Risk parity portfolios with risk factors

Portfolio construction and risk budgeting are the focus of many studies by academics and practitioners. In particular, diversification has spawned much interest and has been defined very differently. In this paper, we analyse a method to achieve portfolio diversification based on the decomposition of the portfolio’s risk into risk factor contributions. First, we expose the relationship between risk factor and asset contributions. Secondly, we formulate the diversification problem in terms of risk factors as an optimization program. Finally, we illustrate our methodology with a real example of building a strategic asset allocation based on economic factors for a pension fund facing liability constraints.     

Long-only equal risk contribution portfolios for CVaR under discrete distributions

Portfolios in which all assets contribute equally to the conditional value-at-risk (CVaR) represent an interesting variation of the popular risk parity investment strategy. This paper considers the use of convex optimization to find long-only equal risk contribution (ERC) portfolios for CVaR given a set of equally likely scenarios of asset returns. We provide second-order conic and non-linear formulations of the problem, which yields an ERC portfolio when CVaR is both positive and differentiable at the optimal solution. We identify sufficient conditions for differentiability and develop a heuristic that obtains an approximate ERC portfolio when the conditions are not satisfied. Computational tests show that the approach performs well compared to non-convex formulations that have been proposed in the literature.     

Sensitivity analysis of VaR and Expected Shortfall for portfolios under netting agreements

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocations when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. Collateral issues are also dealt with. For practical purposes we further provide nonparametric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.     

Bayesian inference for generalized linear mixed models of portfolio credit risk

The aims of this paper are threefold. First, we highlight the usefulness of generalized linear mixed models (GLMMs) in the modelling of portfolio credit default risk. The GLMM-setting allows for a flexible specification of the systematic portfolio risk in terms of observed fixed effects and unobserved random effects, in order to explain the phenomena of default dependence and time-inhomogeneity in historical default data. Second, we show that computational Bayesian techniques such as the Gibbs sampler can be successfully applied to fit models with serially correlated random effects, which are special instances of state space models. Third, we provide an empirical study using Standard and Poor's data on U.S. firms. A model incorporating rating category and sector effects, and a macroeconomic proxy variable for state-of-the-economy suggests the presence of a residual, cyclical, latent component in the systematic risk.     

A multi-factor model for the risk management of portfolios

We propose a methodology for modelling the value at risk of a complex portfolio, based on an extension of the Ho, Stapleton and Subrahmanyam technique. We model the variance-covariance structure of up to seven variables. These could represent four country indices and three exchange rates, for example. In addition, the effect of an arbitrary number of orthogonal factors can be analysed. The system is illustrated by estimating the value at risk for a portfolio of international stocks where the factors are stock market indices and exchange rates, a portfolio of international bonds where the factors are interest rates as well as exchange rates, and a portfolio of interest rate derivatives in different currencies. In this last case, we model a two-factor term structure of interest rates in each of the currencies, valuing the derivatives at a future date using these term structures and the Black model. The model is applied for different fineness of the binomial density and computational accuracy and efficiency are estimated.
G13, G15, G21     

A credit risk model for large dimensional portfolios with application to economic capital

In this paper we develop a multi-period and multi-state portfolio credit risk model which is applicable to large dimensional portfolios like for example retail and mortgage portfolios. The model includes a methodology for estimation and simulation of systematic transition risk through a model for stochastic migration, a methodology for the modelling of recoveries in the case of stochastic collaterals as well as an approach to dimension reduction of the portfolio. One important application of our model is economic capital (EC) and a concept of EC based on the analogy with classical risk theory is introduced and the questions of allocation as well as risk-adjusted pricing based on the allocation of EC are structured and described. The model is illustrated by an extensive numerical example giving a concretization of the model as well as of several of the concepts introduced.     

Non-monotonic pricing kernel and an extended class of mixture of distributions for option pricing

We derive closed form European option pricing formulae under the general equilibrium framework for underlying assets that have an \(N\) -mixture of transformed normal distributions. The component distributions need not belong to the same class but must all be transformed normal. An important implication of our results is that the mixture of distributions is consistent with a “what appears to be abnormal” non-monotonic (asset specific) pricing kernel for the S&P 500 and that the representative agent has a “logical” monotonic decreasing marginal utility. We show that a mixture of two lognormal distributions is sufficient to produce this result and also implied volatility smiles of a wide variety of shapes.     

The selection of multinational equity portfolios: forecasting models and estimation risk

The problem considered is the selection of a portfolio of international assets, particularly the forecasting of the inputs to a selection algorithm. Four models of the asset return generating process are considered, two of which ignore the international nature of the universe of assets, two which exploit it in different ways. Several estimation methods are considered for each component: expected return, variance and covariance of returns. The combinations of model and estimation method are first evaluated in terms of their forecasting performance for the components mentioned for the individual assets. The universe used is the components of the Financial Times Eurotrack 100 Index. Significant differences were found between the forecasting accuracy of the methods considered for each component. In the final stage of the analysis, a comparison of the returns on portfolios chosen using each combination showed a significant difference. The analysis suggests that the choice of estimation method is more critical than the choice of pricing model.     

