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What is the standard error of the difference between the means of the two series of returns?

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Identify the standard error of the sample mean of the paired differences between the two series of returns

 

 

SKU: solviv12

Question 1

(a) A financial planner allocated $10,000.00 between a growth stock and a mid-cap stock in January 2013. As of January 2014, the percentage returns on both investments were 6% and 5.75%, respectively.
Calculate the dollar amounts allocated to each of the investments. Show all working.

Note: The total dollar return as of January 2014 was $588.75.

 

(b) An investment firm agrees to fully redeem $1,102,500 worth of bonds in two years. The firm plans to invest $1,000,000 today.

Calculate the annual compounded rate of interest that is required so the investment firm can redeem the bonds at the agreed value at the end of a 2 year period. Show all working.

 

(c) Suppose you measure the absolute daily percentage rate of return (not annualised) on a particular asset worth $1,000,000 and record this data for one year. You calculate the mean and standard deviation of these numbers as +0.04% and 1.9%.

Assuming a normal probability distribution, what is the probability of an absolute return on any one day of less than 0%? Show all working.

 

Question 2

As an analyst you receive the following regression information:

E(Yt)=β1 + β2Xt was estimated by the Ordinary Least Squares Method as

Yt=227.3642+6.04075Xt

Σ(Xt – X)² = Σ xt² ≈ 33.43

σ² ≈ 35.4198

Degrees of freedom = n – 2 = 10

 

(a) Perform a two-tail test of the null hypothesis H0:β2 = 0 against the alternative H1:β2 ≠ 0 with α = 0.05.

Find the value of the t-statistic and the associated critical value for the test.

Conclude whether the null hypothesis can be rejected at the 5% level.

 

(b) Perform a one-tail test of the null hypothesis H0:β2 = 0 against the alternative H1:β2 > 0 with α = 0.05.

Find the value of the t-statistic and the associated critical value for the test.

Conclude whether the null hypothesis can be rejected at the 5% level.

 

(c) Construct a 95% confidence interval for β2.

Note: You must show all working for each part of the question.

 

Question 3

Bauman and Miller (1994) examine the hypothesis of modern portfolio theory that portfolio returns are positively correlated with risk as measured by beta (systematic risk) and sigma (standard deviation of return). They analyse how well portfolio returns in one market cycle are predicted by portfolio returns, sigma and beta in the previous market cycle, using a regression model. They conclude:

Measuring and ranking the returns of portfolios over stock market cycles is very useful in predicting rankings and returns over the next market cycle; this is generally more useful than employing portfolios’ betas and sigmas for prediction purposes. Predictions of portfolio returns are highly significant in all the market cycles when past returns are used in conjunction with portfolio sigmas, however.

Explain how the figures in the exhibits (below) from the Bauman and Miller paper support their conclusions with regard to:

(a) returns and sigma (from Exhibit 6 and Exhibit 7)

(b) returns and beta (from Exhibit 5 and Exhibit 7)

(c) Sharpe ratio or Treynor ratio (from Exhibit 4)

(d) portfolio returns, sigma, and beta in the previous market cycle (from Exhibit 7)

(e) portfolio returns measured over a full market cycle (from Exhibit 7)

(f) a multiple regression model that uses prior portfolio returns and prior sigma (from Exhibit 7).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 4

You are assigned the task of predicting the stock return for the raw materials sector. You propose that the raw materials sector stock returns are explained by the following four explanatory variables:
• industrial production growth rate
• 10-year Treasury bond yields
• inflation rate
• the spread in yields between BB rated bonds and AAA rated bonds.

 

You run a multiple linear regression using 120 monthly observations (n) with the following results.

 

For the questions below, show full workings and explain the approach you take, including the reasons why particular statistics are used.

(a) From the data provided in the table above, calculate the t-statistic for the four (4) regression coefficients.

 

(b) Which coefficient estimates are statistically significant at the 0.05 level?

 

(c) What is the 95% confidence interval for the industrial production growth rate slope?

 

(d) You make the following predictions:
• constant 1.0%
• industrial production growth rate 0.5%
• 10-year Treasury bond yield 4.0%
• inflation rate 3.0%
• yield spread 5.0%

 

Based on these predictions and the regression results in the table above, what do you predict the raw materials sector return will be?

 

(e)Given the regression results in the table above, what is the expected change in the raw materials sector return in response to a one percentage point decrease in the industrial production growth rate (assuming no change in the remaining explanatory variables)?

 

Resources for Questions 5, 6, and 7

Note: The construction of the model for this assignment is based on a number of underlying assumptions.

