Assignment 2 - Report

# Assignment 2 - Report - gure QQplot of the Sample...

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Figure : Log Price Changes gure : QQplot of the Sample Data versus Standard Normal Measuring Price Risk for an Equity Portfolio using Single Factor Model Value-at-Risk Estimation for iid Multivariate and Univariate Representation Introduction A portfolio composed of 10 companies – Altria, Coco-Cola, Eastman Kodak, Hewlett Packard, Intel Corporation, Microsoft, IBM, McDonald’s, Wal-Mart and Walt Disney - has an initial value of \$10,000,000. The daily price per share and the index value (SP500 and DOW) has been provided for the period January 2002 – December 2008. The objective is to estimate the value-at-risk (VaR) for a 10-day horizon assuming an iid multivariate normal representation of risk factors and assuming a single factor model and iid univariate normal representation of risk factors at a 99 percent confidence level. The Portfolio The daily price per share of each company and the index was used to compute the biweekly log changes in the price. This gives a better picture of the price movements and also makes it easier to compute the VaR for the 10-day horizon. The log price changes for the companies and the index have been presented in Figure 1 . The plots seem to indicate that neither the companies nor the index prices have a normal distribution. This fact can be further established from Table 1 which gives the distribution statistics for 176 data points for all the companies in the portfolio as well as the index. The table also provides two statistical tests to check the normality of the distribution – t-test and the Jarque-Berra (JB) test. The t-test statistic value of zero indicates that we accept the null hypothesis that the sample data is normally distributed with a mean 0 and an unknown variance at the 5 percent significance level. The JB test is a two-sided goodness-of-fit test suitable when a fully-specified null distribution is unknown and its parameters must be estimated. The JB test statistic of 1 seems to reject the null hypothesis that the sample data is normally distributed at the 5 percent significance level. A normal distribution has a skewness of 0 and a kurtosis of 3. A look at the skewness and kurtosis data suggests that most of the companies in the sample data are not normally distributed except for Walmart and to a slight extent Eastman Kodak. Hence, from the plots and the above statistical tests we can conclude that the sample data is not normally distributed.

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-1 -0.5 0 MO log price change -0.1 0 0.1 KO log price change -0.2 -0.1 0 0.1 EK log price change -0.2 -0.1 0 0.1 HWP log price change -0.3 -0.2 -0.1 0 0.1 INTC log price change -0.6 -0.4 -0.2 0 0.2 MSFT log price change -0.2 -0.1 0 0.1 IBM log price change -0.2 -0.1 0 0.1 MCD log price change -0.1 0 0.1 WMT log price change -0.2 -0.1 0 0.1 DIS log price change -0.15 -0.1 -0.05 0 0.05 SP500 log price change -0.15 -0.1 -0.05 0 0.05 DOW log price change
Table : Distributional Statistics MO KO EK HWP INTC MSFT IBM MCD WMT DIS SP500 DOW Max 0.29 0.11 0.17 0.20 0.20 0.21 0.19 0.16 0.11 0.20 0.08 0.07 Mean -0.01 0.00 -0.01 0.00 0.00 -0.01 0.00 0.00 0.00 0.00 0.00 0.00 Min -1.17 -0.16 -0.29 -0.24 -0.34 -0.68 -0.21 -0.21 -0.11 -0.21 -0.16 -0.15 Skew -8.43 -1.03 -0.91 -0.32 -0.96 -4.92 -0.30 -0.71 -0.12 -0.45 -1.10 -0.98 Kurt 97.21 5.93 4.92 4.75 6.29 49.27 6.39 5.70 3.20 5.36 7.09 6.44 N 176 176 176 176 176 176 176 176 176 176 176 176 T-test 0 0 0 0 0 0 0 0 0 0 0 0 JB_Test 1 1 1 1 1 1 1 1 0 1 1 1

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