Lab |

[ E.56.1 ] Properties of the ex-ante performance

[ E.56.2 ] Strong dominance implies arbitrage

[ E.56.3 ] Estimability and monotonicity imply weak dominance consistency

[ E.56.4 ] Translation invariance implies constancy

[ E.56.5 ] Expected value: consistency with second order dominance

[ E.56.6 ] Expected value: consistency with order q dominance

[ E.56.7 ] The negative variance is concave

[ E.56.8 ] The negative variance and the mean-variance trade-off are not consistent with order q dominance

[ E.56.9 ] The negative variance and the mean-variance trade-off are not comonotonic additive

[ E.56.10 ] The negative variance is not risk averse

[ E.56.11 ] The negative variance and the mean-variance trade-off are not super-additive

[ E.56.12 ] Expectation, variance jointly elicitable

[ E.56.13 ] The mean-variance trade-off is translation invariant

[ E.56.14 ] The mean variance trade-off is joint elicitable with the variance

[ E.56.15 ] The fundamental risk quadrangle: subquantile

[ E.56.16 ] Certainty-equivalent: uniqueness

[ E.56.17 ] Certainty-equivalent: estimability

[ E.56.18 ] Certainty-equivalent: monotonicity (increasing utility)

[ E.56.19 ] Certainty-equivalent: consistency with weak dominance (increasing utility)

[ E.56.20 ] Certainty-equivalent: consistency with order q dominance

[ E.56.21 ] Certainty-equivalent: constancy

[ E.56.22 ] Certainty-equivalent: money-equivalence

[ E.56.23 ] Certainty-equivalent: positive homogeneity of degree 1 (power utility)

[ E.56.24 ] Certainty-equivalent: translation invariance (exponential utility)

[ E.56.25 ] Certainty-equivalent: additivity (linear utility)

[ E.56.26 ] Certainty-equivalent: comonotonic additivity (linear utility)

[ E.56.27 ] Certainty-equivalent: risk aversion, risk propensity and risk neutrality

[ E.56.28 ] Relation between Arrow-Pratt risk aversion function and utility function

[ E.56.29 ] Certainty-equivalent and positive affine transformations of the utility function

[ E.56.30 ] Certainty-equivalent (quadratic normal distribution)

[ E.56.31 ] Certainty-equivalent (elliptical distribution)

[ E.56.32 ] The value at risk

[ E.56.33 ] Quantile (VaR) satisfaction measure: estimability

[ E.56.34 ] Quantile (VaR) satisfaction measure: monotonicity

[ E.56.35 ] Quantile (VaR) satisfaction measure: consistency with weak dominance

[ E.56.36 ] Quantile (VaR) satisfaction measure: violation of consistency with order q dominance

[ E.56.37 ] Quantile (VaR) satisfaction measure: constancy

[ E.56.38 ] Quantile (VaR) satisfaction measure: money-equivalence

[ E.56.39 ] Quantile (VaR) satisfaction measure: positive homogeneity of degree 1

[ E.56.40 ] Quantile (VaR) satisfaction measure: translation invariance

[ E.56.41 ] Quantile (VaR) satisfaction measure: violation of super-additivity

[ E.56.42 ] Quantile (VaR) satisfaction measure: comonotonic additivity

[ E.56.43 ] Quantile (VaR) satisfaction measure: violation of concavity and convexity

[ E.56.44 ] Quantile (VaR) satisfaction measure: violation of risk-aversion, risk-seeking and risk-neutrality

[ E.56.45 ] Quantile (VaR) satisfaction measure of normally distributed ex-ante performances satisfies super-additivity

[ E.56.46 ] Central moments of an affine transformation of a multivariate random variable

[ E.56.47 ] Variance of an affine transformation of a multivariate random variable

[ E.56.48 ] Expectation, standard deviation and skewness of a portfolio P&L under lognormality

