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At least 433 records · Page 24Linked to original sources
arXiv2013

Stochastic areas of diffusions and applications in risk theory

In this paper we study the stochastic area swept by a regular time-homogeneous diffusion till a stopping time. This unifies some recent literature in this area. Through stochastic time change we establish a link between the stochastic area and the stopping time of another associated time-homogeneous diffusion. Then we characterize the Laplace transform of the stochastic area in terms of the eigenfunctions of the associated diffusion. We also explicitly obtain the integer moments of the stochastic area in terms of scale and speed densities of the associated diffusion. Specifically we study in detail three stopping times: the first passage time to a constant level, the first drawdown time and the Azema-Yor stopping time. We also study the total occupation area of the diffusion below a constant level. We show applications of the results to a new structural model of default (Yildirim 2006), the Omega risk model of bankruptcy in risk analysis (Gerber, Shiu and Yang 2012), and a diffusion risk model with surplus-dependent tax (Albrecher and Hipp 2007, Li, Tang and Zhou 2013).

q-fin.RM↗
arXiv2013

Optimal insurance purchase strategies via optimal multiple stopping times

In this paper we study a class of insurance products where the policy holder has the option to insure $k$ of its annual Operational Risk losses in a horizon of $T$ years. This involves a choice of $k$ out of $T$ years in which to apply the insurance policy coverage by making claims against losses in the given year. The insurance product structure presented can accommodate any kind of annual mitigation, but we present three basic generic insurance policy structures that can be combined to create more complex types of coverage. Following the Loss Distributional Approach (LDA) with Poisson distributed annual loss frequencies and Inverse-Gaussian loss severities we are able to characterize in closed form analytical expressions for the multiple optimal decision strategy that minimizes the expected Operational Risk loss over the next $T$ years. For the cases where the combination of insurance policies and LDA model does not lead to closed form expressions for the multiple optimal decision rules, we also develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.

q-fin.RM↗
arXiv2013

The impact of systemic risk on the diversification benefits of a risk portfolio

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 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 could be easily generalized for investment risk. We see that, even with a small probability of occurrence, systemic risk can reduce dramatically the diversification benefits. It is clearly revealed via a non-diversifiable term that appears in the analytical expression of the variance of our models. We propose two ways of introducing it and discuss their advantages and limitations. By using both VaR and TVaR to compute the loading, we see that only the latter captures the full effect of systemic risk when its probability to occur is low

q-fin.RM↗
arXiv2014

Dynamic Limit Growth Indices in Discrete Time

We propose a new class of mappings, called Dynamic Limit Growth Indices, that are designed to measure the long-run performance of a financial portfolio in discrete time setup. We study various important properties for this new class of measures, and in particular, we provide necessary and sufficient condition for a Dynamic Limit Growth Index to be a dynamic assessment index. We also establish their connection with classical dynamic acceptability indices, and we show how to construct examples of Dynamic Limit Growth Indices using dynamic risk measures and dynamic certainty equivalents. Finally, we propose a new definition of time consistency, suitable for these indices, and we study time consistency for the most notable representative of this class -- the dynamic analog of risk sensitive criterion.

q-fin.RM↗
arXiv2015

What is the best risk measure in practice? A comparison of standard measures

Expected Shortfall (ES) has been widely accepted as a risk measure that is conceptually superior to Value-at-Risk (VaR). At the same time, however, it has been criticised for issues relating to backtesting. In particular, ES has been found not to be elicitable which means that backtesting for ES is less straightforward than, e.g., backtesting for VaR. Expectiles have been suggested as potentially better alternatives to both ES and VaR. In this paper, we revisit commonly accepted desirable properties of risk measures like coherence, comonotonic additivity, robustness and elicitability. We check VaR, ES and Expectiles with regard to whether or not they enjoy these properties, with particular emphasis on Expectiles. We also consider their impact on capital allocation, an important issue in risk management. We find that, despite the caveats that apply to the estimation and backtesting of ES, it can be considered a good risk measure. As a consequence, there is no sufficient evidence to justify an all-inclusive replacement of ES by Expectiles in applications. For backtesting ES, we propose an empirical approach that consists in replacing ES by a set of four quantiles, which should allow to make use of backtesting methods for VaR. Keywords: Backtesting; capital allocation; coherence; diversification; elicitability; expected shortfall; expectile; forecasts; probability integral transform (PIT); risk measure; risk management; robustness; value-at-risk

q-fin.RM↗
arXiv2013

Extrapolating the term structure of interest rates with parameter uncertainty

Pricing extremely long-dated liabilities market consistently deals with the decline in liquidity of financial instruments on long maturities. The aim is to quantify the uncertainty of rates up to maturities of a century. We assume that the interest rates follow the affine mean-reverting Vasicek model. We model parameter uncertainty by Bayesian distributions over the parameters. The cross-sectional and time series parameters are obtained via the restricted bivariate VAR(1) model. The empirical example shows extremely low confidence in long term extrapolations due to the accumulated effect of the mean-reversion`s behaviour close to the unit root.

q-fin.CP↗
arXiv2013

Systematic and multifactor risk models revisited

Systematic and multifactor risk models are revisited via methods which were already successfully developed in signal processing and in automatic control. The results, which bypass the usual criticisms on those risk modeling, are illustrated by several successful computer experiments.

