以蒙地卡羅模擬法計算具厚尾性質投資組合之風險值

Abstract

風險值(Value-at-Risk，簡稱VaR)目前已成為目前已成為實務上衡量市場風險的重要風險控管工具。VaR代表的意義是：以一金額數字來表達投資組合在特定持有期間內，某一機率百分比(信賴水準)下，受相關市場價格變動時(如：利率、匯率和股價等)，最大可能損失金額為多少。但近年來，隨著市場上信用衍生性商品的交易日漸蓬勃，以及不斷有重大公司破產案發生，有關信用風險領域的議題已經越來越重要，如何有效評估信用風險的影響程度及信用風險控管，將是一個重要議題。 由於VaR具有簡單明瞭的特性，加上目前的廣泛應用，本文同樣把VaR應用在衡量信用風險上，但是具有信用風險之金融資產報酬率或價值的機率分配，有著厚尾(Fat-tailed)性質，即極端值(Extreme Value)實際發生的機率會遠高於常態分配假設下的機率，而且更難以特定的統計分配模型來配置(fit)，所以常態分配假設下所求出的VaR，不適用於含信用風險的投資組合。因此本文參考Nomura在2005年提出的Technical Report，建立出一套方法，使其能以今日市場上既有的資訊，計算一個同時具有市場風險和信用風險投資組合之VaR，並且本文不給定一個特定的統計分配，而是以蒙地卡羅模擬法去估計投資組合的價值的機率分配，進而求出投資組合的VaR。由於含市場風險與信用風險的投資組合，其價值的機率分配具厚尾(Fat-tailed)性質時，稱其VaR為「厚尾風險值」(Fat-tailed VaR，以下簡稱為：FTV)，以FTV代表依據本文方法所建立的VaR。對於同樣成份的投資組合，更可以藉由將不同的權重配置在相同模擬路徑上，進而求出效率前緣(Efficient Frontier)曲線，在曲線上的投資組合，會在相同風險之下，產生最大的報酬，或是在相同報酬下，面臨最低風險。 本文建立的FTV有下列特色：同時考慮到市場風險與信用風險對投資組合的影響；不以特定機率分配模型去配置投資組合價值的機率分配，避免因模型上的選取不當，造成低估或高估投資組合的風險；應用的範圍更為廣泛，例如：可應用FTV求算效率前緣。因此由本文方法計算出的FTV，不僅可作為風險控管上的指標，更可以將此應用在投資組合最佳化上。

The risk management technique known as VaR(Value-at-Risk) has recently become an important tool for measuring the market risk of financial and commodity derivative instruments, and other financial instruments. VaR models measure the loss of a portfolio or an asset that will be exceeded with a specified probability over a specified time horizon. Moreover, the issues of the credit risk has been getting more and more important recently, so what we concern about now is to measure not only the market risk but also the credit risk effectively. In this thesis, we also apply VaR for measuring the credit risk due to its simple property. However, the distribution of the value of a portfolio involved with the credit risk is fat-tailed i.e. the occurrences of the extreme values are more often than normal distribution, so we can’t calculate VaR under the normal distribution. Hence, we refer to a technical report proposed by Nomura in 2005 to establish a method to calculate VaR of a portfolio which is involved with the market risk and credit risk. We use the Monte Carlo simulation to estimate the distribution of the value of a portfolio in order to calculate VaR of the portfolio. We call VaR calculated by the method in this thesis as “FTV” for fat-tailed VaR because of the fat-tailed property of the portfolio. Furthermore, we also apply FTV to construct the efficient frontier. There are some features of FTV we construct here. First, we consider the effects of the market risk and credit risk at the same time. Second, there is no specified statistical distribution used to fit the distribution of the value of a portfolio in our method, so we don’t overestimate or underestimate the risk from the inappropriate choice of distribution models. Finally, we can apply FTV to many areas, such as the risk management, optimization of the portfolio and asset allocation.