Neste trabalho são definidos, discutidos e estimados o Valor em Risco e o Expected Shortfall. Estas são medidas de Risco Financeiro de Mercado muito utilizadas por empresas e investidores para o gerenciamento do risco, aos quais podem estar expostos. O objetivo foi apresentar e utilizar vários métodos e modelos para a estimação dessas medidas e estabelecer qual o modelo mais adequado dentro de determinados cenários. / In this work Value at Risk and Expected Shortfall are defined, discussed and estimated . These are measures heavily used in Financial Market Risk, in particular by companies and investors to manage risk, which they may be exposed. The aim is to present and use several methods and models for estimating those measures and to establish which model is most appropriate in certain scenarios.

Este trabalho consiste em um estudo comparativo de diversos modelos para cálculo do Valor em Risco de um portifólio. São comparados modelos que consideram a série univariada de log-retornos do portifólio versus mo- delos multivariados, que consideram as séries de log-retornos de cada ativo que compõe o portifólio e suas correlações condicionais. Além disso, são testados modelo propostos recentemente, que possuem pouca literatura a respeito, como o PS-GARCH e o VARMA-GARCH. Também propomos um novo modelo, que utiliza o resultado acumulado do portifólio nos últimos dias como variável exógena. Os diferentes modelos são avaliados em termos de sua adequação às exigëncias do Acordo de Basileia e seu impacto financeiro, em um período que inclui épocas de alta volatilidade. De forma geral, não foram notadas grandes diferenças de performance entre modelos univariados e multivariados. Os modelos mais complexos mostraram-se mais eficientes, produzindo resultados satisfatórios inclusive em tempos de crise. / The present work consists of a comparative study of several portfolio Value-at-Risk models. Univariate models, which consider only the portfolio log-returns series, are compared to multivariate models, which consider the log-returns series of each asset individually and their conditional correlations. Additionally, recently proposed models such as PS-GARCH and VARMA-GARCH are tested. We also propose a new model that uses past cumulative returns as exogenous variables. All models are evaluated in terms of their compliance to Basel Accord and financial impact, in period that includes high volatility times. In general, univariate and multivariate models performed similarly. More complex models yielded more accurate results, with satisfactory performance including in crisis periods.

行政院會於民國八十八年十二月十六日通過保險法修正草案，修正草案中針對強化保險業之監理機制與增進保戶大眾之權益係以強化其資本適足性為其修法目標，所採之方法即建立風險基礎資本額制（Risk-based Capital, RBC）。而保險法修正案於民國九十年六月二十六日業已經立法院三讀通過，基於保險法相關條文規定，RBC制度將於民國九十二年中實施。 另一方面，美國經濟、社會及投資環境和台灣不盡相同，若將此制度直接或稍加修改即套用於台灣，將可能造成不切實際與誤導的作用，其結果不僅可能無法有效規範及避免保險公司失卻清償能力，亦可能因而造成龐大的社會成本，反而和當初建立RBC制度之原意背道而馳。因此，本論文即依循台灣產險業之產業特性制訂一套合宜之產險RBC制度，其中包括各個適當之風險項目及所屬之風險係數。 本論文在資產風險部分結合風險值（Value at risk, VaR）來計算資產之風險係數；在準備金風險以及自留保費收入風險則依照美國RBC制度之原始公式重新計算得來，惟準備金風險部分實因資料取得限制無法順利求出，為求模型完整性此部分本論文以財政部草案取代之；而於自留保費收入風險方面是採險種別及公司別。 研究結果發現：台灣產險的風險係數確實和美國產險的風險係數是有相當的差異，並且須根據台灣產業的經驗及配合我國的社會、經濟、投資環境並經由實際的運算才能得到適切的風險係數；而以論文所建立之RBC模型試算於各公司之風險基礎資本比則多有偏低之情形。 / Legislative Yuan has pass the draft of Insurance Law on June 26, 2001. In order to strengthen insurance regulation mechanism and to protect the insureds' benefit, the Risk-based Capital will be implemented in Taiwan Insurance market in 2003. On the other way, the economic environment and investment markets in United State are different from those in Taiwan. If we directly imitate their RBC system in Taiwan, the outcome would be impractical. It not only can't regulate the insurers effectively, but also may cause huge social cost. Therefore, the purpose of the thesis is to establish a suitable risk items and suitable risk factors for Property-Liability insurance in Taiwan by our own empirical data. This study finds that risk factors are significant different between Taiwan and American for Property-Liability insurance industry. The risk factors of the RBC system in Taiwan must depend on our own empirical data. I used the RBC model built in the thesis to test every Property-Liability company in Taiwan, and found that calculated Risk-based Capital ratios were relatively low.

