Multivariate Tweedie lifetimes: the impact of dependence

Systematic longevity risk is increasingly relevant for public pension schemes and insurance companies that provide life benefits. In view of this, mortality models should incorporate dependence between lives. However, the independent lifetime assumption is still heavily relied upon in the risk management of life insurance and annuity portfolios. This paper applies a multivariate Tweedie distribution to incorporate dependence, which it induces through a common shock component. Model parameter estimation is developed based on the method of moments and generalized to allow for truncated observations. The estimation procedure is explicitly developed for various important distributions belonging to the Tweedie family, and finally assessed using simulation.     

Multivariate Cox Hidden Markov models with an application to operational risk

ABSTRACT

Modeling multivariate time-series aggregate losses is an important actuarial topic that is very challenging due to the fact that losses can be serially dependent with heterogeneous dependence structures across loss types and business lines. In this paper, we investigate a flexible class of multivariate Cox Hidden Markov Models for the joint arrival process of loss events. Some of the nice properties possessed by this class of models, such as closed-form expressions, thinning properties and model versatility are discussed in details. We provide the expectation-maximization (EM) algorithm for efficient model calibration. Applying the proposed model to an operational risk dataset, we demonstrate that the model offers sufficient flexibility to capture most characteristics of the observed loss frequencies. By modeling the log-transformed loss severities through mixture of Erlang distributions, we can model the aggregate losses. Finally, out-of-sample testing shows that the proposed model is adequate to predict short-term future operational risk losses.     

Multivariate Shrinkage for Optimal Portfolio Weights

Abstract

This paper proposes a multivariate shrinkage estimator for the optimal portfolio weights. The estimated classical Markowitz weights are shrunk to the deterministic target portfolio weights. Assuming log asset returns to be i.i.d. Gaussian, explicit solutions are derived for the optimal shrinkage factors. The properties of the estimated shrinkage weights are investigated both analytically and using Monte Carlo simulations. The empirical study compares the competing portfolio selection approaches. Both simulation and empirical studies show that the proposed shrinkage estimator is robust and provides significant gains to the investor compared to benchmark procedures.     

International Asset Excess Returns and Multivariate Conditional Volatilities

This paper constructs a multivariate model in relating multi-asset excess returns to their conditional variances. Applying weekly data to investigate the foreign-exchange risk premium, the evidence from a multivariate GARCH model shows that the foreign-exchange excess returns are significantly correlated with economic fundamentals such as the real interest-rate differential, long-short interest-rate spread differential, and equity-premium differential. The evidence also suggests that foreign-exchange excess returns are not independent of the conditional variances of these fundamental variables, supporting the time-varying risk-premium hypothesis.     

Reduction of Value-at-Risk bounds via independence and variance information

We derive lower and upper bounds for the Value-at-Risk of a portfolio of losses when the marginal distributions are known and independence among (some) subgroups of the marginal components is assumed. We provide several actuarial examples showing that the newly proposed bounds strongly improve those available in the literature that are based on the sole knowledge of the marginal distributions. When the variance of the joint portfolio loss is small enough, further improvements can be obtained.     

From BASEL III to BASEL IV and beyond: Expected shortfall and expectile risk measures

The article contributes to the ongoing search for a market risk measure that is both coherent and elicitable. We compare two traditional measures, namely Value-at-Risk and the expected shortfall, with another relatively novel one established on the expectile probability term. Our research is based on five models: Black–Scholes, exponential tempered stable, Heston, Bates and another stochastic volatility model with a tempered stable jump correction. We apply the general Fourier inversion formula to derive closed form formulas for calculating not only the expectile based risk measure but also the Value-at-Risk and the expected shortfall. These models are calibrated by combining nonlinear programming with simulated annealing at a moving window. Additionally, we compare the generated values of the risk measures with the real ones. Last but not least, we modify the expectile based risk measure as well as the expected shortfall by introducing correction coefficients.     

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship. I. Molchanov supported by Swiss National Science Foundation Grant 200020-109217.     

沪深股市风险度量中半参数ES模型的实证检验

目前度量预期不足(Expected Shortfall, ES)的风险技术大多基于参数模型,其建模过程避免不了对收益的分布类型做出假定,但这些分布往往与现实相悖。为此,介绍两种重要半参数模型,即CARE模型和CARES模型,并应用我国2007-2016年上证综合指数与深证成分指数的相关数据评估模型优劣。结果表明：CARES模型与CARE模型在度量我国股市风险中都具有较好的效果,但两者比较,CARES模型明显优于CARE模型。因此,CARES模型能作为我国股市风险度量工具中的一个重要补充。     

Multivariate lifetime distributions for the exponential dispersion family

We consider a general form of a multivariate lifetime model in which dependence is induced via a common shock component. The univariate marginal distributions come from the well-known and widely applied exponential dispersion family that includes the normal, compound-Poisson, gamma and negative binomial distributions. Any combination of truncation or censoring, either left or right, is considered, for which all moments are derived. This allows for the model to be calibrated to any affine transformation of lifetime data.     

