RECOVERING PORTFOLIO DEFAULT INTENSITIES IMPLIED BY CDO QUOTES

We propose a stable nonparametric algorithm for the calibration of “top‐down” pricing models for portfolio credit derivatives: given a set of observations of market spreads for collateralized debt obligation (CDO) tranches, we construct a risk‐neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk‐neutral loss process “closest” to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton–Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct a default intensity process which leads to tranche spreads consistent with the observations. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the previous number of defaults, and offer quantitative evidence for contagion effects in the (risk‐neutral) loss process.     

CORRELATION UNDER STRESS IN NORMAL VARIANCE MIXTURE MODELS

We investigate correlations of asset returns in stress scenarios where a common risk factor is truncated. Our analysis is performed in the class of normal variance mixture (NVM) models, which encompasses many distributions commonly used in financial modeling. For the special cases of jointly normally and t‐distributed asset returns we derive closed formulas for the correlation under stress. For the NVM distribution, we calculate the asymptotic limit of the correlation under stress, which depends on whether the variables are in the maximum domain of attraction of the Fréchet or Gumbel distribution. It turns out that correlations in heavy‐tailed NVM models are less sensitive to stress than in medium‐ or light‐tailed models. Our analysis sheds light on the suitability of this model class to serve as a quantitative framework for stress testing, and as such provides valuable information for risk and capital management in financial institutions, where NVM models are frequently used for assessing capital adequacy. We also demonstrate how our results can be applied for more prudent stress testing.     

PROFIT SHARING IN HEDGE FUNDS

In a new scheme for hedge fund managerial compensation known as the first‐loss scheme, a fund manager uses her investment in the fund to cover any fund losses first; by contrast, in the traditional scheme currently used in most US funds, the manager does not cover investors' losses in the fund. We propose a framework based on cumulative prospect theory to compute and compare the trading strategies, fund risk, and managers' and investors' utilities in these two schemes analytically. The model is calibrated to the historical attrition rates of US hedge funds. We find that with reasonable parameter values, both fund managers' and investors' utilities can be improved and fund risk can be reduced simultaneously by replacing the traditional scheme (with 10% internal capital and 20% performance fee) with a first‐loss scheme (with 10% first‐loss capital and 30% performance fee). When the performance fee in the first‐loss scheme is 40% (a current market practice), however, such substitution renders investors worse off.     

Network valuation in financial systems

We introduce a general model for the balance‐sheet consistent valuation of interbank claims within an interconnected financial system. Our model represents an extension of clearing models of interdependent liabilities to account for the presence of uncertainty on banks' external assets. At the same time, it also provides a natural extension of classic structural credit risk models to the case of an interconnected system. We characterize the existence and uniqueness of a valuation that maximizes individual and total equity values for all banks. We apply our model to the assessment of systemic risk and in particular for the case of stress testing. Further, we provide a fixed‐point algorithm to carry out the network valuation and the conditions for its convergence.     

OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

Risk management with weighted VaR

This article studies the optimal portfolio selection of expected utility‐maximizing investors who must also manage their market‐risk exposures. The risk is measured by a so‐called weighted value‐at‐risk (WVaR) risk measure, which is a generalization of both value‐at‐risk (VaR) and expected shortfall (ES). The feasibility, well‐posedness, and existence of the optimal solution are examined. We obtain the optimal solution (when it exists) and show how risk measures change asset allocation patterns. In particular, we characterize three classes of risk measures: the first class will lead to models that do not admit an optimal solution, the second class can give rise to endogenous portfolio insurance, and the third class, which includes VaR and ES, two popular regulatory risk measures, will allow economic agents to engage in “regulatory capital arbitrage,” incurring larger losses when losses occur.     

GENERALIZATION OF THE DYBVIG–INGERSOLL–ROSS THEOREM AND ASYMPTOTIC MINIMALITY

The long‐term limit of zero‐coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig–Ingersoll–Ross theorem, which says that long‐term spot and forward rates can never fall in an arbitrage‐free model. Extensions of popular interest rate models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous‐time short‐rate models.     

