Value-at-risk capital requirement regulation,risk taking and asset allocation: a mean–variance analysis

In this study, the mean–variance framework is employed to analyze the impact of the Basel value-at-risk (VaR) market risk regulation on the institution's optimal investment policy, the stockholders’ welfare, as well as the tendency of the institution to change the risk profile of the held portfolio. It is shown that with the VaR regulation, the institution faces a new regulated capital market line, which induces resource allocation distortion in the economy. Surprisingly, only when a riskless asset is available does VaR regulation induce the institution to reduce risk. Otherwise, the regulation may induce higher risk, accompanied by asset allocation distortion. On the positive side, the regulation implies an upper bound on the risk the institution takes and it never induces the firm to select an inefficient portfolio. Moreover, when the riskless asset is available, tightening the regulation always increases the amount of maintained eligible capital and decreases risk.     

When more is less: Using multiple constraints to reduce tail risk

Financial institutions suffered large trading losses during the 2007–2009 global financial crisis. These losses cast doubt on the effectiveness of regulations and risk management systems based on a single Value-at-Risk (VaR) constraint. While some researchers have recommended using Conditional Value-at-Risk (CVaR) to control tail risk, VaR remains popular among practitioners and regulators. Accordingly, our paper examines the effectiveness of multiple VaR constraints in controlling CVaR. Under certain conditions, we theoretically show that they are more effective than a single VaR constraint. Furthermore, we numerically find that the maximum CVaR permitted by the constraints is notably smaller than with a single constraint. These results suggest that regulations and risk management systems based on multiple VaR constraints are more effective in reducing tail risk than those based on a single VaR constraint.     

Optimal portfolios with minimum capital requirements

We propose a novel approach to active risk management based on the recent Basel II regulations to obtain optimal portfolios with minimum capital requirements. In order to avoid regulatory penalties due to an excessive number of Value-at-Risk (VaR) violations, capital requirements are minimized subject to a given number of violations over the previous trading year. Capital requirements are based on the recent Basel II amendments to account for the ‘stressed’ VaR, that is, the downside risk of the portfolio under extreme adverse market conditions. An empirical application for two portfolios involving different types of assets and alternative stress scenarios demonstrates that the proposed approach delivers an improved balance between capital requirement levels and the number of VaR exceedances. Furthermore, the risk-adjusted performance of the proposed approach is superior to that of minimum-VaR and minimum-stressed VaR portfolios.     

Model uncertainty and VaR aggregation

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different dependence scenarios on the factors of the portfolio. Besides summarizing the most relevant analytical bounds, including a discussion of their sharpness, we introduce a numerical algorithm which allows for the computation of reliable (sharp) bounds for the VaR of high-dimensional portfolios with dimensions d possibly in the several hundreds. We show that additional positive dependence information will typically not improve the upper bound substantially. In contrast higher order marginal information on the model, when available, may lead to strongly improved bounds. Several examples of practical relevance show how explicit VaR bounds can be obtained. These bounds can be interpreted as a measure of model uncertainty induced by possible dependence scenarios.     

Does the Basle Capital Accord reduce bank fragility? An assessment of the value-at-risk approach

We examine the economic implications arising from a bank using a VaR-constrained mean-variance model for the selection of its trading portfolio as a consequence of the Basle Capital Accord. Surprisingly, we show that when a VaR constraint is imposed, it is plausible that certain banks will end up selecting ‘riskier’ portfolios than they would have chosen in the absence of the constraint. Accordingly, regulators such as the Basle Committee on Banking Supervision should be aware that allowing a bank to use VaR to determine its minimum regulatory capital may increase its fragility. Alternatives to VaR-based bank capital regulation that mitigate or even preclude its perverse implications are presented.     

