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Optimal reinsurance design under distortion risk measures and reinsurer’s default risk with partial recovery

AbstractReinsurers may default when they have to pay large claims to insurers but are unable to fulfill their obligations due to various reasons such as catastrophic events, underwriting losses, inadequate capitalization, or financial mismanagement. This paper studies the problem of optimal reinsurance design from the perspectives of both the insurer and reinsurer when the insurer faces the potential default risk of the reinsurer. If the insurer aims to minimize the convex distortion risk measure of his retained loss, we prove the optimality of a stop-loss treaty when the promised ceded loss function is charged by the expected value premium principle and the reinsurer offers partial recovery in the event of default. For any fixed premium loading set by the reinsurer, we then derive the explicit expressions of optimal deductible levels for three special distortion functions, including the TVaR, Gini, and PH transform distortion functions. Under these three explicit distortion risk measures adopted by the insurer, we seek the optimal safety loading for the reinsurer by maximizing her net profit where the reserve capital is determined by the TVaR measure and the cost is governed by the expectation. This procedure ultimately leads to the Bowley solution between the insurer and the reinsurer. We provide several numerical examples to illustrate the theoretical findings. Sensitivity analyses demonstrate how different settings of default probability, recovery rate, and safety loading affect the optimal deductible values. Simulation studies are also implemented to analyze the effects induced by the default probability and recovery rate on the Bowley solution.

Pure risk, agency conflict, and hedging

This study develops a theoretical model to analyze the asset substitution problem over insurance decisions. We find that agency conflict is related to a firm’s risk level and capital structure. In particular, at the optimal leverage, agency conflict occurs only when the risk level is relatively high, which explains why insurance covenants are typically for significant pure risks. Moreover, when the risk level is specified, agency conflict over insurance decisions occurs within a specific leverage range. This is consistent with the findings of some research on the asset substitution problem over speculative risk choices. In addition, we consider premium loadings and conclude that full hedging is not a firm’s optimal risk management strategy, contributing to the literature on optimal hedging decisions with transaction frictions. Our framework with premium loadings can also explain many insurance phenomena, such as risk retention for small losses and subsidies for catastrophe insurance.

On technical bases and surplus in life insurance

We revisit surplus on general life insurance contracts, represented by Markov models. We classify technical bases in terms of boundary conditions in Thiele's equation(s), allowing more general regulations than Scandinavian-style ‘first-order/second-order’ regimes, and replacing the traditional retrospective policy value. We propose a ‘canonical’ model with three technical bases (premium, valuation, accumulation) and show how each pair of bases defines premium loadings and surplus. Along with a ‘true’ or ‘real-world’ experience basis, this expands fundamental results of Ramlau-Hansen [(1988b). The emergence of profit in life insurance. Insurance: Mathematics and Economics, 7(4), 225–236]. We conclude with two applications: lapse-supported business; and the retrospectively-oriented regime proposed by Møller and Steffensen [(2007). Market-valuation methods in life and pension insurance. Cambridge University Press].

On Correlation Aversion and Insurance Demand

This is a study of decision problems under two-dimensional risk. We use an existing index of absolute correlation aversion to conveniently classify bivariate preferences, with respect to attitudes toward this risk. This classification seems to be more important than whether decision makers are correlation-averse or correlation-seeking for the study of insurance demand when a loss has a multidimensional impact. On this note, we also re-examine Mossin’s theorem under bivariate preferences, where full insurance is preferred with a fair premium, while less than full coverage is preferred with a proportional premium loading. Furthermore, based on the comparative statics of this two-dimensional insurance model for changes in correlation aversion, we derive testable implications about the classification of bivariate utility functions. For the particular case when the two-dimensional risk can be interpreted as risk on income and health, we identify the form of separable utility functions depending on health status and income that is consistent with household disability insurance decisions.

Age penalties and take-up of private health insurance.

