of 10 million) of random variables $X\sim N(0, 1)$, the expected skewness and kurtosis,Īs we will convince ourselves in a second, this is not the case for our IBM return sample. Statistically, VaR is defined as one of the lower quantiles of the distribution of returns that is only exceeded by a certain probability (e.g. As a test you may verify that for a huge sample (e.g. In Python, a correct computation of the sample skewness and kurtosis is possible thanks to the scipy.stats module. Where, in particular, the bottom one reveals five days where the stock experienced 5% or higher daily loss. Where we cover the most recent past 5 years of trading of IBM at NASDAQ. # Fetching Yahoo! Finance for IBM stock dataĭata = web.DataReader("IBM", data_source='yahoo',Ĭp = np.array(data.values) # daily adj-close prices A standard deviation, mean vector, or covariance matrix are all examples of parameters. In the parametric method, the asset returns are assumed to follow a known probability distribution whilst the Monte Carlo method assumes that asset returns are. It may take on real, vector, or matrix values. Formally, a parameter is a function that is applied to a random vector’s probability distribution. # (c) 2015 Pawel Lachowicz, įrom scipy.stats import skew, kurtosis, kurtosistest Parameters describe random vectors much as we might use height or age to describe a person. # The case study of VaR at the high significance levels This function won't return a meaningful value until after a simulation has been run. To obtain the cumulative probability to the left of x 14, for the most recent simulation, use the function RiskXtoP (AB123,14). # Student t Distributed Linear Value-at-Risk Suppose you have an RISK input or output, or even just an Excel formula, in cell AB123. Probability density of the exchange rate. As the first task we have to verify before heading further down this road will be the sample skewness and kurtosis. 5 VaR is defined on the cumulative probability function as a 5 quantile. Within the following case study we will make of use of Yahoo! Finance data provider in order to fetch the price-series of the IBM stock (adjusted close) and transform it into return-series. Normal and Student t VaR for IBM Daily Returns A random vector X can be thought of as an n -dimensional vector of random variables Xi all defined on the same sample space. You should be familiar with random variables and random vectors. If E is an event, we denote its probability Pr ( E ). $$ Having that, we are ready to use this knowledge in practice and to understand, based on data analysis, the best regimes where one should (and should not) apply the Student t VaR as a more “correct” measure of risk.Ģ. We assume familiarity with basic notation and concepts from probability. Within a standard approach, it is computed based on the analytical formula: The problem arises if we investigate a Value-at-Risk (VaR) measure. A rough approximation of the asset return distribution by the Normal distribution becomes often an evident exaggeration or misinterpretations of the facts. One of the most underestimated feature of the financial asset distributions is their kurtosis.
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