You have been provided with data on a number of stocks to build a model for the following questions, although not all stocks are used for all questions. The data to enable you to answer Questions 5 to 7 is contained in ‘Assignment Excel template’ (i.e. FIN236.AS1.10.xls) under ‘Assignment’ in LearningSpace. Data in different tabs is used for different questions.

The model will hold a portfolio of some of these 10 stocks, the returns of which are considered to be random variables. The volatility of the portfolio return is selected as a measure of risk.

Stock price movements are assumed to be related to a total of three explanatory variables (or factors) — size, momentum and value, defined below:

• size: the logarithm of the market capitalisation, which is the number of shares issued multiplied by the share price. LOG or LOG10 Excel functions are applied instead of the LN Excel function

• momentum: the rolling three-month price return, which is the percentage change of the current price from the price three months ago

• value: the earnings yield, which is the percentage earnings per share divided by the price per share.
The monthly data provided for this assignment are for the 96 months from January 2001 to December 2008. The monthly data are split into two periods:

• the in-sample period (84 months from January 2001 to December 2007)

• the out-of-sample period (12 months from January 2008 to December 2008).

In this regression model, monthly returns can be considered as a linear combination of the three explanatory variables and a specific return (random error). Once the model for each stock is estimated, the portfolio risk of the stocks can be calculated simply by using:

• the covariance matrices of the three common factors’ coefficients

• the associated three common factors’ exposures (matrix of observations on the independent variables)

• the covariance matrix of specific returns.

Note: A model-based covariance matrix of total returns is used for risk analysis (rather than the historic data-based covariance matrix) for the 10 stocks.

In order to estimate the covariance matrix of the three factors’ coefficients, a sample of the coefficients of the three common factors from the cross-sectional regressions has to be obtained. Using the ordinary least squares (OLS) method for each month over the in-sample period, 10 stock returns (the dependent variables) are regressed against the associated three factors’ exposures. These cross-sectional regressions are repeated for each month over the in-sample period.

The covariance matrix of the three regression coefficients is estimated based on the monthly regression coefficients over the different months. With the three factors’ exposures, the ‘common factor’ covariance matrix can be calculated at any month. The covariance matrix of specific returns is added to the common factor covariance matrix to obtain the model-based covariance matrix of total returns at any month.

The results obtained are then used to forecast the volatility of monthly returns of each stock and/or a portfolio of stocks at a month over the out-of-sample period, given certain values of the three factors’ exposures.

 

Note:
• No prior knowledge of risk-factor modelling is assumed apart from information from Topics 1 to 4 of FIN236.

• Express returns and their volatilities as percentages to two (2) decimal places.

• Report all elements in the correlation and covariance matrices to six (6) decimal places.

• Be consistent in the use of measurement units for all calculations.

 

Question 5

(a) Determine whether the monthly returns for the 96 months of the data (January 2001 to December 2008 in Tab 5) for BHP and TLS are from the same population (i.e. whether the sample data is drawn from underlying populations that have the same mean and standard deviation) by conducting the following tests:

(i) Test if the mean of paired differences between the two series of returns is zero. Use the Excel function TTEST(BHP,TLS,2,1) for this test.

Do you reject the null hypothesis at 5% level of significance based on a two-sided t-test (from your test result)? Explain your decision.

Identify the sample mean of the paired differences between the two series of returns.

Identify the standard error of the sample mean of the paired differences between the two series of returns.

 

(ii) Test if the difference between the means of the two series of returns is zero. (Assume that BHP and TLS have the same population of price returns.) Use the Excel function TTEST(BHP,TLS,2,2) for this test.

Do you reject the null hypothesis at 5% level of significance based on a two-sided t-test (from your test result)? Explain your decision.

What is the difference between the means of the two series of returns?

What is the standard error of the difference between the means of the two series of returns?

 

(b) (i) Compute the mean (μ) and standard deviation (σ) of monthly returns (x) for the BHP stock price between January 2001 and December 2008. Express your answers as percentages.

Note: Show all relevant workings.

(ii) Using the symbols in part (i) above, write the formula of the Z-scores for x.
What conditions on the distribution of x would enable the Z-scores to follow a standard normal distribution?

(iii) Using your sample estimates of the mean and standard deviation in part (i) above, what is the probability of observing positive monthly returns for BHP?

 

(c) Using the monthly returns for the year 2008 as a random sample for any 12-month period, calculate the following:

(i) the sample mean

(ii) the standard deviation of the sample mean

(iii) the probability that the average monthly returns for BHP over any 12-month period will be positive.

Note: Show all relevant workings.