Financial penalties for delayed enrollment could be useful tools to encourage people to enroll earlier in health insurance markets, but little is known about how effective they are. We use a large administrative dataset for a 10% random sample of all Australian tax-filers to study how people respond to a step-wise age-based penalty, and whether the effect has changed over time. Individuals must pay a 2% premium surcharge for each year they delay enrollment beyond age 31. The penalty stops after 10years of continuous hospital cover. The age-based penalty creates discontinuities in the incentive to insure by age, which we exploit to estimate causal effects. We find that people respond as expected to the initial age-penalty, but not to subsequent penalties. The 2% premium loading results in a 0.78-3.69 percentage points (or 2.1%-9.0%) increase in the take-up rate at age 31. We simulate the penalty impact and implications of potential reforms, and conclude that modest changes around the policy make little difference in the age distribution of insured, premiums or take-up rates. Our study provides important evidence on an understudied area in the literature and offers insights for countries considering financial penalties.

Stackelberg reinsurance chain under model ambiguity

In this paper, we consider a continuous-time version of a reinsurance chain, which is sequentially formed by n+1 companies, with the first company being the primary insurer and the rest being reinsurers. Because of possible model misspecification, all companies are ambiguous about the original risk of the primary insurer. We model each reinsurance contracting problem as a Stackelberg game, in which the assuming reinsurer acts as the leader while the ceding company is the follower. Reinsurance is priced using the mean-variance premium principle and all companies are risk neutral under their own beliefs. We obtain equilibrium indemnities, premium loadings, and distortions in closed form, all of which are proportional to the original risk, with the corresponding proportions decreasing along the chain. We also show that the reinsurance chain with ambiguity aversions in increasing order is optimal from the perspectives of both selfish individual companies and an unselfish central planner.

Calculation of the discount rate for the Russian market in present-day conditions

Subject. This article focuses on the consideration of the algorithm and methodological approaches to assessing the cost of capital. Objectives. The article aims to develop a practical algorithm for assessing the cost of capital and calculating the discount rate based on ruble risk-free assets and Russian financial statistics. Methods. For the study, we used the Weighted Average Capital Cost (WACC) and Capital Asset Pricing (CAPM) models. Results. The article offers a tested step-by-step algorithm for calculating the cost of capital and risk premium loading to it using domestic financial statistics. It also justifies the need to take non-systematic risks out and beyond the weighted average capital cost and offers a deterministic mechanism for their assessment. Conclusions and Relevance. The author-developed algorithm for estimating the cost of capital, as well as methodological recommendations for its use, can be used by domestic investors to make the necessary decisions, as well as by the financial services of Russian companies to build the economic policy of the company. Using the proposed approach may help obtain a more adequate assessment of the cost of capital of Russian companies for domestic investors, thereby improving their competitiveness.

On some effects of dependencies on an insurer’s risk exposure, probability of ruin, and optimal premium loading

We study how the presence of dependencies between risks in a population of prospective insurance customers translates into risk exposure for an insurance company, depending on the company’s market share on the various risks. It turns out that the dependency structure in the insurer’s portfolio may differ significantly from the dependency structure of those risks in the general population, even when policyholders for different risks are selected independently. We obtain an upper bound for the difference between the ruin probability and its estimate based on the company’s portfolio marginal distributions. Under certain conditions, dependencies between risks in the portfolio of a company with small market shares are mild. We characterize the optimal loadings and market shares, assuming generic demand functions for the different risks.

The impact of counterparty risk on the basis risk of industry loss warranties and on (collateralized) reinsurance under (non-)linear dependence structures

PurposeThe purpose of this article is to investigate the impact of counterparty risk on the basis risk of industry loss warranties as well as on reinsurance with and without collateral under different dependence structures. The authors additionally compare the solvency and Sharpe ratio for different premium loadings and contract parameters.Design/methodology/approachThe authors propose a model framework extension to account for the counterparty risk of risk transfer arrangements. Copulas are used to also take into account non-linear dependencies between risk factors, and Monte Carlo simulation is employed to derive numerical results and to conduct sensitivity analyses.FindingsThe authors show that the impact of counterparty risk is particularly pronounced for higher degrees of dependencies and tail dependent losses, i.e. in cases of basis risk levels that appear low if counterparty risk is not considered. With respect to counterparty risk management, the authors find that already partial collateralization limits counterparty and basis risk to more acceptable levels.Practical implicationsThe study results are particularly relevant to practitioners, as insurers may not only underestimate the “true” basis risk of index-linked instruments, but also the effect of counterparty risk of reinsurance contracts along with the consequences for solvency and profitability.Originality/valueThe authors extend existing literature by allowing for the (partial) default of industry loss warranties and reinsurance under different dependence structures. Furthermore, the authors include profitability in addition to risk considerations. The interaction effects between counterparty risk and the basis risk of index-based alternative risk transfer instruments are largely unstudied, despite their considerable relevance in practice.

