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Nowadays insurers have to account for potentially complex dependence between risks. In the field of loss reserving, there are many parametric and non-parametric models attempting to capture dependence between business lines. One common approach has been to use additive background risk models (ABRMs) which provide rich and interpretable dependence structures via a common shock model. Unfortunately, ABRMs are often restrictive. Models that capture necessary features may have impractical to estimate parameters. For example models without a closed-form likelihood function for lack of a probability density function (e.g. some Tweedie, Stable Distributions, etc). We apply a modification of the continuous generalised method of moments (CGMM) of [Carrasco and Florens, 2000] which delivers comparable estimators to the MLE to loss reserving. We examine models such as the one proposed by [Avanzi et al., 2016] and a related but novel one derived from the stable family of distributions. Our CGMM method of estimation provides conventional non-Bayesian estimates in the case where MLEs are impractical., Comment: 42 pages. Preprint of paper from author's PhD thesis

In order to properly manage risk, practitioners must understand the aggregate risks they are exposed to. Additionally, to properly price policies and calculate bonuses the relative riskiness of individual business units must be well understood. Certainly, Insurers and Financiers are interested in the properties of the sums of the risks they are exposed to and the dependence of risks therein. Realistic risk models however must account for a variety of phenomena: ill-defined moments, lack of elliptical dependence structures, excess kurtosis and highly heterogeneous marginals. Equally important is the concern over industry-wide systematic risks that can affect multiple business lines at once. Many techniques of varying sophistication have been developed with all or some of these problems in mind. We propose a modification to the classical individual risk model that allows us to model company-wide losses via the class of Multivariate Stable Distributions. Stable Distributions incorporate many of the unpleasant features required for a realistic risk model while maintaining tractable aggregation and dependence results. We additionally compute the Tail Conditional Expectation of aggregate risks within the model and the corresponding allocations., Comment: 28 pages. Preprint of paper from author's PhD thesis

This study presents two models to optimize pressure management in water distribution networks. The first model forecasts pressure at distribution points and compares predictions with actual data to detect anomalies such as leaks and blockages. Early detection allows for timely interventions, minimizing economic losses and ensuring system sustainability. The second model estimates the necessary inlet pressure based on the influence of various distribution points, ensuring consistent water supply while reducing waste and optimizing resource management. Both models utilize modern machine learning algorithms to enhance the prediction process. The methodology includes the CNN-EMD model, which analyzes historical data collected every 15 minutes over two months to predict future pressures. The Empirical Mode Decomposition (EMD) method identifies fluctuations and anomalies, improving prediction accuracy. The second model combines CNN, EMD, and LSTM techniques to forecast required inlet pressure, emphasizing the impact of distribution points. Results show that the CNN-EMD and CNN-EMD-LSTM models enhance pressure management capabilities, with the first model achieving an anomaly detection accuracy of 85% to 95% and the second model predicting inlet pressure with an average accuracy of 93%. This enables flexible system adjustments and identifies critical factors affecting inlet pressure. In conclusion, advanced machine learning models like CNN-EMD and LSTM significantly improve pressure management in water distribution networks, facilitating early issue identification, ensuring efficient water supply, and optimizing resource management for future generations., Comment: 13 pages, 14 figures

The stability of income payments in a pooled annuity fund is studied. In those funds, members receive a fluctuating income depending on their experienced mortality in exchange for their pension savings. The focus is on describing the influence of different initial savings on the ability of the fund to provide a stable income in retirement. Because of this, members coincide in their characteristics except for their initial savings. We identify a term, which we dub ``implied number of homogeneous members'', that directly links the initial savings to the size of the income fluctuations. Our main contribution is the analysis of this term and the development of a criterion to answer the question of whether or not a given group of same-aged people should pool their funds together., Comment: 31 pages, 7 figures

Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations., Comment: 24 pages, 4 figures

In this paper we consider an optimal investment and reinsurance problem with partially unknown model parameters which are allowed to be learned. The model includes multiple business lines and dependence between them. The aim is to maximize the expected exponential utility of terminal wealth which is shown to imply a robust approach. We can solve this problem using a generalized HJB equation where derivatives are replaced by generalized Clarke gradients. The optimal investment strategy can be determined explicitly and the optimal reinsurance strategy is given in terms of the solution of an equation. Since this equation is hard to solve, we derive bounds for the optimal reinsurance strategy via comparison arguments.

