Background.
The capital asset pricing model (CAPM) is used in finance to determine a theoretically appropriate required rate of return of an asset, where that asset is to be added to an already well-diversified portfolio, given that asset’s non-diversifiable risk. Traditionally, applications of the CAPM use only one variable to describe the returns of a portfolio or stock with the returns of the market as a whole:
〖r_stock-r_f=α_stock+β_stock (r_m-r_f )+u〗_t (1)
In contrast, the Fama–French model uses three variables:
r_stock-r_f=α_stock+β_stock (r_m-r_f )+β_(2 ) SMB+β_3 HML+u_t (2)
r_stock is the stock’s rate of return, r_fis the risk-free return rate, and r_mis the return of the whole stock market. The parameter α_stock is the stock’s ‘alpha’. It measures how much the stock outperforms its ‘theoretical’ predicted returns under the CAPM and β_stockis the stock’s ‘beta’, which measures the stock’s exposure to the overall market. Different stocks will have different parameters.
The Fama-French model contains two additional factors to explain stock returns. SMB “Small market capitalization Minus Big “; measures the historic excess returns of small cap stocks over big caps. HML “High book-to-market ratio (BtM) Minus Low book-to-market ratio” measures the historic excess returns of value stocks (small BtM ratio) over growth stocks (High BtM ratio). These factors are calculated with combinations of portfolios composed by ranked stocks (BtM ranking, Capitalisation ranking) and available historical market data. Historical values are available on Kenneth French’s web page for American stocks. We have supplied Australian data from the ASX in the spreadsheet AssgtData.xls not available there.
The variables are as follows:
r_BHP = Monthly return on BHP stock as observed on the ASX.
r_m = Monthly return on market index, here the All Ordinaries Index, AOI.
SMB = Small market capitalization Minus Big market capitalization factor.
HML = High book-to-market ratio Minus Low book-to-market ratio
You are to assume a risk-free rate of r_f=0.005 per month. Your task is to estimate the Fama-French three factor model using the given data. and determine whether it is any better at explaining the BHP stock returns compared to the market excess returns given by only the All Ordinaries Index.
Please enter all your answers into the provided spaces below.
Part 1: Deriving the Least Squares Estimators
Recall, in Linear Least Squares, we have to estimate the ‘line of best fit’ which minimizes the ‘sum of squared deviations’ of the data. This is equivalent to choosing parameters which minimises the function of two variables (namely SSR):
.
You will use calculus to show that is minimized for the choices:
where and .
You may assume the following summation properties in your solution (see Lecture 5):
If k is a constant, .
Given and for , then .
Differentiating a summation (the ‘derivative of the sums’ equals ‘the sum of the derivatives’):
Q1 (1 mark): In lectures we saw that the ‘total deviation about the mean’ is always zero, i.e. . Expand the summation across the brackets and apply this result to the to prove that: .
Q2 (1 mark) Prove that ∑_(i=1)^n▒〖〖(x〗_i-x ̅)〗^2 =∑_(i=1)^n▒〖x_i 〖(x〗_i-x ̅)〗
Q3 (2 marks): By differentiating the summation, show that when .
Q4 (1 mark): Differentiate the summation with respect to to get a summation expression for .
Q5 (2 marks): Prove that by choosing b0 and b1 to minimize you obtain the least squares estimators, namely:
Q6. (1 marks): Read the supplied data into Eviews. Generate two new variables rBHP_rf and 〖 r〗_m_r_f , which are the stock and market‘excess returns’ respectively, assuming r_f=0.005. Use these to estimate the model given by Equation (1):
〖r_BHP-r_f=B_0+B_1 〖( r〗_m-r_f)+u〗_t
Q7. (1 mark) Comment on the sign of the estimated coefficient B_1, and state whether this is what you expect.
Q8 (1 mark) Formulate and carry out an appropriate hypothesis test, to test whether the excess market returns explain the excess returns of BHP shares. Use the t-statistic approach, at the =0.05 level. Assume the large sample approximation applies.
Q9 (1 mark) Formulate and carry out an appropriate hypothesis test for testing whether BHP’s ‘beta’ is greater than one at the =0.05 level
Q10. (1 mark): Use the data to estimate the Fama-French 3-Factor model given by Equation (2):
r_BHP-r_f=B_0+B_1 〖( r〗_m-r_f)+B_2 SMB+B_3 HML+u_t
Q11 (2 marks) In the Fama-French 3-factor model you estimated, test the following hypotheses about the coefficients B2 and B3. Clearly specify the rejection region if you are using critical values, and clearly state your conclusions. When using p values, calculate and compare your p-values to the test size then state your conclusion. (Hint, assume the Central Limit Theorem Holds)
(a) H0: B_2=0, H1: B_2>0, with =0.05 using the critical-value approach.
(b) H0: B_2=0, H1: B_2<0, with =0.05 using the critical-value approach. (c) H0: B_3=0, H1: B_3>0, with =0.05 using the p-value approach.
(d) H0: B_3=0, H1: B_3<0, with =0.05 using the p-value approach.
Q12 (3 marks) Formulate a joint-hypothesis test to test whether the Fama-French 3-Factor model explains the stock returns better than the model given by Equation (1). Perform the hypothesis test by calculating the homoskedasticity-consistent F-Statistic, using the relevant formula.
Verify your conclusion by performing the Wald test in Eviews and considering p-values. What is your conclusion?
Q13 (3 marks) A Financial Analyst believes that the effect of book-to-market values (HML) on stock returns is twice as great as the effect of market capitalization (SMB). Formulate an appropriate hypothesis test and use re-parametrisation to convert it to a simple t-test to test the assertion. Perform the required regression and paste your Eviews output below. State your conclusion at the 5% level.
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