Designing Annuities with Flexibility Opportunities in an Uncertain Mortality Scenario

We consider annuity designs in which the benefit amount is allowed to fluctuate (up or down), based on a given mortality/longevity experience. This way, guarantees are relaxed in respect of traditional annuity arrangements. On the other hand, while the annuitant is exposed to the risk of a future reduction of the benefit amount because of higher longevity, he/she can immediately take advantage of a lower premium loading, as well as of a future increase of the benefit amount in the case of higher mortality. Flexibility in the annuity design could be welcomed by individuals, as the conservative features of traditional products partly explain their lack of attractiveness in most markets. To further contribute to the flexibility of the product, we suggest a pricing structure based on periodic fees applied to the policy fund, instead of the usual upfront loading at issue. Periodic fees are more suitable to support a revision of the arrangement after issue, which is currently not allowed in traditional annuity products. We show that periodic fees can be introduced by identifying a discount factor to be used for pricing and reserving. We assume stochastic mortality, and we compare alternative mortality/longevity linking solutions, by assessing the periodic fees and other quantities.

Optimal pooling strategies under heterogeneous risk classes

PurposeThe purpose of this paper is to determine optimal pooling strategies from the perspective of an insurer's shareholders underlying a default probability driven premium loading and convex price-demand functions.Design/methodology/approachThe authors use an option pricing framework for normally distributed claims to analyze the net present value for different pooling strategies and contrast multiple risk pools structured as a single legal entity with the case of multiple legal entities. To achieve the net present value maximizing default probability, the insurer adjusts the underlying equity capital.FindingsThe authors show with the theoretical considerations and numerical examples that multiple risk pools with multiple legal entities are optimal if the equity capital must be decreased. An equity capital increase implies that multiple risk pools in a single legal entity are generally optimal. Moreover, a single risk pool for multiple risk classes improves in relation to multiple risk pools with multiple legal entities whenever the standard deviation of the underlying claims increases.Originality/valueThe authors extend previous research on risk pooling by introducing a default probability driven premium loading and a relation between the premium level and demand through a convex price-demand function.

The introduction of lifetime community rating in the Irish private health insurance market: Effects on coverage and plan choice.

In response to declining insurance coverage and adverse selection, in May 2015, Ireland introduced lifetime community rating (LCR) of health insurance premiums to encourage take-up of health insurance at younger ages. LCR requires that late-entry premium loadings be applied to those, aged 35 and over, taking out health insurance for the first time. This analysis exploits quasi-experimental difference-in-differences methods to estimate the effect of LCR regulations on insurance take-up, controlling for potentially confounding factors. Findings reveal that the introduction of LCR increased coverage rates for those aged 35-69 by approximately 2.5 percentage points (p<0.01). However, the impact on coverage was largely concentrated in the 35-54 age cohort. Relative to underlying plan coverage rates, take-up was proportionately greatest for low-cover plans. Ireland is the second country to introduce late-entry premium loadings to its voluntary health insurance market, following Australia in 2000. This analysis adds to the current evidence base and improves understanding of consumer response to these regulations. The introduction of LCR has encouraged insurance take-up by those who may have otherwise postponed take-up to older ages contributing to a reversal in declining overall coverage trends, at least initially. However, the purchase of cheaper plans by relatively younger consumers could be contributing to risk segmentation within the market. Moreover, these cheaper insurance plans tend to offer limited benefits and consumers may still face significant out-of-pocket costs to access private care.

