Capital Allocation for Insurance Companies—What Good IS IT?

In their 2001 Journal of Risk and Insurance article, Stewart C. Myers and James A. Read Jr. propose to use a specific capital allocation method for pricing insurance contracts. We show that in their model framework no capital allocation to lines of business is needed for pricing insurance contracts. In the case of having to cover frictional costs, the suggested allocation method may even lead to inappropriate insurance prices. Beside the purpose of pricing insurance contracts, capital allocation methods proposed in the literature and used in insurance practice are typically intended to help derive capital budgeting decisions in insurance companies, such as expanding or contracting lines of business. We also show that net present value analyses provide better capital budgeting decisions than capital allocation in general.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.     

A revised option pricing formula with the underlying being banned from short selling

An important issue in derivative pricing that hasn't been explored much until very recently is the impact of short selling to the price of an option. This paper extends a recent publication in this area to the case in which a ban of short selling of the underlying alone is somewhat less ‘effective’ than the extreme case discussed by Guo and Zhu [Equal risk pricing under convex trading constraints. J. Econ. Dyn. Control, 2017, 76, 136–151]. The case presented here is closer to reality, in which the effect of a ban on the underlying of an option alone may quite often be ‘diluted’ due to market interactions of the underlying asset with other correlated assets. Under a new assumption that there exists at least a correlated asset in the market, which is allowed to be short sold and thus can be used by traders for hedging purposes even though short selling of the underlying itself is banned, a new closed-form equal-risk pricing formula for European options is successfully derived. The new formula contains two distinguishable advantages; (a) it does not induce any significantly extra burden in terms of numerically computing option values, compared with the effort involved in using the Black–Scholes formula, which is still popularly used in finance industry today; (b) it remains simple and elegant as only one additional parameter beyond the Black–Scholes formula is introduced, to reflect the dilution effect to the ban as a result of market interactions.     

A Framework for Identifying (and Avoiding) Fraudulent Financial Reporting*

This commentary analyzes the relationship of fraud risk assessments to other risk assessments by auditors. The Public Company Accounting Oversight Board notes that this is a problem area of current practice. Effective detection of fraudulent financial reporting requires an integrative accounting/auditing conceptual framework. As a result, this paper is as much about accounting theory as it is about auditing. To simplify the development of such an integrated framework, this paper uses an expanded risk model. This effectively results in a risk perspective on fraudulent financial reporting. There are many potential implications but the major findings are as follows. First, the study identifies the crucial role of benchmarks based on acceptable levels of risk to help differentiate between intentional and unintentional misstatements. Such differentiation is critical to successfully implementing the American Institute of Certified Public Accountants' Statement on Auditing Standards (SAS) No. 99 and international standards ISA Nos. 240, 540, and 700. Second, the paper shows the importance of not allowing the major categories of risks identified here from getting too high. This paper explains the need to set acceptable levels of these risks, either by standard‐setters as a matter of broad policy, or by individual practitioners as part of the terms of specific engagements. I propose that a major factor in the concept of “present fairly” be the acceptable levels of accounting risks that are defined here, especially the risks due to intentional forecast errors. Third, this paper clarifies how the fraud risk of SAS No. 99, and similar international standards, relates to the current audit risk model framework.     

Hierarchical structures in the aggregation of premium risk for insurance underwriting

In the valuation of the Solvency II capital requirement, the correct appraisal of risk dependencies acquires particular relevance. These dependencies refer to the recognition of risk diversification in the aggregation process and there are different levels of aggregation and hence different types of diversification. For instance, for a non-life company at the first level the risk components of each single line of business (e.g. premium, reserve, and CAT risks) need to be combined in the overall portfolio, the second level regards the aggregation of different kind of risks as, for example, market and underwriting risk, and finally various solo legal entities could be joined together in a group.

Solvency II allows companies to capture these diversification effects in capital requirement assessment, but the identification of a proper methodology can represent a delicate issue. Indeed, while internal models by simulation approaches permit usually to obtain the portfolio multivariate distribution only in the independence case, generally the use of copula functions can consent to have the multivariate distribution under dependence assumptions too.

However, the choice of the copula and the parameter estimation could be very problematic when only few data are available. So it could be useful to find a closed formula based on Internal Models independence results with the aim to obtain the capital requirement under dependence assumption.

A simple technique, to measure the diversification effect in capital requirement assessment, is the formula, proposed by Solvency II quantitative impact studies, focused on the aggregation of capital charges, the latter equal to percentile minus average of total claims amount distribution of single line of business (LoB), using a linear correlation matrix.

