The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia

A multi-country test of brief reappraisal interventions on emotions during the COVID-19 pandemic.

The COVID-19 pandemic has increased negative emotions and decreased positive emotions globally. Left unchecked, these emotional changes might have a wide array of adverse impacts. To reduce negative emotions and increase positive emotions, we tested the effectiveness of reappraisal, an emotion-regulation strategy that modifies how one thinks about a situation. Participants from 87 countries and regions (n = 21,644) were randomly assigned to one of two brief reappraisal interventions (reconstrual or repurposing) or one of two control conditions (active or passive). Results revealed that both reappraisal interventions (vesus both control conditions) consistently reduced negative emotions and increased positive emotions across different measures. Reconstrual and repurposing interventions had similar effects. Importantly, planned exploratory analyses indicated that reappraisal interventions did not reduce intentions to practice preventive health behaviours. The findings demonstrate the viability of creating scalable, low-cost interventions for use around the world

Robust pricing in insurance and energy markets

Preise von Verträgen mit riskanten Aspekten sind typischerweise mit spezifischen Unsicherheiten und Wahrscheinlichkeiten von unvorteilhaften Szenarien verbunden. Versicherungsunternehmen tragen Verlustrisiken im Austausch für Prämien, die von den Schadensverteilungen abhängen. Ein weiteres Beispiel, bei dem Risiko gegen einen fixen Preis ausgetauscht wird, sind Swap-Verträge: Strom-Futures können als Swap-Verträge angesehen werden, wobei die variable Komponente Spot-Preise und die fixe Komponente konstante Preise für die Stromlieferung einer längeren Periode darstellen. Das Hauptziel dieser Arbeit ist die Integration von Model-Unsicherheiten in die Preisgestaltung dieser Verträge. Es werden komplexe Strukturen in den unterschiedlichen Bereichen von Energiemärkten untersucht und die Preisgestaltung einer Realoption unter Modell-Ambituität untersucht. Zunächst werden Distortionsprinzipien bei der Preisgestaltung von Versicherung behandelt. Es werden geschlossene Lösungen für optimale Distortionsprämien unter Modell-Ambiguität mit Wasserstein-Distanzen berechnet. In verschiedenen Fällen werden die Verteilungen, die den optimalen Preis erreichen, errechnet. Bezüglich des Distortionsprinzips kann festgehalten werden, dass der Preis für Ambiguität nur vom Ambiguitätsradius und den Distortionsfunktionen abhängt und nicht von der Basis-Verteilung. Zusätzlich wird die Unbeschränktheit der robusten Distortionsprämie charakterisiert. Außerdem werden Distortionsfunktion auf Grund der beobachteten Preise identifiziert. Dafür wird eine Methode vorgeschlagen, um die Distortionsfunktionen nicht-parametrisch in zwei Fällen zu ermitteln: Average value-at-risk und Potenzfunktionen. Im zweiten Teil der Arbeit werden Regeln für die Versicherungspreisgestaltung und Regeln für die Strom-Future-Preisgestaltung gemeinsam betrachtet. Dadurch, dass Elektrizität nicht gespeichert werden kann, untersuchen viele Autoren die empirischen Daten, um die Future-Preise und die Risikoprämien in diesem Markt zu erklären. Die vorhandene Literatur wird erweitert und es wird eine Erklärung der Preisbildung dieser Verträge mit drei unterschiedlichen Komponenten vorgeschlagen: eine Distortionsprämie, einen Korrekturfaktor und eine Ambiguitätsprämie. Diese drei Teile stellen einen allgemeinen Mechanismus von Future-Preisen dar. Es wurde nachgewiesen, dass die Future-Preise mit langer Restlaufzeit zunehmen. Zudem wird ein saisonales Muster der Riskioprämien identifiziert und die Veränderung in der Risikoaversion in Abhängigkeit von den Restlaufzeiten erklärt. Die Ambiguitäts-Prämie ist nicht null und nimmt mit Restlaufzeiten der Base-Future-Verträge zu. Für diese Berechnungen wird ein neues Regime-Switching-Modell für Spot-Preise spezifiziert. Der letzte Teil dieser Arbeit untersucht eine angemessene Bewertung eines Wärmekraftwerks unter Einbeziehung von Modell-Ambiguität. Die verschiedenen Unsicherheiten, die sich auf die erwarteten Profite dieser Realoption auswirken, sind Strompreise, Treibstoffpreise und CO2-Zertifikate. Für das Wärmekraftwerk werden wöchentlich Entscheidungen getroffen, die die Produktion einer gesamten Woche festlegen, obwohl die Unsicherheiten den Profit innerhalb einer Woche beeinflussen können. Zunächst werden die Unsicherheiten in einem Lattice-Prozess diskretisiert und quantisiert. Um unterschiedliche Preise innerhalb einer Woche zu simulieren, wird ein Interpolations-Prozess, genauer ein Bridge-Prozess, vorgeschlagen. Folgend wird eine Distanz zwischen Lattice-Prozessen vorgeschlagen, die geeignet ist dynamische Probleme rückwärts in der Zeit zu lösen. Diese Distanz ist eine uniforme Wasserstein-Distanz mit einer zugrundeliegenden Metrik in Abhängigkeit von den Zuständen des Wärmekraftwerks. Die emprischen Ergebnissen zeigen dass die Produktion konservativer und der Profit umso geringer ist, je größer der Ambiguitätsradius ist. Es wird zwar ein spezifisches Problem gelöst, jedoch können die Ergebnisse auf viele ähnliche mehrstufige Entscheidungsprobleme angewendet werden.Prices of contracts with risky aspects are typically linked to specific uncertainties and probabilities of adverse scenarios. Insurance companies carry the risk of losses in exchange for a premium, which depends on the loss distribution. Another example where risk is exchanged for a fixed price is swap contracts. Electricity futures can be seen as swaps where the floating component are spot prices and the fixed component is a constant price for delivering electricity over a longer period. The primary goal of this thesis is the incorporation of model ambiguity for pricing these contracts. Moreover, we contemplate the complex structure of energy markets. For this reason, we also explore pricing a real option under model ambiguity. First of all, we study the theoretical properties of the distortion principle for insurance pricing. We find closed-form solutions for the optimal distortion premium under model ambiguity using Wasserstein distances. In various cases, we also find the distributions that reach the optimal prices. For the distortion principle, we can conclude that the price to pay for ambiguity only depends on the ambiguity radius and the distortion function, but not on the initial distribution. Additionally, we characterize the unboundedness of the robust distortion premium. Besides, we investigate the identification of distortion functions from observed prices. We propose a method to recover them from simulated prices in two cases: the average value-at-risk and power distortion principle. In the second part of this thesis, we bring together insurance pricing rules and electricity futures pricing rules. Due to the non-storability of electricity, many authors study different rules and empirical results to explain futures prices and the risk premia in this market. We extend the present literature and propose to explain the price formation of these contracts with three different quantities: the distortion premium, a correction factor and an ambiguity premium. [option]. The ambiguity premium is significant and increases with time-to-delivery for base futures. For these calculations, we specify a new regime-switching model for spot prices. [option1] These three factors capture a general mechanism of futures prices. We conclude the magnitude of futures increases with time-to-delivery. In addition, we recover a seasonal pattern of the risk premia and explain the changes in risk aversion depending on time-to-delivery. [option2] These three factors capture main characteristics of futures prices and the risk premia. Among them, we recover a seasonal pattern of the risk premia and explain the changes in risk aversion depending on time-to-delivery. The last part of this thesis studies an appropriate evaluation of a thermal power plant by incorporating model ambiguity. The different uncertainties that affect the expected profits of this real option are electricity prices, fuel prices and CO2 allowances. The power plant takes weekly decisions fixing the production for an entire week, while the uncertainties may affect the profit within weeks. Firstly, we discretize and quantize the uncertainties in a lattice process. To simulate different prices within weeks, we introduce an interpolation process called bridge process. Secondly, we propose a distance between lattice processes, which is tractable for solving dynamic problems backwards in time. This distance is a Wasserstein distance type with an underlying metric dependent on the state of the power plant. Our empirical results show that the larger the ambiguity radius is, the more conservative the production, and the less the achieved profit is. Although we solve a specific problem, our results can be applied to different multistage decision problems

