BUS5SBF: Statistics for Business and Finance - Capital Asset Pricing Model

Solution Code: 1AGFE

Question: Statistics

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Statistics Assignment

Assignment Task

Variables and Data Sources:

PS&P = S&P 500 Price Index This is Standard and Poor index of 500 companies and will be used as market portfolio.

https://finance.yahoo.com/quote/SPY/

PIBM = International Business Machines Stock Price A particular stock we are interested in to determine how it behaves in response to market changes. https://finance.yahoo.com/quote/IBM/
PGM = General Electric Stock Price A particular stock we are interested in to determine how it behaves in response to market changes. https://finance.yahoo.com/quote/GE/
rf =Interest rate on 10 year US-Treasury Note This variable will serve as a risk-free interest rate. We will use this variable to compute excess returns of IBM and GE and Market excess returns. https://finance.yahoo.com/quote/%5ETNX/

Downloading the Data Download monthly data for S&P 500 index, International Business Machines (IBM) Stock Price, General Electric (GE) Stock Price, and US TN (10 year) by clicking the above links and choosing Historical Data for all variables covering the period based on the following criterion. Task 1: Comparison of Stock Returns (50 Marks)

Obtain the line charts for S&P, IBM and GE series (prices) and comment on your observations.

Now obtain returns for these three series (ignoring any of the dividends paid) using thetransformation: ( ) [ ]% ln)ln(100 1- - = t t t P P r .
Obtain the summary statistics together with their histograms to further analyse each of thethree returns distributions. 1
Interpret your results. In particular, explain risk and average return relationship.
Based on your statistics, is IBM relatively riskier than the GE stock?

Suppose you are not convinced with your finding about riskiness of the two stocks based on yourestimates for sample standard deviations and, therefore, you want to test hypothesis whether bothreturns are equally volatile ( 2 20 : GE IBM H ? ? = ).Perform an appropriate hypothesis test using 5% significance level and provide all the steps involved.What do you conclude about returns volatility in this case?
Before investing in one of the two stocks based on higher risk, you further want to determine whetherboth stocks have same population average return.

Perform an appropriate hypothesis test given the sample and report your findings. Which stock will you prefer and why?

Draw the scatter plot of each of the two returns series againstreturn. Also, compute the sample covariance and sample correlation between these two stocks and explain your findings.

1 Monthly return for August 2016 will be calculated as r2016 = 100[ln(PAugust2016)-ln(PJuly2016)] % Task 2: Estimation of CAPM and Hypothesis Testing

Now compute excess return on your preferred stock and excess market return by subtracting the 10-year T-Bill rate from both series. That is,

Excess return on preferred :stock rry= - Excess return on market : = Capital Asset Pricing Model (See last page for more details on CAPM taken from The Basics of Financial Econometrics by Fabozzi et al. (2014)) Interpret the estimated coefficients in relation to the profitability of the Stock and its riskiness in ? 0 + ? 1 urr tM , - tf , + t t = ? 0 + ? 1 t + t ,...,2,1, = 7. a. Estimate the CAPM and report your results. b. Interpret the estim c.Estimate the CAPM and report your results. comparison with the market.

Interpret the value of R2.
Perform the hypothesis test to determine whether your preferred stock is aggressive (a stock is anaggressive stock if the slope coefficient is greater than 1 and is defensive stock if the slope coefficient isless than 1).
Construct 95% confidence interval for the slope coefficient.
Estimate AR (1) model (that is, AutoRegressive model of order 1) using either GE stock price or IBMstock price. Report and interpret your result.

Estimating the CAPM for Mutual Funds (Fabozzi (2014), p.25)

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Solution:

TASK 1The line charts of stock prices are as follows:

The stock prices for S&P has been between $105 to $160 until the Great Recession of 2008-09. The closing share prices dipped heavily to as low as $74. The share prices had earlier soared to above $150 in 2007. There has been a slight rising trend in the share prices of S&P before the Great Recession and a steep rise following it.

IBM share prices have gone through several ups and downs between 2005 and 2009. An upward trend is visible until the Great Recession which is more prominent after that.

GE share prices have been nearly constant before the Great Recession, however, its prices dipped to almost $5 during the Great Recession. A slow recovery is visible after that.

Returns for S&P, IBM, and GE series are:

GE stock prices had provided the highest and the lowest returns in between 2005 and 2009 among these three. IBM had a positive average return rate of 0.57%. The median return for IBM is almost 1%. The graph shows that most of the time IBM stock prices have been increasing as they are above the x-axis.
The summary statistics of the stocks and the market suggest that IBM has the highest average rate of return whereas the GE has mostly been on the losing side. On comparing the median, only IBM both had a positive medium suggesting most of the time the stocks have gained its value. GE had suffered the maximum loss of 35.44% and enjoyed the highest gain of 22.41%. The variations in the GE stocks have been high so the risk is higher in this case. All the three stock returns are however positively skewed suggesting that there have been a few extremely high gains in the stock prices.
The average return suggest that IBM has a positive average rate of return whereas GE has a negative average rate of return. So, one can expect a positive return while buying IBM stocks whereas GE would be expected to offer a negative return on an average. Also, a higher standard deviation of the GE stocks suggests that it is less consistent than IBM in terms of return.
Based on the statistics, higher standard deviation and average negative rate of return for GE suggests that the GE stocks are relatively riskier than the IBM stock.
The Hypothesis test to check whether returns for both the stocks are equally volatile, is conducted as follows:

