We consider the problem of calculating tail probabilities of the returns of linear asset portfolios. As flexible and accurate model for the logarithmic returns we use the $t$-copula dependence structure and marginals following the generalized hyperbolic distribution. Exact calculation of the tail-loss probabilities is not possible and even simulation leads to challenging numerical problems. Applying a new numerical inversion method for the generation of the marginals and importance sampling with carefully selected mean shift we develop an efficient simulation algorithm. Numerical results for a variety of realistic portfolio examples show an impressive performance gain. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

An asset manager's goal is to provide a high return relative the risk taken, and thus faces the challenge of how to choose an optimal portfolio. Many mathematical methods have been developed to achieve a good balance between these attributes and using di erent risk measures. In thisthesis, we test the use of a relatively simple and common approach: the Markowitz mean-variance method, and a more quantitatively demanding approach: the tail optimization method. Using active portfolio based on data provided by the Swedish fund management company Enter Fonderwe implement these approaches and compare the results. We analyze how each method weighs theunderlying assets in order to get an optimal portfolio.

Understanding financial asset return correlation is a key facet in asset allocation and investor’s portfolio optimization strategy. For the last decades, several studies have investigated this relationship between stock and bond returns. But, fewer studies have dealt with multi-asset return dynamics. While initial literature attempted to understand the fundamental pattern of comovements, later studies model the economic state variables influencing such time-varying comovements of primarily stock and bond returns. Research widely acknowledges that return distributions of financial assets are non-normal. When the joint distributions of the asset returns follow a non-elliptical structure, linear correlation fails to provide sufficient information of their dependence structure. In particular two issues arise from this existing empirical evidence. The first is to propose a more reliable alternative density specification for a higher-dimensional case. The second is to formulate a measure of the variables’ dependence structure which is more instructive than linear correlation. In this work I use a time-varying conditional multivariate elliptical and non-elliptical copula to examine the return comovements of three different asset classes: financial assets, commodities and real estate in the US market. I establish the following stylized facts about asset return comovements. First, the static measures of asset return comovements overestimate the asset return comovements in the economic expansion phase, while underestimating it in the periods of economic contraction. Second, Student t-copulas outperform both elliptical and non-elliptical copula models, thus confirming the ii dominance of Student t-distribution. Third, findings show a significant increase in asset return comovements post August 2007 subprime crisis ... [cont.].

Understanding financial asset return correlation is a key facet in asset allocation and investor’s portfolio optimization strategy. For the last decades, several studies have investigated this relationship between stock and bond returns. But, fewer studies have dealt with multi-asset return dynamics. While initial literature attempted to understand the fundamental pattern of comovements, later studies model the economic state variables influencing such time-varying comovements of primarily stock and bond returns. Research widely acknowledges that return distributions of financial assets are non-normal. When the joint distributions of the asset returns follow a non-elliptical structure, linear correlation fails to provide sufficient information of their dependence structure. In particular two issues arise from this existing empirical evidence. The first is to propose a more reliable alternative density specification for a higher-dimensional case. The second is to formulate a measure of the variables’ dependence structure which is more instructive than linear correlation. In this work I use a time-varying conditional multivariate elliptical and non-elliptical copula to examine the return comovements of three different asset classes: financial assets, commodities and real estate in the US market. I establish the following stylized facts about asset return comovements. First, the static measures of asset return comovements overestimate the asset return comovements in the economic expansion phase, while underestimating it in the periods of economic contraction. Second, Student t-copulas outperform both elliptical and non-elliptical copula models, thus confirming the ii dominance of Student t-distribution. Third, findings show a significant increase in asset return comovements post August 2007 subprime crisis ... [cont.].

It is of great importance to find an analytical copula that will represent the empirical lower tail dependence. In this study, the pairwise empirical copula are estimated using data of the S&P 500 stocks during the period 2007-2010.Different optimization methods and measures of dependence have been used to fit Gaussian, t and Clayton copula to the empirical copulas, in order to represent the empirical lower tail dependence. These different measures of dependence and optimization methods with their restrictions, point at different analytical copulas being optimal. In this study the t copula with 5 degrees of freedom is giving the most fulfilling result, when it comes to representing lower tail dependence. The t copula with 5 degrees of freedom gives the best representation of empirical lower tail dependence, whether one uses the 'Empirical maximum likelihood estimator', or 'Equal Ƭ' as an approach.

In 2005 Attilio Meucci presented his article Beyond Black-Litterman: Views on Non-Normal Markets which introduces the copula opinion pooling approach using generic non-normal market assumptions. Copulas and opinion pooling are used to express views on the market which provides a posterior market distribution that smoothly blends an arbitrarily distributed market prior distribution with arbitrarily chosen views. This thesis explains how to use this method in practice and investigates its performance in different investment situations. The method is tested on three portfolios, each showing some different feature. The conclusions that can be drawn are e.g. that the method can be used in many different investment situations in many different ways, implementation and calculations can be made within a few seconds for a large data set and the method could be useful for portfolio managers using mathematical methods. The presented examples together with the method generate reasonable results.

蒙地卡羅模擬是在組合信用風險的管理上相當實用的計算工具。衡量組合信用風險時，必須以適當的模型描述資產間的相依性。常態關聯結構是目前最廣為使用的模型，但實證研究認為 t 關聯結構更適合用於配適金融市場的資料。在本文中，我們採用 Bassamboo et al. (2008) 提出的極值相依模型建立 t 關聯結構用以捕捉資產之間的相關性。同時，為增進蒙地卡羅法之收斂速度，我們以 Chiang et al. (2007) 的重要性取樣法為基礎，將其拓展到極值相依模型下，並提出兩階段的重要性取樣技巧確保使用此方法估計一籃子信用違約時，所有模擬路徑均會發生信用事件。數值結果顯示，所提出的演算法皆達變異數縮減。而在模型自由度較低或是資產池較大的情況下，兩階段的重要性取樣法將會有更佳的估計效率。我們也以同樣的思路，提出用以估計投資組合損失機率的演算法。雖然所提出的演算法經過重要性取樣的技巧後仍無法使得欲估計的事件在所有模擬路徑下都會發生，但數值結果仍顯示所提出的方法估計效率遠遠優於傳統蒙地卡羅法。 / Monte Carlo simulation is a useful tool on portfolio credit risk management. When measuring portfolio credit risk, one should choose an appropriate model to characterize the dependence among all assets. Normal copula is the most widely used mechanism to capture this dependence structure, however, some emperical studies suggest that $t$-copula provides a better fit to market data than normal copula does. In this article, we use extremal depence model proposed by Bassamboo et al. (2008) to construct $t$-copula. We also extend the importance sampling (IS) procedure proposed by Chiang et al. (2007) to evaluate basket credit default swaps (BDS) with extremal dependence and introduce a two-step IS algorithm which ensures credit events always take place for every simulation path. Numerical results show that the proposed methods achieve variance reduction. If the model has lower degree of freedom, or the portfolio size is larger, the two-step IS method is more efficient. Following the same idea, we also propose algorithms to estimate the probability of portfolio losses. Althought the desired events may not occur for some simulations, even if the IS technique is applied, numerical results still show that the proposed method is much better than crude Monte Carlo.