[ E.56.49 ] The expected shortfall as sub-quantile

[ E.56.50 ] Sub-quantile satisfaction measures are monotone

[ E.56.51 ] Sub-quantile satisfaction measures: consistency with weak dominance

[ E.56.52 ] Sub-quantile satisfaction measures: constancy

[ E.56.53 ] Sub-quantile satisfaction measures: money-equivalence

[ E.56.54 ] Sub-quantile satisfaction measures: positive homogeneity of degree 1

[ E.56.55 ] Sub-quantile satisfaction measures: translation invariance

[ E.56.56 ] Sub-quantile satisfaction measure: super additivity

[ E.56.57 ] Sub-quantile satisfaction measures: comonotonic additivity

[ E.56.58 ] Sub-quantile satisfaction measures: risk-aversion

[ E.56.59 ] Sub-quantile satisfaction measures: violation of consistency with order q dominance

[ E.56.60 ] Spectral satisfaction measures: estimability

[ E.56.61 ] Spectral satisfaction measures are monotone

[ E.56.62 ] Spectral satisfaction measures: consistency with weak dominance

[ E.56.63 ] Spectral satisfaction measures: constancy

[ E.56.64 ] Spectral satisfaction measures: money-equivalence

[ E.56.65 ] Spectral satisfaction measures: positive homogeneity of degree 1

[ E.56.66 ] Spectral satisfaction measures: translation invariance

[ E.56.67 ] Spectral satisfaction measures: comonotonic additivity

[ E.56.68 ] Spectral satisfaction measures: violation of consistency with order q dominance

[ E.56.69 ] Spectral satisfaction measures: violation of super-additivity

[ E.56.70 ] Spectral satisfaction measures: violation of concavity and convexity

[ E.56.71 ] Spectral/distortion satisfaction measure weights (scenario-probability distribution)

[ E.56.72 ] Alternative representation of spectral satisfaction measure

[ E.56.73 ] Quantile (VaR) satisfaction measure: distortion function

[ E.56.74 ] Sub-quantile satisfaction measure: distortion function

[ E.56.75 ] The Wang expectation: distortion function

[ E.56.76 ] The Buhlmann expectation is not a distortion expectation

[ E.56.77 ] Equivalence between spectral and distortion satisfaction measures

[ E.56.78 ] Spectral measures as weighted averages of expected shortfalls

[ E.56.79 ] Equivalent definitions of monotonicity

[ E.56.80 ] The mean-lower partial moment root is coherent

[ E.56.81 ] Dual representation of the sub-quantile satisfaction measure in the scenario-probability framework

[ E.56.82 ] Worst possible measure for the expected shortfall in the scenario-probability framework

[ E.56.83 ] Coherent spectral satisfaction measures as distortions

[ E.56.84 ] Coherent spectral satisfaction measures: super-additivity

[ E.56.85 ] Mean-lower partial moment trade-off: violation of comonotonic additivity

[ E.56.86 ] Coherent satisfaction measures: consistency with weak dominance

[ E.56.87 ] Coherent satisfaction measures: constancy

[ E.56.88 ] Coherent satisfaction measures: money-equivalence

[ E.56.89 ] Coherent representation of coherent spectral measures

[ E.56.90 ] Coherent spectral measures: characterization

[ E.56.91 ] Worst case representation of expectiles

[ E.56.92 ] Convex combinations of coherent satisfaction measures

[ E.56.93 ] The Wang distortion expectation

[ E.56.94 ] The proportional hazards distortion expectation

[ E.56.95 ] Cornish-Fisher approximation for spectral satisfaction measures

[ E.56.96 ] Extreme value theory: approximation of the sub-quantile satisfaction measure

[ E.56.97 ] Sub-quantile satisfaction measure weights (scenario-probability distribution)

[ E.56.98 ] Derivative of an indefinite integral

[ E.56.99 ] Relation between the omega and the kappa ratio

[ E.56.100 ] Economic capital

[ E.56.101 ] The Buhlmann expectation is a distortion expectation

[ E.56.102 ] First order approximation of the Buhlmann expectation

[ E.56.103 ] Esscher transform as minimum entropy distribution

[ E.56.104 ] First order approximation of the Esscher expectation

[ E.56.105 ] The Esscher expectation is neither positive homogeneous nor linear

[ E.56.106 ] Buhlmann expectation: linearity

[ E.56.107 ] The utility function as the cdf of a subjective distribution

[ E.56.108 ] The certainty-equivalent as the quantile (VaR)

[ E.56.109 ] The Arrow-Pratt approximation

[ E.56.110 ] Esscher expectation under normality assumption

[ E.56.111 ] Buhlmann expectation under normality assumption

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