Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting

In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk.

A fuzzy risk attitude classification based on prospect theory

Traditional methods do not actually measure peoples’ risk attitude naturally and precisely. Therefore, a fuzzy risk attitude classification method is developed. Since the prospect theory is usually considered as an effective model of decision making, the personalized parameters in prospect theory are firstly fuzzified to distinguish people with different risk attitudes, and then a fuzzy classification database schema is applied to calculate the exact value of risk value attitude and risk be- havior attitude. Finally, by applying a two-hierarchical clas- sification model, the precise value of synthetical risk attitude can be acquired.

Temporal variability of damage potential on roads as a conceptual contribution towards a short-term avalanche risk simulation

The fatality risk caused by avalanches on road networks can be analysed using a long-term approach, resulting in a mean value of risk, and with emphasis on short-term fluctuations due to the temporal variability of both, the hazard potential and the damage potential. In this study, the approach for analysing the long-term fatality risk has been adapted by modelling the highly variable short-term risk. The emphasis was on the temporal variability of the damage potential and the related risk peaks. For defined hazard scenarios resulting from classified amounts of snow accumulation, the fatality risk was calculated by modelling the hazard potential and observing the traffic volume. The avalanche occurrence probability was calculated using a statistical relationship between new snow height and observed avalanche releases. The number of persons at risk was determined from the recorded traffic density. The method resulted in a value for the fatality risk within the observed time frame for the studied road segment. The long-term fatality risk due to snow avalanches as well as the short-term fatality risk was compared to the average fatality risk due to traffic accidents. The application of the method had shown that the long-term avalanche risk is lower than the fatality risk due to traffic accidents. The analyses of short-term avalanche-induced fatality risk provided risk peaks that were 50 times higher than the statistical accident risk. Apart from situations with high hazard level and high traffic density, risk peaks result from both, a high hazard level combined with a low traffic density and a high traffic density combined with a low hazard level. This provided evidence for the importance of the temporal variability of the damage potential for risk simulations on road networks. The assumed dependence of the risk calculation on the sum of precipitation within three days is a simplified model. Thus, further research is needed for an improved determination of the diurnal avalanche probability. Nevertheless, the presented approach may contribute as a conceptual step towards a risk-based decision-making in risk management.

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.

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.

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.

Value of the SYNTAX score for risk assessment in the all-comers population of the randomized multicenter LEADERS (Limus Eluted from A Durable versus ERodable Stent coating) trial

OBJECTIVES: We aimed to assess the predictive value of the SYNTAX score (SXscore) for major adverse cardiac events in the all-comers population of the LEADERS (Limus Eluted from A Durable versus ERodable Stent coating) trial. BACKGROUND: The SXscore has been shown to be an effective predictor of clinical outcomes in patients with multivessel disease undergoing percutaneous coronary intervention. METHODS: The SXscore was prospectively collected in 1,397 of the 1,707 patients enrolled in the LEADERS trial (patients after surgical revascularization were excluded). Post hoc analysis was performed by stratifying clinical outcomes at 1-year follow-up, according to 1 of 3 SXscore tertiles. RESULTS: The 1,397 patients were divided into tertiles based on the SXscore in the following fashion: SXscore8 and 16 (SXhigh) (n=461). At 1-year follow-up, there was a significantly lower number of patients with major cardiac event-free survival in the highest tertile of SXscore (SXlow=92.2%, SXmid=91.1%, and SXhigh=84.6%; p<0.001). Death occurred in 1.5% of SXlow patients, 2.1% of SXmid patients, and 5.6% of SXhigh patients (hazard ratio [HR]: 1.97, 95% confidence interval [CI]: 1.29 to 3.01; p=0.002). The myocardial infarction rate tended to be higher in the SXhigh group. Target vessel revascularization was 11.3% in the SXhigh group compared with 6.3% and 7.8% in the SXlow and SXmid groups, respectively (HR: 1.38, 95% CI: 1.1 to 1.75; p=0.006). Composite of cardiac death, myocardial infarction, and clinically indicated target vessel revascularization was 7.8%, 8.9%, and 15.4% in the SXlow, SXmid, and SXhigh groups, respectively (HR: 1.47, 95% CI: 1.19 to 1.81; p<0.001). CONCLUSIONS: The SXscore, when applied to an all-comers patient population treated with drug-eluting stents, may allow prospective risk stratification of patients undergoing percutaneous coronary intervention. (LEADERS Trial Limus Eluted From A Durable Versus ERodable Stent Coating; NCT00389220).

A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk Process

In this article we propose a bootstrap test for the probability of ruin in the compound Poisson risk process. We adopt the P-value approach, which leads to a more complete assessment of the underlying risk than the probability of ruin alone. We provide second-order accurate P-values for this testing problem and consider both parametric and nonparametric estimators of the individual claim amount distribution. Simulation studies show that the suggested bootstrap P-values are very accurate and outperform their analogues based on the asymptotic normal approximation.