Value at risk and conditional extreme value theory via markov regime switching models

This study develops a new conditional extreme value theory‐based (EVT) model that incorporates the Markov regime switching process to forecast extreme risks in the stock markets. The study combines the Markov switching ARCH (SWARCH) model (which uses different sets of parameters for various states to cope with the structural changes for measuring the time‐varying volatility of the return distribution) with the EVT to model the tail distribution of the SWARCH processed residuals. The model is compared with unconditional EVT and conditional EVT‐GARCH models to estimate the extreme losses in three leading stock indices: S&P 500 Index, Hang Seng Index and Hang Seng China Enterprise Index. The study found that the EVT‐SWARCH model outperformed both the GARCH and SWARCH models in capturing the non‐normality and in providing accurate value‐at‐risk forecasts in the in‐sample and out‐sample tests. The EVTSWARCH model, which exhibits the features of measuring the volatility of a heteroscedastic financial return series and coping with the non‐normality owing to structural changes, can be an alternative measure of the tail risk. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:155–181, 2008     

Hedge Fund Styles and their Contagion from the Equity Market

We examine the dynamic contagion process of the equity market on 10 hedge fund styles. We investigate the contagion mechanism for each style using single equation error correction and latent factor models. We find that the contagion effects of the equity market on each style index depend specifically on the fund style strategy. We demonstrate that certain fund styles are more prone to contagion from the equity market than others. Our results help illuminate the relative effectiveness of a particular strategy under certain market conditions and provide insights into the long‐standing controversy around the efficient market hypothesis.     

Robust XVA

We introduce an arbitrage‐free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the exact return rate of her counterparty's bond, but she knows it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super‐replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the closeout value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.     

Multiple Ratings Model of Defaultable Term Structure

A new approach to modeling credit risk, to valuation of defaultable debt and to pricing of credit derivatives is developed. Our approach, based on the Heath, Jarrow, and Morton (1992) methodology, uses the available information about the credit spreads combined with the available information about the recovery rates to model the intensities of credit migrations between various credit ratings classes. This results in a conditionally Markovian model of credit risk. We then combine our model of credit risk with a model of interest rate risk in order to derive an arbitrage‐free model of defaultable bonds. As expected, the market price processes of interest rate risk and credit risk provide a natural connection between the actual and the martingale probabilities.     

Risk Management of Nonstandard Basket Options with Different Underlying Assets

Basket options are among the most popular products of the new generation of exotic options. They are particularly attractive because they can efficiently and simultaneously hedge a wide variety of intrinsically different financial risks and are flexible enough to cover all the risks faced by firms. Oddly, the existing literature on basket options considers only standard baskets where all underlying assets are of the same type and hedge the same kind of risk. Moreover, the empirical implementation of basket‐option models remains in its early stages, particularly when the baskets contain different underlying assets. This study focuses on various steps for developing sound risk management of basket options. We first propose a theoretical model of a nonstandard basket option on commodity price with stochastic convenience yield, exchange rate, and domestic and foreign zero‐coupon bonds in a stochastic interest rate setting. We compare the hedging performance of the extended basket option containing different underlying assets with that of a portfolio of individual options. The results show that the basket strategy is more efficient. We apply the maximum likelihood method to estimate the parameters of the basket model and the correlations between variables. Monte Carlo simulations are conducted to examine the performance of the maximum likelihood estimator in finite samples of simulated data. A real‐data study for a nonfinancial firm is presented to illustrate ways practitioners could use the extended basket option. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:299‐326, 2013     

SENSITIVITY ANALYSIS OF NONLINEAR BEHAVIOR WITH DISTORTED PROBABILITY

In this paper, we propose a sensitivity‐based analysis to study the nonlinear behavior under nonexpected utility with probability distortions (or “distorted utility” for short). We first discover the “monolinearity” of distorted utility, which means that after properly changing the underlying probability measure, distorted utility becomes locally linear in probabilities, and the derivative of distorted utility is simply an expectation of the sample path derivative under the new measure. From the monolinearity property, simulation algorithms for estimating the derivative of distorted utility can be developed, leading to gradient‐based search algorithms for the optimum of distorted utility. We then apply the sensitivity‐based approach to the portfolio selection problem under distorted utility with complete and incomplete markets. For the complete markets case, the first‐order condition is derived and optimal wealth deduced. For the incomplete markets case, a dual characterization of optimal policies is provided; a solvable incomplete market example with unhedgeable interest rate risk is also presented. We expect this sensitivity‐based approach to be generally applicable to optimization problems involving probability distortions.     