q-fin.RM↗
arXiv2013

Hedging Against the Interest-rate Risk by Measuring the Yield-curve Movement

By adopting the polynomial interpolation method, we propose an approach to hedge against the interest-rate risk of the default-free bonds by measuring the nonparallel movement of the yield-curve, such as the translation, the rotation and the twist. The empirical analysis shows that our hedging strategies are comparable to traditional duration-convexity strategy, or even better when we have more suitable hedging instruments on hand. The article shows that this strategy is flexible and robust to cope with the interest-rate risk and can help fine-tune a position as time changes.

q-fin.PM↗
arXiv2013

Bankruptcy Risk Induced by Career Concerns of Regulators

We introduce a model in which a regulator employs mechanism design to embed her human capital beta signal(s) in a firm's capital structure, in order to enhance the value of her post career change indexed executive stock option contract with the firm. We prove that the agency cost of this revolving door behavior increases the firm's financial leverage, bankruptcy risk, and affects estimation of firm value at risk (VaR).

q-fin.RM↗
arXiv2014

Optimal consumption and portfolio choice with ambiguity

We consider optimal consumption and portfolio choice in the presence of Knightian uncertainty in continuous-time. We embed the problem into the new framework of stochastic calculus for such settings, dealing in particular with the issue of non-equivalent multiple priors. We solve the problem completely by identifying the worst--case measure. Our setup also allows to consider interest rate uncertainty; we show that under some robust parameter constellations, the investor optimally puts all his wealth into the asset market, and does not save or borrow at all.

q-fin.PM↗
arXiv2014

Optimal Hedging for Fund & Insurance Managers with Partially Observable Investment Flows

All the financial practitioners are working in incomplete markets full of unhedgeable risk-factors. Making the situation worse, they are only equipped with the imperfect information on the relevant processes. In addition to the market risk, fund and insurance managers have to be prepared for sudden and possibly contagious changes in the investment flows from their clients so that they can avoid the over- as well as under-hedging. In this work, the prices of securities, the occurrences of insured events and (possibly a network of) the investment flows are used to infer their drifts and intensities by a stochastic filtering technique. We utilize the inferred information to provide the optimal hedging strategy based on the mean-variance (or quadratic) risk criterion. A BSDE approach allows a systematic derivation of the optimal strategy, which is shown to be implementable by a set of simple ODEs and the standard Monte Carlo simulation. The presented framework may also be useful for manufactures and energy firms to install an efficient overlay of dynamic hedging by financial derivatives to minimize the costs.

q-fin.CP↗
arXiv2014

Law-invariant risk measures: extension properties and qualitative robustness

We characterize when a convex risk measure associated to a law-invariant acceptance set in $L^\infty$ can be extended to $L^p$, $1\leq p<\infty$, preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concrete examples including risk measures based on expected utility, max-correlation risk measures, and distortion risk measures.

q-fin.RM↗
arXiv2014

Capital adequacy tests and limited liability of financial institutions

The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. The objective of this paper 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 financial 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 emphasize the advantages of dealing with surplus-invariant acceptance sets as the primary object rather than directly with risk measures, such as loss-based and excess-invariant risk measures, which have been recently studied by Cont, Deguest, and He (2013) and by Staum (2013), respectively.

q-fin.RM↗
arXiv2015

Quasi-Hadamard differentiability of general risk functionals and its application

We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a specifically chosen "tangent space". We show that this notion of quasi-Hadamard differentiability yields both strong laws and limit theorems for the asymptotic distribution of the plug-in estimators. Our results can be regarded as a contribution to the statistics and numerics of risk measurement and as a case study for possible refinements of the functional delta method through fine-tuning the underlying notion of differentiability

math.ST↗
arXiv2014

Risk aggregation and stochastic claims reserving in disability insurance

We consider a large, homogeneous portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic-demographic environment. Using a conditional law of large numbers, we establish the connection between claims reserving and risk aggregation for large portfolios. Further, we derive a partial differential equation for moments of present values. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for solving the PDEs very efficiently. Finally, we give a numerical example where moments of present values of disability annuities are computed using finite difference methods and Monte Carlo simulations.

q-fin.RM↗
arXiv2015

On the Measurement of Economic Tail Risk

This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the only risk measures that satisfy a set of economic axioms for the Choquet expected utility and the statistical property of elicitability (i.e. there exists an objective function such that minimizing the expected objective function yields the risk measure) are the mean functional and the median shortfall, which is the median of tail loss distribution. Elicitability is important for backtesting. We also extend the result to address model uncertainty by incorporating multiple scenarios. As an application, we argue that median shortfall is a better alternative than expected shortfall for setting capital requirements in Basel Accords.

q-fin.RM↗
arXiv2014

Why free markets die: An evolutionary perspective

Company mergers and acquisitions are often perceived to act as catalysts for corporate growth in free markets systems: it is conventional wisdom that those activities lead to better and more efficient markets. However, the broad adoption of this perception into corporate strategy is prone to result in a less diverse and more unstable environment, dominated by either very large or very small niche entities. We show here that ancestry, i.e. the cumulative history of mergers, is the key characteristic that encapsulates the diverse range of drivers behind mergers and acquisitions, across a range of industries and geographies. A long-term growth analysis reveals that entities which have been party to fewer mergers tend to grow faster than more highly acquisitive businesses.

q-fin.GN↗