Portfolio selection with a drawdown constraint

When identifying optimal portfolios, practitioners often impose a drawdown constraint. This constraint is even explicit in some money management contracts such as the one recently involving Merrill Lynch’ management of Unilever’s pension fund. In this setting, we provide a characterization of optimal portfolios using mean–variance analysis. In the absence of a benchmark, we find that while the constraint typically decreases the optimal portfolio’s standard deviation, the constrained optimal portfolio can be notably mean–variance inefficient. In the presence of a benchmark such as in the Merrill Lynch–Unilever contract, we find that the constraint increases the optimal portfolio’s standard deviation and tracking error volatility. Thus, the constraint negatively affects a portfolio manager’s ability to track a benchmark.     

Mean–variance portfolio selection with ‘at-risk’ constraints and discrete distributions

We examine the impact of adding either a VaR or a CVaR constraint to the mean–variance model when security returns are assumed to have a discrete distribution with finitely many jump points. Three main results are obtained. First, portfolios on the VaR-constrained boundary exhibit (K + 2)-fund separation, where K is the number of states for which the portfolios suffer losses equal to the VaR bound. Second, portfolios on the CVaR-constrained boundary exhibit (K + 3)-fund separation, where K is the number of states for which the portfolios suffer losses equal to their VaRs. Third, an example illustrates that while the VaR of the CVaR-constrained optimal portfolio is close to that of the VaR-constrained optimal portfolio, the CVaR of the former is notably smaller than that of the latter. This result suggests that a CVaR constraint is more effective than a VaR constraint to curtail large losses in the mean–variance model.     

The Risk Map: A new tool for validating risk models

This paper presents a new method to validate risk models: the Risk Map. This method jointly accounts for the number and the magnitude of extreme losses and graphically summarizes all information about the performance of a risk model. It relies on the concept of a super exception, which is defined as a situation in which the loss exceeds both the standard Value-at-Risk (VaR) and a VaR defined at an extremely low probability. We then formally test whether the sequences of exceptions and super exceptions are rejected by standard model validation tests. We show that the Risk Map can be used to validate market, credit, operational, or systemic risk estimates (VaR, stressed VaR, expected shortfall, and CoVaR) or to assess the performance of the margin system of a clearing house.     

Downside risk and portfolio diversification in the euro-zone equity markets with special consideration of the crisis period

This study examines the Value-at-Risk for ten euro-zone equity markets individually and also divided into two groups: PIIGS (Portugal, Italy, Ireland, Greece and Spain) and the Core (Austria, Finland, France, Germany and the Netherlands), employing four VaR estimation and evaluation methods considered over the full period and the pre- and post-global crisis subperiods 1 and 2. The backtesting results are also evaluated according to the Basel capital requirements. The results demonstrate that the CEVT methods meet all the statistical criteria the best for most individual equity indices over the full period, but these results over the two subperiods for those two methods are mixed, compared to those the DPOT methods. Moreover, the two optimal group portfolios of the PIIGS and the Core as well as the grand portfolio that combines the ten indices do not show much diversification benefits. The PIIGS portfolio selects Spain's IBEX only, while that of the Core opts for Austria's ATX only in the full period and subperiod 1. However, Germany's DAX overwhelmingly dominates both the Core and the Grand portfolios in subperiod 2.     

Stress testing credit risk: The Great Depression scenario

By employing Moody’s corporate default and rating transition data spanning the last 90 years we explore how much capital banks should hold against their corporate loan portfolios to withstand historical stress scenarios. Specifically, we will focus on the worst case scenario over the observation period, the Great Depression. We find that migration risk and the length of the investment horizon are critical factors when determining bank capital needs in a crisis. We show that capital may need to rise more than three times when the horizon is increased from 1 year, as required by current and future regulation, to 3 years. Increases are still important but of a lower magnitude when migration risk is introduced in the analysis. Further, we find that the new bank capital requirements under the so-called Basel 3 agreement would enable banks to absorb Great Depression-style losses. But, such losses would dent regulatory capital considerably and far beyond the capital buffers that have been proposed to ensure that banks survive crisis periods without government support.     

