Extreme spectral risk measures: An application to futures clearinghouse margin requirements

This paper applies the extreme-value (EV) generalised pareto distribution to the extreme tails of the return distributions for the S&P500, FT100, DAX, Hang Seng, and Nikkei225 futures contracts. It then uses tail estimators from these contracts to estimate spectral risk measures, which are coherent risk measures that reflect a user’s risk-aversion function. It compares these to VaR and expected shortfall (ES) risk measures, and compares the precision of their estimators. It also discusses the usefulness of these risk measures in the context of clearinghouses setting initial margin requirements, and compares these to the SPAN measures typically used.     

A simplified approach to measuring bond duration

Because interest rates vary over time, the realized return on a fixed-income investment will depend on the price at which the instrument is ultimately liquidated and the rate at which interim cash flows are reinvested. This variation in realized return, known as interest-rate risk, should be addressed by both individual and institutional investors. Tools for measuring the impact and adjusting for the effects of interest rate changes on fixed-income instrument performance have long been available with duration and its companion adjustment factor, convexity. In this article, a simplified alternative to the traditional complex duration calculation is developed and demonstrated. Thus, anyone who can calculate a bond price can quickly estimate the interest rate risk associated with a bond as well as calculate the expected bond price change for a given change in market yield-to-maturity.     

On the coherence of expected shortfall

Expected shortfall (ES) in several variants has been proposed as remedy for the deficiencies of value-at-risk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss distributions have discontinuities. In this case even the coherence property of ES can get lost unless one took care of the details in its definition. We compare some of the definitions of ES, pointing out that there is one which is robust in the sense of yielding a coherent risk measure regardless of the underlying distributions. Moreover, this ES can be estimated effectively even in cases where the usual estimators for VaR fail.     

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.     

On the Use of Portfolio Risk Models and Capital Requirements in Emerging Markets: The Case of Argentina

A portfolio-based model (CreditRisk+ of Credit Suisse FirstBoston) and recent Central Bank of Argentina credit bureau dataare used to estimate whether current capital and provisioningregulations match actual risks. Arguing that provisions shouldcover expected losses and that capital requirements should coverpotential losses beyond expected losses subject to some statisticallevel of tolerance, the article assesses how well actual capitaland provisioning requirements match the estimated requirementsgiven by the model. Actual provisioning requirements were foundto be close to implied levels of expected losses. The estimateof potential losses was found to be highly sensitive to theassumptions of the model, especially the parameter relatingthe volatility of a loan's rate of default to its mean value.This volatility parameter cannot be estimated accurately withthe credit bureau data because of the short time span covered,so proxy data were used to estimate it, and two values aroundthat estimate were tried. The difficulty of estimating thiscritical parameter implies that the results should only be regardedas suggestive. Moreover, the methodology only seeks to estimatecredit risk and not interest rate risk or exchange rate risk,nor does it fully take into account the indirect effects ofinterest rates and exchange rates on credit risk. As recentevents in Argentina have demonstrated, estimating credit riskalong these lines should be thought of as just one tool in attemptingto assess the appropriate level of bank provisions and capital.     

Estimation of tail‐related risk measures in the Indian stock market: An extreme value approach

The purpose of the study is to estimate tail-related risk measures using extreme value theory (EVT) in the Indian stock market. The study employs a two stage approach of conditional EVT originally proposed by McNeil and Frey (2000) to estimate dynamic Value at Risk (VaR) and expected shortfall (ES). The dynamic risk measures have been estimated for different percentiles for negative and positive returns. The estimates of risk measures computed under different quantile levels exhibit strong stability across a range of the selected thresholds, implying the accuracy and reliability of the estimated quantile based risk measures.     

Unbiased estimation of expected return using CAPM

Estimation of expected return is required for many financial decisions. For example, an estimate for cost of capital is required for capital budgeting and cost of equity estimates are needed for performance evaluation based on measures such as EVA. Estimates for expected return are often based on the Capital Asset Pricing Model (CAPM), which states that expected excess return (expected return minus the risk-free rate) is equal to the asset's sensitivity to the world market portfolio (β) times the risk premium on the “world market portfolio” (the market risk premium). Since the world market portfolio, by definition, contains all assets in the world, it is not observable. As a result, an estimate for expected return is commonly obtained by taking an estimate for β based on some index (as a proxy for the world market portfolio) and an estimate for the market risk premium based on a potentially different index and multiplying them together. In this paper, it is shown that this results in a biased estimate for expected return. This is undesirable since biased estimates lead to misallocation of funds and biased performance measures. It is also shown in this paper that the straightforward procedure suggested by Fama and MacBeth [J. Financ. Econ. 1 (1974) 43] results in an unbiased estimate for expected return. Further from the analysis done, it follows that, for an unbiased estimate, it does not matter what proxy is used, as long as it is used correctly an unbiased estimate for expected return results.     

