Collecting, Manipulating, And Preparing Data For Statistical Inference

The report discusses about the volatility and risk of stock market, return and market return. Usually, it is observed that stock market is volatile and unsteady (Berenson et al. 2012). Therefore, it is hard for evaluating the market performance. The report focuses to indicate the method of evaluating the company’s Price indexes through its market price movements. The price indexes for Boeing and International Business Machines (IBM) has been chosen for demonstrating (Freed, Bergquist and Jones 2014). However, the historical price movements of the individual price indexes from 1^st December 2010 and 31^st May 2016 cannot depict the appropriate outputs. Therefore, the historical index prices of S&P500 index and the 10 years’ US Treasury Bill are involved in the evaluation method for computing the effective outcomes. S&P500 index represents the summarisation of the market and the 10 years’ US Treasury Bill refers the risk-free return of the market.

The method of price index evaluation is divided in some parts. They are the compared and their market returns are calculated in the first part. In the next part, hypotheses are tested and CAPM is calculated using linear regression model. The whole evaluation is incorporated based on Capital Asset Pricing Model. 

The movement of the price indexes for a defined time-period depicts trends of price indexes. The first line chart involves all the three types of trend lines of price indexes. The second, third and fourth line charts involve the line charts individually. The price indexes of IBM and BA in the line charts are shown below: 

It could be inferred from the above line charts that the price indexes of both S&P500 and BA have increased from 01/12/2010 to 31/05/2016. The IBM price index has increased and then decreased within this period. It has better stationary trend in case of IBM price index than BA price index.

Summary Statistics:

The average return of Boeing (BA) is greater than average returns of IBM (1.0139883>0.071483586). The risk is determined by standard deviation of returns of close rates of price index. The risk in terms of standard deviation shows that Boeing return is more volatile than IBM return (5.9876504>5.052276003).

The risk is relatively greater for Boeing price return for its greater variability in terms of standard deviation.    

Jerque-Bera test is carried out for testing the normality of price indexes that are Boeing and IBM.

The Jerque-Bera test statistic (JB) is given as-

JB = n *

Firstly, the JB test statistics of both the price indexes are calculated. For BA price return and IBM price return, they are 16.7353157 and 15.0915299. Then applying significant test statistic, we have tested Chi-square tests at 5% level of significance (χ2 (0.05, 2) = 5.99). For both one and two-tail Chi-square tests, Boeing and IBM price returns failed to attain normality. Hence, none of the price returns is normally distributed at 95% confidence limit.

A one-sample t-test determines whether the average price return of Boeing Close return (BA) is at least 3%.   The t-statistic is - . The t-statistic is 1.324921544. At 5% level of significance, we reject the null hypothesis of average price return greater than or equal to 0.03 as T_0.05 < T_cric.

Therefore, the average price return of Boeing is not at least 3%.

The riskiness of returns of two price returns could be more effectively compared by F-test of two samples variances. The F-test for comparing the riskiness of the price returns of IBM and GE are conducted here.

Hypotheses:

Null hypothesis (H₀): σ₁² = σ₂²

Alternative hypothesis (H_A): σ₁² ≠ σ₂²

The F value for two-tail test is computed as F = F_{1-α/2, N1-1, N2-1}

Here, α=0.05, N₁-1=64 and N₂-1=64.

The risk associated with each of the two price returns is compared with the help of F-statistic. The calculated F-statistics (F = is 1.404554937.

For Boeing and IBM price returns, p-value of the F-statistic is 0.088449703. It is greater than 0.05. The null hypothesis is accepted at 5% level of significance.

Hence, it could be depicted that level of volatility of the two price returns for the given period are almost equal to each other (Groebner et al. 2008).

The average return is indicated by the mean of returns of the price returns. Hence, for comparing the average return of Boeing (BA) and IBM price returns, two sample z-test (for unequal samples) and two sample t-test (for equal samples) can be conducted on the calculated returns of the two price returns.

Hypotheses:

Null hypothesis (H0): μ_BA = μ_IBM

      Alternative hypothesis (H_A): μ_BA ≠ μ_IBM

The z-statistic is given as z and t-statistic is given as .

Z-test of equality of means of two samples:

For comparing the average returns of each of the two investing price returns, a z-test is applied. The variances are known for each of the price returns. The calculated z-statistic is 0.9699. The p-value for two-tail z-statistic is 0.332 (>0.05). Therefore, we can reject the null hypothesis of equality of averages of returns of two price returns at 5% level of significance.

Two-sample t-test of equality of means for unequal variances:

The t-test assuming equal variances of BA and IBM price returns gives the t-statistic 0.96991968. The p-value of the two-tail t-test is found to be 0.333974621. The level of significance is 5%, which is lesser than calculated p-value. Therefore, we cannot reject the null hypothesis of equality of averages of both the price returns.

