form worst-case target semi-variance by including distributional shape information, crucial for finance (symmetry) and insurance (non-negativity) applications. We demonstrate that all results are applicable to robust portfolio selection, where the closed-form formulas greatly simplify the calculations for optimal robust portfolio selections, either through explicit forms or via easily solvable optimization problems. Risk sharing focuses on the redistribution of total risk among agents in a specific way. In contrast to the traditional risk sharing rules, Chapter 4 introduces a new risk sharing framework - anonymized risk sharing, which requires no information on preferences, identities, private operations, and realized losses from the individual agents. We establish an axiomatic theory based on four axioms of fairness and anonymity within the context of anonymized risk sharing. The development of this theory provides a solid foundation for further explorations on decentralized and digital economy including peer-to-peer (P2P) insurance, revenue sharing of digital contents and blockchain mining pools. Hypothesis testing plays a vital role not only in statistical inference but also in risk management, particularly in the backtesting of risk measures. In Chapter 5, we address the problem of testing conditional mean and conditional variance for non-stationary data using the recent emerging concept of e-statistics. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. To demonstrate the practical application of these methodologies, empirical studies using financial data are conducted under several settings.
Risk Measures and Capital Allocation Principles for Risk Management
(University of Waterloo, 2016-09-21) wang ying; Cai, Jun
Risk measures (or premium principles) and capital allocation principles play a signi cant role in risk management. Regulators and companies in the nancial markets usually adopt an appropriate risk measure, for example, Value-at-Risk (VaR) or Tail Value-at-Risk (TVaR), to determine the benchmarks. However, these risk measures are determined from the loss functions with constant weights, not random weight functions. This thesis proposes new approaches to determine risk measures from two perspectives. Firstly, we will generalize the de nition of the tail subadditivity for distortion risk measures; we de ne the generalized GlueVaR (a linear combination of VaR and TVaRs) to approach any coherent distortion risk measure. Secondly, we will research the risk measures (or premium principles) and capital allocation principles based on the loss functions with random weight functions. The new reinsurance premium principles are derived similarly to the new risk measures. The two thresholds for the weight in the loss function can be employed by reinsurance companies as benchmarks when pricing the reinsurance products. The capital allocation principles derived based on the weighted loss functions are both mathematically and economically reasonable. Many of the risk measures and allocation principles, including the new risk measures, can be covered by this model. The results of this thesis have not only uni ed many of the risk measures and capital allocation principles, but also provided new and practical models.
Risk Measures of Stop-loss and Limited Loss Random Variables under Model Uncertainty with Applications in Insurance
(University of Waterloo, 2023-09-07) Yin, Mingren; Cai, Jun; Liu, Fangda
In this thesis, our focus is on the optimization of reinsurance design, accounting for the influence of model uncertainty. The following chapters outline our approach: In Chapter 2, we identify the worst-case distributions for both insurers and reinsurers by assuming that insurers and reinsurers respectively have their own uncertainty sets. These distributions are structured to maximize their respective shares of the total loss, assessed by a distortion risk measure. We consider a reinsurance contract structured as a stop-loss treaty with a deductible. Our uncertainty sets adopt traditional two-moment characteristics, incorporated with distance constraints defined using Wasserstein distance. We provide numerical solutions for the worst-case distributions in a general scenario, along with analytical solutions for cases when uncertainty sets only have constraints on the first two moments of the underlying loss random variable. Based on that, we find the optimal stop-loss reinsurance policy from the perspective of the insurer taking model uncertainty into account. In Chapter 3, we assume that uncertainty sets of insurers and reinsurers are defined only by Wasserstein distance. We consider the worst-case risk measures of limited stoploss functions and determine the worst-case distributions for both insurers and reinsurers under limited stop-loss reinsurances. In addition, by conducting numerical experiments, we explore how the limits and deductibles of limited stop-loss reinsurances impact worst-case risk measures for both parties. Moving into Chapter 4, we integrate the notion of distribution ambiguity into a negotiation framework, specifically Pareto optimality. Through numerical experiments based on results presented in Chapters 2 and 3, we investigate how the negotiation power between parties influences the equilibrium point. Concluding our study, the final chapter outlines potential directions for future research and development, building upon the foundation laid out in this work.
Risk Sharing and Risk Aggregation via Risk Measures
(University of Waterloo, 2017-04-21) Liu, Haiyan; Cai, Jun; Wang, Ruodu
Risk measures have been extensively studied in actuarial science in the guise of premium calculation principles for more than 40 years, and recently, they have been the standard tool for financial institutions in both calculating regulatory capital requirement and internal risk management. This thesis focuses on two topics: risk sharing and risk aggregation via risk measures. The problem of risk sharing concerns the redistribution of a total risk among agents using risk measures to quantify risks. Risk aggregation is to study the worst-case value of aggregate risks over all possible dependence structures with given marginal risks. On the first topic, we address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. The family of RVaR includes the Value-at-Risk (VaR) and the Expected Shortfall (ES), the two popular and competing regulatory risk measures, as special cases. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the Pareto-optimal risk sharing problem is solved through explicit construction. We also study risk sharing in a competitive market and obtain an explicit Arrow-Debreu equilibrium. Robustness and comonotonicity of optimal allocations are investigated, and several novel advantages of ES over VaR from the perspective of a regulator are revealed. Reinsurance, as a special type of risk sharing, has been studied extensively from the perspective of either an insurer or a reinsurer. To take the interests of both parties into consideration, we study Pareto optimality of reinsurance arrangements under general model settings. We give the necessary and sufficient conditions for a reinsurance contract to be Pareto-optimal and characterize all such optimal contracts under more general model assumptions. Sufficient conditions that guarantee the existence of the Pareto-optimal contracts are obtained. When the losses of an insurer and a reinsurer are measured by the ES risk measures, we obtain the explicit forms of the Pareto-optimal reinsurance contracts under the expected value premium principle. On the second topic, we first study the aggregation of inhomogeneous risks with a special type of model uncertainty, called dependence uncertainty, in individual risk models. We establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with dependence uncertainty. Then, we bring the well studied dependence uncertainty in individual risk models into collective risk models. We study the worst-case values of the VaR and the ES of the aggregate loss with identically distributed individual losses, under two settings of dependence uncertainty: (i) the counting random variable and the individual losses are independent, and the dependence of the individual losses is unknown; (ii) the dependence of the counting random variable and the individual losses is unknown. Analytical results for the worst-case values of ES are obtained. For the loss from a large portfolio of insurance policies, the asymptotic equivalence of VaR and ES is established, and approximation errors are obtained under the two dependence settings.

Browse

Browsing Statistics and Actuarial Science by Author "Cai, Jun"