Solvency Constraints Research Articles

Optimal reinsurance design under solvency constraints

We consider the optimal risk transfer from an insurance company to a reinsurer. The problem formulation considered in this paper is closely connected to the optimal portfolio problem in finance, with some crucial distinctions. In particular, the insurance company's surplus is here (as is routinely the case) approximated by a Brownian motion, as opposed to the geometric Brownian motion used to model assets in finance. Furthermore, risk exposure is dialled ‘down’ via reinsurance, rather than ‘up’ via risky investments. This leads to interesting qualitative differences in the optimal designs. In this paper, using the martingale method, we derive the optimal design as a function of proportional, non-cheap reinsurance design that maximises the quadratic utility of the terminal value of the insurance surplus. We also consider several realistic constraints on the terminal value: a strict lower boundary, the probability (Value at Risk) constraint, and the expected shortfall (conditional Value at Risk) constraints under the P and Q measures, respectively. In all cases, the optimal reinsurance designs boil down to a combination of proportional protection and option-like protection (stop-loss) of the residual proportion with various deductibles. Proportions and deductibles are set such that the initial capital is fully allocated. Comparison of the optimal designs with the optimal portfolios in finance is particularly interesting. Results are illustrated.

An insurer's optimal strategy towards a new independent business

In this paper, we investigate the optimal decision making of an insurer towards a new insurable business, whose risk is independent of the existing risk faced by the insurer. We assume that the insurer, with the objective of maximizing the expected utility of its final wealth, together with the solvency constraint and the availability of reinsurance as a risk transfer mechanism, is deciding if it is viable to underwrite a new insurance business risk. If this new business is underwritten, it is shown that a stop-loss reinsurance contract is optimal when the solvency risk is quantified by the conditional value at risk. If the regulatory regime changes to the value at risk, the optimal reinsurance form becomes more complicated; it can be either stop-loss or two-layer under the assumption that the new risk has a strictly decreasing probability density function. Numerical examples are provided to illuminate the insurer's decision making and the optimal form of the reinsurance strategy.

Responsible investments in life insurers’ optimal portfolios under solvency constraints

Sozial verantwortliches Investieren (Socially Responsible Investing, SRI) gewinnt an den Finanzmärkten aus verschiedenen Gründen zunehmend an Bedeutung, beispielsweise aufgrund der sich abzeichnenden Auswirkungen des Klimawandels oder des Strebens nach einer Netto-Null-Wirtschaft. Bestehende SRI-Ansätze haben Umwelt-, Sozial- und Governancekriterien (ESG Kriterien) als weitere Dimension in die Portfolioauswahl einbezogen; diese Ansätze konzentrieren sich allerdings auf klassische Investoren und berücksichtigen nicht spezifische Aspekte von Versicherungsunternehmen. In unserem Beitrag betrachten wir das Problem der Aktienportfolioselektion von Lebensversicherungsunternehmen. Neben dem Aktienrisiko berücksichtigt unser Modell auch andere wichtige Marktrisikokategorien von Versicherern, nämlich das Zinsrisiko und das Kreditrisiko. In Übereinstimmung mit den üblichen Standards in der Solvabilitätsregulierung messen wir das Risiko anhand der Solvenzquote, d.h. anhand des Verhältnisses zwischen dem marktbasierten Eigenkapital des Versicherers und den Solvenzkapitalanforderungen aller modellierten Risikokategorien. Hierauf aufbauend verwenden wir eine Modifikation der Portfolioselektionstheorie nach Markowitz, indem wir die Solvenzquote als Maß für das Abwärtsrisiko nutzen, um mögliche optimale Portfolios in einer dreidimensionalen (Risiko, Rendite und ESG) Kapitalallokationsebene zu identifizieren. Wir stellen fest, dass bei einer gegebenen Solvenzquote Aktienportfolios mit einem moderaten ESG-Niveau zu einer höheren erwarteten Rendite führen können als solche mit einem niedrigen ESG-Niveau. Ein sehr ehrgeiziges ESG-Niveau verringert jedoch die erwartete Rendite. Aufgrund der Besonderheiten in ihrem Geschäftsmodell können die Auswirkungen des ESG-Niveaus auf die erwartete Rendite von Lebensversicherern erheblich von den entsprechenden Auswirkungen für klassische Investoren abweichen.