由於許多資產報酬率的分配呈現厚尾的現象，因此，本文探討將最低報酬要求限制條件加入傳統的平均數╱變異數模型中，考慮在分配已知的情形下，假設資產報酬率的分配為t分配及常態分配，來求取最適的資產配置；在分配未知的情形下，利用古典Bootstrap法、移動區塊Bootstrap法及定態Bootstrap法的抽樣方法來模擬資產報酬率的分配形式，並利用模擬的資產報酬率分配求出最適的資產配置。 同時，本文亦探討資產配置在風險管理上的運用，當分配已知時，若對分配參數的估計正確，則使用的最低要求報酬率就是此資產配置的涉險值，反之，若對參數的估計錯誤時，會對資產配置產生很大的影響及風險管理上的不正確；當分配未知時，利用模擬方法來產生分配，則使用的最低要求報酬率可看成是此資產配置的涉險值。 實證部分選取資料分成本國及全球，研究發現對於何種分配或模擬方法的資產配置績效最好？沒有一定的結論。其原因是各種分配或模擬方法皆必須視資料的性質而定，因此，本論文的貢獻僅在建議使用厚尾分配及利用模擬方法，來符合資產報酬率呈現厚尾的現象，並利用此分配，以期在考慮最低報酬要求限制條件下的資產配置更為精確。 / The distributions of many asset returns tend to be fat-tail. This paper attempts to add the shortfall constraint in Mean-Variance Analysis. When the distribution is known, we find the optimal asset allocation under student-t distribution and normal distribution. On the other hand, we use Classical Bootstrap, Moving Block Bootstrap, and Stationary Bootstrap to stimulate the distribution of asset return, and to obtain the optimal asset allocation. We also examine the risk management of asset allocation. When we use the correct estimators of parameters under the known distribution, the threshold in shortfall constraint is the value-at-risk in asset allocation. Otherwise, if using the wrong estimators, we get the incorrect asset allocation and the improper risk management. When the distribution is unknown, using simulation to generate the distribution, the value-at-risk is the threshold. The empirical study is conducted in two parts, domestic and global asset allocation. The results cannot point out which distributions and simulations are suitable. They depend on the data’s property. The contribution of this paper is to introduce some methods to fit the fat-tail behavior of asset return in asset allocation.

