# Abstracts

**Surplus-invariant capital adequacy tests and their risk measures**

*Santiago Moreno-Bromberg (University of Zurich, Switzerland)*

Tuesday June 3, 14:00-14:30 | session 2.3 | Risk Measures | room EF

The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. Our objective is to investigate, in the context of bounded financial positions, the class of surplus-invariant acceptance sets. These are characterized by the fact that acceptability does not depend on the positive part, or surplus, of a capital position. We argue that surplus invariance is a reasonable requirement from a regulatory perspective, because it focuses on the interests of liability holders of a fi nancial institution. We provide a dual characterization of surplus-invariant, convex acceptance sets, and show that the combination of surplus invariance and coherence leads to a narrow range of capital adequacy tests, essentially limited to scenario-based tests. Finally, we analyze the relationship between surplus-invariant acceptance sets and loss-based and excess-invariant risk measures.

**Liquidity-adjusted risk measures**

*Stefan Weber (Leibniz Universität Hannover, Germany)*

Tuesday June 3, 14:30-15:00 | session 2.3 | Risk Measures | room EF

Liquidity risk is an important type of risk, especially during times of crises. As observed by Acerbi and Scandolo (2008), it requires adjustments to classical portfolio valuation and risk measurement. Main drivers are two dimensions of liquidity risk, namely price impact of trades and limited access to financing. The key contribution of the current presentation is the construction of both single- and multi-objective liquidity-adjusted risk measures that can naturally be interpreted as capital requirements. In the multi-dimensional setting, portfolios are modeled in physical units as originally suggested by Kabanov (1999). Numerical case studies illustrate how price impact and limited access to financing influence the liquidity-adjusted risk measurements.

**Conditional Systemic Risk Measures**

*Thilo Meyer-Brandis (University of Munich, Germany)*

Tuesday June 3, 15:00-15:30 | session 2.3 | Risk Measures | room EF

The recent financial crisis has demonstrated that systemic risk due to the interconnectedness of financial-market participants - such as financial institutions, insurers, governments and, even, regulators themselves - can dramatically amplify the consequences of isolated shocks to financial systems and pose a serious threat to prosperity and social stability. The traditional approach to risk control in financial mathematics is to apply risk measures to single institutions. However, this strategy fails to capture systemic risk because it treats institutions as if they were in isolation, and recent literature in financial mathematics has started to develop various approaches to rectify this deficiency in traditional risk management strategies. Some examples are the Conditional Value at Risk [Adrian and Brunnermeier, 2009] or the Conditional Expected Shortfall [Acharya et al., 2010] where the risk of the system, measured as the sum of the single institutions' P&L figures, is analyzed conditional on a stress scenario of a specific institution, or vice versa. However, while for a portfolio manager summing the single P&L figures is appropriate, this aggregation of risk factors might be problematic to measure systemic risk in a financial system. In [Chen et al., 2013] an axiomatic approach to (unconditional) risk measures for a system is introduced that considers more appropriate aggregation of risk factors than taking the sum. In our work we extend the approach in [Chen et al., 2013] along different lines. In particular, we will introduce and characterize conditional quasi-convex systemic risk measures that can be decomposed into a classical (conditional) risk measure applied to a (conditional) aggregation of risk factors. Further, we will consider consistency issues of such risk measures and discuss some examples. Also, contrary to [Chen et al., 2013] where a finite probability space is considered, we will work on a general probability space.

**Conditional Weighted Expected Shortfall, Conditional Distortion Risk Measures, and Application to Risk Capital Allocation**

*Uwe Schmock (Vienna University of Technology, Austria)*

Tuesday June 3, 15:30-16:00 | session 2.3 | Risk Measures | room EF

Based on the concepts of measurable upper envelopes and conditional lower quantiles, we define conditional distortion risk measures via a stochastic integral representation. We show various properties of conditional distortion risk measures, including coherence with respect to distortion processes with concave paths. Conditional weighted expected shortfall arises as a special case of conditional distortion risk measures. We also give a definition via an explicit density on a modelling setup with stochastic levels, involving generalised conditional expectations based on sigma-integrability.

Then we prove several properties and give several alternative representations of conditional (weighted) expected shortfall. Furthermore, we point out the link to dynamic risk measures and show a supermartingale property.

In the next step we introduce contributions to conditional weighted expected shortfall and prove several properties. In particular, it is possible to derive the contribution of a subportfolio to the whole portfolio in order to be able to identify main risks. Conditional weighted expected shortfall includes beta- and alpha-value-at-risk as special cases. We end with some applications including a time series example.