Prognostic value of microvascular invasion in predicting the cancer specific survival and risk of metastatic disease in renal cell carcinoma: a multicenter investigation

While microvascular invasion is an accepted risk factor in various cancers, its prognostic role in renal cell carcinoma is still unclear. Therefore, a large multicenter study examining the experience of 5 international institutions was performed to evaluate the prognostic value of microvascular invasion in the occurrence of metastases and cancer specific survival.

Value of age, creatinine, and ejection fraction (ACEF score) in assessing risk in patients undergoing percutaneous coronary interventions in the 'All-Comers' LEADERS trial

Background— The age, creatinine, and ejection fraction (ACEF) score (age/left ventricular ejection fraction+1 if creatinine >2.0 mg/dL) has been established as an effective predictor of clinical outcomes in patients undergoing elective coronary artery bypass surgery; however, its utility in “all-comer” patients undergoing percutaneous coronary intervention is yet unexplored. Methods and Results— The ACEF score was calculated for 1208 of the 1707 patients enrolled in the LEADERS trial. Post hoc analysis was performed by stratifying clinical outcomes at the 1-year follow-up according to ACEF score tertiles: ACEFlow ≤1.0225, 1.0225< ACEFmid ≤1.277, and ACEFhigh >1.277. At 1-year follow-up, there was a significantly lower number of patients with major adverse cardiac event–free survival in the highest tertile of the ACEF score (ACEFlow=92.1%, ACEFmid=89.5%, and ACEFhigh=86.1%; P=0.0218). Cardiac death was less frequent in ACEFlow than in ACEFmid and ACEFhigh (0.7% vs 2.2% vs 4.5%; hazard ratio=2.22, P=0.002) patients. Rates of myocardial infarction were significantly higher in patients with a high ACEF score (6.7% for ACEFhigh vs 5.2% for ACEFmid and 2.5% for ACEFlow; hazard ratio=1.6, P=0.006). Clinically driven target-vessel revascularization also tended to be higher in the ACEFhigh group, but the difference among the 3 groups did not reach statistical significance. The rate of composite definite, possible, and probable stent thrombosis was also higher in the ACEFhigh group (ACEFlow=1.2%, ACEFmid=3.5%, and ACEFhigh=6.2%; hazard ratio=2.04, P<0.001). Conclusions— ACEF score may be a simple way to stratify risk of events in patients treated with percutaneous coronary intervention with respect to mortality and risk of myocardial infarction.

Comparative analysis of the value of allogeneic hematopoietic stem-cell transplantation in acute myeloid leukemia with monosomal karyotype versus other cytogenetic risk categories

To determine to what extent allogeneic hematopoietic stem-cell transplantation (alloHSCT) quantitatively reduces relapse in acute myeloid leukemia with monosomal karyotype (MK-AML) compared with alternative postremission therapy and how it compares with other cytogenetic subcategories.

Added value of coronary artery calcium score as an adjunct to gated SPECT for the evaluation of coronary artery disease in an intermediate-risk population

The coronary artery calcium (CAC) score is a readily and widely available tool for the noninvasive diagnosis of atherosclerotic coronary artery disease (CAD). The aim of this study was to investigate the added value of the CAC score as an adjunct to gated SPECT for the assessment of CAD in an intermediate-risk population. METHODS: Seventy-seven prospectively recruited patients with intermediate risk (as determined by the Framingham Heart Study 10-y CAD risk score) and referred for coronary angiography because of suspected CAD underwent stress (99m)Tc-tetrofosmin SPECT myocardial perfusion imaging (MPI) and CT CAC scoring within 2 wk before coronary angiography. The sensitivity and specificity of SPECT alone and of the combination of the 2 methods (SPECT plus CAC score) in demonstrating significant CAD (>/=50% stenosis on coronary angiography) were compared. RESULTS: Forty-two (55%) of the 77 patients had CAD on coronary angiography, and 35 (45%) had abnormal SPECT results. The CAC score was significantly higher in subjects with perfusion abnormalities than in those who had normal SPECT results (889 +/- 836 [mean +/- SD] vs. 286 +/- 335; P < 0.0001). Similarly, with rising CAC scores, a larger percentage of patients had CAD. Receiver-operating-characteristic analysis showed that a CAC score of greater than or equal to 709 was the optimal cutoff for detecting CAD missed by SPECT. SPECT alone had a sensitivity and a specificity for the detection of significant CAD of 76% and 91%, respectively. Combining SPECT with the CAC score (at a cutoff of 709) improved the sensitivity of SPECT (from 76% to 86%) for the detection of CAD, in association with a nonsignificant decrease in specificity (from 91% to 86%). CONCLUSION: The CAC score may offer incremental diagnostic information over SPECT data for identifying patients with significant CAD and negative MPI results.