A generalized Nash equilibrium problem arising in banking regulation: An existence result with Tarski's theorem

When hit with an adverse shock, banks that do not comply with capital regulation sell risky assets to satisfy their solvency constraint. When financial markets are imperfectly competitive, this naturally gives rise to a GNEP. We consider a new framework with an arbitrary number of banks and assets, and show that Tarski's theorem can be used to prove the existence of a Nash equilibrium when markets are sufficiently competitive. We also prove the existence of ϵ-Nash equilibria.

On the surplus management of funds with assets and liabilities in presence of solvency requirements

In this paper, we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu [(2003). Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends. North American Actuarial Journal 7(3), 37–56]. We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint [as in Paulsen (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 7(4), 457–473] is consequently imposed and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. The existence and uniqueness of the optimal strategies are shown. Results are illustrated.

Optimal Asset Allocation Subject to Withdrawal Risk and Solvency Constraints

This paper investigates the optimal asset allocation of a financial institution whose customers are free to withdraw their capital-guaranteed financial contracts at any time. In accounting for the asset-liability mismatch risk of the institution, we present a general utility optimization problem in a discrete-time setting and provide a dynamic programming principle for the optimal investment strategies. Furthermore, we consider an explicit context, including liquidity risk, interest rate, and credit intensity fluctuations, and show by numerical results that the optimal strategy improves both the solvency and asset returns of the institution compared to a standard institutional investor’s asset allocation.

Периодическая процентная ставка по кредиту модели Нэша с ограничением платежеспособности Базель II

This paper introduces multi–period loan interest rate Nash game models in the banking sector under regulatory solvency constraints. By taking solvency constraint as Basel II and modelling economic condition as AR(1) process, we obtain results regarding the existence of loan interest rate equilibrium. A sensitivity analysis for the solvency constraint model and some numerical results are presented.

Responsible Investments in Life Insurers' Optimal Portfolios under Solvency Constraints

Responsible Investments in Life Insurers' Optimal Portfolios under Solvency Constraints

INSURANCE VALUATION: A TWO-STEP GENERALISED REGRESSION APPROACH

AbstractCurrent approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

Enhancing an insurer's expected value by reinsurance and external financing

In this paper, we analyze a decision-making problem for an insurer with limited liability, who is subject to a solvency constraint and wants to maximize the expected value through reinsurance purchase and external financing. We impose mild conditions on the reinsurance premium principle, which are the axioms of law invariance, risk loading and preserving the convex order. These three axioms are satisfied by all the widely used premium principles, except the Esscher principle, listed in Young (2004). If value at risk (VaR) or conditional value at risk (CVaR) is adopted to calculate the insurer's regulatory capital, we show that the optimal reinsurance can be in the form of two layers. The optimal reinsurance form is reduced to one layer once the premium principle further satisfies a weak condition. In particular, under Wang's, expected value, variance and Dutch premium principles, we derive the insurer's optimal strategies of reinsurance and financing, which are obtained explicitly under the VaR risk measure but have to be solved numerically for the CVaR risk measure. Results indicate that the insurer has three types of optimal strategies: the reinsurance only strategy, external financing only strategy, and mixed strategy. Moreover, the insurer's attitude toward ceding the risk through reinsurance and increasing capital through external financing is often greatly affected by the regulatory regime, the financing cost and the reinsurance price.

