Natural frequencies and Bayesian reasoning: the impact of formal education and problem context

Research suggests that children and adults are able to solve Bayesian problems when natural frequencies are used, but most studies have involved only students or academics. The present research focuses on random samples from the general population. Experiments 1 and 2 showed that the natural-frequency format resulted in more Bayesian responses than the probability format. Results of Experiment 3 showed that when the task is not framed as a medical problem and the numbers are not too difficult, about 40% of the general population could correctly solve the problems. More importantly, the natural-frequency format resulted in a substantial increase in correct answers in the lower level of formal education group. Experiment 4 suggests that the context of a problem influences people’s ability to solve a Bayesian problem. Participants were twice as likely to solve a social problem compared with a medical problem. The results of the four experiments show that natural frequency resulted in much better estimates than probabilities. It should be emphasized, however, that even when natural frequencies were used, a majority of participants were not able to solve the problems correctly.     

Improving consistency in interpreting SFAS 5 probability phrases

Statement of Financial Accounting Standard No. 5, Accounting for Contingencies (SFAS No. 5), relies on verbal probability phrases to guide recognition or disclosure decisions for loss contingencies. One of the challenges facing accountants is that verbal probability terms are vague and may have multiple meanings; thus, different accountants may interpret the same probability phrase differently. Given this background, our study addresses the difficulty of interpreting verbal probability phrases and explores a simple way to improve judgment quality. Evidence from our experiment suggests that supplementing verbal probabilities with their corresponding numerical values reduces interpersonal variability in interpreting SFAS No. 5 terms.     

Mood and Judgment of Subjective Probabilities: Evidence from the U.S. Index Option Market

Numerous psychological studies show that weather conditions affect people'smood and that mood states are correlated with people's subjective evaluationof future probabilities. In this paper, a new approach is developed and assetmarket data are employed to test the mood-subjective probability relation. Cloudcover and precipitation volume serve as two mood proxies. Our statistical analysissuggests that bad mood states are characterized by investors placing higher probabilitieson adverse events.     

On a risk model with dependence between interclaim arrivals and claim sizes

We consider an extension to the classical compound Poisson risk model for which the increments of the aggregate claim amount process are independent. In Albrecher and Teugels (2006 Albrecher, H. and Teugels, J. 2006. Exponential behavior in the presence of dependence in risk theory. Journal of Applied Probability, 43(1): 257–273. [Crossref], [Web of Science ®] , [Google Scholar]), an arbitrary dependence structure among the interclaim time and the subsequent claim size expressed through a copula is considered and they derived asymptotic results for both the finite and infinite-time ruin probabilities. In this paper, we consider a particular dependence structure among the interclaim time and the subsequent claim size and we derive the defective renewal equation satisfied by the expected discounted penalty function. Based on the compound geometric tail representation of the Laplace transform of the time to ruin, we also obtain an explicit expression for this Laplace transform for a large class of claim size distributions. The ruin probability being a special case of the Laplace transform of the time to ruin, explicit expressions are therefore obtained for this particular ruin related quantity. Finally, we measure the impact of the various dependence structures in the risk model on the ruin probability via the comparison of their Lundberg coefficients.     

The gambler’s fallacy fallacy (fallacy)

The gambler’s fallacy is the irrational belief that prior outcomes in a series of events affect the probability of a future outcome, even though the events in question are independent and identically distributed. In this paper, we argue that in the standard account of the gambler’s fallacy, the gambler’s fallacy fallacy can arise: the irrational belief that all beliefs pertaining to the probabilities of sequences of outcomes constitute the gambler’s fallacy, when, in fact, they do not. Specifically, the odds of the probabilities of some sequences of outcomes can be epistemically rational in a given decision-making situation. Not only are such odds of probabilities of sequences of outcomes not the gambler’s fallacy, but they can be implemented as a simple heuristic for avoiding the gambler’s fallacy in risk-related decision-making. However, we have to be careful not to fall prey to a variant of the gambler’s fallacy, the gambler’s fallacy fallacy (fallacy), in which we do not calculate odds for the probabilities of sequences that matter, but rather simply believe that the raw probability for the occurrence of a sequence of outcomes is the probability for the last outcome in that sequence.     