On the other hand, this formula produces the correct result only for a restricted class of distributions, while it may underestimate the diversification effect.

In this paper we present an alternative method, based on the idea to adjust that formula with proper calibration factors (proposed by Sandström (2007)) and appropriately extended with the aim to consider very skewed distribution too.

In the last part considering different non-life multi-line insurers, we compare the capital requirements obtained, for only premium risk, applying the aggregation formula to the results derived by elliptical copulas and hierarchical Archimedean copulas.     

Statistical Independence and Fractional Age Assumptions

Abstract

This paper considers in some detail the issue of statistical independence of the curtate future lifetime and the fractional part of the future lifetime of a general status.

Statistical independence is often employed in actuarial contexts, primarily because it leads to simple relationships between quantities of interest and statistical information that is of a discrete nature, such as a life table. The uniform distribution of deaths (UDD) assumption is the most commonly used because of its simplicity and intuitive appeal, but it can be somewhat restrictive. For example, all deaths or withdrawals may be assumed to be at a particular point in the year such as the middle; assumptions of this type are often made in a multiple decrement context. This paper attempts to unify these assumptions and extend their applicability in an actuarial context.

The conditions for independence need to be stated carefully, and the last-survivor status is cited as an example in which failure to do so can lead to erroneous conclusions.

The fractional independence (Fl) assumption is defined, and it is demonstrated that many of the formulas for life table functions that hold under the more restrictive UDD assumption are extended easily to the general Fl case. The simple relationship under UDD between insurances payable on other than an annual mode and those payable at the end of the year of death is extended to the Fl case as well. These results are then used to obtain results for annuities and reserves, again generalizing UDD relationships. It is then demonstrated that many contingent probabilities in the multiple life context are exactly the same under the Fl assumption as under the more restrictive UDD assumption. Finally, a very general result that holds in the multiple decrement context is shown to hold under the Fl assumption.     

“Bonus-Malus Systems: The European and Asian Approach to Merit-Rating”, Jean Lemaire,January 1998

Abstract

This paper presents an integration-by-parts proof of the Hattendorff theorem in the general fully continuous insurance model. The proof motivates a derivation of the theorem in the general fully discrete insurance model. Increments of a martingale over disjoint time intervals are uncorrelated random variables; the paper explains that the Hattendorff theorem can be viewed as an application of this result. A notable feature of the paper is the extensive use of the indicator function.     

OPTION PRICING AND THE ARBITRAGE PRICING THEORY

This paper applies the arbitrage pricing theory to option pricing. Under certain distribution assumptions or the assumption that there is only one common factor, the underlying asset of an option is the sole risky factor that explains its expected return. Based upon this relationship, a new and simple option-pricing formula is derived, and some important existing option-pricing formulae are reproduced. Empirical results show that the new formula performs as well as the Black-Scholes formula.     

Empirical Bayes credibility

Abstract

A credibility estimator is Bayes in the restricted class of linear estimators and may be viewed as a linear approximation to the (unrestricted) Bayes estimator. When the structural parameters occurring in a credibility formula are replaced by consistent estimators based on data from a collective of similar risks,we obtain an empirical credibility estimator, which is a credibility counterpart of empirical Bayes estimators. Empirical credibility estimators are proposed under various model assumptions, and sufficient conditions for asymptotic optimality are established.     

What is the correct meaning of implied volatility?

This paper presents a closed-form solution for the valuation of European options under the assumption that the excess returns of an underlying asset follow a diffusion process. In light of our model, the implied volatility computed from the Black–Scholes formula should be viewed as the volatility of excess returns rather than as the volatility of gross returns. Using the SPX and the OMX options data, we test whether implied volatility obtained from Black-Scholes option price explains the volatilities of excess returns better than gross returns, even though the result is not statistically significant.     

The perception of risk among unauthorized migrants in Ghana

Abstract

In recent years, the number of unauthorized migrants trying to reach Europe by boat has increased. In an attempt to curtail this trend, European countries have increasingly deployed information campaigns regarding the risks involved in unauthorized migration. The assumption behind these campaigns is that a strong relation exists between the prevalence of high-risk migration and a lack of awareness of the involved dangers, and that the dissemination of ‘objective’ information would cause prospective migrants to refrain from leaving. Although regarded as a crucial tool in the management of migration, there is little evidence on their effectiveness. Based on four weeks of fieldwork in Ghana, this qualitative study has sought to fill this lacuna. By concentrating on both the circulation and perception of the available risk information in Ghana, this article explores the ways in which notions of risk affect the motivation and decision of unauthorized migrants to engage in high-risk behaviour. Thereby, it will be argued that unauthorized migrants are not the ignorant actors they are often portrayed to be, but that they make their decisions carefully and within a particular vulnerable, socioeconomic and cultural context.     