(Ir)rational explorers in the financial jungle

In recent decades, several scholars have formalised Minsky’s profound insight that increasing financial fragility accompanies periods of economic stability. It must be noted, however, that incisive assessments of the role of expectations formation with heterogeneous agents has been provided only by those contributions focusing on stock-market prices. Macroeconomic models dealing with debt dynamics, on the other hand, have not yet presented such an account. It is our purpose to fill this gap in the literature by formalising switches between different heuristics in a small-scale evolutionary agent-based model where solvency aspects matter. Our system generates time-series that reproduce important empirical stylised facts such as fat tails and asymmetric skewness. We show how changes in risk perception are amplified in an asymmetric way by the right-skewed solvency constraint, leading to output fluctuations compatible with the possibility of crashes. Moreover, while the destabilising role of extrapolative behaviour is part of conventional wisdom, we discuss the conditions under which fundamentalists, the existence of resource constraints, and the time horizon of the economic unit may also lead to instability.

Loan interest rate Nash models with solvency constraints in the banking sector

ABSTRACT This paper attempts to study loan interest rate Nash game models in the banking sector under regulatory solvency constraints. By taking solvency constraints as Basel I, Basel II, and Expected Shortfall (ES), we obtain results regarding the existence of loan interest rate equilibrium. A sensitivity analysis for solvency models and some numerical results are presented. Numerical results show that the weighted loan interest rate of the Mongolian banking system is consistent with the base case of the theoretical weighted loan interest rate corresponding to the Nash equilibrium.

Assessing Financial Resource Capability on Insurance Claims Management in Nigeria: The Moderating Role of Information Technology

Abstract This study examines the extent to which financial resource capability impacts claims management in the Nigerian insurance industry and also attempts to determine if this process is moderated by information technology. Given the numerous litigations arising from claims default, the Nigerian insurance sector has earned itself a bad reputation with the consequent customer apathy. However, it is also noteworthy that most of these insurance companies do not appear to possess the financial capacity to meet claims obligations as they arise, perhaps, due to low capitalisation, poor risk assessment and solvency constraints. The research is a quantitative design that utilises the survey strategy. It is predicated on a philosophical foundation of positivism and ontological orientation of objectivism. 17 insurance companies were included in the study using the stratified sampling technique. 280 questionnaires were distributed to the 17 sampled companies out of which 235 were returned and found usable for the study. Data was analysed using the Andy Hayes Process v3.3 for regression. Findings from the study revealed that financial resources and information technology have statistically significant relationship with claims management but the relationship between financial resources and claims management is not significantly moderated by information technology.

The impact of shareholders’ limited liability on risk- and value-based management

In this paper, we analyze the consequences of shareholders’ limited liability for the risk- and value-based investment decisions made by a nonlife insurer under solvency constraints. We consider an insurance company with shareholders who are not responsible for debts exceeding the amount of their stake. This insurer aims to maximize the shareholder value based on preference functions while simultaneously controlling for the ruin probability. We show that the optimal asset allocation given this limited liability is riskier in terms of the standard deviation and in terms of the corresponding ruin probability than in the case of full liability. Moreover, the limited liability sets an incentive to follow an investment strategy that exploits the maximum admissible ruin probability level and even to select inefficient risk–return combinations. Since the limited liability problem arises if customers cannot monitor the activity – or, more precisely, the solvency – level of the insurer, and thus cannot condition their demand based on the risk, this study emphasizes the importance of public reporting as required by the Solvency and Financial Condition Report of Solvency II

UNIQUE MARKOV EQUILIBRIUM UNDER LIMITED COMMITMENT

AbstractI develop a recursive method for the characterization of competitive equilibrium under limited commitment with not‐too‐tight solvency constraints. The reputational mechanism is fragile, as it sustains constrained efficiency as well as a large set of constrained inefficient equilibria. However, I establish that the only strongly ergodic Markov equilibrium with trade is constrained efficient. The method relies on a planning program along with the theory of monotone concave operators. It is suitable for other applications to macroeconomics and dynamic contracts.

