An axiomatic characterization of capital allocations of coherent risk measures

An axiomatic definition of coherent capital allocations is given. It is shown that coherent capital allocations defined by the proposed axiom system are closely linked to coherent risk measures. More precisely, the associated risk measure of a coherent capital allocation is coherent and, conversely, for every coherent risk measure there exists a coherent capital allocation.     

Vector-valued coherent risk measures

We define (d,n)-coherent risk measures as set-valued maps from into satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner et al. [2]. We then discuss the aggregation issue, i.e., the passage from valued random portfolio to valued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided.Received: February 2004, Mathematics Subject Classification (2000): 91B30, 46E30JEL Classification: D81, G31     

Performance ratio-based coherent risk measure and its application

Utilizing a specific acceptance set, we propose in this paper a general method to construct coherent risk measures called the generalized shortfall risk measure. Besides some existing coherent risk measures, several new types of coherent risk measures can be generated. We investigate the generalized shortfall risk measure’s desirable properties such as consistency with second-order stochastic dominance. By combining the performance evaluation with the risk control, we study in particular the performance ratio-based coherent risk (PRCR) measures, which is a sub-class of generalized shortfall risk measures. The PRCR measures are tractable and have a suitable financial interpretation. Based on the PRCR measure, we establish a portfolio selection model with transaction costs. Empirical results show that the optimal portfolio obtained under the PRCR measure performs much better than the corresponding optimal portfolio obtained under the higher moment coherent risk measure.     

Two-sided coherent risk measures and their application in realistic portfolio optimization

By using a different derivation scheme, a new class of two-sided coherent risk measures is constructed in this paper. Different from existing coherent risk measures, both positive and negative deviations from the expected return are considered in the new measure simultaneously but differently. This innovation makes it easy to reasonably describe and control the asymmetry and fat-tail characteristics of the loss distribution and to properly reflect the investor’s risk attitude. With its easy computation of the new risk measure, a realistic portfolio selection model is established by taking into account typical market frictions such as taxes, transaction costs, and value constraints. Empirical results demonstrate that our new portfolio selection model can not only suitably reflect the impact of different trading constraints, but find more robust optimal portfolios, which are better than the optimal portfolio obtained under the conditional value-at-risk measure in terms of diversification and typical performance ratios.     

Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.     

Spectral Risk Measures: Properties and Limitations

Spectral risk measures (SRMs) are risk measures that take account of user risk-aversion, but to date there has been little guidance on the choice of utility function underlying them. This paper addresses this issue by examining alternative approaches based on exponential and power utility functions. A number of problems are identified with both types of spectral risk measure. The general lesson is that users of spectral risk measures must be careful to select utility functions that fit the features of the particular problems they are dealing with, and should be especially careful when using power SRMs.

On risk management problems related to a coherence property

Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.     

A framework to measure integrated risk

A framework underlying various models that measure the credit risk of a portfolio is extended in this paper to allow the integration of credit risk with a range of market risks using Monte Carlo simulation. A structural model is proposed that allows interest rates to be stochastic and provides closed-form expressions for the market value of a firm's equity and its probability of default. This model is embedded within the integrated framework and the general approach illustrated by measuring the risk of a foreign exchange forward when there is a significant probability of default by the counterparty. For this example moving from a market risk calculation to an integrated risk calculation reduces the expected future value of the instrument by an amount that could not be calculated using the common pre-settlement exposure technique for estimating the credit risk of a derivative.     

Optimization of a firm's capital structure: A quantitative approach based on a probabilistic prognosis of risk and time of bankruptcy

This article develops a new probabilistic approach to the problem of optimization of a firm's capital structure. The main idea of the approach is straightforward. As a possible firm's bankruptcy is the principal factor restricting the amount of borrowed capital, we assess the probabilities of bankruptcy at various time horizons in the future dependent on the proportion of debt capital and other indices of a firm's current financial position and then calculate how these probabilities influence the firm's value.We identify a set of factors determining conditions of existence and the value of the optimal debt/equity ratio. These include the characteristics of a firm's debt (proportion of short-term component of the debt, cost of service, and maturity horizons of long-term component), characteristics of a firm's ability to pay the debt, and some macroeconomic factors.We represent dependencies of optimal debt/equity ratio and gains in a firm's value on the main influencing factors.The approach is based on real data of real firms and does not use superfluously formalized models. We believe it can be used in practical capital structure decisions although specific calculations must be fulfilled for each firm that needs such decision.     

