Standard Risk Measures Research Articles

Capital Requirements and Optimal Investment with Solvency Probability Constraints

Quantifying the economic capital and optimally allocating it into portfolios of financial instruments are two key topics in the asset/liability management (ALM) of an insurance company. In general these problems are studied in the literature by minimizing standard risk measures such as the value at risk (VaR) and the conditional value at risk (CVaR). Motivated by Solvency II regulations, we introduce a novel risk-return'' optimization problem to solve for the optimal required capital and the portfolio structure when the ruin probability is used as an insurance solvency constraint. The proposed approach relies on a semi-parametric setting using scenario-based asset returns and different parametric assumptions for the liability distribution. Extensive simulations are provided to assess the sensitivity and robustness of our solutions relative to model factors.

Risk vs. Anomaly: A New Methodology Applied to Accruals

Research suggesting the existence of the accrual anomaly runs into the issue that risk serves as a competing explanation for abnormal returns. This paper proposes a novel approach to distinguish between risk and anomaly explanations for the negative association between accruals and returns. The intuition is that high risk stocks should experience relatively high and low returns more often than low risk stocks. Thus, a variable that has the opposite correlations with high returns than with low returns is unlikely to capture risk, which points toward an anomaly. The paper implements this perspective via two logistic regressions predicting relatively high and low returns. Controlling for standard risk measures, we document that low accruals increase the probability of large positive returns, but reduce the likelihood of large negative returns. This finding is inconsistent with the prediction that accruals reflect risk and supports the hypothesis that the accrual “anomaly” is indeed an anomaly.

An Economic Evaluation of the Model Risk for Risk Models

The recent experience from the global financial crisis has raised serious doubts about the accuracy of standard risk measures as a tool to quantify extreme downward risks. Standard risk measures are subject to a “model risk” due to the specification and estimation uncertainty. We propose a general adjustment of the Value-at-Risk to compute risk measures robust to the model risk. The proposed procedure aims empirically adjusting the imperfect quantile estimate assessing the good quality of VaR models such as frequency exceptions, independence of violations and magnitude of violations. Based on a long sample of U.S. data, we find an inverse U-shape relation between VaR model errors and the horizon: corrections (for model errors) are higher for short-term horizons but are also increasing for long-term horizons. We also provide a fair comparison between the main risk models using the same metric that corresponds to model risk required corrections.

Housing Risk and Return: Evidence from a HousingAsset-Pricing Model

Case, Cotter, and Gabriel investigate the risk–return relationship in the determination of housing asset pricing. In so doing, they evaluate the behavioral hypotheses advanced by Case and Shiller in studies of boom and post-boom housing markets. The authors specify a multifactor housing assetpricing model and evaluate whether the market factor and other measures of risk, including idiosyncratic risk, momentum, and MSA size effects, have explanatory power for metropolitan- specific housing returns. Further, they test the robustness of the asset pricing results to inclusion of controls for socioeconomic variables commonly represented in the house price literature, including changes in employment, affordability, and foreclosure incidence, and find a sizable and statistically significant influence of the market factor on MSA house price returns. Moreover, they show that market betas have varied substantially over time. Results are largely robust to the inclusion of other explanatory variables, including standard measures of risk and other housing market fundamentals. Additional tests of model validity using the Fama–MacBeth framework offer further strong support of a positive risk-and-return relationship in housing. Case, Cotter, and Gabriel’s findings are supportive of the application of a housing investment risk–return framework in explanation of the variation in the metro-area cross section and time series of U.S. house price returns. Further, results strongly corroborate the Case–Shiller survey research, which indicates the importance of speculative forces in the determination of U.S. housing returns. <b>TOPICS:</b>Real estate, analysis of individual factors/risk premia, equity portfolio management

The Performance of Johnson Distributions for ComputingValue at Risk and Expected Shortfall

Option pricing plainly depends on the probability distribution of the underlying asset return, at least the portion of the distribution for which the option is in the money. Risk management also depends on the return distribution, but standard risk measures like value at risk (VaR) and expected shortfall (ES) concentrate only on its tails. While the lognormal may be (arguably) adequate for modeling option values, empirically, “risk” is inherently tied up with fat-tailed return processes. This result has led to interest in methods to approximate an unknown distribution by matching its first four moments. The Cornish–Fisher and Gram–Charlier expansions are frequent choices. Simonato argues, however, that these techniques do not actually work very well because the range of densities for which the approximations are valid is quite limited. For example, the density approximation may have negative portions, even when skewness and kurtosis seem quite reasonable. He proposes adopting the Johnson family of densities instead, which also uses four parameters to match the first four moments of an empirical distribution. A simulation study shows that with Johnson distributions, tail fitting is accurate and available over the full range of parameter values. <b>TOPICS:</b>Options, VAR and use of alternative risk measures of trading risk, tail risks

Financial Innovation and a Universally Applicable Distinction between Accruals and Realization Taxation

Existing accruals/realization tax borderlines differ by jurisdiction and are generally incapable of dealing consistently with many innovative structured financial instruments, particularly hybrid instruments and those incorporating both fixed and contingent returns. This article develops a new analytical approach to defining the accruals/realization tax borderline, which does not disturb, and is consistent with, the existing realizations treatment of shares and the accruals treatment of bonds. The approach relies on a single, standard measure of risk, the annualized volatility of returns. The approach provides a unique solution for all innovative financial instruments and could have application in all jurisdictions.