Cash management using multi-stage stochastic programming

We consider a cash management problem where a company with a given financial endowment and given future cash flows minimizes the Conditional Value at Risk of final wealth using a lower bound for the expected terminal wealth. We formulate the optimization problem as a multi-stage stochastic linear program (SLP). The company can choose between a riskless asset (cash), several default- and option-free bonds, and an equity investment, and rebalances the portfolio at every stage. The uncertainty faced by the company is reflected in the development of interest rates and equity returns. Our model has two new features compared to the existing literature, which uses no-arbitrage interest rate models for the scenario generation. First, we explicitly estimate a function for the market price of risk and change the underlying probability measure. Second, we simulate scenarios for equity returns with moment-matching by an extension of the interest rate scenario tree.     

Interpreting long-horizon estimates in predictive regressions

This paper analyzes the asymptotic properties of long-horizon estimators under both the null hypothesis and an alternative of predictability. Asymptotically, under the null of no predictability, the long-run estimator is an increasing deterministic function of the short-run estimate and the forecasting horizon. Under the alternative of predictability, the conditional distribution of the long-run estimator, given the short-run estimate, is no longer degenerate and the expected pattern of coefficient estimates across horizons differs from that under the null. Importantly, however, under the alternative, highly endogenous regressors, such as the dividend–price ratio, tend to deviate much less than exogenous regressors, such as the short interest rate, from the pattern expected under the null, making it more difficult to distinguish between the null and the alternative.     

Distortion Risk Measures and Economic Capital

Abstract

To provide incentive for active risk management, it is argued that a sound coherent distortion risk measure should preserve some higher degree stop-loss orders, at least the degree-three convex order. Such risk measures are called tail-preserving risk measures. It is shown that, under some common axioms and other plausible conditions, a tail-preserving coherent distortion risk measure identifies necessarily with the Wang right-tail measure or the expected value measure. This main result is applied to derive an optimal economic capital formula.     

Applying importance sampling for estimating coherent credit risk contributions

A Monte Carlo simulation method based on importance sampling is applied to the problem of determining individual risk contributions of the obligors in a credit portfolio. The effectiveness of the method is benchmarked against standard Monte Carlo techniques and the asymptotic optimality of the method is proved. The risk measure adopted is expected shortfall, a particualr coherent risk measure. The concept of a coherent risk spectrum is discussed on the basis of some numerical examples.     

EXPONENTIAL DURATION: A MORE ACCURATE ESTIMATION OF INTEREST RATE RISK

We develop a new method to estimate the interest rate risk of an asset. This method is based on modified duration and is always more accurate than traditional estimation with modified duration. The estimates by this method are close to estimates using traditional duration plus convexity when interest rates decrease. If interest rates rise, investors will suffer larger value declines than predicted by traditional duration plus convexity estimate. The new method avoids this undesirable value overestimation and provides an estimate slightly below the true value. For risk‐averse investors, overestimation of value declines is more desirable and conservative.     

A Bayesian approach for more reliable tail risk forecasts

This paper demonstrates that existing quantile regression models used for jointly forecasting Value-at-Risk (VaR) and expected shortfall (ES) are sensitive to initial conditions. Given the importance of these measures in financial systems, this sensitivity is a critical issue. A new Bayesian quantile regression approach is proposed for estimating joint VaR and ES models. By treating the initial values as unknown parameters, sensitivity issues can be dealt with. Furthermore, new additive-type models are developed for the ES component that are more robust to initial conditions. A novel approach using the open-faced sandwich (OFS) method is proposed which improves uncertainty quantification in risk forecasts. Simulation and empirical results highlight the improvements in risk forecasts ensuing from the proposed methods.     