 

(d) The ‘Q5d’ tab in the Excel spreadsheet provides the summary statistics of 84 regressions based on cross-sectional data for each month between January 2001 and December 2007. The following summary statistics are shown in the spreadsheet:

• multiple R

• R-square

• adjusted R-square

• standard error

• F

• significance of F.

 

(i) Describe how each of the statistics above has been applied from January 2001 to December 2007.

(ii) Identify the months in which the F-test rejects the null hypothesis of the non-existence of the linear model at 5% level.

(iii) Based on the summary statistics of all regression results for each month of the 84-month in-sample period provided:

• identify the average, minimum and maximum of the six (6) summary statistics from January 2001 to December 2007

• comment on the stability of the goodness-of-fit of the regression model

• explain whether the signs of the t-statistics are relevant.

 

(e) Using the estimated coefficients of the regression model for each month based on the cross-sectional data between January 2001 and December 2007 provided in the ‘Q5e’ tab in the Excel spreadsheet, calculate the covariance matrix for the 84 monthly coefficients of the in-sample period (also known as the common factor covariance matrix).

From your Excel results, paste a copy of the annualised covariance matrix of the coefficients in the Word document of your assignment (and in the format below).

 

(f) (i) Consider the following notations:

Ri = return of stock i for a particular month

Xi1 = exposure of stock i to the constant unit of 1

Xi2 = exposure of stock i to the size

Xi3 = exposure of stock i to the momentum

Xi4 = exposure of stock i to the value

f1 = the intercept term

f2 = the regression coefficient of size

f3 = the regression coefficient of momentum

f4 = the regression coefficient of value.

 

Complete the formula for the specific returns (or residual term) of stock i for a particular month, Ui, given the information above and the linear regression model. What do the specific returns attempt to measure?

 

(ii) Calculate the monthly specific returns (i.e. the residuals of the regressions) for CBA, WES and BHP for the 84-month in-sample period. To do this calculation, use the formula derived from part (i), the data given in the notations in part (i), and the data in the ‘Q5f’ tab of the Excel spreadsheet.

From your Excel results, paste a copy of the specific returns for CBA, WES and BHP for the months of January 2001 and December 2007 only in the Word document of your assignment (and in the format below).

 

 

(g) Calculate the historical variance of the monthly specific returns for CBA, WES and BHP for the 84-month in-sample period.

Note: The specific returns are assumed to be uncorrelated so that the off-diagonal elements of the covariance matrix are zero.

From your Excel results, paste a copy of the monthly specific returns covariance matrix for CBA, WES and BHP in the Word document of your assignment (and in the format below). See the ‘Q5g-h-i’ tab of the Excel spreadsheet.

 

(h) From your Excel results, paste a copy of the annualised specific returns covariance matrix for CBA, WES and BHP in the Word document of your assignment (and in the format below). See the ‘Q5g-h-i’ tab of the Excel spreadsheet.

 

(i) From your Excel results, paste a copy of the annualised specific returns volatilities for CBA, WES and BHP in the Word document of your assignment (and in the format below). See the ‘Q5g-h-i’ tab of the Excel spreadsheet.

 

Question 6

The data below is provided in the ‘Q6’ tab of the Excel spreadsheet and is the calculated monthly stock returns for CBA, WES and BHP for each month of the 12-month out-of-sample period.

 

From the results above, calculate the annualised volatilities for CBA, WES and BHP and paste your results in the Word document of your assignment (in the format below).

Note: Annualised volatility is the annualised standard deviation of monthly returns.

Save your Excel workings in the lower table on the tab labelled ‘Q6’.

 

Question 7

Answer the questions below, providing reasons and valid arguments to demonstrate your understanding of regression limitations and how to manage them in practice.

(a) Describe the four (4) conditions that must be satisfied for a regression analysis to be valid.

 

(b) Describe two (2) ways of dealing with heteroscedasticity.

 

(c) Calculate the rolling 12-month volatility of price returns for each of the 10 stocks over the in-sample period using data in the ‘Q7’ tab in the Excel spreadsheet.

Note: The rolling 12-month volatility at any month is the annualised standard deviation using the monthly returns of the preceding 12 months.

Save your Excel workings for part (c) as a tab labelled ‘7c’.

 

(d) Draw a graph of the results in part (c). Based on the graph, describe the relationship between volatilities of stocks over time.

 

(e) Explain the Durbin-Watson test and how its results should be interpreted.

 

(f) Why would a transformation of the dependent variable be used? Describe what sorts of transformations are commonly used.

 

(g) Explain four (4) steps in a backward elimination stepwise regression approach.

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