极值风险价值和预期损失的估计应用:以影子银行和A股市场为例

在检测金融时间序列极值的风险价值和预期损失时,由于金融时间序列具有尖峰和厚尾的特征,首先利用广义帕累托分布的安全阈值模型Hill图观测.当样本数据不多时,利用Gibbs抽样和蒙特卡洛模拟马尔科夫链来检验参数的拟合效果.最后,对从Hill图观测出的极值情况基于MCMC和MLE方法估计其风险价值和预期损失.本文以中国的影子银行规模、上证指数、上证成交量时间序列为例,检测了三者的极值风险值和预期损失,经过比较发现:上证成交量极值风险更大,影子银行极值风险相对较小.     

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.     

VaR模型及其在寿险公司风险管理中的应用

随着保险产品的创新、竞争的加剧及金融市场的发展,我国寿险公司面临的风险将更为复杂和多样,如何对这些风险进行有效管理是寿险公司面临的重要难题,而VaR模型作为当前国际金融风险管理和金融监管的主流方法,含有丰富的风险管理思想,故对VaR模型进行研究并探讨其在我国寿险公司风险管理中的应用具有一定的现实意义。本文分析了寿险公司应用VaR模型时需注意的问题,并指出我国寿险公司建立基于VaR的风险管理体系需在公司治理结构、风险管理组织架构、风险管理技术、风险数据库等方面加以完善。     

An analytic approach to credit risk of large corporate bond and loan portfolios

We derive an analytic approximation to the credit loss distribution of large portfolios by letting the number of exposures tend to infinity. Defaults and rating migrations for individual exposures are driven by a factor model in order to capture co-movements in changing credit quality. The limiting credit loss distribution obeys the empirical stylized facts of skewness and heavy tails. We show how portfolio features like the degree of systematic risk, credit quality and term to maturity affect the distributional shape of portfolio credit losses. Using empirical data, it appears that the Basle 8% rule corresponds to quantiles with confidence levels exceeding 98%. The limit law's relevance for credit risk management is investigated further by checking its applicability to portfolios with a finite number of exposures. Relatively homogeneous portfolios of 300 exposures can be well approximated by the limit law. A minimum of 800 exposures is required if portfolios are relatively heterogeneous. Realistic loan portfolios often contain thousands of exposures implying that our analytic approach can be a fast and accurate alternative to the standard Monte-Carlo simulation techniques adopted in much of the literature.     

Optimal dividend control for a generalized risk model with investment incomes and debit interest

This paper investigates dividend optimization of an insurance corporation under a more realistic model, which takes into consideration refinancing or capital injections. The model follows the compound Poisson framework with credit interest for positive reserve and debit interest for negative reserve. Ruin occurs when the reserve drops below the critical value. The company controls the dividend pay-out dynamically with the objective to maximize the expected total discounted dividends until ruin. We show that the optimal strategy, is a band strategy and it is optimal to pay no dividends when the reserve is negative.     

A model for tax advantages of portfolios with many assets

Taxable portfolios present challenges for optimization models with even a limited number of assets. Holding many assets, however, has a distinct tax advantage over holding few assets. In this paper, we develop a model that takes an extreme view of a portfolio as a continuum of assets to gain the broadest possible advantage from holding many assets. We find the optimal strategy for trading in this portfolio in the absence of transaction costs and develop bounding approximations on the optimal value. We compare the results in a simulation study to a portfolio consisting only of a market index and show that the multi-asset portfolio’s tax advantage can lead either to significant consumption or bequest increases.     

Kernel-type estimators for the distortion risk premiums of heavy-tailed distributions

A new kernel-type estimator for the distortion risk premiums of heavy-tailed losses is introduced. Using a least-squares approach, a bias-reduced version of this estimator is proposed. Under suitable assumptions, the asymptotic normality of the given estimators is established. A small simulation study, to illustrate the performance of our method, is carried out.     

Derivative activities and the risk of international banks: A market index and VaR approach

The results of a comparison of international banks using a three-factor multi-index model and a modified value-at-risk (VaR) analysis indicate that the use of options increases the interest rate beta for all banks, while both interest rate and currency swaps generally reduce risk. The results are the strongest and the most consistent for U.S. dealer banks, followed by European banks, and then Japanese banks. Furthermore, the evidence suggests that the VaR approach to risk management can effectively be used by both domestic as well as international banks, although the results appear to be somewhat sensitive to the regulatory environment in which the bank operates.     

Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions

Financial returns typically display heavy tails and some degree of skewness, and conditional variance models with these features often outperform more limited models. The difference in performance may be especially important in estimating quantities that depend on tail features, including risk measures such as the expected shortfall. Here, using recent generalizations of the asymmetric Student-t and exponential power distributions to allow separate parameters to control skewness and the thickness of each tail, we fit daily financial return volatility and forecast expected shortfall for the S&P 500 index and a number of individual company stocks; the generalized distributions are used for the standardized innovations in a nonlinear, asymmetric GARCH-type model. The results provide evidence for the usefulness of the general distributions in improving fit and prediction of downside market risk of financial assets. Constrained versions, corresponding with distributions used in the previous literature, are also estimated in order to provide a comparison of the performance of different conditional distributions.     

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.