The effect of downside risk reduction on UK equity portfolios included with Managed Futures Funds

The concept of asymmetric risk estimation has become more widely applied in risk management in recent years with the increased use of Value-at-risk (VaR) methodologies. This paper uses the n-degree lower partial moment (LPM) models, of which VaR is a special case, to empirically analyse the effect of downside risk reduction on UK portfolio diversification and returns. Data on Managed Futures Funds are used to replicate the increasingly popular preference of investors for including hedge funds and fund-of-funds type investments in the UK equity portfolios. The result indicates, however that the potential benefits of fund diversification may deteriorate following reductions in downside risk tolerance levels. These results appear to reinforce the importance of risk (tolerance) perception, particularly downside risk, when making decisions to include Managed Futures Funds in UK equity portfolios as the empirical analysis suggests that this could negatively affect portfolio returns.     

Capital incentives and adequacy for securitizations

This paper analyzes the capital incentives and adequacy of financial institutions for asset portfolio securitizations. The empirical analysis is based on US securitization rating and impairment data. The paper finds that regulatory capital rules for securitizations may be insufficient to cover implied losses during economic downturns such as the Global Financial Crisis. In addition, the rating process of securitizations provides capital arbitrage incentives for financial institutions and may further reduce regulatory capital requirements. These policy-relevant findings assume that the ratings assigned by rating agencies are correct and can be used to build a test for the ability of Basel capital regulations to cover downturn losses.     

Value-at-risk-based risk management: optimal policies and asset prices

This article analyzes optimal, dynamic portfolio and wealth/consumptionpolicies of utility maximizing investors who must also managemarket-risk exposure using Value-at-Risk (VaR). We find thatVaR risk managers often optimally choose a larger exposure torisky assets than non-risk managers and consequently incur largerlosses when losses occur. We suggest an alternative risk-managementmodel, based on the expectation of a loss, to remedy the shortcomingsof VaR. A general-equilibrium analysis reveals that the presenceof VaR risk managers amplifies the stock-market volatility attimes of down markets and attenuates the volatility at timesof up markets.     

Nonlinear portfolio selection using approximate parametric Value-at-Risk

As the skewed return distribution is a prominent feature in nonlinear portfolio selection problems which involve derivative assets with nonlinear payoff structures, Value-at-Risk (VaR) is particularly suitable to serve as a risk measure in nonlinear portfolio selection. Unfortunately, the nonlinear portfolio selection formulation using VaR risk measure is in general a computationally intractable optimization problem. We investigate in this paper nonlinear portfolio selection models using approximate parametric Value-at-Risk. More specifically, we use first-order and second-order approximations of VaR for constructing portfolio selection models, and show that the portfolio selection models based on Delta-only, Delta–Gamma-normal and worst-case Delta–Gamma VaR approximations can be reformulated as second-order cone programs, which are polynomially solvable using interior-point methods. Our simulation and empirical results suggest that the model using Delta–Gamma-normal VaR approximation performs the best in terms of a balance between approximation accuracy and computational efficiency.     

Time-consistency in managing a commodity portfolio: A dynamic risk measure approach

We address the problem of managing a storable commodity portfolio, that includes physical assets and positions in spot and forward markets. The vast amount of capital involved in the acquisition of a power plant or storage facility implies that the financing period stretches over a period of several quarters or years. Hence, an intertemporally consistent way of optimizing the portfolio over the planning horizon is required. We demonstrate the temporal inconsistency of static risk objectives based on final wealth and advocate the validity in our setting of a new class of recursive risk measures introduced by Epstein and Zin [Epstein, G., Zin, S., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 57 (4) 937–969] and Wang [Wang, T., 2000. A class of dynamic risk measures University of British Columbia]. These risk measures provide important insights on the trade-offs between date-specific risks (i.e., losses occurring at a point in time) and time-duration risks represented by the pair (return, risk) over a planning horizon; in a number of situations, they dramatically improve the efficiency of static risk objectives, as exhibited in numerical examples.     