Financial Development and Total Factor Productivity Growth: Evidence from China

Abstract

In this article, we estimate the effect of China’s regional financial development on total factor productivity (TFP) growth using large provincial panel data for the years 1990 to 2009. Using the nonparametric stochastic frontier data envelopment approach, we analyze how financial development is related to efficiency improvement and technological progress, the two components of TFP. The study shows that Chinese financial development plays a significant role in promoting TFP growth via technological progress rather than efficiency change. The faster the financial development takes place, the better it could correct the mismatch of resource allocation, thus promoting TFP growth. The results imply that China needs to both further optimize the allocation of financial resources and perfect the regional financial system.     

On the Equivalence between the APV and the wacc Approach in a Growing Leveraged Firm

While in a steady state framework the choice between the wacc approach ( Modigliani‐Miller, 1963 ) and the adjusted present value (APV) approach ( Myers, 1974 ) is irrelevant since the two approaches provide the same result, however, in a growing firm context the wacc equation seems to be inconsistent with the APV result. In this paper we propose a simple model to evaluate the tax savings in a growing firm in order to show under which assumptions the two approaches lead to the same results. We demonstrate that the use of the wacc model in a steady‐growth scenario gives rise to some unusual assumptions with regard to the discount rates to be used in calculating tax shields. We show that the widely used wacc formula, if used, as it is in most cases, in a growth context, implies that a) debt tax shield related to already existing debt are discounted using k_d; b) debt tax shield related to new debt, due to company's growth, are discounted, according to a mixed procedure, using both k_u and k_d. We discuss the inconsistency of such a discounting procedure and the preferred features of the APV approach.     

An explanatory analysis of perceived risk decision weights (perceived-risk attitudes) and perceived benefit decision weights (perceived-benefit attitudes) in risk-value models

Abstract

All decision making requires a trade-off between risks and values. While Markowitz defined risk as the variance of returns (thus reasoning that investors should consider it as undesirable), the more general risk–value framework allows risk to be defined as a person’s subjective judgments. Psychological risk–return models go further, decomposing observed behavior (risk taking) into two processes: (1) a judgment of benefits and risks and (2) a trade-off between perceived benefits and perceived risks, with a person-specific willingness to trade-off units of returns (benefits) for units of risk, conceptualized as attitude toward perceived risk (PRA) and attitude toward perceived benefits (PBA). PRA and PBA describe the degree to which people find perceived risks and benefits attractive, all other things being equal, and are assumed to be relatively stable across situations and domains. We test this assumption in an empirical study, checking the temporal stability of PRA and PBA (using the a Domain-Specific Risk-Taking [DOSPERT] scale ) and the cross-task stability of PBA (performing comparisons between the DOSPERT and the Columbia Card Task[CCT]). Finally, we explain both PRA and PBA using the Big Five personality dimensions and Stimulating–Instrumental Risk Inventory (SIRI), showing that PBA weights increase with openness to experience, while the negative effect of perceived risk on risk taking (PRA) increases with conscientiousness and decreases with stimulating risk taking. The results show that PBA and PRA can be treated as traits which, in some instances at least, are stable across time and tasks, and which can be partially explained by personality, providing a link to the idea of a personality dependent ‘ideal point’ for risk preference.     

Ulrich Beck: exploring and contesting risk

Abstract

While risk research normally understands risk as an entity that can be calculated using statistics and probabilities – and which therefore also can become the object of insurance technology – it is the production of new, non-calculable risks and therefore also risks that cannot be insured against, which is at the centre of Ulrich Beck’s risk society theory. The article examines Beck’s conceptualization of risk and discusses how he has clarified and further refined the concept since publishing Risk Society in 1986. The article shows, first, how Beck understands risk as an entity that is neither danger nor risk in the traditional sense but rather something in between, which he refers to as ‘man-made disasters’ and ‘new risks’. The discussion then addresses Beck’s position in relation to the ontological status of risk, which is an intermediate position between realism and constructivism. The non-calculability and non-insurability of the new risks are also examined. The article discusses what it means that the new risks are not visible and the significance of non-knowledge for how we understand them. Finally, the new conditions of existence for politics, states and individuals are outlined in the aftermath of Beck’s risk society theory. The article concludes with a discussion of the analytical potential of the theory.     

