Credibilistic risk aversion

In the probabilistic risk aversion approach, risks are presumed as random variables with known probability distributions. However, in some practical cases, for example, due to the absence of historical data, the inherent uncertain characteristic of risks or different subject judgements from the decision-makers, risks may be hard or not appropriate to be estimated with probability distributions. Therefore, the traditional probabilistic risk aversion theory is ineffective. Thus, in order to deal with these cases, we suggest measuring these kinds of risks as fuzzy variables, and accordingly to present an alternative risk aversion approach by employing credibility theory. In the present paper, first, the definition of credibilistic risk premium proposed by Georgescu and Kinnunen [Fuzzy Inf. Eng., 2013, 5, 399–416] is revised by taking the initial wealth into consideration, and then a general method to compute the credibilistic risk premium is provided. Secondly, regarding the risks represented with the commonly used LR fuzzy intervals, a simple calculation formula of the local credibilistic risk premium is put forward. Finally, in a global sense, several equivalent propositions for comparative risk aversion under the credibility measurement are provided. Illustrated examples are presented to show the applicability of the theoretical findings.     

On Mixed and Compound Mixed Poisson Distributions

Recursive formulae are derived for the evaluation of the t-th order cumulative distribution function and the t-th order tail probability of compound mixed Poisson distributions in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. Also, some general results are derived for the evaluation of the t-th order moments of stop-loss transforms. The recursions can be applied for the exact evaluation of the probability function, distribution function, tail probability and stop-loss premium of compound mixed Poisson distributions and the corresponding mixed Poisson distributions. Several examples are also presented.     

Investability and Firm Value

We study how investability, or openness to foreign equity investors, affects firm value in a sample of over 1,400 firms from 26 emerging markets. We find that, on average, investability is associated with a 9% valuation premium (as measured by Tobin's q). This significant valuation premium persists in firm‐fixed effects regressions, although the magnitude and robustness of the premium is somewhat lower. Analysis of the components of Tobin's q shows that firms that become investable experience significant increases in both market values and physical investment. These effects are strongest for firms that face country‐level or firm‐level financial constraints prior to becoming investable.     

Decomposing the VIX: Implications for the predictability of stock returns

The VIX index is not only a volatility index but also a polynomial combination of all possible higher moments in market return distribution under the risk-neutral measure. This paper formulates the VIX as a linear decomposition of four fundamentally different elements: the realized variance (RV), the variance risk premium (VRP), the realized tail (RT), and the tail risk premium (TRP), respectively. Using an innovative and nonparametric tail risk measure, we find that approximately one-third of the VIX's formation is attributed to the TRP. In addition to VRP, RT and TRP are crucial components for predicting future returns on equity portfolios.     

Probability of ruin with variable premium rate

Abstract

The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to vary. Section 9 gives some comments on the possible effect of “dangerousness” of a claim size distribution on ruin probability.     

Bayesian and Bühlmann credibility for phase-type distributions with a univariate risk parameter

Credibility theory is a statistical tool to calculate the premium for the next period based on past claims experience and the manual rate. Each contract is characterized by a risk parameter. A phase-type (or PH) random variable, which is defined as the time until absorption in a continuous-time Markov chain, is fully characterized by two sets of parameters from that Markov chain: the initial probability vector and transition intensity matrix. In this article, we identify an interpretable univariate risk parameter from amongst the many candidate parameters, by means of uniformization. The resulting density form is then expressed as an infinite mixture of Erlang distributions. These results are used to obtain a tractable likelihood function by a recursive formula. Then the best estimator for the next premium, i.e. the Bayesian premium, as well as its approximation by the Bühlmann credibility premium are calculated. Finally, actuarial calculations for the Bühlmann and Bayesian premiums are investigated in the context of a gamma prior, and illustrated by simulated data in a series of examples.     

Folded and log-folded-t distributions as models for insurance loss data

A rich variety of probability distributions has been proposed in the actuarial literature for fitting of insurance loss data. Examples include: lognormal, log-t, various versions of Pareto, loglogistic, Weibull, gamma and its variants, and generalized beta of the second kind distributions, among others. In this paper, we supplement the literature by adding the log-folded-normal and log-folded-t families. Shapes of the density function and key distributional properties of the ‘folded’ distributions are presented along with three methods for the estimation of parameters: method of maximum likelihood; method of moments; and method of trimmed moments. Further, large and small-sample properties of these estimators are studied in detail. Finally, we fit the newly proposed distributions to data which represent the total damage done by 827 fires in Norway for the year 1988. The fitted models are then employed in a few quantitative risk management examples, where point and interval estimates for several value-at-risk measures are calculated.     