Multivariate models for operational risk

Böcker and Klüppelberg [Risk Mag., 2005, December, 90–93] presented a simple approximation of OpVaR of a single operational risk cell. The present paper derives approximations of similar quality and simplicity for the multivariate problem. Our approach is based on the modelling of the dependence structure of different cells via the new concept of a Lévy copula.     

Large portfolio losses

This paper provide a large-deviations approximation of the tail distribution of total financial losses on a portfolio consisting of many positions. Applications include the total default losses on a bank portfolio, or the total claims against an insurer. The results may be useful in allocating exposure limits, and in allocating risk capital across different lines of business. Assuming that, for a given total loss, the distress caused by the loss is larger if the loss occurs within a smaller time period, we provide a large-deviations estimate of the likelihood that there will exist a sub-period of the future planning period during which a total loss of the critical severity occurs. Under conditions, this calculation is reduced to the calculation of the likelihood of the same sized loss over an initial time interval whose length is a property of the portfolio and the critical loss level.Received: March 2003Mathematics Subject Classification: 60F10, 91B28, 91B28JEL Classification: G21, G22, G33Amir Dembo is with the Department of Statistics, Stanford University. His research was partially supported by NSF grant #DMS-0072331. Jean-Dominique Deuschel is with the Department of Mathematics, Technische Universität, Berlin. His research was partially supported by DFG grant #663/2-3 and DFG FZT 86. Darrell Duffie is with the Graduate School of Business, Stanford University. We are extremely grateful for research assistance by Nicolae Gârleanu and Gustavo Manso, for conversations with Michael Gordy, and for comments from Michael Stutzer, Peter Carr, David Heath, and David Siegmund.     

Systemic risk and bank consolidation: International evidence

This paper analyzes the systemic risk effects of bank mergers to test the “concentration-fragility” hypothesis. We use the marginal expected shortfall as well as the lower tail dependence between a bank’s stock returns and a relevant bank sector index to capture the merger-related change in an acquirer’s contribution to systemic risk. In our empirical analysis of a dataset of international domestic and cross-border mergers, we find clear evidence for a significant increase in the merging banks’, the combined banks’ as well as their competitors’ contribution to systemic risk following mergers, thus confirming the “concentration-fragility” hypothesis.     

Tranching and Rating

In this paper we analyse the source and magnitude of marketing gains from selling structured debt securities at yields that reflect only their credit ratings, or specifically at yields on equivalently rated corporate bonds. We distinguish between credit ratings that are based on probabilities of default and ratings that are based on expected default losses. We show that subdividing a bond issued against given collateral into subordinated tranches can yield significant profits under the hypothesised pricing system. Increasing the systematic risk or reducing the total risk of the bond collateral increases the profits further. The marketing gain is generally increasing in the number of tranches and decreasing in the rating of the lowest rated tranche.     

A note on generalized distortion risk measures

A generalized distortion risk measure is introduced as power of the mean absolute deviation power of a distorted random variable with respect to a location parameter. This class of risk measures extends both the distortion risk measure by Wang and Denneberg and the class of financial risk measures by Pedersen and Satchell, which itself contains the class of Stone. Integral representations and a stop–loss order preserving property of a special up-side risk measure are derived.     

Kernel-type estimators for the distortion risk premiums of heavy-tailed distributions

A new kernel-type estimator for the distortion risk premiums of heavy-tailed losses is introduced. Using a least-squares approach, a bias-reduced version of this estimator is proposed. Under suitable assumptions, the asymptotic normality of the given estimators is established. A small simulation study, to illustrate the performance of our method, is carried out.     

On a risk measure inspired from the ruin probability and the expected deficit at ruin

In this paper, we study a risk measure derived from ruin theory defined as the amount of capital needed to cope in expectation with the first occurrence of a ruin event. Specifically, within the compound Poisson model, we investigate some properties of this risk measure with respect to the stochastic ordering of claim severities. Particular situations where combining risks yield diversification benefits are identified. Closed form expressions and upper bounds are also provided for certain claim severities.     

A Generalisation of the Mean-Variance Analysis

In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above this reference point. We also suppose that the decision maker generally distorts the objective probabilities. First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This 'mean-partial moments' utility generalises not only mean-variance utility of Tobin and Markowitz, but also mean-semivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present a solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure which generalises the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker's skewness preferences have first-order impact on risk measurement even when the risk is small.