JZ Assignment Page 1 of 5

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1 JZ Assignment Page 1 of 5 Data: This paper retrieved data by using WinORSai. The data used in this paper include: BAC (Bank of America) daily normal returns and log returns (in %) ( ) ^GSPC (Standard and Poor s 500 Index daily normal and log returns (in %) ( ) BAC monthly average normal returns and log returns (in %) as well as BAC and ^GSPC monthly sum normal returns ( ) 3 Month Treasury Bill interest rate in secondary market ( ) Objective/method: To understand the normal returns and log returns. To learn how to use WinORSai to get the descriptive statistics and how to analyze these statistical results. To study how to use T-test to decide the characteristics of the stock returns distribution. To learn the JB test and use JB test to determine if the stock returns is normally distributed. To understand the excess returns and determine if the excess returns are positive and negative based on the hypothesis testing. 1a: Table 1: Descriptive Statistics for the Daily Simple Returns (%) of BAC and ^GSPC ( ) Stats BAC Normal GSPC Normal Obs<>Miss Sum Mean S.D SE of Mean Variance t-value p-value Skewness-A Kurtosis Min Value Max Value

2 Normal Returns (%) JZ Assignment Page 2 of 5 1b Table 2: Descriptive Statistics for the Daily Log Returns (%) of BAC and ^GSPC ( ) Stats BAC Log GSPC Log Obs<>Miss Sum Mean S.D SE of Mean Variance t-value p-value Skewness-A Kurtosis Min Value Max Value a H 0: μ = 0 Based on table 2, p = >>0.05, we accept H 0. The mean of the log returns of BAC stock is NOT statistically different from zero. 1b H 0: μ = 0 Based on table 2, p = >>0.05, we accept H 0. The mean of the normal returns of S&P 500 is NOT statistically different from zero. 1c. Line Chart of the BAC Stock Normal Returns in BAC (Bank of America) Stock Normal Returns in Time

3 Log Returns (%) JZ Assignment Page 3 of 5 Line Chare of the BAC Stock Log Returns in BAC (Bank of America) Stock Log Returns in Time a. H 0: S(r) = 0 (S(r) is the skewness of the stock log return.) H 1: S(r) 0 ˆ( ) SrT 6/ = / 697 = This gives a p value about (derived from excel), which is >>0.05. So, we accept H 0. The daily log returns of BAC stock are NOT significantly skewed at the 5% level. 2b. H 0: K(r) -3 = 0 (K(r)-3 is the excess kurtosis of the stock log return.) H 1: K(r) -3 0 Kr ˆ ( ) / T 24 / 697 = This gives a p value about (derived from excel), which is << So, we reject H 0. The daily log return of BAC stock has significantly excess kurtosis at 5% level. In practice, this means the log returns of BAC stock has heavy tails, implying that the distribution puts more mass on the tails than a normal distribution does. 2c. H 0: BAC stock log returns is normally distributed H 1: BAC stock log returns is not normally distributed Sˆ ( r) ( Kˆ ( r) 3) ( 0.132) ( ) JB= 6 / T 24 / T 6 / / = This gives a p value about (derived from excel), which is << So, we reject H 0. The daily log return of BAC stock is NOT normally distributed at 5% level.

4 JZ Assignment Page 4 of 5 4. Table 4: Descriptive Statistics for Monthly Sum Normal Excess Returns for BAC and S&P500 ( ) Stats Excess returns for BAC Excess returns for ^GSPC Obs<>Miss Sum Mean S.D SE of Mean Variance t-value p-value Skewness-A Kurtosis Min Value Max Value a. For BAC excess returns: H 0: μ = 0 Based on table 4, p = >>0.05, we accept H 0. The mean of the BAC excess returns is NOT statistically different from zero. For S&P500 excess returns: H 0: μ = 0 Based on table 2, p = >>0.05, we accept H 0. That is to say, the mean of the S&P500 excess returns is NOT statistically different from zero. 4b. Based on the table, the mean is 0.003, which is positive. H 0: μ>0 H 1: μ 0 ˆ S / n = / 261 = According to one tail T-distribution testing, P(t>0.195) = >> 0.05, so we accept H 0. This means that the mean of the S&P500 is positive. 4c. H 0: S(r) = 0 (or the monthly excess returns of the stock is symmetric.) H 1: S(r) 0 (or the monthly excess returns of the stock is not symmetric.)

5 JZ Assignment Page 5 of 5 ˆ( ) SrT 6/ = / 261 = This gives a p value about , which is << So, we reject H 0. The monthly excess returns of the BAC stock are significantly skewed at the 5% level. 4d. H 0: K(r) -3 = 0 (or the stock monthly excess returns do not have heavy tails.) H 1: K(r) -3 0 (or the stock monthly excess returns do have heavy tails.) Kr ˆ ( ) = / T 24 / 261 This gives a p value about , which is << So, we reject H 0. The monthly excess return of BAC stock has significantly excess kurtosis at 5% level. In practice, this means the log returns of BAC stock has heavy tails, implying that the distribution puts more mass on the tails than a normal distribution does.