ON THE OPTIMAL COMBINATION OF ANNUITIES AND TONTINES

AbstractTontines, retirement products constructed in such a way that the longevity risk is shared in a pool of policyholders, have recently gained vast attention from researchers and practitioners. Typically, these products are cheaper than annuities, but do not provide stable payments to policyholders. This raises the question whether, from the policyholders' viewpoint, the advantages of annuities and tontines can be combined to form a retirement plan which is cheaper than an annuity, but provides a less volatile retirement income than a tontine. In this article, we analyze and compare three approaches of combining annuities and tontines in an expected utility framework: the previously introduced “tonuity”, a product very similar to the tonuity which we call “antine” and a portfolio consisting of an annuity and a tontine. We show that the payoffs of a tonuity and an antine can be replicated by a portfolio consisting of an annuity and a tontine. Consequently, policyholders achieve higher expected utility levels when choosing the portfolio over the novel retirement products tonuity and antine. Further, we derive conditions on the premium loadings of annuities and tontines indicating when the optimal portfolio is investing a positive amount in both annuity and tontine, and when the optimal portfolio turns out to be a pure annuity or a pure tontine.

On the Optimal Combination of Annuities and Tontines

Tontines, retirement products constructed in such a way that the longevity risk is shared in a pool of policyholders, have recently gained vast attention from researchers and practitioners. Typically, these products are cheaper than annuities, but do not provide stable payments to policyholders. This raises the question whether, from the policyholders' viewpoint, the advantages of annuities and tontines can be combined to form a retirement plan which is cheaper than an annuity, but provides a less volatile retirement income than a tontine. In this article, we analyze and compare three approaches of combining annuities and tontines in an expected utility framework: The previously introduced tonuity, a product very similar to the tonuity which we call and a portfolio consisting of an annuity and a tontine. We show that the payoffs of a tonuity and an antine can be replicated by a portfolio consisting of an annuity and a tontine. Consequently, policyholders achieve higher expected utility levels when choosing the portfolio over the novel retirement products tonuity and antine. Further, we derive conditions on the premium loadings of annuities and tontines indicating when the optimal portfolio is investing a positive amount in both annuity and tontine, and when the optimal portfolio turns out to be a pure annuity or a pure tontine.

Modeling the Frequency and Severity of Auto Insurance Claims Using Statistical Distributions

Claims experience in non-life insurance is contingent on random eventualities of claim frequency and claim severity. By design, a single policy may possibly incur more than one claim such that the total number of claims as well as the total size of claims due on any given portfolio is unpredictable. For insurers to be able to settle claims that may occur from existing portfolios of policies at some future time periods, it is imperative that they adequately model historical and current data on claims experience; this can be used to project the expected future claims experience and setting sufficient reserves. Non-life insurance companies are often faced with two challenges when modeling claims data; selecting appropriate statistical distributions for claims data and establishing how well the selected statistical distributions fit the claims data. Accurate evaluation of claim frequency and claim severity plays a critical role in determining: An adequate premium loading factor, required reserve levels, product profitability and the impact of policy modifications. Whilst the assessment of insurers’ actuarial risks in respect of their solvency status is a complex process, the first step toward the solution is the modeling of individual claims frequency and severity. This paper presents a methodical framework for choosing a suitable probability model that best describes automobile claim frequency and loss severity as well as their application in risk management. Selected statistical distributions are fitted to historical automobile claims data and parameters estimated using the maximum likelihood method. The Chi-square test is used to check the goodness-of-fit for claim frequency distributions whereas the Kolmogorov-Smirnov and Anderson-Darling tests are applied to claim severity distributions. The Akaike information criterion (AIC) is used to choose between competing distributions. Empirical results indicate that claim severity data is better modeled using heavy-tailed and skewed distributions. The lognormal distribution is selected as the best distribution to model the claim size while negative binomial and geometric distributions are selected as the best distributions for fitting the claim frequency data in comparison to other standard distributions.

