Elicitable distortion risk measures: A concise proof

Elicitability has recently been discussed as a desirable property for risk measures. Kou and Peng (2014) showed that an elicitable distortion risk measure is either a Value-at-Risk or the mean. We give a concise alternative proof of this result, and discuss the conflict between comonotonic additivity and elicitability.

Risk measures with the CxLS property

In the present contribution, we characterise law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (Math. Finance 16:419–441, 2006), we show that these risk measures can be identified with a class of generalised shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (Math. Finance, 2014, http://onlinelibrary.wiley.com/doi/10.1111/mafi.12080/abstract) and Bellini and Bignozzi (Quant. Finance 15:725–733, 2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterisation of robustness for convex risk measures in terms of a weak notion of mixture continuity.

Predicting the onset of psychosis in patients at clinical high risk: practical guide to probabilistic prognostic reasoning

Prediction of psychosis in patients at clinical high risk (CHR) has become a mainstream focus of clinical and research interest worldwide. When using CHR instruments for clinical purposes, the predicted outcome is but only a probability; and, consequently, any therapeutic action following the assessment is based on probabilistic prognostic reasoning. Yet, probabilistic reasoning makes considerable demands on the clinicians. We provide here a scholarly practical guide summarising the key concepts to support clinicians with probabilistic prognostic reasoning in the CHR state. We review risk or cumulative incidence of psychosis in, person-time rate of psychosis, Kaplan-Meier estimates of psychosis risk, measures of prognostic accuracy, sensitivity and specificity in receiver operator characteristic curves, positive and negative predictive values, Bayes’ theorem, likelihood ratios, potentials and limits of real-life applications of prognostic probabilistic reasoning in the CHR state. Understanding basic measures used for prognostic probabilistic reasoning is a prerequisite for successfully implementing the early detection and prevention of psychosis in clinical practice. Future refinement of these measures for CHR patients may actually influence risk management, especially as regards initiating or withholding treatment.

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.

Choosing a random distribution with prescribed risks

We describe several simulation algorithms that yield random probability distributions with given values of risk measures. In case of vanilla risk measures, the algorithms involve combining and transforming random cumulative distribution functions or random Lorenz curves obtained by simulating rather general random probability distributions on the unit interval. A new algorithm based on the simulation of a weighted barycentres array is suggested to generate random probability distributions with a given value of the spectral risk measure.

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.

Coherence and Elicitability

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile-based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law-invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law-invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.

Higher order elicitability and Osband’s principle

A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance, this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the “correct” quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.

Risk assessment and preventive measures

A prerequisite for preventive measures is to diagnose erosive tooth wear and to evaluate the different etiological factors in order to identify persons at risk. No diagnostic device is available for the assessment of erosive defects. Thus, they can only be detected clinically. Consequently, erosion not diagnosed in the early stage may render timely preventive measures difficult. In order to assess the risk factors, patient should record their dietary intake for a distinct period of time. Then a dentist can determine the erosive potential of the diet. Particularly, patients with more than four dietary acid intakes have a higher risk for erosion when other risk factors (such as holding the drink in the mouth) are present. Regurgitation of gastric acids (reflux, vomiting, alcohol abuse, etc.) is a further important risk factor for the development of erosion which has to be taken into account. Based on these analyses, an individually tailored preventive program may be suggested to the patients. It may comprise dietary advice, optimization of fluoride regimes, stimulation of salivary flow rate, use of buffering medicaments and particular motivation for nondestructive toothbrushing habits with a low abrasive toothpaste. The frequent use of fluoride gel and fluoride solution in addition to fluoride toothpaste offers the opportunity to reduce somewhat abrasion of tooth substance. It is also advisable to avoid abrasive tooth cleaning and whitening products, since they may remove the pellicle and may render teeth more susceptible to erosion. Since erosion, attrition and abrasion often occur simultaneously all causative components must be taken into consideration when planning preventive strategies.

Can quantitative EEG measures predict clinical outcome in subjects at Clinical High Risk for psychosis? A prospective multicenter study

BACKGROUND Prediction studies in subjects at Clinical High Risk (CHR) for psychosis are hampered by a high proportion of uncertain outcomes. We therefore investigated whether quantitative EEG (QEEG) parameters can contribute to an improved identification of CHR subjects with a later conversion to psychosis. METHODS This investigation was a project within the European Prediction of Psychosis Study (EPOS), a prospective multicenter, naturalistic field study with an 18-month follow-up period. QEEG spectral power and alpha peak frequencies (APF) were determined in 113 CHR subjects. The primary outcome measure was conversion to psychosis. RESULTS Cox regression yielded a model including frontal theta (HR=1.82; [95% CI 1.00-3.32]) and delta (HR=2.60; [95% CI 1.30-5.20]) power, and occipital-parietal APF (HR=.52; [95% CI .35-.80]) as predictors of conversion to psychosis. The resulting equation enabled the development of a prognostic index with three risk classes (hazard rate 0.057 to 0.81). CONCLUSIONS Power in theta and delta ranges and APF contribute to the short-term prediction of psychosis and enable a further stratification of risk in CHR samples. Combined with (other) clinical ratings, EEG parameters may therefore be a useful tool for individualized risk estimation and, consequently, targeted prevention.

