Fair Premium Research Articles

The Optimal Insurance Coverage With Default Risk

This articles re-examines a standard result on the demand for insurance (full coverage with a fair premium and partial coverage with a loaded premium') in the presence of a default risk. It is established that the optimal insurance coverage is always partial coverage if the default is total, irrespective of whether the premium is fair or loaded. For partial default, it is demonstrated in both the fair premium case and the loaded premium case that the optimal insurance coverage might be either full coverage or partial coverage for coinsurance, deductible insurance and upper-limit insurance.

On Three Standard Results in the Theory of Insurance Demand

Abstract This article re-examines three standard results in the theory of insurance demand: (i) full coverage with a fair premium and partial coverage with an unfair premium; (ii) insurance is an inferior good under decreasing absolute risk aversion (DARA) and (iii) insurance may be a Giffen good under DARA. It has been shown recently that (i) holds for the class of insurance policies in which maximum coverage fully covers the potential loss. We show that whether (i) holds beyond this class of policies is indeterminate. In addition, we employ a unified framework to investigate the effects of changes in initial wealth and price. In particular, we show that both (ii) and (iii) hold for a certain class of insurance policies which include all commonly-used types of policies. The result also provides a unified treatment of several results in the extant literature.

On Three Standard Results in the Theory of Insurance Demand

This article re-examines three standard results in the theory of insurance demand: (i) full coverage with a fair premium and partial coverage with an unfair premium; (ii) insurance is an inferior good under decreasing absolute risk aversion (DARA) and (iii) insurance may be a Giffen good under DARA. It has been shown recently that (i) holds for the class of insurance policies whose maximum coverage fully covers the potential loss. We show that whether (i) holds beyond this class of policies is indeterminate. In addition, we employ a unified framework to investigate the effects of changes in initial wealth and price. In particular, we show that both (ii) and (iii) hold for a certain class of insurance policies which include all commonly-used types of policies. The result also provides a unified treatment of several results in the extant literature.

Valuation of risk-based premium of DB pension plan with terminations

This paper concentrates on the premium valuation of pension insurance provided by the Pension Benefit Guaranty Corporation (PBGC). The PBGC provides a defined benefit pension sponsor with coverage in case that the pension fund fails to make pension payments as promised or that the plan sponsor does not stay in business any more. In practice, both the pension fund and the sponsor assets play a critical role in fulfilling the commitment of pension payments, and thereby it is not reasonable to isolate the risk of distress termination of the sponsor assets from that of the premature termination of the pension fund. Different from previous works in which the premature termination of the pension fund and the distress termination of the sponsor assets are analyzed separately, our model examines the situation in which retirees suffer the risk of two types of terminations at the same time. We evaluate the risk-based fair premium under the framework that the pension fund and the sponsor assets are correlated and subject to the risk of the involuntary termination (i.e., premature termination) and the distress termination, respectively. In this framework, we manage to obtain closed-form pricing formulas. Our model is more practical because of the realistic design of termination schemes. Numerical simulations are also carried out to demonstrate our findings. Our numerical experiments validate that a variable rate premium is more appropriate for the PBGC to implement.

Political connections, government capital injection, and deposit insurance premium

PurposeThe purpose of this paper is to develop a capped barrier option framework to consider the politically preferential treatment for bank loans incentivized by government capital injections and calculate loan-risk sensitive insurance premiums.Design/methodology/approachThis paper takes a capped barrier option approach to the market valuation of the equity of the bank and the liability of the deposit insurer. The cap demonstrates the dynamics of a politically connected borrowing firm’s asset and highlights the truncated nature of loan payoffs. The barrier addresses that default can occur at any time before the maturity date. The bank participating in a government capital injection program is required to fund the politically connected firm that has preferential access to financing.FindingsPolitical connection as such makes the bank more prone to risk taking at a reduced interest margin, produces greater safety for the bank owing to government capital injections, and leads to increasing the fair deposit insurance premium. The positive effect of political connection on the deposit insurance premium, which ignores the cap and the barrier yields significant over-estimation.Originality/valueThe study on the politically connected borrowing firm shows that political connection is likely to affect the distressed bank’s performance, yielding the political-connection cost of a reduced bank interest margin and the political-connection benefit of a reduced bank equity risk, contributing the literature on political connection and bank bailout.