Bounded rationality and the adoption of weather index insurance

PurposeThe purpose of this paper is to check which role-bounded rationality might play as an explanation for farmers’ missing willingness to adopt weather index insurance (WII). WII is an innovative risk management instrument that causes low administration and regulation costs. Moreover, index insurance is plagued neither by moral hazard nor by adverse selection. Nonetheless, WII has been little used to date in agriculture.Design/methodology/approachAn extra-laboratory experiment in the form of a multi-period, single-person business simulation game is conducted with farmers as experimental subjects to investigate the reasons for the low willingness to adopt WII.FindingsFirst, the demand for WII decreases as the premium loading increases. Second, a transparent communication of the loading reduces demand, indicating that farmers refrain from transactions if they feel that the other party earns (too) much money. Third, communicating to farmers that the index insurance has been subsidized raises demand even though insurance costs in terms of loading are kept constant. This can be taken as an indication that farmers interpret subsidies as a signal for profitable action.Originality/valueUsing an experimental approach and going beyond observational research, this study investigates the prominent question of farmers’ risk management and innovation adoption behavior, and, in particular, the behavioral effect of subsidies. Using a randomized controlled trial, the real behavior of real subjects with real incentives is studied under controlled experimental conditions. Compared to prior studies, the external validity of the experiment is improved by recruiting farmers instead of a convenience group of students.

Loss Risk Management: Perspectives from Corporate Finance

We develop a theoretical framework to analyze the relationship between loss risk management and firm value by focusing on insurance. In the model, the option analogy for equity implies that a higher insurance coverage decreases equity value, which further increases endogenous bankruptcy threshold chosen by shareholders. This plays a key role in the trade-off between tax benefits and bankruptcy costs. We derive that insurance can significantly increase firm value. Moreover, firm value shows monotonically increasing or U-shaped relationship with respect to insurance coverage, which implies optimal insurance coverage is full or none. By considering frictions of loss risk management (e.g., premium loadings in insurance practice), we can explain many phenomena in insurance practice, such as, incomplete insurance coverage, risk retention for small losses and subsidies for catastrophe insurance.

Some mathematical aspects of price optimisation

Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely (a) construction of an optimal renewal tariff subject to business and technical constraints, and (b) determination of an optimal allocation of certain premium loadings. We consider both continuous and discrete optimisation and then present several algorithmic suboptimal solutions. Additionally, we explore some simulation techniques. Several illustrative examples show both the complexity and the importance of the optimisation approach.

Detailed Analysis of The Laffer Effect at Turkish Social Security Administration Premium Revenue

This study is an attempt to analyse and deep the social security premium rate/load as a factor that influences the premium revenues for the Turkish Social Security Administration by using the Laffer curve logic and to identify relation and the premium load that maximizes premium revenues and assessing the revenue increase for the Administration. Turkish Social Security system separates the employees in three main groups: Indıvıduals working on service contract who are subject to SSK (Social Insurance Administration), Individuals working on their own names and accounts who are subject to Bag-Kur (Craftsmen And Artisans And Other Self-Employed Social Insurance Institution) and public employees or civil servants who are subject to ES (Retirement Fund General Directorate) before the social security reform. The previous studies; “Model Proposal for Investigating and Increasing the Social Security Administration’s Premium Collection Revenue” analysed the relationship between the premium revenue and premium rate/load for all the employees, make a total analyse for premium rate and revenue and “The Laffer Effect at Turkish Social Security Administration’s Premium Revenue” analysed the most crowded group of the Turkish Social Security system SSK.In this study it will be focused on Bag-Kur and ES premium revenue. The monthly data for the period between January 2009 and December 2012 were used in this study.&#x0D; Keywords: Premium Rate; Premium Revenue; Earnings Based On Premium; The Laffer Curve&#x0D;

Optimal consumption, portfolio, and life insurance policies under interest rate and inflation risks

This paper solves the optimal life insurance, consumption, and portfolio decisions of a wage earner before retirement under interest rate and inflation risks. The wage earner’s preferences are represented by the stochastic differential utility, which separates the coefficient of relative risk aversion from the elasticity of intertemporal substitution (EIS). The wage earner’s life insurance demand is affected by the volatile interest rates and inflation. The optimal life insurance demand decreases with the level of nominal interest rates. Under an assumption of deterministic nominal income, the demand for life insurance would not be affected by the level of inflation. However, if the wage earner’s income is indexed to inflation, the life insurance demand would increase with the level of inflation. Furthermore, under investment opportunities with greater volatilities, wage earners who optimally allocate their wealth to the financial market benefit more from financial investments and cut their demand for life insurance. An analysis of EIS and risk aversion on life insurance demand shows that the demand for life insurance over the planning horizon increases with the measure of relative risk aversion but decreases with EIS. Optimal consumption is affected by the insurance premium load and the direction depends on the size of EIS relative to unity.