Capital Structure, Product Market Dynamics, and the Boundaries of the Firm

We study how interactions between financing and investment decisions can shape firm boundaries in dynamic product markets. In particular, we model a new product market opportunity as a growth option and ask whether it is best exploited by a large incumbent firm (Integration) or by a separate, specialized firm (Non-Integration). Starting from a standard theoretical framework, in which value-maximizing corporate investment and financing decisions are jointly determined, we show that Integration best protects assets in place value, while Non-Integration best protects the value of the growth option and maximizes financial flexibility. These forces drive different organizational equilibria depending on firm and product market characteristics. In particular, we show that increases in standard measures of cash flow risk predict exploitation of new opportunities by specialized firms, while increases in product market risk (i.e., the risk of preemption by competitors) predict exploitation by incumbents. We also show that alliances organized as licensing agreements or revenue sharing contracts sometimes better balance the different sources of value, and thus may dominate more traditional forms of organization. These key results arise from the dynamic interaction of the new opportunity’s option-like features with realistic competitive forces.The appendices for this paper are available at the following URL: http://ssrn.com/abstract=2443868

Hybrid Tail Risk and Expected Stock Returns: When Does the Tail Wag the Dog?

We introduce a new, hybrid measure of stock return tail covariance risk, motivated by the under-diversified portfolio holdings of individual investors, and investigate its cross-sectional predictive power. Our key innovation is that this covariance is measured across the left tail states of the individual stock return distribution, not across those of the market return as in standard systematic risk measures. We document a positive and significant relation between hybrid tail covariance risk (H-TCR) and expected stock returns, with an annualized premium of 9%, in contrast to the insignificant or negative results for purely stock-specific or systematic tail risk measures.

Housing Risk and Return: Evidence from a Housing Asset-Pricing Model

This paper investigates the risk-return relationship in determination of housing asset pricing. In so doing, the paper evaluates behavioral hypotheses advanced by Case and Shiller (1988, 2002, 2009) in studies of boom and post-boom housing markets. The paper specifies and tests a housing asset pricing model (H-CAPM), whereby expected returns of metropolitan-specific housing markets are equated to the market return, as represented by aggregate US house price time-series. We augment the model by examining the impact of additional risk factors including aggregate stock market returns, idiosyncratic risk, momentum, and Metropolitan Statistical Area (MSA) size effects. Further, we test the robustness of H-CAPM results to inclusion of controls for socioeconomic variables commonly represented in the house price literature, including changes in employment, affordability, and foreclosure incidence. Consistent with the traditional CAPM, we find a sizable and statistically significant influence of the market factor on MSA house price returns. Moreover we show thatmarket betas have varied substantially over time. Also, we find the basic housing CAPM results are robust to the inclusion of other explanatory variables, including standard measures of risk and other housing market fundamentals. Additional tests of the validity of the model using the Fama-MacBeth framework offer further strong support of a positive risk and return relationship in housing. Our findings are supportive of the application of a housing investment risk-return framework in explanation of variation in metro-area cross-section and time-series US house price returns. Further, results strongly corroborate Case-Shiller behavioral research indicating the importance of speculative forces in the determination of U.S. housing returns.

RISK HORIZON AND REBALANCING HORIZON IN PORTFOLIO RISK MEASUREMENT

This paper analyzes portfolio risk and volatility in the presence of constraints on portfolio rebalancing frequency. This investigation is motivated by the incremental risk charge (IRC) introduced by the Basel Committee on Banking Supervision. In contrast to the standard market risk measure based on a 10‐day value‐at‐risk calculated at 99% confidence, the IRC considers more extreme losses and is measured over a 1‐year horizon. More importantly, whereas 10‐day VaR is ordinarily calculated with a portfolio’s holdings held fixed, the IRC assumes a portfolio is managed dynamically to a target level of risk, with constraints on rebalancing frequency. The IRC uses discrete rebalancing intervals (e.g., monthly or quarterly) as a rough measure of potential illiquidity in underlying assets. We analyze the effect of these rebalancing intervals on the portfolio’s profit and loss distribution over a risk‐measurement horizon. We derive limiting results, as the rebalancing frequency increases, for the difference between discretely and continuously rebalanced portfolios; we use these to approximate the loss distribution for the discretely rebalanced portfolio relative to the continuously rebalanced portfolio. Our analysis leads to explicit measures of the impact of discrete rebalancing under a simple model of asset dynamics.