Delay-Dependent Robust Stability Analysis for Premium-Reserve Models in an Arbitrary Regime Switching Discrete-Time Framework

AbstractIn the insurance industry, the actuarial team experiences significant challenges for pricing a competitive but also fair premium and keeping an accurate level of reserve, which leads inevit...

Spatial variability in regional scale drought index insurance viability across Australia’s wheat growing regions

Wheat is key global food crop that is heavily influenced by climatic variability. There has been extensive research on improving forecasts and management practices to minimise climate related yield losses, but less on how to handle yield losses caused by climate variability. We investigated whether index insurance could be used to manage climate related losses, specifically from winter rainfall drought for wheat crops in Australia. We utilised 31 years of yield data from 15 of Australia’s key wheat producing regions. The winter rainfall index was developed and tested using generalised additive regression models, allowing for non-linear effects. Models with the winter rainfall index explained significant variation in wheat yields in each of the regions assessed. Wheat yield models had cross-validated R2 values > 0.5 for two-thirds of the 15 regions modelled and best explained wheat yields in the Mallee, Western Australia (cross-validated R2 of 0.70). Calculated fair premiums ranged from $8.62 to $77.1 AUD/ha, while maximum liability was $59.25 to $212.12 AUD/ha. Throughout the eastern most wheat growing regions the winter rainfall index was consistently inefficient (i.e. not beneficial). In contrast, the winter rainfall index was financially efficient (i.e. beneficial) in the western wheat regions of eastern Australia and parts of Western Australia, with benefits of up to $97 AUD/ha and loss reductions of $9 AUD/ha. The spatial variability in insurance efficiency was explained by rainfall variance. As rainfall variance increased the efficiency of the winter rainfall index insurance for wheat decreased. Our findings have two important policy implications; (1) in areas where climate change is anticipated to increase rainfall variability risk-transfer options, such as index insurance, may become less viable and as such policies that support the development of index insurance without acknowledging or adjusting for variability in its benefit could lead to inefficient outcomes for both government and agricultural producers; and (2) where index rainfall insurance is not efficient then greater emphasise may need to be placed on developing alternate types of index insurance (e.g. using satellites) and/or on risk-management and climate adaptation strategies that minimise losses.

Remarks on the Mossin Theorem

We establish several new results regarding the Mossin Theorem under both nonrandom initial wealth and random initial wealth. For the nonrandom initial wealth case, we show that the Mossin Theorem holds for any constrained class of insurance contracts whose maximum coverage provides full coverage of the potential loss. This result not only settles an open conjecture, but also provides a unified treatment for extant varieties of the Mossin Theorem. For the random initial wealth case, we give a thorough study of the upper-limit insurance. In particular, we show that (1) for a fair premium, the Generalized Mossin Theorem for coinsurance does not hold for upper-limit insurance, and (2) for an unfair premium, partial insurance will always be optimal, regardless of the risk preference of the individual and the dependence structure between the random loss and the random initial wealth.

Risk management activities of a non-industrial private forest owner with a bivariate utility function

In this paper, we propose to analyse the choice of risk management activity made by a non-industrial private forest owner who derives utility from consu mption and from the sentimental value of the forest that bears a risk of disaster. We consider a bivari ate utility function depending on consumption and sentimental value of forest. In this context, we an alyse insurance and/or self-insurance decisions. We show that, under fair premium, full insurance is op timal only if the cross derivative of the utility function equals zero. Under-insurance and over-insu rance may also be optimal depending on the sign of this cross derivative. We also show that, under a positive loading factor, optimal partial insuranc e is validated only if the cross derivative is positive; otherwise full insurance may be optimal even with a loading insurance. We also observe that risk aversi on increases the level of insurance demand and self insurance activity, extending this standard result obtained with an univariate utility function to a bivariate utility function. Moreover, when the forest owner can simultaneously insure and invest in self-insurance activity, full insurance is never optimal if the cross derivative is positive. Finally, we prove that insurance and s elfinsurance may be substitutes, and if preferences ar e separable and exhibit decreasing absolute risk aversion, then insurance and self-insurance are alw ays considered as substitutes.

