The limits of granularity adjustments

We provide a rigorous proof of granularity adjustment (GA) formulas to evaluate loss distributions and risk measures (value-at-risk) in the case of heterogenous portfolios, multiple systematic factors and random recoveries. As a significant improvement with respect to the literature, we detail all the technical conditions of validity and provide an upper bound of the remainder term for finite portfolio sizes. Moreover, we deal explicitly with the case of general loss distributions, possibly with masses. For some simple portfolio models, we prove empirically that the granularity adjustments do not always improve the infinitely granular first-order approximations. This stresses the importance of checking some conditions of regularity before relying on such techniques. Smoothing the underlying loss distributions through random recoveries or exposures improves the GA performances in general.     

Risk factor contributions in portfolio credit risk models

Determining contributions to overall portfolio risk is an important topic in risk management. For positions (instruments and sub-portfolios), this problem has been well studied, and a significant theory built, around the calculation of marginal contributions. We consider the problem of determining the contributions to portfolio risk of risk factors. This cannot be addressed through an immediate extension of techniques for position contributions, since the portfolio loss is a nonlinear function of the risk factors. We employ the Hoeffding decomposition of the portfolio loss into a sum of terms depending on the factors. This decomposition restores linearity, but includes terms arising from joint effects of groups of factors. These cross-factor terms provide information to risk managers, since they can be viewed as best hedges of the portfolio loss involving instruments of increasing complexity. We illustrate the technique on multi-factor portfolio credit risk models, where systematic factors represent industries, geographical sectors, etc.     

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.     

Asset allocation: How much does model choice matter?

This paper analyzes the optimal portfolio decision of a CRRA investor in models with stochastic volatility and stochastic jumps. The investor follows a buy-and-hold strategy in the stock, the money market account, and one additional derivative. We show that both the type of the model and the structure of the risk premia have a significant impact on the optimal portfolio, on the utility gain from having access to derivatives, and on whether the investor prefers to trade OTM or ATM options. We also show that model mis-specification results in significant utility losses. Omitting jumps in volatility can be devastating, in particular if the investor chooses the seemingly optimal OTM put options. A misestimation of the structure of the risk premia has a less devastating effect, but can still lead to a loss of around 4% in the annual certainty equivalent return.     

Combining equilibrium,resampling, and analyst’s views in portfolio optimization

This paper proposes the use of a portfolio optimization methodology which combines features of equilibrium models and investor’s views as in Black and Litterman (1992), and also deals with estimation risk as in Michaud (1998). In this way, our combined methodology is able to meet the needs of practitioners for stable and diversified portfolio allocations, while it is theoretically grounded on an equilibrium framework. We empirically test the methodology using a comprehensive sample of developed countries fixed income and equity indices, as well as sub-samples stratified by geographical region, time period, asset class and risk level. In general, our proposed combined methodology generates very competitive portfolios when compared to other methodologies, considering three evaluation dimensions: financial efficiency, diversification, and allocation stability. By generating financially efficient, stable, and diversified portfolio allocations, our methodology is suitable for long-term investors such as Central Banks and Sovereign Wealth Funds.     

Hedging structured credit products during the credit crisis: A horse race of 10 models

Pricing and hedging structured credit products poses major challenges to financial institutions. This paper puts several valuation approaches through a crucial test: How did these models perform in one of the worst periods of economic history, September 2008, when Lehman Brothers went under? Did they produce reasonable hedging strategies? We study several bottom-up and top-down credit portfolio models and compute the resulting delta hedging strategies using either index contracts or a portfolio of single-name CDS contracts as hedging instruments. We compute the profit-and-loss profiles and assess the performances of these hedging strategies. Among all 10 pricing models that we consider the Student-t copula model performs best. The dynamical generalized-Poisson loss model is the best top-down model, but this model class has in general problems to hedge equity tranches. Our major finding is however that single-name and index CDS contracts are not appropriate instruments to hedge CDO tranches.     

A systematic approach to multi-period stress testing of portfolio credit risk

We propose a new method for analysing multi-period stress scenarios for portfolio credit risk more systematically than in current macro stress tests. The plausibility of a scenario is quantified by its distance from an average scenario. For a given level of plausibility, we search systematically for the most adverse scenario. This ensures that no plausible scenario will be missed. We show how this method can be applied to some models already in use by practitioners. While worst case search requires numerical optimisation we show that we can work with reasonably good linear approximations to the portfolio loss function. This makes systematic multi-period stress testing computationally efficient and easy to implement. Applying our approach to data from the Spanish loan register we show that, compared to standard stress test procedures, our method identifies more harmful scenarios that are equally plausible.     

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.     