確保保險公司的清償能力一直是保險監理的重心。在所有施行的保險清償監理工具中，風險基礎資本（Risk-Based Capital, RBC）是目前為止最先進的代表。然銀行監理機關已經推薦涉險值（Value at Risk, VaR）系統為資本適足要求的工具，因此涉險值有很大的潛力成為下一代的保險資本適足要求工具，雖然尚未施行。由於保險監理的重要性以及RBC和VaR在其中扮演重要的角色，兩者相對上的精確性是我們所感興趣的。 本篇論文的目的是實際去比較RBC及VaR在破產預測上的相對精確性。我們以美國1995到1998年產險公司的實際清償記錄，用型1及型2錯誤檢視RBC及VaR的破產預測能力。RBC的數據直接從產險公司報給NAIC的年報上就可取得，而VaR的數據來自於我們所建立的現金流量模擬模型。既然RBC的數據是實際的數據，而VaR的估計值也是基於公司實際的財務數據而來，我們能以實例展現VaR相較於RBC的財務預警能力。 我們的結果顯示RBC沒有任何財務預警能力，換句話說，沒有一個破產公司的RBC值小於0.7（監理機關可以根據這個值關掉公司）。另一方面，VaR有較好的財務預警能力，但是它同時也會使許多財務健全的公司必須接受許多沒有必要的檢查。我們VaR模型的整體正確分類能力只比隨意分類稍微好一些。 雖然結果並不如原先預期的好，我們仍然對VaR成為保險監理工具抱持樂觀的態度，因為它是目前為止最嚴密也最先進的風險管理工具。我們認為這些結果可以藉由修正不適當的假設後獲得改善，未來研究可以先朝這個方向努力。 / Assuring insurance company solvency has always been the focal point of insurance regulation. Among the employed solvency regulation methods, RBC represents the currently state-of-the-art capital adequacy requirement. Bank regulators already advocated the use of VaR systems in capital adequacy requirements. Value at risk thus has great potential to be the next-generation capital adequacy regulation, although not implemented yet. Because of the importance of solvency regulation as well as the key role played in that regulation by RBC and VaR, the relative accuracy of RBC and VaR is of great interest. The purpose of this research is to empirically compare the relative effectiveness of RBC and VaR in predicting insolvency. Through the solvency record of property-casualty insurers in the United States from 1995 to 1998, we examine the Type I and Type II error of VaR and RBC in predicting insolvency. The RBC figures are readily available from the annual statement since 1994 and the VaR values come from a simulation model that we build up. Since the RBC figures are the “real” numbers and the VaR estimates also base on the companies’ real financial positions, our research will demonstrate how VaR is compared to RBC in early warning for real cases. Our result shows that RBC doesn’t have any prediction power. In other words, none of the bankrupt insurers has a RBC ratio lesser than 0.7, the threshold according to which the regulator can seize the company. On the other hand, VaR has good early warning ability, but also leads the regulator to take too much unnecessary actions on solvent companies. The overall ability of correct classification of our model is just a little stronger than arbitrary classification. Although our results are not as good as we expect, we are still optimistic about the use of VaR, the currently most comprehensive and advanced approach of risk management, as an insurance solvency regulation tool. We attribute the unsatisfactory outcome to some assumptions that may be inappropriate. Further researches can move toward this aspect.

本論文的目的在比較標準法和風險值法（VaR）於外匯部位資本計提數額上的差異。在VaR法方面，本篇採用變異數-共變異數法、歷史模擬法以及極端值法等三種衡量方法，並利用回溯測試（backtest）對三種方法預測風險的能力做一檢測。標準法是指財政部規定的資本計提標準方法。 本篇論文實證結果發現用VaR法所計提的資本數額是依標準法所需計提數額的一半。也就是說依標準法提列會造成過多的資金成本。另外，從安全性的角度觀之，經過回溯測試，發現採取歷史模擬法或極端值法則是值得信賴的資本計提的方法。反之，變異數-共變異數法會有低估的現象。但因計算極端值法所需要的資料過於龐大，建議使用歷史模擬法，如此相對於標準法將可省下可觀的資金成本。 第一章 研究動機與目的…………………………………1 第二章 國內外資本適足的規定…………………………3 第一節 資本適足規定(BIS)的發展……………………3 第二節 台灣相關法令規定……………………………6 第三章 文獻探討……………………………………… 10 第四章 研究方法與模型……………………………… 14 第一節 VaR模型…………………………………… 14 第二節 回溯測試…………………………………… 24 第五章 實證分析……………………………………… 28 第一節 實證資料介紹……………………………… 28 第二節 實證結果…………………………………… 29 第六章 結論…………………………………………… 42 參考文獻……………………………………………………44

指數型基金透過追蹤指數來提供投資人被動管理的投資策略，因而成為保守投資人的熱門投資工具。本論文的目的在於建立一個追蹤指數的同時也能有效控管損失的指數型基金。在此目標下，該基金面臨到的不單是追蹤指數的績效，還有降低資產配置風險的問題。有鑑於此，本論文融合兩種下方風險的概念：指數追蹤的下方偏差(downside absolute deviation)以及條件風險值(conditional value-at-risk, CVaR)。針對兩者間的規避程度分別分配其權重，並以該基金之平均報酬大於指數的平均報酬作為限制條件，經由改寫下方偏差與離散化CVaR後得到一個新的線性規劃模型。本論文以台灣50指數與恆生指數的歷史資料做為實證探討的對象，驗證使用本線性規劃模型所建立之指數型基金的效能。 / Index fund has become popular in these days among the conservative investors since it provides a passive investment strategy. The main purpose of this paper is to create an index fund which can replicate the performance of a broad-based index of stocks and has the ability to control the loss efficiently at the same time. For this purpose, the index fund we build confronts with not only the performance of index tracking, but also lowering the level of the risk of assets allocation. In order to accomplish the goal, we combine two concepts of downside risk: downside absolute deviation and conditional value-at-risk (CVaR). Under the constraint of average portfolio return being greater than average index return, and assign weights according to the degree of evasion to each of the risks, a linear programming model is formulated by rewriting downside absolute deviation and discretizing CVaR. The results obtained from the computational experience on Taiwan 50 index and Hang Seng index are provided for testing the efficiency of this model.