Conditional Properties of Hedge Funds: Evidence from Daily Returns*

Using daily returns on a set of hedge fund indices, we study (i) the properties of the indices' conditional density functions and (ii) the presence of asymmetries in conditional correlations between hedge fund indices and other investments and between hedge fund indices themselves. We use the SNP approach to obtain estimates of conditional densities of hedge fund returns and then proceed to examine their properties. In general, a nonparametric GARCH(1,1) model appears to provide the best fit for all strategies. We find that the conditional third and fourth moments are significantly affected by changes in the current volatility of returns on hedge fund indices. We examine changes in the conditional probability of tail events and report significant changes in the probability of extreme events when the conditioning information changes. These results have important implications for models of hedge fund risk that rely on probability of tail events. We formally test for the presence of asymmetries in conditional correlations to determine if there is contagion between hedge funds and other investments and between various hedge fund indices in extreme down markets versus extreme up markets. We generally do not find strong evidence in support of asymmetric correlations.     

An Actuarial Premium Pricing Model for Nonnormal Insurance and Financial Risks in Incomplete Markets

Abstract

A model for pricing insurance and financial risks, based on recent developments in actuarial premium principles with elliptical distributions, is developed for application to incomplete markets and heavy-tailed distributions. The pricing model involves an application of a generalized variance premium principle from insurance pricing to the pricing of a portfolio of nontraded risks relative to a portfolio of traded risks. This pricing model for a portfolio of insurance or financial risks reflects preferences for features of the distributions other than mean and variance, including kurtosis. The model reduces to the Capital Asset Pricing Model for multinormal portfolios and to a form of the CAPM in the case where the traded and nontraded risks have the same elliptical distribution.     

Risk aversion and the yield of corporate debt

This paper develops a model to estimate the implied default probability of corporate bonds. The model explicitly considers the risk averse behavior of investors to provide a more precise framework for estimating the implied default probability. A Kalman filter method is used to estimate time-varying risk premium associated with the investor's risk aversion. The results of nonlinear regressions indicate that previous risk-neutrality models consistently overestimate the implied default rates of corporate bonds. The results also suggest that investors may have been adequately compensated for investment in risky bonds.     

Finanzielle Anreize durch Erfahrungstarifierung in der gesetzlichen Unfallversicherung

The statutory accident insurance in Germany exhibits in comparison with the other social insurance institutions an extensive premium differentiation. Especially by the use of the Beitragsausgleichsverfahren (a form of experience rating) and the resulting secondary premium differentiation incentives should be generated to increase the prevention activities. In this contribution the effects of the Beitragsausgleichsverfahren of the Butcher's Accident Prevention & Insurance Association are analyzed. It is shown that the average financial burden between high and low risks differs only very limited and the extent of the premium differentiation also depends one the companies size.     

Some results on moments and cumulants

Abstract

In the present paper we discuss various results related to moments and cumulants of probability distributions and approximations to probability distributions. As the approximations are not necessarily probability distributions themselves, we shall apply the concept of moments and cumulants to more general functions. Recursions are deduced for moments and cumulants of functions in the form R_k [a, b] as defined by Dhaene & Sundt (1996). We deduce a simple relation between the De Pril transform and the cumulants of a function. This relation is applied to some classes of approximations to probability distributions, in particular the approximations of Hipp and De Pril.     

Semi-Parametric Specification Tests for Discrete Probability Models

Loss functions play an important role in analyzing insurance portfolios. A fundamental issue in the study of loss functions involves the selection of probability models for claim frequencies. In this article, we propose a semi‐parametric approach based on the generalized method of moments (GMM) to solve the specification problems concerning claim frequency distributions. The GMM‐based testing procedure provides a general framework that encompasses many specification problems of interest in actuarial applications. As an alternative approach to the Pearson χ² and other goodness‐of‐fit tests, it is easy to implement and should be of practical use in applications involving selecting and validating probability models with complex characteristics.     

Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

We consider a risk process R _t where the claim arrival process is a superposition of a homogeneous Poisson process and a Cox process with a Poisson shot noise intensity process, capturing the effect of sudden increases of the claim intensity due to external events. The distribution of the aggregate claim size is investigated under these assumptions. For both light-tailed and heavy-tailed claim size distributions, asymptotic estimates for infinite-time and finite-time ruin probabilities are derived. Moreover, we discuss an extension of the model to an adaptive premium rule that is dynamically adjusted according to past claims experience.     

Continuity Estimates for Ruin Probabilities

A method of continuity analysis of ruin probabilities with respect to variation of parameters governing risk processes is proposed. It is based on the representation of the ruin probability as the stationary probability of a reversed process. We apply Kartashov's technique designed for continuity analysis of stationary distributions of general Markov chains in order to obtain desired continuity estimates. The method is illustrated by the Sparre Andersen and Markov modulated risk models.     

What Cost of Capital Should You Use? The Market Has an Answer

The most widely used means of estimating a company's cost of equity capital is the Capital Asset Pricing Model (CAPM). But as a growing number of academics and practitioners have suggested, use of the CAPM produces estimates that often fail to reflect the risks of the companies as perceived by current and potential investors. The authors' work, together with other research, also suggests that the cost of equity produced by the CAPM is often too high. To the extent this is so, companies are discounting investment projects at rates of return that may be leading them to pass up value‐adding opportunities. The authors advocate the use of a simple and practical alternative to the CAPM that does not use either an assumed market risk premium or a beta. It uses instead an equity premium that is implied by the current market price of a company's stock and, as such, is implicitly derived from investors' assessments of the firm's risk that are reflected in that price. More specifically, the alternative approach solves for the internal rate of return that equates the present value of expected future cash flows to the current market price. In support of this approach, studies have shown that such market‐implied measures are better predictors than CAPM‐based estimates of future stock returns, both at the individual‐firm and aggregate market levels.     

Diversification and generalized tracking errors for correlated non‐normal returns

Abstract

The probability distribution for the relative return of a portfolio constructed from a subset n of the assets from a benchmark, consisting of N assets whose returns are multivariate normal, is completely characterized by its tracking error. However, if the benchmark asset returns are not multivariate normal then higher moments of the probability distribution for the portfolio's relative return are not related to its tracking error. We discuss the convergence of generalized tracking error measures as the size of the subset of benchmark assets increases. Assuming that the joint probability distribution for the returns of the assets is symmetric under their permutations we show that increasing n makes these generalized tracking errors small (even though n « N). For n » 1 the probability distribution for the portfolio's relative return is approximately symmetric and strongly peaked about the origin. The results of this paper generalize the conclusions of Dynkin et al (Dynkin L, Hyman J and Konstantinovsky V 2002 Sufficient Diversification in Credit Portfolios (Lehman Brothers Fixed Income Research)) to more general underlying asset distributions.     

Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.     

The Tail Stein's Identity with Applications to Risk Measures

In this article, we examine a generalized version of an identity made famous by Stein, who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works followed to extend the lemma to the larger class of elliptical distributions. The lemma has had many applications in statistics, finance, insurance, and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the tail portion of the distribution of a random variable. Understanding the tails of a distribution is very important in actuarial science and insurance. Our article therefore introduces the concept of the “tail Stein's identity” to the case of any random variable defined on an appropriate probability space with a Lebesgue density function satisfying certain regularity conditions. We also examine this “tail Stein's identity” to the class of discrete distributions. This extended identity allows us to develop recursive formulas for generating tail conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate tail conditional moments. This holds a large promise for applications in risk management.     

An Asymptotic Theory for Estimating Beta-Pricing Models Using Cross-Sectional Regression

Without the assumption of conditional homoskedasticity, a general asymptotic distribution theory for the two-stage cross-sectional regression method shows that the standard errors produced by the Fama–MacBeth procedure do not necessarily overstate the precision of the risk premium estimates. When factors are misspecified, estimators for risk premiums can be biased, and the t -value of a premium may converge to infinity in probability even when the true premium is zero. However, when a beta-pricing model is misspecified, the t -values for firm characteristics generally converge to infinity in probability, which supports the use of firm characteristics in cross-sectional regressions for detecting model misspecification.