Forced Portfolio Liquidation

We study the problem of a leveraged investor that is forced to unwind a significant fraction of its portfolio in a collection of illiquid markets. It is shown that markets may become disrupted in response to a relatively small liquidity shock. As a consequence, the probability of default can be much higher than suggested by standard risk measures. We also study the impact of successful liquidation on relative asset prices. Our analysis suggests that effective risk management of leveraged financial entities should focus on the entity's potential to generate emergency cash-flows net of third-party claims for liquidity.

Portfolio crash testing: making sense of extreme event exposures

The topic of extreme events is becoming ever more important for risk management. Stress testing is a technique that is explicitly designed to deal with extreme shocks; however, its methodology and place in the risk process is often unclear to risk managers. This paper addresses some common misconceptions about stress testing and provides methodology for its incorporation into the risk process as a supplement to risk measures such as value-at-risk and tracking error. Two stress testing models are presented and empirically validated on actual extreme periods. Both are based on multivariate normal distributions conditional on a factor shock, differing only in the way that covariances are estimated. The first model uses temporal weighting commonly used in the risk model construction; the other uses event weighting, which assigns a higher weight to extreme events that are similar to the factor shock specified. The key conclusion is that the time weighted model performs better in moderate or semi-expected shocks, while the event weighted model performs better in more extreme and unexpected shocks like the Long-Term Capital Management crisis, 9/11 terrorist attacks, and the Fall 2008 financials-led meltdown. The event weighted model, which is designed to reflect the rise in correlations and variances during extreme markets, produces a more conservative estimate of return impacts. Our results support the conclusion that stress testing can be a very valuable addition to standard risk measures.

Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled (‘tilted’) one-period convex risk measures, using a d -dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete ‘driver’ characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.

Performance of DJIA Stocks Using Fundamental Indexation

During the decade beginning on December 31, 1999, an equal-weighted portfolio of the Dow Jones Industrial Average (DJIA) stocks outperformed the market capitalization and priceweighted portfolios, with ending values of $1,020, $731, and $777 on the $1,000 originally invested, respectively. The net income and free cash flows-weighted portfolios performed the best with identical ending values of $1,082. The portfolio with the lowest standard deviation measure of total risk was weighted on net income (14.26%) and the portfolio with the highest total risk was weighted on total assets (27.78%). All portfolios had their worst performance in 2008, while the best performance varied across portfolios but were limited to the years 2003, 2006 and 2009.

Systemic risk, financial contagion and financial fragility

Although it is hard to arrive at a widely accepted definition for Systemic Risk; it is generally acknowledged that it is the risk of the occurrence of an event that threatens the well functioning of the system of interest (financial, payments, banking, etc.) sometimes to the point of making its operation impossible. We model systemic risk with two main components: a random shock that weakens one or more financial institutions and a transmission mechanism which transmits and possibly exacerbates such negative effects to the rest of the system. Our model could be conceptually represented by a network already described in previous works. In this work we show how is possible to estimate the distribution of losses for the banking system with our model. Additionally, we show how it is possible to separate the distribution of losses into two components: the losses incurred by the initial shock and the losses resulting from the contagion process. Finally, once the distribution is estimated, we can derive standard risk measures for the system as a whole. Another important contribution of this work is that we can follow the evolution of certain risk measures like the expected loss or the CVaR in order to evaluate if the system is becoming more or less risky, in fact, more or less fragile. Additionally, we can decompose the distribution of losses of the whole banking system into the systemic and the contagion elements and we can determine if the system is more prone to experience contagious difficulties during a certain period of time.

Accounting for risk of non linear portfolios

The presence of non linear instruments is responsible for the emergence of non Gaussian features in the price changes distribution of realistic portfolios, even for Normally distributed risk factors. This is especially true for the benchmark Delta Gamma Normal model, which in general exhibits exponentially damped power law tails. We show how the knowledge of the model characteristic function leads to Fourier representations for two standard risk measures, the Value at Risk and the Expected Shortfall, and for their sensitivities with respect to the model parameters. We detail the numerical implementation of our formulae and we emphasize the reliability and efficiency of our results in comparison with Monte Carlo simulation.

