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# 金融代写|金融衍生品代写Financial Derivatives代考|FM360 STRESS TESTING AND BACK TESTING

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## 金融代写|金融衍生品代写Financial Derivatives代考|STRESS TESTING AND BACK TESTING

In addition to calculating a VaR, many companies carry out what is known as a stress test of their portfolio. Stress testing involves estimating how the portfolio would have performed under some of the most extreme market moves seen in the last 10 to 20 years.
For example, to test the impact of an extreme movement in U.S. equity prices, a company might set the percentage changes in all market variables equal to those on October 19, 1987 (when the S\&P 500 moved by 22.3 standard deviations). If this is considered to be too extreme, the company might choose January 8,1988 (when the S\&P 500 moved by 6.8 standard deviations). To test the effect of extreme movements in U.K. interest rates, the company might set the percentage changes in all market variables equal to those on April 10, 1992 (when 10-year bond yields moved by 7.7 standard deviations).

Stress testing can be considered as a way of taking into account extreme events that do occur from time to time but that are virtually impossible according to the probability distributions assumed for market variables. A five-standard-deviation daily move in a market variable is one such extreme event. Under the assumption of a normal distribution, it happens about once every 7,000 years, but, in practice, it is not uncommon to see a five-standard-deviation daily move once or twice every 10 years. Whatever the method used for calculating VaR, an important reality check is back testing. It involves testing how well the VaR estimates would have performed in the past. Suppose that we are calculating a 1-day $99 \%$ VaR. Back testing would involve looking at how often the loss in a day exceeded the 1-day $99 \%$ VaR calculated for that day. If this happened on about $1 \%$ of the days, we can feel reasonably comfortable with the methodology for calculating VaR. If it happened on, say, $7 \%$ of days, the methodology is suspect.

## 金融代写|金融衍生品代写Financial Derivatives代考|PRINCIPAL COMPONENTS ANALYSIS

One approach to handling the risk arising from groups of highly correlated market variables isprincipal components analysis. This takes historical data on movements in the market variablesand attempts to define a set of components or factors that explain the movements. The approach is best illustrated with an example. The market variables we will consider are tenU.S. Treasury rates with maturities between three months and 30 years.results produced by Frye for these market variables using 1,543 daily observations between 1989 and 1995.u The first column in Table 16.3 shows the maturities of the rates that were considered. The remaining ten columns in the table show the ten factors (or principal components) describing the rate moves. The first factor, shown in the column labeled PCI, corresponds to a roughly parallel shift in the yield curve. When we have one unit of that factor, the three-month rate increases by 0.21 basis points, the six-month rate increases by 0.26 basis points, and so on. The second factor is shown in the column labeled PC2. It corresponds to a “twist” or “steepening” of the yield curve. Rates between 3 months and 2 years move in one direction; rates between 3 years and 30 years move in the other direction. The third factor corresponds to a “bowing” of the yield curve. Rates at the short end and long end of the yield curve move in one direction; rates in the middle move in the other direction. The interest rate move for a particular factor is known as factorloading. In our example, the first factor’s loading for the three-month rate is 0.21 .15 Because there are ten rates and ten factors, the interest rate changes observed on any given day can always be expressed as a linear sum of the factors by solving a set of ten simultaneous equations. The quantity of a particular factor in the interest rate changes on a particular day is known as the factor score for that day.

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。