Inference:

According to the price return averages and price return standard deviations (risk), an equality is established. Hence, we cannot draw firm decision to choose any one price returns between BA and IBM. Hence, we further proceed with both of them. Next, we are willing to excess price return, excess market return and CAPM of both the price returns. With the help of these, we can find the volatility of both the price returns. The preferable price return would be distinguished after that.

The Capital Asset Pricing Model (CAPM) is known as CAPM, which is one of the fundamental models in the financial field. The CAPM elaborates variability in the rate of return (r_t) as a function of the rate of return on a market portfolio (r_M,t) consisting all publicly traded price returns. Usually, the rate of return of any price return can be measured using opportunity cost that is the return on a risk free asset (r_f,t). The difference between the return and risk free rate is known as “risk premium” as it is the reward or punishment for performing a risky investment (Peirson et al. 2014). In accordance to CAPM, the risk premium on a security (r_t –r_f,t) is proportional to the risk premium on the market portfolio (r_M,t – r_f,t). According to CAPM,

(r_t –r_f,t) = β_M*(r_M,t – r_f,t)   ……………….(1)

Equation (1) is called economic model as it describes association between excess price returns and excess market return.

The CAPM beta is crucial from the viewpoints of investors as it discloses the volatility of market price returns. Particularly, the bête (slope) measures the sensitivity of variation of given return of security in the whole price market. Value of beta defines whether the price return is a defensive, a neutral price index or an aggressive price index. Including an intercept (β₀) and an error term (u_t) in the model, we have a simple linear regression model –

(r_t - r_f,t) = β₀ + β_M (r_M,t - r_f,t) +u_t    ………………..(2)

Boeing (BA) Excess return:

IBM Excess return: 

The calculated β-value for Boeing excess return and Excess market return is 1.11931665. The calculated β-value for IBM excess return and Excess market return is 0.720411483. The calculated β-values define that BA price indexes are 111.93% less volatile than the market, whereas the volatility level of IBM compared to the market is 72.04%. Therefore, it can be stated that Boeing (BA) is highly volatile than IBM. Therefore, Boeing (BA) is considered to be more profitable than IBM price returns.

The linear regression tables describe that the values of R² of BA and IBM are 0.40482754 and 0.237985555. The R² indicates the relationship of the dependent variable with the independent variable. Hence, from the values of multiple R² of the two price returns it could be stated that Boeing (BA) excess return (40.48%) is more associated than the association of IBM (23.80%).

Confidence Interval of IBM Price Return:

1. For Boeing (BA) price return, slope (β₁) = 1.11931665, Standard Error = 170989356, d.f. = 64, t-value = 6.546119. Hence, the 95% confidence interval for the slope coefficient would be (0.777621689, 1.461011607).
2. For IBM price return, slope (β₁) = 0.720411483, Standard Error = 162411454, d.f. = 64, t-value = 4.435718. Hence, the 95% confidence interval for the slope coefficient would be (0.3958581, 1.04496487). 

The testing of aggressiveness of the excess price returns needs the following hypothesis:

Null hypothesis (H₀): β₁ = 1

Alternative hypothesis (H₁): β₁ < 1

For BA price returns, β₁ is 1.11931665 along with the standard error (SE) 0.170989356. The “residual degrees of freedom” is 63 and calculated p-value is 0.0. Hence, t = β₁/ SE = 6.546119.

For IBM price indexes, β₁ is 0.720411483 along with the standard error (SE) 0.162411454. The “residual degrees of freedom” is 63 and calculated p-value is 0.0. Hence, t = β₁/ SE = 4.435718.

For both the excess price returns, the p-values are positive t-value and equal degrees of freedom 64. The 95% confidence intervals for beta values of both BA and IBM price returns are (0.777621689, 1.461011607) and (0.3958581, 1.04496487). The confidence intervals near to 0 refers more neutral nature for price excess return. The confidence intervals of t-statistics indicate that IBM price return is more neutral (Moffett, Stonehill and Eiteman 2014).

IBM Ecess Price return residual plot: 

The method of ordinary least squares (OLS) helps to establish the normality with diagram. The error terms in the model are graphically shown in normal probability plot. It shows that the error terms are not following normal distributions for IBM price indexes. The distributions of residual values are not symmetric for both the market return values.  

Besides, we perform a Jarque-Bera test for examining the normality of the residual values. The JB statistic of IBM (18.21) refers that normality of residual values of the regression is rejected at 5% level of significance. 

Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Basic business statistics: Concepts and applications. Pearson Higher Education AU.

Freed, N., Bergquist, T., & Jones, S. (2014). Understanding business statistics. John Wiley & Sons.

Groebner, D.F., Shannon, P.W., Fry, P.C. and Smith, K.D., 2008. Business statistics. Pearson Education.

Moffett, M. H., Stonehill, A. I., & Eiteman, D. K. (2014). Fundamentals of multinational finance. Pearson.

Peirson, G., Brown, R., Easton, S., & Howard, P. (2014). Business finance. McGraw-Hill Education Australia.