Index insurance benefits agricultural producers exposed to excessive rainfall risk

Managing the risks of climate variability on crop production is central to ensuring financially viable farming systems and sustainable food production. Insurance provides a mechanism to manage and transfer climate risks. However, traditional multi-peril crop insurance (MCPI) is often too expensive and so other methods, such as index insurance, are being explored as a cheaper way to insure farmers against climate induced crop losses. Here we investigate the potential financial benefits of index insurance (protecting against excessive rainfall) for agricultural producers, namely sugar cane farmers in Tully, northern Australia. We used 80 years of historical climate and yield data to develop an excessive rainfall index. The index was developed and tested using generalized additive regression models (allowing for non-linear effects) and quantile regression, which allows relationships with lower quantiles (i.e. low yield events) to be assessed. From the regression models we derived relationships between the excessive rainfall index and sugar cane yield losses that were converted to insurance fair premiums (i.e. premiums that cover expected losses). Finally, we used efficiency analysis, based on Conditional Tail Expectation (CTE), Certainty Equivalence of Revenue (CER) and Mean Root Square Loss (MRSL), to quantify financial benefits to farmers if they purchased excessive rainfall index insurance. The regression model predicted sugar cane yields well (cross-validated R2 of 0.65). The efficiency analysis indicated there could be financial benefit to sugar cane farmers if they were to use excessive rainfall index insurance. Index insurance (based on the assumption of a fair premium) could make farmers better off by $269.85 AUD/ha on average in years with excessive rainfall (i.e. years with rainfall over the 95th percentile). Index insurance could offer a viable method for managing the financial risks posed by excessive rainfall for sugar cane producers in northern Australia. We are not aware of any other study demonstrating the potential benefits of excessive rainfall index insurance in the literature, but our results suggest this type of insurance may be viable for sugar cane producers, and other crops, in parts of the world where extreme rainfall poses a risk to the financial sustainability of production.

How to fight against southern pine beetle epidemics: An insurance approach

AbstractThe southern pine beetle (SPB) is among the leading biological agents killing southern pine species in the eastern United States. In light of recognized spatiotemporal autocorrelation in SPB outbreaks, we devise a spatiotemporal block bootstrapping method that can be applied to analyze spatiotemporally dependent infestations. We also identify the relevant risk determinants and evaluate their impacts on the frequency of SPB outbreaks. For example, we find forest type, climate, and natural disasters like storm and forest management are all significantly associated with SPB risks. Using the results of a statistical model, we design a county‐level group index insurance plan that generates estimates of actuarially fair premium rates for timber stands containing southern pine species. Given that no government‐provided compensation scheme for SPB epidemics currently exists, application of this new insurance product could reduce forest owners losses. Our study offers an approach to analyzing and protecting against risks of other destructive pests affecting the timber sector.

David and Goliath: On the US Investment Grade Credit Risk Premium

We analyze the US Corporate Investment Grade (IG) credit risk premium, as represented by the average excess return of US corporate IG bonds over duration-matched US Treasuries. Over the period January 1973 through April 2018, this credit excess return averaged 60 bps p.a. (implying a stand-alone Information Ratio of 0.16), which is deemed unattractively low. Breaking with standard practice to analyze this credit excess return in isolation, we instead propose a portfolio perspective that does justice to corporate bonds as “package deals”, offering exposure to both interest rate risk and credit risk. Because of diversification effects within this portfolio, not the stand-alone credit risk exposure is relevant but its contribution to total bond risk. As a result, the total risk premium on corporate bonds is a composite risk premium, composed of a pure asset risk premium (the term premium stemming from the “Treasury part”) and a factor risk premium which can only be earned in tandem with the term premium (reflecting the diversified credit risk contribution). Accepting the term premium as fair, we can use the relative risk contributions to solve for the implied credit risk premium. From a macro view, this implied credit risk premium is the fair premium that investors would require as compensation for bearing the credit risk exposure contained in the aggregate market value of outstanding US IG bonds. We find that the implied credit risk premium was 44 bps, which is well below the historical average credit excess return of 60 bps. From our package deal perspective, the historical credit risk premium is not small at all.