Dilatation monotone risk measures are law invariant

We prove that on an atomless probability space, every dilatation monotone convex risk measure is law invariant. This result, combined with the known ones, shows the equivalence between dilatation monotonicity and important properties of convex risk measures such as law invariance and second-order stochastic monotonicity. We would like to thank Johannes Leitner for helpful discussions. The second author made contributions to this paper while being affiliated to Heriot-Watt University and would like to express special thanks to Mark Owen, whose project (EPSRC grant no. GR/S80202/01) supported this research.     

Environmental social controls and capital investments: Australian evidence

Environmental social controls (ESCs) such as mandatory disclosure, regulations, subsidies, and stakeholder opinion are intended to improve firm environmental performance. This paper reports ESC importance to Australian financial managers in making capital investment decisions. A decision‐making experiment showed managers to be most responsive to stakeholder opinion (42 per cent), followed by subsidization (26 per cent) and regulatory cost (22 per cent). Mandatory disclosure has very little influence (10 per cent). ESC interaction effects are limited so coordination of ESC policy is not a primary concern. High degrees of managerial self‐insight suggest policy changes would be enhanced by close consultations with the managers involved.     

The effects of board gender diversity on a firm's risk strategies

We study whether board gender diversity (BGD) affects corporate risk strategies. Specifically, we investigate the association between BGD and firms’ reputation risk and financial risk. Using S&P data from 1997 to 2013, we find that BGD is negatively associated with tax avoidance, suggesting firms with gender‐diverse boards are more cautious about potential reputation risks associated with aggressive tax strategies. However, we find that BGD is positively associated with firms’ financial risk. The combined findings illustrate that BGD aligns a firm's risk exposure closer to risk‐neutral shareholders’ preferences by reducing reputation risk exposure while enabling necessary financial risk exposure.     

Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

Political uncertainty and a firm's credit risk: Evidence from the international CDS market

Using a large sample of firms with single-name credit default swap (CDS) contracts in 30 countries, we document the evidence that political uncertainty, proxied by national election dummy, is positively related to firm-level credit risk. Specifically, this positive relation is more pronounced for the firms that have no political connection or poor international diversification, and in the countries with higher political uncertainty and lower investor protections. Further, by using a difference-in-differences approach, we find evidence to support idiosyncratic volatility and debt rollover channels through which political uncertainty affects the credit risk of individual firm.     

Dynamic risk measures: Time consistency and risk measures from BMO martingales

Time consistency is a crucial property for dynamic risk measures. Making use of the dual representation for conditional risk measures, we characterize the time consistency by a cocycle condition for the minimal penalty function. Taking advantage of this cocycle condition, we introduce a new methodology for the construction of time-consistent dynamic risk measures. Starting with BMO martingales, we provide new classes of time-consistent dynamic risk measures. These families generalize those obtained from backward stochastic differential equations. Quite importantly, starting with right-continuous BMO martingales, this construction naturally leads to paths with jumps.      

The impact of imposing capital requirements on systemic risk

This paper examines the impact of imposing capital requirements on systemic risk. We use a static model on financial institutions’ risk-taking behavior to quantify the systemic risk in the cross-sectional dimension in both regulated and unregulated systems. Although imposing a capital requirement can lower individual risk, it simultaneously enhances systemic linkage within the system. By using a proper systemic risk measure combining both individual risk and systemic linkage, we show that systemic risk in a regulated system can be higher than that in an unregulated system. In addition, we analyze a sufficient condition under which the systemic risk in a regulated system is always lower.     

Computing present values: Capital budgeting done correctly

This paper shows that the standard textbook formula for computing the present value of a future random cash flow – the discounted expected value – is formally incorrect and can generate significant errors when used to compute present values. The correct present value method is provided as well as a simple adjustment to the textbook formula which can be used to obtain an approximation to the correct value.     

Corporate bond default risk: A 150-year perspective

We study corporate bond default rates using an extensive new data set spanning the 1866-2008 period. We find that the corporate bond market has repeatedly suffered clustered default events much worse than those experienced during the Great Depression. For example, during the railroad crisis of 1873-1875, total defaults amounted to 36% of the par value of the entire corporate bond market. Using a regime-switching model, we examine the extent to which default rates can be forecast by financial and macroeconomic variables. We find that stock returns, stock return volatility, and changes in GDP are strong predictors of default rates. Surprisingly, however, credit spreads are not. Over the long term, credit spreads are roughly twice as large as default losses, resulting in an average credit risk premium of about 80 basis points. We also find that credit spreads do not adjust in response to realized default rates.