Null Hypothesis, H0: ?2IBM = ?2IBM (Claim) Alternative Hypothesis, H1: ?2IBM ? ?2IBM Significance level = 5% or .05 We will conduct an F-test for the equality of two variances. Calculations: Sample size of IBM stocks = 59 Sample size of GE stocks = 59 Numerator degrees of Freedom = 59 – 1 = 58 Denominator degrees of Freedom = 59 – 1 = 58 From the F Distribution table, for ?/2 = .025, DFN = 58, and DFD = 58, critical F = 1.546 Rejection Rule: F test statistics > F critical Test statistics = 1222 Variance of GE is higher than IBM, so 12=0.0101 And, 22=0.0042 So, F-test statistics = 0.01010.0042=2.3853 Result: Since the F test statistics > F critical, we reject the null hypothesis. Inference: At .05 level of significance, there is enough evidence to reject the claim that both returns are equally volatile. GE stocks appear to be more volatile than IBM as was found earlier. So, I would rather prefer IBM stocks than GE stocks.

The hypothesis test to check whether both stocks, IBM and GE, offer the same population average return, is conducted as below:

Null Hypothesis, H0: µIBM=µGE (Claim) Alternative Hypothesis, H1: µIBMµGE Significance level = 5% or .05 Population standard deviation are unknown for both the samples. Samples are independent in a way that the stock prices are considered for two different stocks. We will conduct an independent sample t-test for the equality of two means, assuming unequal variances. Calculations: Sample size of IBM stocks = 59 Sample size of GE stocks = 59 Average return for IBM stocks = 0.00572 Average return for GE stocks = –0.01475 Sample standard deviation for IBM stocks returns = 0.06497 Sample standard deviation for GE stocks returns = 0.10034

Result: Since the t test statistics is not greater than the critical t of 1.9842, we fail to reject the null hypothesis. Inference: At .05 level of significance, there is not enough evidence to reject the claim that both the stocks offer the same population average return. GE stocks appeared to be more volatile than IBM as was found earlier. So, I would prefer IBM stocks than GE stocks. Nothing certain can be determined from this hypothesis test for equality of means. IBM vs the Market Rate of Return is shown in the chart below:

The market rate of return has been quite volatile over the period and so have been the IBM stock returns. GE vs Market Rate of Return is shown in the chart below:

The market rate of return has been quite volatile over the period and so have been the GE stock returns, especially during and after the Great Depression. Market rate of return and GE returns appear to follow the same trend. The sample covariance between IBM and GE stock returns, computed through Excel, is 0.002414 which indicates a positive relationship between the two stocks. Correlation between IBM and GE stock returns, computed through Excel, is 0.370378 which indicates a weak positive relationship between the two stocks. TASK 2

My preferred stock is IBM as it is less volatile and risky than GE and it also provides a positive return. Excess return on IBM stock and on market suggests that T-Bill rate are usually higher during the 5-year period (2005-2009). Both excess over IBM and excess over market are negative on average with values around -3.53% and -4% respectively.
CAPM model is estimated as:

IBM Stock Return = 0.00652-0.80889Market Return The results were calculated by running a regression analysis over the IBM stock return and Market return using Excel, with IBM return as the dependent variable. The intercept of the equation is .00652 which suggest that when the market return is 0%, IBM returns will be 0.65%. The slope of the equation is -0.8089 which suggest that as market return rises, the IBM stock return decreases and for every rise in market return by .01, IBM returns decreases by .81 The R2 value computed is 0.3552. This means that 35.52% of variations in the IBM stock returns is explained by the market returns. Hypothesis Test: Null Hypothesis, H0: ?1 ? 1 Alternative Hypothesis, H1: ?1 > 1 Standard Error = 0.0526 Degrees of Freedom = n-2=59-2=57 Significance Level = 0.05 It is a right tailed-test. Critical t = 2.003 Rejection region: T statistics > Critical t Test statistics = 1-1se=-0.8089-10.0526=-34.37 Since the test statistic does not lie in the rejection region, we fail to reject the null hypothesis. There is not enough evidence to support the claim that the stock is aggressive. The slope is negative which is less than 1, so the IBM stock is defensive. 95% confidence interval for the slope coefficient is (-1.0980, -0.5198) which was computed as a part of regression analysis. The confidence interval entirely lies below 0 which suggest that the slope is significant. The Auto Regressive Model for IBM stock price is: IBM Stock Price, yt=5.1757+0.9539yt-1. The auto regressive model suggests that IBM stock prices are positively related to previous year’s prices.

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