Risk Perception and Demand for Risk Mitigation in Transport: A Comparison of Lay People,Politicians and Experts

This paper aims at examining risk perception, worry and demand for risk mitigation in transport and to compare judgements made by lay people, politicians and experts. The results are based on three questionnaire surveys carried out during autumn and winter 2004. The first study involved a representative sample of the Norwegian population (n = 1716), the second sample a group of Norwegian politicians (n = 146) and the third a group of experts on transport safety (n = 26). Studies carried out previously (Sjöberg, 1998a Sjöberg, L. 1998a. Worry and risk perception. Risk Analysis, 18(1): 85–93.  [Google Scholar], 1999 Sjöberg, L. 1999. Consequences of perceived risk: demand for risk mitigation. Journal of Risk Research, 2(2): 129–149.  [Google Scholar]) have given support to the idea that consequences are more important for demands of risk mitigation than probability assessments. In the present study it is hypothesised that this may be because they are associated with worry and it is also proposed that worry relates more strongly to demands for risk mitigation than evaluation of consequences. The results of SEM‐modelling showed that worry was a stronger and more significant predictor of demands for risk mitigation compared to consequences and worry mediated the effect of consequences. Probability assessment was a totally insignificant predictor. In accordance with previous studies, the results showed that experts demanded less risk reduction than lay people and politicians. The results indicate that this is because they stress the probability more than the other two groups.     

Insurance premiums and default risk in mutual insurance

Abstract

Two types of default risk are discussed in the article: The traditional “probability of ruin” (insurer being unable to meet his obligations) and a “perceived probability of ruin” (the probability of the insured being affected by ruin). The explicit relationship between these probabilities on the actuarial loading factors of a mutual insurer were developed. The explicit mathematical formulae obtained for these complex relationships were followed also by numerical results. A second concept presented in the paper is related to the idea of actuarially fair premiums. It is shown that the premium must also be a function of the payments of the other insured as well as their claim distributions, reflecting thereby the simultaneity and mutual dependence of the insured.     

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [(1994a) Journal of Finance, 49, 1213–1252; (1994b) Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley] or Leland and Toft [(1996) Journal of Finance, 51(3), 987–1019] endogenous structural approaches, as formalized by Hilberink and Rogers [(2002) Finance and Stochastics, 6(2), 237–263], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Lévy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models.      

Contingent means in multi-life models

Multi-life models are useful in actuarial science for studying life contingency. Contingent probabilities are well-understood by most actuaries and are discussed extensively in the existing actuarial literature. However, the mean of a life in a multi-life model involving order of deaths is often found to be rather challenging to interpret by most actuaries who do not understand measure-theoretic probability. Standard textbooks on actuarial science or statistics do not elaborate on the correct interpretation of contingent means, leaving the actuaries at risk of making a blunder. This paper presents the correct interpretation both heuristically and rigorously using a non-measure-theoretic language, so that actuaries will be aware of some common misconceptions and avoid pitfalls in their work. The primary audience of this paper is practicing actuaries, actuarial students and actuarial educators. So we have given several actuarial applications. We hope that applied statisticians also will find this paper useful.     

Exact solutions for ruin probability in the presence of an absorbing upper barrier

Abstract

An explicit solution for the probability of ruin in the presence of an absorbing upper barrier was developed by Segerdahl (1970) for the particular case in which both the interoccurrence times between successive claims and the single claim amounts follow an exponential distribution with unit mean. In this paper we show that his method of solution may be extended to produce explicit solutions for two more general types of single claim amount distribution. These are the gamma distribution, denoted γ(a), where a is an integer, and the mixed exponential distribution. Comparisons are drawn between this approach when the upper barrier tends to infinity, and the classical solution for ruin probability in these particular cases given in Cramér (1955).     

The effect of interventions on realignment probabilities

The expected future change of the exchange rate within its currency band and the expected realignment rate are estimated using the Regime–Switching Model. There exists an unobserved variable s_t, which characterises the equilibrium state of the expected future change of the exchange rate within its currency band at any time t with certain probabilities. Different values of s_t correspond to states with high and low risk of realignment, respectively. The probabilities of switching between one regime and another depend on central bank intervention in the foreign exchange market. Daily data on intervention by Norges Bank are used. The data contain relatively few actual realignments, and the sample distribution of realignments may not be representative enough to capture the discrete changes in the exchange rate caused by a non-zero subjective probability of realignment (even when no realignment has in fact taken place). This causes the very well known peso problem in the estimation.     