<p>The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.</p>

The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.

The main objective of this thesis is to investigate risk neutral and risk averse approaches to multistage stochastic programming with applications to hydrothermal operation planning problems. The purpose of hydrothermal system operation planning is to define an operation strategy which, for each stage of the planning period, given the system state at the beginning of the stage, produces generation targets for each plant. This problem can be formulated as a large scale multistage stochastic linear programming problem. The energy rationing that took place in Brazil in the period 2001/2002 raised the question of whether a policy that is based on a criterion of minimizing the expected cost (i.e. risk neutral approach) is a valid one when it comes to meet the day-to-day supply requirements and taking into account severe weather conditions that may occur. The risk averse methodology provides a suitable framework to remedy these deficiencies. This thesis attempts to provide a better understanding of the risk averse methodology from the practice perspective and suggests further possible alternatives using robust optimization techniques. The questions investigated and the contributions of this thesis are as follows. First, we suggest a multiplicative autoregressive time series model for the energy inflows that can be embedded into the optimization problem that we investigate. Then, computational aspects related to the stochastic dual dynamic programming (SDDP) algorithm are discussed. We investigate the stopping criteria of the algorithm and provide a framework for assessing the quality of the policy. The SDDP method works reasonably well when the number of state variables is relatively small while the number of stages can be large. However, as the number of state variables increases the convergence of the SDDP algorithm can become very slow. Afterwards, performance improvement techniques of the algorithm are discussed. We suggest a subroutine to eliminate the redundant cutting planes in the future cost functions description which allows a considerable speed up factor. Also, a design using high performance computing techniques is discussed. Moreover, an analysis of the obtained policy is outlined with focus on specific aspects of the long term operation planning problem. In the risk neutral framework, extreme events can occur and might cause considerable social costs. These costs can translate into blackouts or forced rationing similarly to what happened in 2001/2002 crisis. Finally, issues related to variability of the SAA problems and sensitivity to initial conditions are studied. No significant variability of the SAA problems is observed. Second, we analyze the risk averse approach and its application to the hydrothermal operation planning problem. A review of the methodology is suggested and a generic description of the SDDP method for coherent risk measures is presented. A detailed study of the risk averse policy is outlined for the hydrothermal operation planning problem using different risk measures. The adaptive risk averse approach is discussed under two different perspectives: one through the mean-$avr$ and the other through the mean-upper-semideviation risk measures. Computational aspects for the hydrothermal system operation planning problem of the Brazilian interconnected power system are discussed and the contributions of the risk averse methodology when compared to the risk neutral approach are presented. We have seen that the risk averse approach ensures a reduction in the high quantile values of the individual stage costs. This protection comes with an increase of the average policy value - the price of risk aversion. Furthermore, both of the risk averse approaches come with practically no extra computational effort and, similarly to the risk neutral method, there was no significant variability of the SAA problems. Finally, a methodology that combines robust and stochastic programming approaches is investigated. In many situations, such as the operation planning problem, the involved uncertain parameters can be naturally divided into two groups, for one group the robust approach makes sense while for the other the stochastic programming approach is more appropriate. The basic ideas are discussed in the multistage setting and a formulation with the corresponding dynamic programming equations is presented. A variant of the SDDP algorithm for solving this class of problems is suggested. The contributions of this methodology are illustrated with computational experiments of the hydrothermal operation planning problem and a comparison with the risk neutral and risk averse approaches is presented. The worst-case-expectation approach constructs a policy that is less sensitive to unexpected demand increase with a reasonable loss on average when compared to the risk neutral method. Also, we comp are the suggested method with a risk averse approach based on coherent risk measures. On the one hand, the idea behind the risk averse method is to allow a trade off between loss on average and immunity against unexpected extreme scenarios. On the other hand, the worst-case-expectation approach consists in a trade off between a loss on average and immunity against unanticipated demand increase. In some sense, there is a certain equivalence between the policies constructed using each of these methods.