A conditional asset-pricing model with the optimal orthogonal portfolio

This paper employs a conditional asset-pricing model based on the optimal orthogonal portfolio approach to construct a factor portfolio that embodies all the latent factors important for pricing a given set of test assets. The advantage of this portfolio to the anomaly related mimicking portfolios is its ability to separate out the components of average return that are not related to the return covariation. The performance of this portfolio is evaluated against several conventional factors, using both cross-sectional and time-series regression approaches, as well as the Hansen and Jagannathan (1997) distance measure. Its strong out-of-sample results indicate that our suggested methodology may have important applications in risk management, portfolio selection and performance evaluation.     

Utilitarianism and fairness in portfolio positioning

The paper introduces the theory of optimal positioning of financial products. It is illustrated in the context of long-term intertemporal portfolio allocation and can be applied for example to asset allocation funds. We embed this problem in location theory: the portfolio is optimized within the investors’risk aversion dimension. For the CRRA utility functions, we compute explicitly the distance functions. For the first (utilitarian criterion), the average utility of the investors is maximized. For the second one (fairness criterion), the choice of portfolio is optimized so that the average monetary loss due to the lack of customization is minimized. Given the distribution of investors’ risk aversion, we provide a solution method and an algorithm to optimally position standardized portfolio along one of these two criteria.     

International portfolio selection with exchange rate risk: A behavioural portfolio theory perspective

This paper analyzes international portfolio selection with exchange rate risk based on behavioural portfolio theory (BPT). We characterize the conditions under which the BPT problem with a single foreign market has an optimal solution, and show that the optimal portfolio contains the traditional mean–variance efficient portfolio without consideration of exchange rate risk, and an uncorrelated component constructed to hedge against exchange rate risk. We illustrate that the optimal portfolio must be mean–variance efficient with exchange rate risk, while the same is not true from the perspective of local investors unless certain conditions are satisfied. We further establish that international portfolio selection in the BPT with multiple foreign markets consists of two sequential decisions. Investors first select the optimal BPT portfolio in each market, overlooking covariances among markets, and then allocate funds across markets according to a specific rule to achieve mean–variance efficiency or to minimize the loss in efficiency.     

Dynamic hedge fund portfolio construction: A semi-parametric approach

In this article, we evaluate alternative optimization frameworks for constructing portfolios of hedge funds. We compare the standard mean–variance optimization model with models based on CVaR, CDaR and Omega, for both conservative and aggressive hedge fund investment strategies. In order to implement the CVaR, CDaR and Omega optimization models, we propose a semi-parametric methodology, which is based on extreme value theory, copula and Monte Carlo simulation. We compare the semi-parametric approach with the standard, non-parametric approach, used to compute CVaR, CDaR and Omega, and the benchmark parametric approach, based on both static and dynamic mean–variance optimization. We report two main findings. The first is that the CVaR, CDaR and Omega models offer a significant improvement in terms of risk-adjusted portfolio performance over the parametric mean–variance model. The second is that semi-parametric estimation of the CVaR, CDaR and Omega models offers a very substantial improvement over non-parametric estimation. Our results are robust to the choice of target return, risk limit and estimation sample size.     

Optimal portfolios when volatility can jump

We consider an asset allocation problem in a continuous-time model with stochastic volatility and jumps in both the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. The demand for jump risk includes a hedging component, which is not present in models without volatility jumps. We further show that the introduction of derivative contracts can have substantial economic value. We also analyze the distribution of terminal wealth for an investor who uses the wrong model, either by ignoring volatility jumps or by falsely including such jumps, or who is subject to estimation risk. Whenever a model different from the true one is used, the terminal wealth distribution exhibits fatter tails and (in some cases) significant default risk.     

Can VAR models capture regime shifts in asset returns? A long-horizon strategic asset allocation perspective

It is often suggested that through a judicious choice of predictors that track business cycles and market sentiment, simple vector autoregressive (VAR) models could produce optimal strategic portfolio allocations that hedge against the bull and bear dynamics typical of financial markets. However, a distinct literature exists that shows that nonlinear econometric frameworks, such as Markov switching (MS), are also natural tools to compute optimal portfolios in the presence of stochastic good and bad market states. In this paper we examine whether simple VARs can produce portfolio rules similar to those obtained under MS, by studying the effects of expanding both the order of the VAR and the number/selection of predictor variables included. In a typical stock-bond strategic asset allocation problem, we compute the out-of-sample certainty equivalent returns for a wide range of VARs and compare these measures of performance with those typical of nonlinear models for a long-horizon investor with constant relative risk aversion. We conclude that most VARs cannot produce portfolio rules, hedging demands, or (net of transaction costs) out-of-sample performances that approximate those obtained from equally simple nonlinear frameworks. We also compute the improvement in realized performance that may be achieved adopting more complex MS models and report this may be substantial in the case of regime switching ARCH.     