Dividend optimization for jump–diffusion model with solvency constraints

Belhaj (2010) established that a barrier strategy is optimal for the dividend problem under jump–diffusion model. However, if the optimal dividend barrier level is set too low, then the bankruptcy probability may be too high to be acceptable. This paper aims to address this issue by taking the solvency constrain into consideration. Precisely, we consider a dividend payment problem with solvency constraint under a jump–diffusion model. Using stochastic control and PIDE, we derive the optimal dividend strategy of the problem.

Consumption insurance with advance information

This paper investigates whether assuming that households possess advance information on their income shocks helps to overcome the difficulty of standard models to understand consumption insurance in the US. As our main result, we find that the quantitative relevance of advance information crucially depends on the structure of insurance markets. For a realistic amount of advance information, a complete markets model with endogenous solvency constraints due to limited commitment explains several key consumption insurance measures better than existing models without advance information. In contrast, when advance information is integrated into a standard incomplete markets model, it affects household consumption‐saving decisions too little to bridge the gap between the model and the data and can induce counterfactual correlations between current consumption growth and future income growth.

An Individual Risk Model for Premium Calculation Based on Quantile: A Comparison between Generalized Linear Models and Quantile Regression

This article deals with the use of quantile regression and generalized linear models for a premium calculation based on quantiles. A premium principle is a functional that assigns a usually loaded premium to any distribution of claims. The loaded premium is generally greater than the expected value of the loss and the difference is considered to be a risk margin or a safety loading. The failure of a right charge of individual risk rate exposes the insurer to adverse selection and, consequently, to deteriorating financial results. The article aim is to define the individual pure premium rates and the corresponding risk margin in balance with some profit or solvency constraints. We develop a ratemaking process based on a two-part model using quantile regression and a gamma generalized linear model, respectively, in order to estimate the claim’s severity quantiles. Generalized linear models focus on the estimation of the mean of the conditional loss distribution but have some drawbacks in assessing distribution moments other than the mean and are very sensitive to outliers. On the contrary, quantile regression exceeds these limits leading to the estimate of conditional quantiles and is more accurate for a better measurement of the variability of a risk class. The proposed methodology for premium calculation allows us to find further limits of the generalized linear model as the solution we found, under specific assumptions for a generalized linear model, is equivalent to the application of the expected value premium principle. Finally, the methodology we suggest for the two-part quantile regression allows reducing the practical issue of overmodeling and over-parameterization with respect to other proposals on the same topic.

Dynamic Set-Up for Designing Optimal Reinsurance Contracts

In this paper, we consider a dynamic set-up for designing optimal reinsurance contracts. The main objective is to maximise the lifetime dividends of an insurance company. We study three problems. First, we consider a general dividend maximisation problem with just budget constraints; second, we add solvency conditions to the previous problem; and finally, we consider a problem with solvency constraints and a particular type of dividend functions. We show that the optimal solutions of the first and second problems do not evolve over time and can be found by solving a static problem, whereas the third problem's optimal solution evolves over time and depends on the problem specification and the state variables. We find the dual problem of the third problem and show that the optimal reinsurance contract always has a multilayer structure. Last, we consider some examples and show that in many cases, the contract is indeed a two-layer policy.

Taking Regulation Seriously: Fire Sales Under Solvency and Liquidity Constraints

We build a framework for modelling fire sales where banks face both liquidity and solvency constraints and choose which assets to sell in order to minimise liquidation losses. Banks constrained by the leverage ratio prefer to first sell assets that are liquid and held in small amounts, while banks constrained by the risk-weighted capital ratio and the liquidity coverage ratio need to trade off assets’ liquidity with their regulatory weights. We calibrate the model to the UK banking system, and find that banks’ optimal liquidation strategies translate into moderate fire-sale losses even for extremely large solvency shocks. By contrast, severe funding shocks can generate significant losses. Thus models focusing exclusively on solvency risk may significantly underestimate the extent of contagion via fire sales. Moreover, when studying combined funding and solvency shocks, we find complementarities between the two shocks’ effects that cannot be reproduced by focusing on either shock in isolation.