Privatization and globalization: An empirical analysis

This paper examines the link between globalization-measured by foreign direct investment (FDI) and foreign portfolio investment (FPI) – and privatization of state-owned enterprises, in a multi-country sample that focuses on developing countries. We hypothesize that privatization has an effect on FDI/FPI as the process of fostering private sector participation is often accompanied by liberalization reforms, and by allocations of substantial shares of newly privatized firms to foreign investors. Similarly, we expect FDI/FPI to foster privatization efforts as new capital inflows, technology and managerial skills that accompany FDI/FPI make the environment more prone to competition, providing governments with incentives to privatize inefficient firms that need to be turned around. This relation is assessed in two ways, first in a dynamic panel using a generalized method of moments approach, and second through panel causality tests. We find a bi-directional positive relation between privatization proceeds and globalization, particularly in the case of FDI.     

Portfolio credit-risk optimization

This paper evaluates several alternative formulations for minimizing the credit risk of a portfolio of financial contracts with different counterparties. Credit risk optimization is challenging because the portfolio loss distribution is typically unavailable in closed form. This makes it difficult to accurately compute Value-at-Risk (VaR) and expected shortfall (ES) at the extreme quantiles that are of practical interest to financial institutions. Our formulations all exploit the conditional independence of counterparties under a structural credit risk model. We consider various approximations to the conditional portfolio loss distribution and formulate VaR and ES minimization problems for each case. We use two realistic credit portfolios to assess the in- and out-of-sample performance for the resulting VaR- and ES-optimized portfolios, as well as for those which we obtain by minimizing the variance or the second moment of the portfolio losses. We find that a Normal approximation to the conditional loss distribution performs best from a practical standpoint.     

Minimizing CVaR and VaR for a portfolio of derivatives

Value at risk (VaR) and conditional value at risk (CVaR) are frequently used as risk measures in risk management. Compared to VaR, CVaR is attractive since it is a coherent risk measure. We analyze the problem of computing the optimal VaR and CVaR portfolios. We illustrate that VaR and CVaR minimization problems for derivatives portfolios are typically ill-posed. We propose to include cost as an additional preference criterion for the CVaR optimization problem. We demonstrate that, with the addition of a proportional cost, it is possible to compute an optimal CVaR derivative investment portfolio with significantly fewer instruments and comparable CVaR and VaR. A computational method based on a smoothing technique is proposed to solve a simulation based CVaR optimization problem efficiently. Comparison is made with the linear programming approach for solving the simulation based CVaR optimization problem.     

Optimal tax-timing and asset allocation when tax rebates on capital losses are limited

This article studies the portfolio problem with realization-based capital gain taxation when limited amounts of losses qualify for tax rebate payments, as is the case under current US tax law. When the tax rate applicable to realized losses exceeds that on realized capital gains, it can be optimal to realize capital gains immediately and pay capital gain taxes to regain the option to use potential future losses against a higher tax rate. This incentive adds an entirely new and as yet unstudied dimension to the portfolio problem. It causes risk averse investors to hold more equity and attain higher welfare levels than is the case when trading under a tax system that seeks to collect the same amount of taxes, but does not allow for tax rebate payments. This is because the benefit to these investors from having their losses subsidized is greater than the suffering from having profits taxed at a higher rate.     

The level and quality of Value-at-Risk disclosure by commercial banks

In this paper we study both the level of Value-at-Risk (VaR) disclosure and the accuracy of the disclosed VaR figures for a sample of US and international commercial banks. To measure the level of VaR disclosures, we develop a VaR Disclosure Index that captures many different facets of market risk disclosure. Using panel data over the period 1996–2005, we find an overall upward trend in the quantity of information released to the public. We also find that Historical Simulation is by far the most popular VaR method. We assess the accuracy of VaR figures by studying the number of VaR exceedances and whether actual daily VaRs contain information about the volatility of subsequent trading revenues. Unlike the level of VaR disclosure, the quality of VaR disclosure shows no sign of improvement over time. We find that VaR computed using Historical Simulation contains very little information about future volatility.