Economic capital for nonperforming loans

A portfolio of nonperforming loans requires economic capital. We present two models for forecasting the portfolio loss and its probability distribution. In the first model, the loss for each nonperforming loan entails a change in provision over the risk horizon. The risk determinants are the single-name concentration, measured by the Herfindahl–Hirschmann index, as well as a systematic factor and the idiosyncratic risk. Our second model allows for interportfolio diversification with a portfolio of performing loans because banks typically own both performing and nonperforming loans. In this model, the nonperforming loan is identified with its systematic risk. Both models allow for closed-form expressions of economic capital and for the capital charge of the single loan. We calibrate the macroeconomic model parameters statistically with a loss panel; the microeconomic parameters depend on the portfolio. The portfolio risk for nonperforming loans mainly depends on the volatility of the systematic economic factor. The dependence becomes more pronounced when interportfolio diversification is taken into account. The magnitude of interportfolio diversification is also marked. Finally, we calculate regulatory capital charges according to Basel II for past-due loans. The regulatory charges are on average smaller than our economic charges and, additionally, take the volatility of economic activity into account only implicitly.     

The impact of terrorism on financial markets: An empirical study

The main focus of this paper is to study empirically the impact of terrorism on the behavior of stock, bond and commodity markets. We consider terrorist events that took place in 25 countries over an 11-year time period and implement our analysis using different methods: an event-study approach, a non-parametric methodology, and a filtered GARCH-EVT approach. In addition, we compare the effect of terrorist attacks on financial markets with the impact of other extreme events such as financial crashes and natural catastrophes. The results of our analysis show that a non-parametric approach is the most appropriate method among the three for analyzing the impact of terrorism on financial markets. We demonstrate the robustness of this method when interest rates, equity market integration, spillover and contemporaneous effects are controlled. We show how the results of this approach can be used for investors’ portfolio diversification strategies against terrorism risk.     

证券投资组合损失率概率评价模型的开发与实证检验

通过建立证券投资组合损失率概率评价模型,定义损失率概率为单个资产或投资组合的损失率超过市场平均损失率的概率,用于对投资组合的风险进行评价。以某基金为例,计算得到该基金的损失率概率为39.74%,表明该基金有39.74%的概率损失率超过市场平均损失率。以证券投资组合损失率概率最小为目标函数,采用遗传算法进行求解,得出该基金投资组合损失率概率最小时的投资权重系数,最小损失率概率为36.01%,与该基金公司的投资组合相比,损失率概率降低了3.73%,实例证明该模型是降低证券投资组合风险的一种实用方法。     

SYMMETRIC VERSUS ASYMMETRIC CONDITIONAL COVARIANCE FORECASTS: DOES IT PAY TO SWITCH?

Volatilities and correlations for equity markets rise more after negative returns shocks than after positive shocks. Allowing for these asymmetries in covariance forecasts decreases mean‐variance portfolio risk and improves investor welfare. We compute optimal weights for international equity portfolios using predictions from asymmetric covariance forecasting models and a spectrum of expected returns. Investors who are moderately risk averse, have longer rebalancing horizons, and hold U.S. equities benefit most and may be willing to pay around 100 basis points annually to switch from symmetric to asymmetric forecasts. Accounting for asymmetry in both variances and correlations significantly lowers realized portfolio risk.     

Nonlinearly weighted convex risk measure and its application

We propose a new class of risk measures which satisfy convexity and monotonicity, two well-accepted axioms a reasonable and realistic risk measure should satisfy. Through a nonlinear weight function, the new measure can flexibly reflect the investor’s degree of risk aversion, and can control the fat-tail phenomenon of the loss distribution. A realistic portfolio selection model with typical market frictions taken into account is established based on the new measure. Real data from the Chinese stock markets and American stock markets are used for empirical comparison of the new risk measure with the expected shortfall risk measure. The in-sample and out-of-sample empirical results show that the new risk measure and the corresponding portfolio selection model can not only reflect the investor’s risk-averse attitude and the impact of different trading constraints, but can find robust optimal portfolios, which are superior to the corresponding optimal portfolios obtained under the expected shortfall risk measure.     

An analysis of portfolio selection with background risk

This paper investigates the impact of background risk on an investor’s portfolio choice in a mean–variance framework, and analyzes the properties of efficient portfolios as well as the investor’s hedging behaviour in the presence of background risk. Our model implies that the efficient portfolio with background risk can be separated into two independent components: the traditional mean–variance efficient portfolio, and a self-financing component constructed to hedge against background risk. Our analysis also shows that the presence of background risk shifts the efficient frontier of financial assets to the right with no changes in its shape. Moreover, both the composition of the hedge portfolio and the location of the efficient frontier are greatly affected by a number of background risk factors, including the proportion of background assets in total wealth and the correlation between background risk and financial risk.