We discuss recent advances in the mathematical quantification of financial risk. The standard approach in terms of Value at Risk has serious deficiencies. This has motivated a systematic analysis of risk measures which satisfy some minimal requirements of coherence and consistency. Our focus will be on the basic structure theorems for convex risk measures, on the role of law-invariance, and on ...

One of the major problem faced by banks is how to manage the risk exposure in large portfolios. According to Basel II regulation banks has to measure the risk using Value-at-Risk with confidence level 99%. However, this regulation does not specify the way to calculate Valueat-Risk. The easiest way to calculate Value-at-Risk is to assume that portfolio returns are normally distributed. Altough, ...

The purpose of this paper is to review the recent derivatives security research involving liquidity risk and to summarize its implications for practical risk management. The literature supports three general conclusions. The first is that the classical option price is "on average" true, even given liquidity risk. Second, it is well known that although the classical (theoretical) option hedge ca...

In the Lee-Carter framework, future survival probabilities are random variables with an intricate distribution function. In large homogeneous portfolios of life annuities, Value-at-Risk or Conditional Tail Expectation of the total yearly payout of the company are approximately equal to the corresponding quantities involving random survival probabilities. This paper aims to derive some bounds in...

Risk diversification is the basis of insurance and investment. It is thus crucial to study the effects that could limit it. One of them is the existence of systemic risk that affects all of the policies at the same time. We introduce here a probabilistic approach to examine the consequences of its presence on the risk loading of the premium of a portfolio of insurance policies. This approach co...

Financial institutions define their marginal cost of risk on the basis of the gradients of arbitrarily chosen risk measures. We reverse this approach by calculating the marginal cost for a profit-maximizing firm with risk-averse counterparties, and then identifying the risk measure delivering the correct marginal cost. The resulting measure is a weighted average of three parts, each correspondi...

In this paper, we analyze the consequences of bank regulation on the size of the real sector. In particular, we address the question whether exogenous shocks on the return-risk characteristics of the technology and on the equity of the real sector are intensified or damped by a value-at-risk constraint on the credit portfolio of a bank. We consider a one-period model with three risk-averse agen...

We study the discrete optimization problem under the distributionally robustframework. We optimize the Entropic Value-at-Risk, which is a coherentrisk measure and is also known as Bernstein approximation for the chanceconstraint. We propose an efficient approximation algorithm to resolve theproblem via solving a sequence of nominal problems. The computationalresults show...

Tail conditional expectations refer to the expected values of random variables conditioning on some tail events and are closely related to various coherent risk measures. In the univariate case, the tail conditional expectation is asymptotically proportional to the value-at-risk, a popular risk measure. The focus of this paper is on asymptotic relations between the multivariate tail conditional...

Abstract: In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall the notion of conditional, dynamic and time-consistent risk measures. We link the latter with its iterated pr...