Approximation of aggregate and extremal losses within the very heavy tails framework

The loss distribution approach is one of the three advanced measurement approaches to the Pillar I modeling proposed by Basel II in 2001. In this paper, one possible approximation of the aggregate and maximum loss distribution in the extremely low frequency/high severity case is given, i.e. the case of infinite mean of the loss sizes and loss inter-arrival times. In this study, independent but not identically distributed losses are considered. The minimum loss amount is considered increasing over time. A Monte Carlo simulation algorithm is presented and several quantiles are estimated. The same approximation is used for modeling the maximum and aggregate worldwide economy losses caused by very rare and very extreme events such as 9/11, the Russian rouble crisis, and the U.S. subprime mortgage crisis. The model parameters are fit on a data sample of operational losses. The respective aggregate and extremal loss quantiles are calculated.     

Cyclical correlations, credit contagion, and portfolio losses

We model aggregate credit losses on large portfolios of financial positions contracted with firms subject to both cyclical default correlation and direct default contagion processes. Cyclical correlation is due to the dependence of firms on common economic factors. Contagion is associated with the local interaction of firms with their business partners. We provide an explicit normal approximation of the distribution of portfolio losses. We quantify the relation between the variability of global economic fundamentals, strength of local firm interaction, and the fluctuation of losses. We find that cyclical oscillations in fundamentals dominate average losses, while local interaction causes additional fluctuations of losses around their average. The strength of the contagion-induced loss variability depends on the complexity of the business partner network.     

Multivariate Cox Hidden Markov models with an application to operational risk

ABSTRACT

Modeling multivariate time-series aggregate losses is an important actuarial topic that is very challenging due to the fact that losses can be serially dependent with heterogeneous dependence structures across loss types and business lines. In this paper, we investigate a flexible class of multivariate Cox Hidden Markov Models for the joint arrival process of loss events. Some of the nice properties possessed by this class of models, such as closed-form expressions, thinning properties and model versatility are discussed in details. We provide the expectation-maximization (EM) algorithm for efficient model calibration. Applying the proposed model to an operational risk dataset, we demonstrate that the model offers sufficient flexibility to capture most characteristics of the observed loss frequencies. By modeling the log-transformed loss severities through mixture of Erlang distributions, we can model the aggregate losses. Finally, out-of-sample testing shows that the proposed model is adequate to predict short-term future operational risk losses.     

Time-varying idiosyncratic risk and aggregate consumption dynamics

Long-term earnings losses for displaced workers are large and counter-cyclical. Similarly, the skewness of earnings growth rates is strongly pro-cyclical. This paper presents an incomplete markets business cycle model in which idiosyncratic risk varies over time in accordance with these empirical findings. These dynamics of idiosyncratic risk give rise to a cyclical precautionary savings motive that substantially raises the volatility of aggregate consumption growth. According to the model, idiosyncratic risk spiked during the Great Recession, leading to a substantial decline in aggregate consumption.     

Loss Modeling Using Spreadsheet-Based Simulation

With the dramatic increase in the speed of personal computers and steep decline in the cost of computing, simulation has become one of the standard tools in the risk manager's toolbox and should now become one of the standard tools in the risk management and insurance student's toolbox. This teaching note aims to facilitate this process by showing how to create and run a simulation in a spreadsheet environment, and interpret simulation results to gain insight and understanding about a real-world problem. Specifically, this teaching note provides step-by-step instruction for simulating the present value of payments for losses occurring within a 1-year policy period. Losses are covered by an aggregate excess of loss treaty. The uncertainty lies in the frequency and severity of losses as well as in claim processing time, and also in the discount rate for calculating the present value of loss payments.     

Management by the Numbers: A Formal Approach to Deriving Informational and Distributional Properties of “Unmanaged” Earnings

We explore the theoretical relation between earnings and market returns as well as the properties of earnings frequency distributions under the assumption that managers use unbiased accounting information to sequentially decide on real options their firms have and report generated earnings truthfully, with the market pricing the firm based on those reported earnings. We generate benchmarks against which empirically observed earnings‐returns relations and aggregate earnings distributions can be evaluated. This parsimonious model shows a coherent set of results: reported losses are less persistent than reported gains, decision making diminishes the S‐shaped market response to earnings and earnings relate to returns asymmetrically in the way documented by Basu [1997]. Furthermore, the implied frequency distribution of aggregate earnings is neither symmetric nor necessarily single‐peaked. Instead, it may exhibit a kink at zero and look similar to the plots reported by Burgstahler and Dichev [1997]. However, within our model, none of these phenomena are due to reporting noise, bias, or some undesirable strategic managerial behavior. They are the natural consequences of using past earnings as the basis for value increasing managerial decision making that in turn generates the future earnings on which future decisions will be based.     