We generalize Quasi-Linear Means by restricting to the tail of the risk distribution and show that this can be a useful quantity in risk management since it comprises in its general form the Value at Risk, the Tail Value at Risk and the Entropic Risk Measure in a unified way. We then investigate the fundamental properties of the proposed measure and show its unique features and implications in the risk measurement process. Furthermore, we derive formulas for truncated elliptical models of losses and provide formulas for selected members of such models.

This survey gives an introduction to monetary measures of risk as monotone and cash additive functions on spaces of univariate random variables. Primal and dual representation results as well as several examples are discussed. Principal ways to construct risk measures are given and extensions to more general situations indicated., Comment: 55 references

Starting from the global financial crisis to the more recent disruptions brought about by geopolitical tensions and public health crises, the volatility of risk in financial markets has increased significantly. This underscores the necessity for comprehensive risk measures capable of capturing the complexity and heightened fluctuations in market volatility. This need is crucial not only for new financial assets but also for the traditional financial market in the face of a rapidly changing financial environment and global landscape. In this paper, we consider the risk measures on a special space $L^{p(\cdot)}$, where the variable exponent $p(\cdot)$ is no longer a given real number as in the conventional risk measure space $L^{p}$, but rather a random variable reflecting potential fluctuations in volatility within financial markets. Through further development of axioms related to this class of risk measures, we also establish dual representations for them.

Under Solvency II the computation of capital requirements is based on value at risk (V@R). V@R is a quantile-based risk measure and neglects extreme risks in the tail. V@R belongs to the family of distortion risk measures. A serious deficiency of V@R is that firms can hide their total downside risk in corporate networks, unless a consolidated solvency balance sheet is required for each economic scenario. In this case, they can largely reduce their total capital requirements via appropriate transfer agreements within a network structure consisting of sufficiently many entities and thereby circumvent capital regulation. We prove several versions of such a result for general distortion risk measures of V@R-type, explicitly construct suitable allocations of the network portfolio, and finally demonstrate how these findings can be extended beyond distortion risk measures. We also discuss why consolidation requirements cannot completely eliminate this problem. Capital regulation should thus be based on coherent or convex risk measures like average value at risk or expectiles.

In this article we consider the parameter risk in the context of internal modelling of the reserve risk under Solvency II. We discuss two opposed perspectives on parameter uncertainty and point out that standard methods of classical reserving focusing on the estimation error of claims reserves are in general not appropriate to model the impact of parameter uncertainty upon the actual risk of economic losses from the undertakings's perspective. Referring to the requirements of Solvency II we assess methods to model parameter uncertainty for the reserve risk by comparing the probability of solvency actually attained when modelling the solvency risk capital requirement based on the respective method to the required confidence level. Using the simple example of a normal model we show that the bootstrapping approach is not appropriate to model parameter uncertainty according to this criterion. We then present an adaptation of the approach proposed in \cite {froehlich2014}. Experimental results demonstrate that this new method yields a risk capital model for the reserve risk achieving the required confidence level in good approximation.

In this paper, we discuss the Cram\'er-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift $a$ and volatility $\sigma> 0.$ By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/\sigma^2 > 1$. More importantly, without this assumption, we show that the probability of ruin is certain for all initial capital $u$, if $2a/\sigma^2 \le 1$.

This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions., Comment: 30 pages

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional sigma-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for different notions of time consistency. In particular we show how bubbles may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure., Comment: 40 pages

This paper is concerned with existence and uniqueness of M-solutions of backward stochastic Volterra integral equations (BSVIEs for short), which Lipschitz coefficients are allowed to be random, which generalize the results in [15]. Then a class of continuous time dynamic dynamic coherent risk measures is derived, allowing the riskless interest rate to be random, which is different from the case in [15].

It is known, but perhaps not well-known, that when the mortality is assumed to be of Gompertz-Makeham-type, the expected remaining life-length and the commutation functions used for calculating the expected values of various types of life insurances can be expressed with an incomplete gamma function with a negative shape parameter. This is not of much use if ones software cannot calculate these values. The aim of this note is to show that one can express the commutation functions using only the exponential function, the (ordinary) gamma function and the gamma distribution function, which are all implemented in common statistical and spreadsheet software. This eliminates the need to evaluate the commutation functions and expected remaining life-length with numerical integration., Comment: 4 pages