The Impact of Longevity Risk on the Price of Life Insurance with the Accidental Option (Type AI and ADI)

Regardless of the type of insurance the basic part of the insurance premium is the pure premium, often referred to as net premium, which is based on the risk of death of the insured, averaged over the entire insurance period. The net premium is therefore the fund amount that the insured should pay in order for the insurer to possess sufficient funds to cover all benefits during the contract period. In accordance with the principles of actuarial mathematics the valuation of standard life insurance is based on the equivalence principle and takes into account the risk of death and also the change in time value of money, i.e. actuarial risk, which is estimated by the insurer at the time of signing the contract. Such method of calculating the net premium does not ensure the protection of collected funds against aggregated and individual longevity risk, which may negatively affect long-term financial stability of insurers as well as the level of financial security of the insured. Hence it is necessary to, on the one hand precisely valuate the trend of further life expectancy, and on the other there is a need to modify the methods of calculating premium rates and their adjustment in the duration of the contract. Therefore the article includes premium calculations that take into account the extended actuarial risk that encompasses longevity risk and its effect on their amount. This problem is the aim of the analysis carried out in the article. That is to say, taking into account the extended actuarial risk that encompasses longevity risk, a fair premium formula including the need for a risk adjustment was presented, followed by a premium calculation based on the example of life insurance with the AI and ADI option, as well as a look into the effect of change in longevity risk on their amount.

An Actuarial Approach to Reload Option Valuation for a Non-tradable Risk Assets under Jump-diffusion Process and Stochastic Interest Rate

We use an actuarial approach to estimate the valuation of the reload option for a non-tradable risk asset under the jump-diffusion processes and Hull-White interest rate. We verify the validity of the actuarial approach to the European vanilla option for non-tradable assets. The formulas of the actuarial approach to the reload option are derived from the fair premium principle and the obtained results are arbitrage. Numerical experiments are conducted to analyze the effects of different parameters on the results of valuation as well as their differences from those obtained by the no-arbitrage approach. Finally, we give the valuations of the reload options under different parameters.

Reinsurance versus securitization of catastrophe risk

We provide a novel explanation for the low volume of securitization in catastrophe risk transfer using a signaling model. Relative to securitization, reinsurance features lower adverse selection costs because reinsurers possess superior underwriting resources than ordinary capital market investors. Reinsurance premia, however, reflect markups over actuarially fair premia due to the additional costs of underwriting. Insurers’ risk transfer choices trade off the costs and benefits of reinsurance relative to securitization. In equilibrium, low risks are transferred via reinsurance, while intermediate and high risks are transferred via partial and full securitization, respectively. An increase in the loss size increases the trigger risk level above which securitization is chosen. Hence, catastrophe exposures, which are characterized by lower probabilities and higher severities, are more likely to be retained or reinsured rather than securitized.

WSKAŹNIKOWE UBEZPIECZENIE NADWYŻKI BEZPOŚREDNIEJ W ROLNICTWIE – IDENTYFIKACJA KLUCZOWYCH WYZWAŃ

The paper attempts at assessment of usefulness of the index-based insurance of gross margin compared to traditional production insurances. The analysis used FADN data and was limited to the example of winter wheat. The conducted simulations showed that the category of gross margin is characterized by higher variability than yields or prices, thus the costs of its insurance expressed as fair premium would be higher than the costs of traditional production insurance. However, the major problem in case of index-based insurances is still the basic risk related to the possibility that part of the insured will not receive compensation even though they incurred losses. The conducted analyses showed that the assumption of the index basing on the average drop in yields in a voivodeship would result in major percentage of errors as regards payment and refusal of payment of the compensation compared to individual insurance. Structuring of the system of index-based insurances would require collecting data – to construct indices – from areas of much smaller territorial coverage.