DYNAMIC CDO TERM STRUCTURE MODELING

This paper provides a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T, x) ‐bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, and zero else. Necessary and sufficient conditions on the stochastic term structure movements for the absence of arbitrage are given. Background market risk as well as feedback contagion effects of the loss process are taken into account. Moreover, we show that any exogenous specification of the volatility and contagion parameters actually yields a unique consistent loss process and thus an arbitrage‐free family of (T, x) ‐bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of doubly stochastic affine term structure models.     

Estimating the Cost‐of‐Equity Capital Using Empirical Asset Pricing Models

Empirical asset pricing models seek to capture characteristic‐based patterns in the cross‐section of average stock returns. I propose a new approach for constructing these models, and investigate its performance with respect to estimating the cost‐of‐equity capital. Using a model that accounts for the cross‐sectional relation between five characteristics and average stock returns, I obtain cost‐of‐equity estimates that outperform those produced by the Fama‐French five‐factor model in out‐of‐sample tests. Because the proposed approach builds directly on standard cross‐sectional regression techniques, it provides complete flexibility in choosing the firm characteristics used to formulate the cost‐of‐equity estimates.     

WORKFORCE AGEING AND THE SUBSTITUTION OF LABOUR: THE ROLE OF SUPPLY AND DEMAND OF LABOUR IN AUSTRIA

We study the sensitivity of projected economic productivity (output per worker) with respect to alternative projections of labour supply and alternative assumptions on the substitutability of workers at different ages. We show that in a pure labour economy assuming imperfect substitution of workers at different ages implies an increase in relative productivity during the next two decades. For a decreasing or hump‐shaped age‐specific productivity profile a negative tradeoff between an increasing labour force at older ages and aggregate productivity results. A decline in productivity can be attenuated by adjusting labour force participation rates to levels currently observed in Nordic countries.     

When in peril, retrench: Testing the portfolio channel of contagion

The impact of past gains and losses on international investors' risk aversion is an important factor in the propagation of financial shocks across countries. We first present a stylized model illustrating how changes in investors' risk aversion affect portfolio decisions and stock prices. We then examine empirically the behavior of international mutual funds. When funds' returns are below average, they reduce their exposure to countries in which they were overweight and vice versa. An index of “financial interdependence” that reflects the extent to which countries share overexposed funds helps explain the pattern of stock market comovement across countries and the pattern of contagion during crises.     

OPTIMAL INSURANCE DESIGN UNDER RANK‐DEPENDENT EXPECTED UTILITY

We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank‐dependent expected utility (RDEU) theory with a concave utility function and an inverse‐S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.     

Reverse stress testing: Scenario design for macroprudential stress tests

We propose a systematic algorithmic reverse-stress testing methodology to create “worst case” scenarios for regulatory stress tests by accounting for losses that arise from distressed portfolio liquidations. First, we derive the optimal bank response for any given shock. Then, we introduce an algorithm which systematically generates scenarios that exploit the key vulnerabilities in banks' portfolio holdings and thus maximize contagion despite banks' optimal response to the shock. We apply our methodology to data of the 2016 European Banking Authority (EBA) stress test, and design worst case scenarios for the portfolio holdings of European banks at the time. Using spectral clustering techniques, we group 10,000 worst-case scenarios into twelve geographically concentrated families. Our results show that even though there is a wide range of different scenarios within these 12 families, each cluster tends to affect the same banks. An “Anna Karenina” principle of stress testing emerges: Not all stressful scenarios are alike, but every stressful scenario stresses the same banks. These findings suggest that the precise specification of a scenario is not of primal importance as long as the most vulnerable banks are targeted and sufficiently stressed. Finally, our methodology can be used to uncover the weakest links in the financial system and thereby focus supervisory attention on these, thus building a bridge between macroprudential and microprudential stress tests.     

Monetary Union Viability: A Criterion Proposal

We propose a model that represents the dynamic behaviour of a monetary union comprising two countries whose natural interest rates are initially unequal. This initial disparity and the subsequent application of a common monetary policy generate different national inflation rates and lead to losses of competitiveness, foreign deficits, and the indebtedness of one country with respect to the other. We propose as a viability criterion for the modelled monetary union a combination of non‐explosive foreign debt and the ability of the central bank to neutralize the contracting effects of taking on additional debt to avoid falling into a liquidity trap.