We develop a theory for valuing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We apply our method to value life annuities. One result of our paper is that the value of the life annuity is {\it identical} to the upper good deal bound of Cochrane and Sa\'{a}-Requejo (2000) and of Bj\"{o}rk and Slinko (2006) applied to our setting. A second result of our paper is that the value per contract solves a {\it linear} partial differential equation as the number of contracts approaches infinity. One can represent the limiting value as an expectation with respect to an equivalent martingale measure (as in Blanchet-Scalliet, El Karoui, and Martellini (2005)), and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market's price of mortality risk., Comment: Keywords: Stochastic mortality; pricing; annuities; Sharpe ratio; non-linear partial differential equations; market price of risk; equivalent martingale measures

In this paper we give an explicit expression for the local time of the classical risk process and associate it with the density of an occupational measure. To do so, we approximate the local time by a suitable sequence of absolutely continuous random fields. Also, as an application, we analyze the mean of the times $s \in [0,T]$ such that $0\leq X_{s} \leq X_{s+\epsilon} $ for some given $\epsilon>0$., Comment: 15 pages

In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cram\'{e}r-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time., Comment: 2 figures, 19pp

We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. (2005) for such processes with Erlang inter-claim times. We illustrate our result in the cases of gamma and mixed exponential inter-claim time distributions., Comment: 12 pages, 2 figures

We derive recursions for the probability distribution of random sums by computer algebra. Unlike the well-known Panjer-type recursions, they are of finite order and thus allow for computation in linear time. This efficiency is bought by the assumption that the probability generating function of the claim size be algebraic. The probability generating function of the claim number is supposed to be from the rather general class of D-finite functions.

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can interpret the limiting price as an expectation with respect to an equivalent martingale measure. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. We present a numerical example to illustrate our results, along with the corresponding algorithms.

We develop a theory for pricing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We prove that our ensuing valuation formula satisfies a number of desirable properties. For example, we show that it is subadditive in the number of contracts sold. A key result is that if the hazard rate is stochastic, then the risk-adjusted survival probability is greater than the physical survival probability, even as the number of contracts approaches infinity., Comment: JEL Classification: G13; G22; C60

We introduce, in continuous time, an axiomatic approach to assign to any financial position a dynamic ask (resp. bid) price process. Taking into account both transaction costs and liquidity risk this leads to the convexity (resp. concavity) of the ask (resp. bid) price. Time consistency is a crucial property for dynamic pricing. Generalizing the result of Jouini and Kallal, we prove that the No Free Lunch condition for a time consistent dynamic pricing procedure (TCPP) is equivalent to the existence of an equivalent probability measure $R$ that transforms a process between the bid process and the ask process of any financial instrument into a martingale. Furthermore we prove that the ask price process associated with any financial instrument is then a $R$-supermartingale process which has a cadlag modification. Finally we show that time consistent dynamic pricing allows both to extend the dynamics of some reference assets and to be consistent with any observed bid ask spreads that one wants to take into account. It then provides new bounds reducing the bid ask spreads for the other financial instruments.

We consider a trader who wants to direct his portfolio towards a set of acceptable wealths given by a convex risk measure. We propose a black-box algorithm, whose inputs are the joint law of stock prices and the convex risk measure, and whose outputs are the numerical values of initial capital requirement and the functional form of a trading strategy to achieve acceptability. We also prove optimality of the obtained capital., Comment: 17 pages

Working in a continuous time setting, we extend to the general case of dynamic risk measures continuous from above the characterization of time consistency in terms of ``cocycle condition'' of the minimal penalty function. We prove also the supermartingale property for general time consistent dynamic risk measures. When the time consistent dynamic risk measure (continuous from above) is normalized and non degenerate, we prove, under a mild condition, that the dynamic risk process of any financial instrument has a cadlag modification. This condition is always satisfied in case of continuity from below., Comment: 23 pages

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 abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the 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., Comment: 26 pages. Substantially revised version with a number of new results added

We propose a pricing technique based on coherent risk measures, which enables one to get finer price intervals than in the No Good Deals pricing. The main idea consists in splitting a liability into several parts and selling these parts to different agents. The technique is closely connected with the convolution of coherent risk measures and equilibrium considerations. Furthermore, we propose a way to apply the above technique to the coherent estimation of the Greeks.