Can universal access be achieved in a voluntary private health insurance market? Dutch private insurers caught between competing logics.

For almost a century, the Netherlands was marked by a large market for voluntary private health insurance alongside state-regulated social health insurance. Throughout this period, private health insurers tried to safeguard their position within an expanding welfare state. From an institutional logics perspective, we analyze how private health insurers tried to reconcile the tension between a competitive insurance market pressuring for selective underwriting and actuarially fair premiums (the insurance logic), and an upcoming welfare state pressuring for universal access and socially fair premiums (the welfare state logic). Based on primary sources and the extant historiography, we distinguish six periods in which the balance between both logics changed significantly. We identify various strategies employed by private insurers to reconcile the competing logics. Some of these were temporarily successful, but required measures that were incompatible with the idea of free entrepreneurship and consumer choice. We conclude that universal access can only be achieved in a competitive individual private health insurance market if this market is effectively regulated and mandatory cross-subsidies are effectively enforced. The Dutch case demonstrates that achieving universal access in a competitive private health insurance market is institutionally complex and requires broad political and societal support.

Chinese write-down bonds and bank capital structure

In this paper, we use a structural model to investigate a bank capital structure that contains deposits, straight bonds, Write-Down (WD) bonds and equity. We first explicitly give the default boundaries and the values of a deposit, straight bond, WD bond, equity and bank asset, and then use a numerical example to demonstrate the relations among leverage, deposits, WD bonds and bank value. Our results show that value-maximizing banks select the ratio of deposit, straight bond and WD debt so that endogenous default is consistent with exogenous bank closure. The bank increases its leverage by swapping both deposits and straight bonds for WD bonds. And the issuance of WD bonds not only reduces the expected bankruptcy loss and credit spread of straight bonds, but also improves the bank value. This indicates that WD bonds do help to stabilize banks. We also study the role of deposit insurance and the Chinese Financial Stability Bureau (FSB), and give a closed-form expression for the fair insurance premium. Lastly, to check the robustness of our results, we do the sensitivity analysis and investigate the effect of three sets of exogenous parameters on bank capital structure: WD parameters, bank business features, closure rules and insurance subsidy, and obtain some practically significant implications.

Fair Valuation of L\xe9vy-Type Drawdown-Drawup Contracts with General Insured and Penalty Functions

In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative Lévy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity p until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium p for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of Lévy processes and rely on a theory of optimal stopping.

Usage‑Based Insurance jako instrument przywracania równowagi na polskim rynku ubezpieczeń komunikacyjnych

Celem artykułu jest weryfikacja możliwości redukcji negatywnego wpływu asymetrii informacji na rynek ubezpieczeń komunikacyjnych przez wdrożenie produktów typu Usage‑Based Insurance. Autor dokonuje analizy ekonomicznej, uzyskując przewidywania modeli rynku przy zastosowaniu trzech przykładowych strategii wdrożenia rozwiązań telematycznych. Wykazuje, że nie każdy ze scenariuszy doprowadzi do otrzymania pożądanego stanu równowagi i wyłącznie prawidłowy wybór strategii rynkowego wdrożenia rozwiązań telematycznych pozwoli ubezpieczonym z grupy niskiego ryzyka otrzymać pełną ochronę przy składce sprawiedliwej. Będzie to stanowić poprawę w sensie Pareto w porównaniu do ich położenia w warunkach występowania asymetrii informacji i może zostać osiągnięte przez wdrożenie produktów wykorzystujących dane z tzw. czarnych skrzynek lub urządzeń typu dongle.