The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia

The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia

Distributionally robust optimization with multiple time scales: valuation of a thermal power plant

The valuation of a real option is preferably done with the inclusion of uncertainties in the model, since the value depends on future costs and revenues, which are not perfectly known today. The usual value of the option is defined as the maximal expected (discounted) profit one may achieve under optimal management of the operation. However, also this approach has its limitations, since quite often the models for costs and revenues are subject to model error. Under a prudent valuation, the possible model error should be incorporated into the calculation. In this paper, we consider the valuation of a power plant under ambiguity of probability models for costs and revenues. The valuation is done by stochastic dynamic programming and on top of it, we use a dynamic ambiguity model for obtaining the prudent minimax valuation. For the valuation of the power plant under model ambiguity we introduce a distance based on the Wasserstein distance. Another highlight of this paper is the multiscale approach, since decision stages are defined on a weekly basis, while the random costs and revenues appear on a much finer scale. The idea of bridging stochastic processes is used to link the weekly decision scale with the finer simulation scale. The applicability of the introduced concepts is broad and not limited to the motivating valuation problem

Distributionally robust optimization with multiple time scales: valuation of a thermal power plant

The valuation of a real option is preferably done with the inclusion of uncertainties in the model, since the value depends on future costs and revenues, which are not perfectly known today. The usual value of the option is defined as the maximal expected (discounted) profit one may achieve under optimal management of the operation. However, also this approach has its limitations, since quite often the models for costs and revenues are subject to model error. Under a prudent valuation, the possible model error should be incorporated into the calculation. In this paper, we consider the valuation of a power plant under ambiguity of probability models for costs and revenues. The valuation is done by stochastic dynamic programming and on top of it, we use a dynamic ambiguity model for obtaining the prudent minimax valuation. For the valuation of the power plant under model ambiguity we introduce a distance based on the Wasserstein distance. Another highlight of this paper is the multiscale approach, since decision stages are defined on a weekly basis, while the random costs and revenues appear on a much finer scale. The idea of bridging stochastic processes is used to link the weekly decision scale with the finer simulation scale. The applicability of the introduced concepts is broad and not limited to the motivating valuation problem

The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia.© The Author(s) 201