Measuring Corpotate Liquidity Risk

This article presents the outline of a framework for evaluating liquidity risk (at the corporate level) with risk measures that are intuitive and economically relevant. In particular, the risk measures are designed explicitly to show the effectiveness of a company's risk management program in helping the firm to (1) avoid financial distress or default and (2) ensure its ability to undertake all strategic investments. For managers attempting to quantify liquidity risks, this paper proposes that the risk measures have two important features: One is to make the liquidity risk estimate depend on some measure of the firm's balance sheet strength, one that reflects the role of the balance sheet as a risk buffer. The second is to provide a useful estimate of the opportunity costs associated with a given liquidity shortage—one that reflects the value of the investment opportunities that liquidity problems could jeopardize. The author illustrates the application of the proposed risk measures with an example of a company evaluating a hedging strategy designed to accompany a substantial increase in its investment budget. Using the risk measures discussed in this paper, the author shows how to assess the effectiveness of a proposed hedge in terms of its expected ability to reduce costly cash shortfalls in scenarios in which the firm's debt capacity is also expected to be depleted.     

Coherent and convex monetary risk measures for unbounded càdlàg processes

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all càdlàg processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted càdlàg processes that can be extended to coherent or convex monetary risk measures on the space of all adapted càdlàg processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded càdlàg processes induced by a so called m-stable set.Due to errors during the typesetting process, this article was published incorrectly in Finance Stoch 9(3):369–387 (2005). The address of the first author was printed incorrectly, and in the whole paper the angular brackets were misprinted as [ ]. The complete corrected article is given here. The online version of the original paper can be found at: http://dx.doi.org/10.1007/s00780-004-0150-7     

The National Flood Insurance Program: Is It Financially Sound?

This article uses data provided by the Federal Emergency Management Agency, which implements the NFIP, to estimate the difference between annual premiums and expected costs associated for the program as a whole and for inland and coastal regions. In addition, we examine the role of discounts, cross‐subsidies, and FEMA's method of setting what it considers to be full‐risk rates in explaining the outcomes that we observe.     

A NEW MODEL FOR ESTIMATING RISK PREMIUMS (ALONG WITH SOME EVIDENCE OF THEIR DECLINE)

This article uses the Gordon growth model with a novel method for forecasting dividend growth rates to estimate the equity cost and associated risk premium for a sample of Canadian telecommunications companies. The results suggest that the Telco risk premium has declined significantly since the early 1980s. Moreover, this decline is accentuated when measured over long Canada yields, rather than over similarly taxed, longterm preferred yields.
The findings of this study also suggest that the inverse relationship between utility risk premium and market interest rates reported by studies of U.S. utilities does not hold in Canada. If anything, the Telco risk premium has tended to vary directly with the level of market interest rates, with risk premium falling along with the general decline in rates. The main reason for these results seems to be the significant increase in interest rate risk, which has caused the long Canada yield to be a very poor proxy for the longterm, riskfree rate.
One practical implication of this finding is that companies that estimate equity costs as a premium over longterm government bond yields are probably seriously overestimating the cost.     

Interest rate risk and optimal design of mortgage instruments

This article develops and analyzes a simple expected utility model for interest rate risk and mortgage choice. The model demonstrates how the risks of interest rate changes should be allocated between borrowers and lenders through varying mortgage payments. In general, we conclude that full protection against interest rate risk, as a normative guideline, is likely to be suboptimal for the typical household. Our results show that the optimal design of adjustable rate mortgages should include an interest rate CAP provision.An earlier version of this article was presented at the American Real Estate and Urban Economics Association Meetings, Atlanta, Georgia, December 27–30, 1989.     

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.     

Nonparametric Estimation of Expected Shortfall

The expected shortfall is an increasingly popular risk measurein financial risk management and it possesses the desired sub-additivityproperty, which is lacking for the value at risk (VaR). We considertwo nonparametric expected shortfall estimators for dependentfinancial losses. One is a sample average of excessive losseslarger than a VaR. The other is a kernel smoothed version ofthe first estimator (Scaillet, 2004 Mathematical Finance), hopingthat more accurate estimation can be achieved by smoothing.Our analysis reveals that the extra kernel smoothing does notproduce more accurate estimation of the shortfall. This is differentfrom the estimation of the VaR where smoothing has been shownto produce reduction in both the variance and the mean squareerror of estimation. Therefore, the simpler ES estimator basedon the sample average of excessive losses is attractive forthe shortfall estimation.