How Much Do Taxes Discourage Incorporation?

The double taxation of corporate income should discourage firms from incorporating. We investigate the extent to which the aggregate allocation of assets and taxable income in the United States between corporate and noncorporate firms responds to the size of this tax distortion during the period 1959–1986. In theory, profitable firms should shift out of the corporate sector when the tax distortion is large, and conversely for firms with tax losses. Our empirical results provide strong support for these forecasts, and imply that the resulting excess burden equals 16 percent of business tax revenue.     

A Theoretical Framework for Incorporating Scenarios into Operational Risk Modeling

In this paper, I introduce a theoretically justified framework that incorporates scenario analysis into operational risk modeling. The basis for the framework is the idea that only worst-case scenarios contain valuable information about the tail behavior of operational losses. In addition, worst-case scenarios introduce a natural order among scenarios that makes possible a comparison of the ordered scenario losses with the corresponding quantiles of the severity distribution that research derives from historical losses. Worst-case scenarios contain information that enters the quantification process in the form of lower bound constraints on the specific quantiles of the severity distribution. The framework gives rise to several alternative approaches to incorporating scenarios.     

Optimal Capital Allocation Principles

This article develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimization argument, requiring that the weighted sum of measures for the deviations of the business unit's losses from their respective allocated capitals be minimized. The approach is fair insofar as it requires capital to be close to the risk that necessitates holding it. The approach is additionally very flexible in the sense that different forms of the objective function can reflect alternative definitions of corporate risk tolerance. Owing to this flexibility, the general framework reproduces several capital allocation methods that appear in the literature and allows for alternative interpretations and possible extensions.     

Unbounded liabilities,capital reserve requirements and the taxpayer put option

When firms access unbounded liability exposures and are granted limited liability, then an all equity firm holds a call option, whereby it receives a free option to put losses back to the taxpayers. We call this option the taxpayer put, where the strike is the negative of the level of reserve capital at stake in the firm. We contribute by (i) valuing this taxpayer put, and (ii) determining the level for reserve capital without a reference to ratings. Reserve capital levels are designed to mitigate the adverse incentives for unnecessary risk introduced by the taxpayer put at the firm level. In our approach, the level of reserve capital is set to make the aggregate risk of the firm externally acceptable, where the specific form of acceptability employed is positive expectation under a concave distortion of the cash flow distribution. It is observed that, in the presence of the taxpayer put, debt holders may not be relied upon to monitor risk as their interests are partially aligned with equity holders by participating in the taxpayer put. Furthermore, the taxpayer put leads to an equity pricing model associated with a market discipline that punishes perceived cash shortfalls.     

Estimating Portfolio Credit Losses in Downturns

This paper suggests formulas able to capture potential strong connection among credit losses in downturns without assuming any specific distribution for the variables involved. We first show that the current model adopted by regulators (Basel) is equivalent to a conditional distribution derived from the Gaussian Copula (which does not identify tail dependence). We then use conditional distributions derived from copulas that express tail dependence (stronger dependence across higher losses) to estimate the probability of credit losses in extreme scenarios (crises). Next, we use data on historical credit losses incurred in American banks to compare the suggested approach to the Basel formula with respect to their performance when predicting the extreme losses observed in 2009 and 2010. Our results indicate that, in general, the copula approach outperforms the Basel method in two of the three credit segments investigated. The proposed method is extendable to other differentiable copula families and this gives flexibility to future practical applications of the model.     

Anatomy of a bail-in

To mitigate potential contagion from future banking crises, the European Commission recently proposed a framework which would provide for the bail-in of bank creditors in the event of failure. In this study, we examine this framework retrospectively in the context of failed European banks during the global financial crisis. Empirical findings suggest that equity and subordinated bond holders would have been the main losers from the €535 billion impairment losses realized by failed European banks. Losses attributed to senior debt holders would, on aggregate, have been proportionally small, while no losses would have been imposed on depositors. Cross-country analysis, incorporating stress-tests, reveals a divergence of outcomes with subordinated debt holders wiped out in a number of countries, while senior debt holders of Greek, Austrian and Irish banks would have required bail-in.     

