Valeria Bignozzi

On elicitable risk measures

Informally, a statistical functional T on a set of probability measures M on the real line is elicitable if it can be defined as the minimizer of a suitable expected scoring function. The simplest examples of elicitable functionals are the mean that minimizes a quadratic score and the quantiles that are the set of minimizers of a piecewise linear score. Despite this notion finds its roots in decision theory, see for example Savage (1971), the term ‘elicitable functional’ seems to have been introduced in Osband (1985). See also Lambert et al. (2008), Gneiting (2011) and the reference therein for an early history of the notion of elicitability. We refer to the recent paper of Gneiting (2011) for the formal definition. Let T : M → 2 be a possibly set-valued functional, where 2 is the power set of R. T is elicitable relative to the class M if there exists a scoring function S :

Elicitable Risk Measures

A statistical functional is elicitable if it can be defined as the minimizer of a suitable expected scoring function (see Gneiting (2011), Ziegel (2013) and the references therein). With financial applications in view, we suggest a slightly more restrictive definition than Gneiting (2011), and we derive several necessary conditions. For monetary risk measures, we show that elicitability leads to a subclass of the shortfall risk measures introduced in Follmer and Schied (2002). In the coherent case, we show that the only elicitable risk measures are the expectiles. Further, we provide an alternative proof of the result in Ziegel (2013) that the only coherent comonotone elicitable risk measure is the expected loss.

How Superadditive Can a Risk Measure Be

In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extremeaggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex shortfall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management.

Parameter Uncertainty and Residual Estimation Risk

The notion of residual estimation risk is introduced in order to study the impact of parameter uncertainty on capital adequacy, for a given risk measure and capital estimation procedure. Residual estimation risk is derived by applying the risk measure on a portfolio consisting of a random loss and a capital estimator, reecting the randomness inherent in the data. Residual risk thus equals the additional amount of capital that needs to be added to the portfolio to make it acceptable. We propose modied capital estimation procedures, based on parametric bootstrapping and on predictive distributions, which tend to increase capital requirements, by compensating for parameter uncertainty and leading to a residual risk close to zero. In the particular case of location-scale families of distributions, the analysis simplies substantially and a capital estimator can always be found that leads to a residual risk of exactly zero.

Model Uncertainty in Risk Capital Measurement

The required solvency capital for a financial portfolio is typically given by a tail risk measure such as Value-at-Risk. Estimating the value of that risk measure from a limited, often small, sample of data gives rise to potential errors in the selection of the statistical model and the estimation of its parameters. We propose to quantify the effectiveness of a capital estimation procedure via the notions of residual estimation risk and estimated capital risk. It is shown that for capital estimation procedures that do not require the specification of a model (eg historical simulation) or for worst-case scenario procedures the impact of model uncertainty is substantial, while capital estimation procedures that allow for multiple candidate models using Bayesian methods, partially eliminate model error. In the same setting, we propose a way of quantifying model error that allows to disentangle the impact of model uncertainty from that of parameter uncertainty. We illustrate these ideas by simulation examples considering standard loss and return distributions used in banking and insurance.

Characterization and Construction of Sequentially Consistent Risk Measures

In dynamic risk measurement the problem emerges of assessing the risk of a financial position at different times. Sufficient conditions are provided for conditional coherent risk measures, in order that the requirements of acceptance, rejection and sequential consistency are satisfied. It is shown that these conditions are often violated for standard methods of updating. A method is consequently proposed for constructing a sequentially consistent risk measure, which entails the modification of the set of probability measures used to obtain the risk assessment at an initial time. This is demonstrated for the coherent entropic risk measure and for the class of Choquet risk measures, which generalizes the well-known TVaR. Finally we consider the situation where the term of risk exposures is longer than the time horizon used in solvency assessment. Then, regulation such as Solvency II requires replacing the financial position itself with its fair value at the time horizon. We show that in this setting acceptance consistency can be preserved, though the same is not true about rejection consistency.

Risk Measures Based on Benchmark Loss Distributions

We introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow for a range of confidence levels that depend on the loss magnitude. The key ingredient is a benchmark loss distribution (BLD), i.e.~a function that associates to each potential loss a maximal acceptable probability of occurrence. The corresponding risk measure, called Loss VaR (LVaR), determines the minimal capital injection that is required to align the loss distribution of a risky position to the target BLD. By design, one has full flexibility in the choice of the BLD profile and, therefore, in the range of relevant quantiles. Special attention is given to piecewise constant functions and to tail distributions of benchmark random losses, in which case the acceptability condition imposed by the BLD boils down to first-order stochastic dominance. We provide a comprehensive study of the main finance theoretical and statistical properties of LVaR with a focus on their comparison with VaR and ES. Merits and drawbacks are discussed and applications to capital adequacy, portfolio risk management and catastrophic risk are presented.

Robust and Pareto Optimality of Insurance Contracts

The optimal insurance problem represents a fast growing topic that explains the most efficient contract that an insurance player may get. The classical problem investigates the ideal contract under the assumption that the underlying risk distribution is known, i.e. by ignoring the parameter and model risks. Taking these sources of risk into account, the decision-maker aims to identify a robust optimal contract that is not sensitive to the chosen risk distribution. We focus on Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)-based decisions, but further extensions for other risk measures are easily possible. The Worst-case scenario and Worst-case regret robust models are discussed in this paper, which have been already used in robust optimisation literature related to the investment portfolio problem. Closed-form solutions are obtained for the VaR Worst-case scenario case, while Linear Programming (LP) formulations are provided for all other cases. A caveat of robust optimisation is that the optimal solution may not be unique, and therefore, it may not be economically acceptable, i.e. Pareto optimal. This issue is numerically addressed and simple numerical methods are found for constructing insurance contracts that are Pareto and robust optimal. Our numerical illustrations show weak evidence in favour of our robust solutions for VaR-decisions, while our robust methods are clearly preferred for CVaR-based decisions.