Risk assessment and causal preventive measures.

A prerequisite for preventive measures is to diagnose erosive tooth wear and to evaluate the different etiological factors in order to identify persons at risk. No diagnostic device is available for the assessment of erosive defects. Thus, they can only be detected clinically. Consequently, erosion not diagnosed at an early stage may render timely preventive measures difficult. In order to assess the risk factors, patients should record their dietary intake for a distinct period of time. Then a dentist can determine the erosive potential of the diet. A table with common beverages and foodstuffs is presented for judging the erosive potential. Particularly, patients with more than 4 dietary acid intakes have a higher risk for erosion when other risk factors are present. Regurgitation of gastric acids is a further important risk factor for the development of erosion which has to be taken into account. Based on these analyses, an individually tailored preventive program may be suggested to the patients. It may comprise dietary advice, use of calcium-enriched beverages, optimization of prophylactic regimes, stimulation of salivary flow rate, use of buffering medicaments and particular motivation for nondestructive toothbrushing habits with an erosive-protecting toothpaste as well as rinsing solutions. Since erosion and abrasion often occur simultaneously, all of the causative components must be taken into consideration when planning preventive strategies but only those important and feasible for an individual should be communicated to the patient.

Clinically relevant quality measures for risk factor control in primary care: a retrospective cohort study

BACKGROUND Assessment of the proportion of patients with well controlled cardiovascular risk factors underestimates the proportion of patients receiving high quality of care. Evaluating whether physicians respond appropriately to poor risk factor control gives a different picture of quality of care. We assessed physician response to control cardiovascular risk factors, as well as markers of potential overtreatment in Switzerland, a country with universal healthcare coverage but without systematic quality monitoring, annual report cards on quality of care or financial incentives to improve quality. METHODS We performed a retrospective cohort study of 1002 randomly selected patients aged 50-80 years from four university primary care settings in Switzerland. For hypertension, dyslipidemia and diabetes mellitus, we first measured proportions in control, then assessed therapy modifications among those in poor control. "Appropriate clinical action" was defined as a therapy modification or return to control without therapy modification within 12 months among patients with baseline poor control. Potential overtreatment of these conditions was defined as intensive treatment among low-risk patients with optimal target values. RESULTS 20% of patients with hypertension, 41% with dyslipidemia and 36% with diabetes mellitus were in control at baseline. When appropriate clinical action in response to poor control was integrated into measuring quality of care, 52 to 55% had appropriate quality of care. Over 12 months, therapy of 61% of patients with baseline poor control was modified for hypertension, 33% for dyslipidemia, and 85% for diabetes mellitus. Increases in number of drug classes (28-51%) and in drug doses (10-61%) were the most common therapy modifications. Patients with target organ damage and higher baseline values were more likely to have appropriate clinical action. We found low rates of potential overtreatment with 2% for hypertension, 3% for diabetes mellitus and 3-6% for dyslipidemia. CONCLUSIONS In primary care, evaluating whether physicians respond appropriately to poor risk factor control, in addition to assessing proportions in control, provide a broader view of the quality of care than relying solely on measures of proportions in control. Such measures could be more clinically relevant and acceptable to physicians than simply reporting levels of control.

Law and science: risk assessment and risk management in the WTO agreement on the application of sanitary and phytosanitary measures

Assessing and managing risks relating to the consumption of food stuffs for humans and to the environment has been one of the most complex legal issues in WTO law, ever since the Agreement on Sanitary and Phytosanitary Measures was adopted at the end of the Uruguay Round and entered into force in 1995. The problem was expounded in a number of cases. Panels and the Appellate Body adopted different philosophies in interpreting the agreement and the basic concept of risk assessment as defined in Annex A para. 4 of the Agreement. Risk assessment entails fundamental question on law and science. Different interpretations reflect different underlying perceptions of science and its relationship to the law. The present thesis supported by the Swiss National Research Foundation undertakes an in-depth analysis of these underlying perceptions. The author expounds the essence and differences of positivism and relativism in philosophy and natural sciences. He clarifies the relationship of fundamental concepts such as risk, hazards and probability. This investigation is a remarkable effort on the part of lawyer keen to learn more about the fundamentals based upon which the law – often unconsciously – is operated by the legal profession and the trade community. Based upon these insights, he turns to a critical assessment of jurisprudence both of panels and the Appellate Body. Extensively referring and discussing the literature, he deconstructs findings and decisions in light of implied and assumed underlying philosophies and perceptions as to the relationship of law and science, in particular in the field of food standards. Finding that both positivism and relativism does not provide adequate answers, the author turns critical rationalism and applies the methodologies of falsification developed by Karl R. Popper. Critical rationalism allows combining discourse in science and law and helps preparing the ground for a new approach to risk assessment and risk management. Linking the problem to the doctrine of multilevel governance the author develops a theory allocating risk assessment to international for a while leaving the matter of risk management to national and democratically accountable government. While the author throughout the thesis questions the possibility of separating risk assessment and risk management, the thesis offers new avenues which may assist in structuring a complex and difficult problem