First-order optimization algorithms via inertial systems with Hessian driven damping

For controlled discrete-time stochastic processes we introduce a new class of dynamic risk measures, which we call process-based. Their main feature is that they measure risk of processes that are functions of the history of a base process. We introduce a new concept of conditional stochastic time consistency and we derive the structure of process-based risk measures enjoying this property. We show that they can be equivalently represented by a collection of static law-invariant risk measures on the space of functions of the state of the base process. We apply this result to controlled Markov processes and we derive dynamic programming equations. We also derive dynamic programming equations for multistage stochastic programming with decision-dependent distributions.

     

Structural Electricity Models and Asymptotically Normal Estimators to Quantify Parameter Risk

ABSTRACT

We estimate a structural electricity (multi-commodity) model based on historical spot and futures data (fuels and power prices, respectively) and quantify the inherent parameter risk using an average value at risk approach (‘expected shortfall’). The mathematical proofs use the theory of asymptotic statistics to derive a parameter risk measure. We use far in-the-money options to derive a confidence level and use it as a prudent present value adjustment when pricing a virtual power plant. Finally, we conduct a present value benchmarking to compare the approach of temperature-driven demand (based on load data) to an ‘implied demand approach’ (demand implied from observable power futures prices). We observe that the implied demand approach can easily capture observed electricity price volatility whereas the estimation against observable load data will lead to a gap, because – amongst others – the interplay of demand and supply is not captured in the data (i.e., unexpected mismatches).     

Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled (‘tilted’) one-period convex risk measures, using a d-dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete ‘driver’ characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.     

Risk-adjusted probability measures in portfolio optimization with coherent measures of risk

We consider the problem of optimizing a portfolio of n assets, whose returns are described by a joint discrete distribution. We formulate the mean–risk model, using as risk functionals the semideviation, deviation from quantile, and spectral risk measures. Using the modern theory of measures of risk, we derive an equivalent representation of the portfolio problem as a zero-sum matrix game, and we provide ways to solve it by convex optimization techniques. In this way, we reconstruct new probability measures which constitute part of the saddle point of the game. These risk-adjusted measures always exist, irrespective of the completeness of the market. We provide an illustrative example, in which we derive these measures in a universe of 200 assets and we use them to evaluate the market portfolio and optimal risk-averse portfolios.     

Solvency II,regulatory capital,and optimal reinsurance: How good are Conditional Value-at-Risk and spectral risk measures?

We study the problem of optimal reinsurance as a means of risk management in the regulatory framework of Solvency II under Conditional Value-at-Risk and, as its natural extension, spectral risk measures. First, we show that stop-loss reinsurance is optimal under both Conditional Value-at-Risk and spectral risk measures. Spectral risk measures thus constitute a more general class of suitable regulatory risk measures than specific Conditional Value-at-Risk. At the same time, the established type of stop-loss reinsurance can be maintained as the optimal risk management strategy that minimizes regulatory capital. Second, we derive the optimal deductibles for stop-loss reinsurance. We show that under Conditional Value-at-Risk, the optimal deductible tends towards restrictive and counter-intuitive corner solutions or “plunging”, which is a serious objection against its use in regulatory risk management. By means of the broader class of spectral risk measures, we are able to overcome this shortcoming as optimal deductibles are now interior solutions. Especially, the recently discussed power spectral risk measures and the Wang risk measure are shown to avoid any plunging. They yield a one-to-one correspondence between the risk parameter and the optimal deductible and, thus, provide economically plausible risk management strategies.     

Conditional,Non-Homogeneous and Doubly Stochastic Compound Poisson Processes with Stochastic Discounted Claims

In this paper, we study the conditional, non-homogeneous and doubly stochastic compound Poisson process with stochastic discounted claims. We derive the moment generating functions of these risk processes and find their inverses, numerically or analytically, by using their corresponding characteristic functions. We then compare their distributions and some risk measures as the VaR and TVaR, and we examine the case where there is a possible dependence between the occurrence time and the severity of the claim.     

Dynamics of multivariate default system in random environment

We consider a multivariate default system where random environmental information is available. We study the dynamics of the system in a general setting of enlargement of filtrations and adopt the point of view of change of probability measures. We also make a link with the density approach in the credit risk modelling. Finally, we present a martingale characterization result with respect to the observable information filtration on the market.     

Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

Risk measuring under liquidity risk

We present a general framework for measuring the liquidity risk. The theoretical framework defines risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a security or a portfolio of securities cannot be sold or bought without causing changes in prices. The risk measures are decomposed into two terms, one measuring the risk of the future value of a given position in a security or a portfolio of securities and the other the initial cost of this position. Within the framework of coherent risk measures, the risk measures applied to the random part of the future value of a position in a determinate security are increasing monotonic and convex cash sub-additive on long positions. The contrary, in certain situations, holds for the sell positions. By using convex risk measures, we apply our framework to the situation in which large trades are broken into many small ones. Dual representation results are obtained for both positions in securities and portfolios. We give many examples of risk measures and derive for each of them the respective capital requirement. In particular, we discuss the VaR measure.     

Risk preferences on the space of quantile functions

We propose a novel approach to quantification of risk preferences on the space of nondecreasing functions. When applied to law invariant risk preferences among random variables, it compares their quantile functions. The axioms on quantile functions impose relations among comonotonic random variables. We infer the existence of a numerical representation of the preference relation in the form of a quantile-based measure of risk. Using conjugate duality theory by pairing the Banach space of bounded functions with the space of finitely additive measures on a suitable algebra \(\varSigma \) , we develop a variational representation of the quantile-based measures of risk. Furthermore, we introduce a notion of risk aversion based on quantile functions, which enables us to derive an analogue of Kusuoka representation of coherent law-invariant measures of risk.     

Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information

In this paper, based on equilibrium control law proposed by Björk and Murgoci (2010), we study an optimal investment and reinsurance problem under partial information for insurer with mean–variance utility, where insurer’s risk aversion varies over time. Instead of treating this time-inconsistent problem as pre-committed, we aim to find time-consistent equilibrium strategy within a game theoretic framework. In particular, proportional reinsurance, acquiring new business, investing in financial market are available in the market. The surplus process of insurer is depicted by classical Lundberg model, and the financial market consists of one risk free asset and one risky asset with unobservable Markov-modulated regime switching drift process. By using reduction technique and solving a generalized extended HJB equation, we derive closed-form time-consistent investment–reinsurance strategy and corresponding value function. Moreover, we compare results under partial information with optimal investment–reinsurance strategy when Markov chain is observable. Finally, some numerical illustrations and sensitivity analysis are provided.     

Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.     

A risk-averse newsvendor with law invariant coherent measures of risk

For general law invariant coherent measures of risk, we derive an equivalent representation of a risk-averse newsvendor problem as a mean-risk model. We prove that the higher the weight of the risk functional, the smaller the order quantity. Our theoretical results are confirmed by sample-based optimization.     

Equilibrium threshold strategies in observable queueing systems with setup/closedown times

This paper considers two types of setup/closedown policies: interruptible and insusceptible setup/closedown policies. When all customers are served exhaustively in a system under the interruptible setup/closedown policy, the server shuts down (deactivates) by a closedown time. When the server reactivates since shutdown, he needs a setup time before providing service again. If a customer arrives during a closedown time, the service is immediately started without a setup time. However, in a system under the insusceptible setup/closedown policy, customers arriving in a closedown time can not be served until the following setup time finishes. For the systems with interruptible setup/closedown times, we assume both the fully and almost observable cases, then derive equilibrium threshold strategies for the customers and analyze the stationary behavior of the systems. On the other hand, for the systems with insusceptible setup/closedown times, we only consider the fully observable case. We also illustrate the equilibrium thresholds and the social benefits for systems via numerical experiments. As far as we know, there is no work concerning equilibrium behavior of customers in queueing systems with setup/closedown times.     

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments.     

Is mortality or interest rate the most important risk in annuity models? A comparison of sensitivity analysis methods

Demographic and financial factors are key risk-drivers for insurance companies and pension funds. This paper proposes a systematic investigation for deepening our understanding how these risk drivers affect the annuity cost. We employ local and global sensitivity methods. For local sensitivity, we derive closed form expressions for the differential importance measures of perturbed annuities and connect them to the entropy of the annuity cost. For global sensitivity, we compare variance-based, moment-independent sensitivity measures and Shapley effects. In particular, moment-independent sensitivity measures and Shapley effects are compared for the first time in the case of dependent risk factors. Our framework encompasses and extends several previous results on the sensitivity analysis of annuity models. From a methodological viewpoint, the techniques compared in this paper can support analysts in building annuity models and in verifying the impact of risk drivers in their models. Numerical results using the U.S. 1990 and the U.K. 1990–1994 mortality tables show that the demographic factor is the most important risk source in low-interest rate contexts. However, when uncertainty on the two risk sources is taken into account, the financial factor becomes the global key-driver of risk. Also, interactions among the two factors appear quantitatively significant.     

Univariate and multivariate measures of risk aversion and risk premiums

This paper develops univariate and multivariate measures of risk aversion for correlated risks. We derive Rubinstein's measures of risk aversion from the risk premiums with correlated random initial wealth and risk. It is shown that these measures are not only consistent with those for uncorrelated or independent risks, but also have the corresponding local properties of the Arrow-Pratt measures of risk aversion. Thus Rubinstein's measures of risk aversion are the appropriate extension of the Arrow-Pratt measures of risk aversion in the univariate case. We also derive a risk aversion matrix from the risk premiums with correlated initial wealth and risk vectors. This matrix measure is the multivariate version of Rubinstein's measures and is also the generalization of Duncan's results for non-random initial wealth. The univariate and multivariate measures of risk aversion developed in this paper are applied to portfolio theory in Li and Ziemba [15].This research was partially supported by the National Research Council of Canada.     

Optimal Cross Hedging of Insurance Derivatives

Abstract

We consider insurance derivatives depending on an external physical risk process, for example, a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover, we check the optimal investment strategies for standard admissibility criteria. Finally, we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.     

Subregular recourse in nonlinear multistage stochastic optimization

We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. We allow for nonlinear dynamics and general objective functionals, including dynamic risk measures. We study causal operators describing the dynamics of the system and derive the Clarke subdifferential for a penalty function involving such operators. Then we introduce the concept of subregular recourse in nonlinear multistage stochastic optimization and establish subregularity of the resulting systems in two formulations: with built-in nonanticipativity and with explicit nonanticipativity constraints. Finally, we derive optimality conditions for both formulations and study their relations.

     

A Minkowskian theory of observation: Application to uncertainty and fuzziness

This paper examines various ways to introduce subjectivity in the measures of uncertainty. In the first part, by using a simple physical remark concerning discrete entropy in Shannon sense, we derive a so-called ‘complete discrete entropy’ which provides a unified approach to discrete and continuous entropy, and applies directly to variables which involves both probability and possibility.In the second part, by using three elementary axioms, we derive a Minkowskian theory of observation which holds when the observable is a pair (syntax, semantics) and which involves a parameter which is directly related to the subjectivity of the observer. This model is then applied to the observation of uncertainty, transinformation and membership, in which case it provides a new approach to fuzzy number.