Risk Management Lessons from ‘Knock‐in Knock‐out’ Option Disaster*

AbstractCurrency knock‐in knock‐out (KIKO) options had been widely used for hedging exchange rate risks in Korean financial markets. However, as the Korean won moved in an unexpected direction during the global financial crisis period of 2007 and 2008, the hedging instruments incurred huge losses to the option holders. In this paper, we analyze the event from the viewpoint of risk assessment and management. We find that, first, if the option holders had assessed the risk levels with and without the KIKO options by using standard risk measures like value‐at‐risk or conditional value‐at‐risk, then many KIKO option contracts would not have been justifiable from the beginning. Second, having a proper view on the exchange rate dynamics turned out to be crucial for risk assessment and management. If the companies had a proper view instead of a myopic view on the exchange rate movement, then the KIKO options might not have been chosen. Finally, ‘hedge‐and‐forget’ behavior proved to be very costly and reckless. If the companies had continuously assessed and managed their risks, then the losses from the KIKO options could have been significantly mitigated. Some relevant pricing issues are also investigated. We find that most KIKO option contracts under study might not be significantly overpriced. However, potential impacts of the possible mispricing could be considerable in some cases. Nonetheless, the risk management failure proved to be more important for the KIKO option losses than the possible mispricing problem.

Revisiting Modern Discounting of Risky Cash Flows: Imbedding Uncertainty Resolution Functions into Risk-Adjusted Discount Rates and Certainty Equivalent Factors (Revised)

The paper revisits the prevailing RADR-based approach to cash flows valuation, indicates its restrictions and failures and suggests flexible methodology of discounting cash flows by means of certainty equivalents with implied instantaneous measure of risk consistent with specific uncertainty resolution function. The RADR discounting techniques do not allow estimate values of risky initial cash outflows or differed nonrecurring cash flows with known standard deviation or more sophisticated measure of risk, since they use periodic measure of risk according to CAPM, its modifications or ex-ante approach. However that is not an insolvable problem, but a restriction of RADR approach. The Certainty Equivalents approach to discounting is easily made consistent with the basic assumptions of RADR and may be fine-tuned to incorporate Equity Risk Premium as a basis for CEQ factors and when properly applied provides the same results as RADR-based discounting, meanwhile overcoming the above-mentioned shortcoming. However the most hard critics of RADR concern the fact that it implicitly produces rigid pattern of risk increasing over time, which is a special case without sufficient evidence to be overwhelming in practice. Particularly, the paper demonstrates that for a remote lottery cash flow there would be instantaneous standard deviation at the moment of payment, which is hardly possible to convert to periodic standard deviation and the discounting at a RADR is surely a mistake, since the risk of such cash flow is fixed over time. This is an extreme example of cash flow with specific uncertainty resolution pattern for which RADR-based discounting is inconsistent and provided strongly distorted results. The paper also presents the formula to calculate RADR-implied standard deviation and finds it being two times higher than the standard deviation of identical cash flows explicitly expressed by Wiener process, i.e. square root of time. Suggested methodology addresses using the arbitrary Uncertainty Resolution function to express risk at a given moment, for instance such as Wiener stochastic process. It’s shown how to embed the Uncertainty Resolution into Certainty Equivalents Method of discounting and thus have a possibility to evaluate cash flows according to their explicit risk pattern. The paper also proves the Propositions for discounting cash flows with different time-remote risk patterns. Specifically it substantiates the recursive algorithm to CE-based valuations and differentiates the natural risk of cash flows and risk of future revaluation of current estimates, which is jointly presented in CAPM.

Is there a Distress Risk Anomaly? Corporate Bond Spread as a Proxy for Default Risk

The standard measures of distress risk ignore the fact that firm defaults are correlated and that some defaults are more likely to occur in bad times. We use risk premium computed from corporate credit spreads to measure a firm’s exposure to systematic variation in default risk. Unlike previously used measures that proxy for a firm’s physical probability of default, credit spreads proxy for a risk-adjusted default probability and thereby explicitly account for the non-diversifiable component of distress risk. In contrast to prior findings in the literature, we find that stocks that have higher credit risk premia, that is stocks with higher systematic default risk exposures, have higher expected equity returns which are largely explained by the market factor. We confirm the robustness of these results by using an alternative systematic default risk factor for firms that do not have bonds outstanding. Consistent with the theoretical result in George and Hwang (2010), we also show that firms react to increases in their systematic default risk exposures by reducing their leverage, leading to lower physical probabilities of distress. Our results show no evidence of firms with high systematic default risk exposure delivering anomalously low returns.

VaR and expected shortfall: a non-normal regime switching framework

We have developed a regime switching framework to compute the Value at Risk and Expected Shortfall measures. Although Value at Risk as a risk measure has been criticized by some researchers for lack of subadditivity, it is still a central tool in banking regulations and internal risk management in the finance industry. In contrast, Expected Shortfall is coherent and convex, so it is a better measure of risk than Value at Risk. Expected Shortfall is widely used in the insurance industry and has the potential to replace Value at Risk as a standard risk measure in the near future. We have proposed regime switching models to measure value at risk and expected shortfall for a single financial asset as well as financial portfolios. Our models capture the volatility clustering phenomenon and variance-independent variation in the higher moments by assuming the returns follow Student-t distributions.