On additivity of tail comonotonic risks

ABSTRACT

As perceived from daily experience together with numerous empirical studies, the multivariate risks demonstrate a strong coherence in the extremal dependence structure especially over the course of financial turmoil or industrial accidents and outbreaks. Under this motivating paradigm, we show the universal asymptotic additivity under upper tail comonotonicity, as the probability level approaching to 1, for Value-at-Risk and Conditional Tail Expectation for a portfolio of fixed number of risks, in which each marginal risk could be any one having a finite endpoint or belonging to one of the three max domains of attraction. Our obtained results do not require the tail equivalence assumption as needed in the existing literature. This resolves a lasting problem in quantitative risk management and covers most distributions commonly encountered in practice.     

“Relative Importance of Risk Sources in Insurance Systems”, Edward W. Frees,April 1998

Abstract

This paper considers a Sparre Andersen collective risk model in which the distribution of the interclaim time is that of a sum of n independent exponential random variables; thus, the Erlang(n) model is a special case. The analysis is focused on the function φ(u), the expected discounted penalty at ruin, with u being the initial surplus. The penalty may depend on the deficit at ruin and possibly also on the surplus immediately before ruin. It is shown that the function φ(u) satisfies a certain integro-differential equation and that this equation can be solved in terms of Laplace transforms, extending a result found in Lin (2003). As a consequence, a closed-form expression is obtained for the discounted joint probability density of the deficit at ruin and the surplus just before ruin, if the initial surplus is zero. For this formula and other results, the roots of Lundberg’s fundamental equation in the right half of the complex plane play a central role. Also, it is shown that φ(u) satisfies Li’s (2003) renewal equation. Under the assumption that the penalty depends only on the deficit at ruin and that the individual claim amount density is a combination of exponential densities, a closed-form expression for φ(u) is derived. In this context, known results of the Cauchy matrix are useful. Surprisingly, certain results are best expressed in terms of divided differences, a topic deleted from the actuarial examinations at the end of last century.     

Accounting for Carbon Emission Allowances in the European Union: In Search of Consistency

Abstract

With the commencement of Phase III of the European Union Emissions Trading System (EU ETS) in 2013, it is projected that approximately one-half of emission allowances will be acquired through auctioning and the provision of free allocations to installations will be substantially tightened. As a result, it is likely that many companies will hold purchased (as opposed to freely allocated or gratis) allowances and will have more significant liabilities under the scheme. The accounting treatment of emission allowances has therefore become more relevant and the lack of uniformity in practice that resulted after the withdrawal of IFRIC 3 is now a more pressing concern. This study uses content analysis to examine disclosed accounting policies of companies with significant emission liabilities under the EU ETS and identifies three more common approaches adopted to date. These can be generally described as the following: (i) a net liability approach, based on the classification of allowances as intangibles but only showing an emission liability when it exceeds the free allocation; (ii) an approach broadly based on IFRIC 3 (recognising the free allocation at fair value and a corresponding gross liability under the EU ETS); and (iii) an approach based on inventory classification, with free allocations given at nil value. The diversity in these treatments highlights the need for guidance from the International Accounting Standards Board.     

On the short-maturity behaviour of the implied volatility skew for random strike options and applications to option pricing approximation

In this paper, we propose a general technique to develop first- and second-order closed-form approximation formulas for short-maturity options with random strikes. Our method is based on a change of numeraire and on Malliavin calculus techniques, which allow us to study the corresponding short-maturity implied volatility skew and to obtain simple closed-form approximation formulas depending on the derivative operator. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches for two-asset and three-asset spread options such as Kirk’s formula or the decomposition method presented in Alòs et al. [Energy Risk, 2011, 9, 52–57]. This methodology is not model-dependent, and it can be applied to the case of random interest rates and volatilities.     

Optimal and Simple,Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement

Abstract

The increasing risk of poverty in retirement has been well documented; it is projected that current and future retirees’ living expenses will significantly exceed their savings and income. In this paper, we consider a retiree who does not have sufficient wealth and income to fund her future expenses, and we seek the asset allocation that minimizes the probability of financial ruin during her lifetime. Building on the work of Young (2004) and Milevsky, Moore, and Young (2006), under general mortality assumptions, we derive a variational inequality that governs the ruin probability and optimal asset allocation. We explore the qualitative properties of the ruin robability and optimal strategy, present a numerical method for their estimation, and examine their sensitivity to changes in model parameters for specific examples. We then present an easy-to-implement allocation rule and demonstrate via simulation that it yields nearly optimal ruin probability, even under discrete portfolio rebalancing.