This paper deals with applications of coherent risk measures to pricing in incomplete markets. Namely, we study the No Good Deals pricing technique based on coherent risk. Two forms of this technique are presented: one defines a good deal as a trade with negative risk; the other one defines a good deal as a trade with unusually high RAROC. For each technique, the fundamental theorem of asset pricing and the form of the fair price interval are presented. The model considered includes static as well as dynamic models, models with an infinite number of assets, models with transaction costs, and models with portfolio constraints. In particular, we prove that in a model with proportional transaction costs the fair price interval converges to the fair price interval in a frictionless model as the coefficient of transaction costs tends to zero. Moreover, we study some problems in the ``pure'' theory of risk measures: we present a simple geometric solution of the capital allocation problem and apply it to define the coherent risk contribution. The mathematical tools employed are probability theory, functional analysis, and finite-dimensional convex analysis., Comment: 4 figures

We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks: - coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price of oil, S&P500 index, or the credit spread; - risk contributions and factor risk contributions, which provide a coherent alternative to the sensitivity coefficients. We also propose two particular classes of coherent risk measures called Alpha V@R and Beta V@R, for which all the objects described above admit an extremely simple empirical estimation procedure. This procedure uses no model assumptions on the structure of the price evolution. Moreover, we consider the problem of the risk management on a firm's level. It is shown that if the risk limits are imposed on the risk contributions of the desks to the overall risk of the firm (rather than on their outstanding risks) and the desks are allowed to trade these limits within a firm, then the desks automatically find the globally optimal portfolio.

This paper is the continuation of "Pricing with coherent risk" and deals with further applications of coherent risk measures to problems of finance. First, we study the optimization problem. Three forms of this problem are considered. Furthermore, the results obtained are applied to the optimality pricing. Again three forms of this technique are considered. Finally, we study the equilibrium problem both in the unconstrained and in the constrained forms. We establish the equivalence between the global and the competitive optima and give a dual description of the equilibrium. Moreover, we provide an explicit geometric solution of the constrained equilibrium problem. Most of the results are presented on two levels: on a general level the results have a probabilistic form; for a static model with a finite number of assets, the results have a geometric form., Comment: 12 figures

We consider the problem of decomposing monetary risk in the presence of a fully traded market in {\it some} risks. We show that a mark-to-market approach to pricing leads to such a decomposition if the risk measure is time-consistent in the sense of Delbaen., Comment: 17 pages

In this paper the authors study a model for the optimal operation of a bank or insurance company which was recently introduced by Peura and Keppo. The model generalizes a previous one of Milne and Robertson by allowing the bank to raise capital as well as to pay out dividends. Optimal operation of the bank is determined by solving an optimal control problem. In this paper it is shown that the solution of the optimal control problem proposed by Peura and Keppo exists for all values of the parameters and is unique., Comment: 21 pages

Let \{X_1, X_2, ...\} be a sequence of independent and identically distributed positive random variables of Pareto-type with index \alpha>0 and let \{N(t); t\geq 0\} be a counting process independent of the X_i's. For any fixed t\geq 0, define T_{N(t)}:=\frac{X_1^2 + X_2^2 + ... + X_{N(t)}^2} {(X_1 + X_2 + ... + X_{N(t)})^2} if N(t)\geq 1 and T_{N(t)}:=0 otherwise. We derive limiting distributions for T_{N(t)} by assuming some convergence properties for the counting process. This is even achieved when both the numerator and the denominator defining T_{N(t)} exhibit an erratic behavior (\mathbb{E}X_1=\infty) or when only the numerator has an erratic behavior (\mathbb{E}X_1<\infty and \mathbb{E}X_1^2=\infty). Thanks to these results, we obtain asymptotic properties pertaining to both the sample coefficient of variation and the sample dispersion., Comment: 21 pages

Over-the-counter derivatives have contributed significantly to the effectiveness and efficiency of the international financial system but also entail significant counterparty credit risk. Collateralization is one of the most important and widespread credit risk mitigation techniques used in derivatives transactions. However, the relevant decisions are often made in an ad-hoc manner, without reference to an analytical framework. Very little academic research has addressed the quantitative analysis of collateralization for contingent credit risk control. The issue of mark-to-market timing becomes important for reducing credit exposure of illiquid and long term derivative contracts due to the difficulty and cost of marking to market. the goal of this research is to propose a framework for minimizing the potential credit exposure of collateralized derivative transactions by optimizing mark-to-market timing., Comment: 21 pages, 6 figures, presented at the 11th International Conference on Computing in Economics and Finance

This seems to be the first English translation of this paper from the French original, ``Sur les rentes viageres''. In the paper, Euler gives a general formula for calculating the price of a life annuity that yields a certain amount per year, assuming the annuity manager can get a 5 percent return, for people of different ages. He also gives formulas to calculate the price of annuities that only start to pay out a certain number of years after they are purchased. He gives many numerical examples, giving tables for the prices of annuities for annuitants up to 90 years old., Comment: 9 pages, English translation of ``Sur les rentes viageres'', Memoires de l'academie des sciences de Berlin 16 (1767), 165-175