Conditional conservatism and disaggregated bad news indicators in accrual models

Conditional conservatism is an integral but often unmodeled part of the normal accrual process. The standard economic determinants of accruals contain information about unrealized losses. We argue that accountants recognize these unrealized losses as disaggregated write-downs for small asset pools. Modeling disaggregated impairments yields new economic insights about accruals and improved accrual models. We predict that accrual conservatism manifests as a sum of asymmetries for a vector of news indicators, rather than as an asymmetry for a scalar aggregate news proxy. We argue that more detailed segment-level and quarterly indicators have an incremental effect on annual firm-level accruals. We also predict a dynamic effect of successive loss indicators because accountants look for consistent patterns in these variables. Empirical results for U.S. firms support our predictions. The asymmetries in accruals are consistent with conservatism in validation tests. We also document improved statistical power and type I error in earnings management tests.     

Mortality-Indexed Annuities Managing Longevity Risk Via Product Design

Abstract

Longevity risk has become a major challenge for governments, individuals, and annuity providers in most countries. In its aggregate form, the systematic risk of changes to general mortality patterns, it has the potential for causing large cumulative losses for insurers. Since obvious risk management tools, such as (re)insurance or hedging, are less suited for managing an annuity provider’s exposure to this risk, we propose a type of life annuity with benefits contingent on actual mortality experience.

Similar adaptations to conventional product design exist with investment-linked annuities, and a role model for long-term contracts contingent on actual cost experience can be found in German private health insurance. By effectively sharing systematic longevity risk with policyholders, insurers may avoid cumulative losses.

Policyholders also gain in comparison with a comparable conventional annuity product: Using a Monte Carlo simulation, we identify a significant upside potential for policyholders while downside risk is limited.     

Ex ante loss control by insurers: Public interest for higher profit

This article examines the incentives of an insurer to modify loss distributions prior to the sale of insurance. While actions such as lobbying Congress for mandatory airbags in automobiles are undertaken by insurers for the stated purpose of reducing the aggregate loss in society, they also change the nature of the risk being insured and, hence, affect the profitability of insurance sales. For the case of loss prevention (reducing the probabilty of a loss), insurers do not always have an incentive to invest in loss control. For loss reduction (reducing the severity of any loss that does occur), the incentive is to reduce the size of small losses while simultaneously increasing the size of large losses. Venezian Associates     

On The Expected Discounted Penalty function for Lévy Risk Processes

Abstract

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes.

In this paper we work out this extension to a generalized risk model driven by a nondecreasing Lévy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian, and generalized inverse Gaussian processes. Expressions for the Gerber-Shiu function are given in some of these special cases, and numerical illustrations are provided.     

Manufacturing versus Distribution: Implicit Pricing of Real Property Characteristics by Submarket

This article develops and tests the idea that the industrial real estate market is an aggregate market consisting of at least two submarkets—manufacturing and distribution. While there is no observable difference in implicit pricing of most industrial property characteristics across these two submarkets, some property-characteristic implicit prices do differ. Therefore, manufacturing and distribution submarket property-pricing functions are best estimated in aggregate, while making allowances for variability of coefficients on some property characteristics. For this sample of 331 industrial property sales from the southeastern region of the United States, the two submarkets vary with regard to implicit pricing of building volume, below-average building condition, site area, and dock-high doors.     

On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo is considered. We study a generalization of the Gerber–Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin; and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure, where the claim severities follow a combination of exponentials, is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.     

Bank profitability during recessions

This paper contributes to the literature on the relation between bank profitability and economic activity. When allowing for stronger co-movement of bank profit with economic activity during deep recessions, we find a much larger impact of output growth on bank profitability than commonly found in the literature. Among the different components of bank profit, loan losses are the main driver of this result. We also find long-term interest rates in previous years to be important determinants of bank profit in times of high economic growth. Our findings are robust to the use of aggregate or individual bank data.     

Does the market dole out collective punishment? An empirical analysis of industry,geography, and Arthur Andersen’s reputation

Arthur Andersen’s reputation was tarnished following news that its Houston office had shredded documents related to the auditing of energy giant Enron. Earlier studies documented widespread spillover of the reputation effect, suggesting a strong commonality in Big 5 audit practices. We examine whether the market is more discriminating in its assessments. We focus on the roles industry specialization of auditors and the geography of clients’ audit offices play in accounting for the contagion. Our results are supportive of investors who differentiate audit practices by industry and who account for the location of the specific office where the audit work is done. We find that losses suffered by energy firms or firms located close to Houston are equivalent to approximately 90% of the aggregate abnormal losses suffered by Big 5 clients. Our evidence suggests the possibility of more localized impact of accounting scandals and supports accounting regulations targeted at individual industries.