Expert opinion and probabilistic volcanic risk assessment

We present data from an international survey of scientists working at volcanic observatories concerning eruption likelihoods. The scientists were asked a range of questions using different types of phrasing. The data suggest that the phrasing of questions affects the ways in which probabilities are estimated. In total, 71% of respondents (N = 70) exhibited some form of inconsistency in their answers between and/or within different question formats. The data also allow for an analysis of the use of scaling in probabilistic assessment, and the use of quantitative versus verbal risk measurements. However, some respondents were uncomfortable with providing any numerical probability estimate, perhaps suggesting that they considered the uncertainty too high for meaningful judgements to be made.     

Moody's correlated binomial default distributions for inhomogeneous portfolios

This paper generalizes Moody's correlated binomial default distribution for homogeneous (exchangeable) credit portfolios, which was introduced by Witt, to the case of inhomogeneous portfolios. We consider two cases of inhomogeneous portfolios. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of inhomogeneity. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized into several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than intra-sector correlation. We study the maximum values of the inter-sector default correlation. Our generalization method can be applied to any correlated binomial default distribution model that has explicit relations to the conditional default probabilities or conditional default correlations, e.g. Credit Risk⁺, implied default distributions. We also compare some popular CDO pricing models from the viewpoint of the range of the implied tranche correlation.     

Parisian ruin probability with a lower ultimate bankrupt barrier

The paper deals with a ruin problem, where there is a Parisian delay and a lower ultimate bankrupt barrier. In this problem, we will say that a risk process get ruined when it stays below zero longer than a fixed amount of time ζ > 0 or goes below a fixed level ?a. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we identify the Laplace transform of the ruin probability in terms of so-called q-scale functions. We find its Cramér-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.     

Loss persistence and returns in the UK

In this study, we examine whether estimated loss reversal probabilities are fully reflected in UK stock market prices. Overall, we provide evidence of varying degrees and types of loss firm mispricing with respect to estimated loss reversal probabilities. In particular, a significant and positive relationship between loss reversal probability and annual returns is found only for firms with higher trading costs. When looking at monthly returns, however, especially for the financial statement release month subsequent to the loss year, a significant and positive relationship is found for all firms. Thus, the evidence is consistent with UK market participants not fully incorporating relevant information into the pricing of loss firms and, as a consequence, being surprised by the content of the earnings for many or all UK loss firms.     

Non‐Monotonicity of the Tversky‐Kahneman Probability‐Weighting Function: A Cautionary Note

Cumulative Prospect Theory has gained a great deal of support as an alternative to Expected Utility Theory as it accounts for a number of anomalies in the observed behavior of economic agents. Expected Utility Theory uses a utility function and subjective or objective probabilities to compare risky prospects. Cumulative Prospect Theory alters both of these aspects. The concave utility function is replaced by a loss‐averse utility function and probabilities are replaced by decision weights. The latter are determined with a weighting function applied to the cumulative probability of the outcomes. Several different probability weighting functions have been suggested. The two most popular are the original proposal of Tversky and Kahneman and the compound‐invariant form proposed by Prelec. This note shows that the Tversky‐Kahneman probability weighting function is not increasing for all parameter values and therefore can assign negative decision weights to some outcomes. This in turn implies that Cumulative Prospect Theory could make choices not consistent with first‐order stochastic dominance.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.     

Risky asset pricing based on safety first fund management

We study a portfolio selection model based on Kataoka's safety-first criterion (KSF model in short). We assume that the market is complete but without risk-free asset, and that the returns are jointly elliptically distributed. With these assumptions, we provide an explicit analytical optimal solution for the KSF model and obtain some geometrical properties of the efficient frontier in the plane of probability risk degree z _α and target return r _α. We further prove a two-fund separation and tangency portfolio theorem in the spirit of the traditional mean-variance analysis. We also establish a risky asset pricing model based on risky funds that is similar to Black's zero-beta capital asset pricing model (CAPM, for short). Moreover, we simplify our risky asset pricing model using a derivative risky fund as a reference for market evaluation.