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:

(1)In contrast, the Fama–French model uses three variables:

(2)is the stock’s rate of return, is the risk-free return rate, and is the return of the whole stock market. The parameter is the stock’s ‘alpha’. It measures how much the stock outperforms its ‘theoretical’ predicted returns under the CAPM and is 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. “**S**mall market capitalization **M**inus **B**ig “; measures the historic excess returns of small cap stocks over big caps. “**H**igh book-to-market ratio (BtM) **M**inus**L**ow 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 real Australian data from the ASX in the spreadsheet ** AssgtData.xlsx**.The variables are defined as follows:

** = **Monthly return on BHP stock as observed on the ASX.

** = M**onthly return on market index, here the **All Ordinaries Index**, AOI.

** = S**mall market capitalization **M**inus **B**ig market capitalization factor.

** = H**igh book-to-market ratio **M**inus**L**ow book-to-market ratio

You are to assume a risk-free rate of per month. Your task is to estimate the Fama-French three factor model using the given data, andto determine whether it is any better at explaining the BHP stock returns compared to the market excess returns given by the All Ordinaries Index.

**Please enter all your answers into the provided spaces below. **

**Part 1: Deriving the Least Squares Estimators**

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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):

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You will use calculus to show that is minimized for the choices:

where and .

*Revise the online videos explaining the Rules of Summation for the Week 5 lecture, particularly for the summation properties described below. You may assume the following summation properties in your assignment solution:*

- If
*k*is a constant, . - Given and for , then .
- Differentiating a summation (the ‘derivative of the sums’ equals ‘the sum of the derivatives’):

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Q5 (4 marks): Prove that by choosing b_{0} and b_{1} to minimize you obtain the least squares estimators, namely:

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Q11 **(4 marks) **In the Fama-French 3-factor model you estimated, test the following hypotheses about the coefficientsB_{2} and B_{3}. 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) H_{0}: , H_{1}: , with a=0.05 using the critical-value approach.

(b) H_{0}: , H_{1}: , with a=0.05 using the critical-value approach.

(c) H_{0}: , H_{1}: , with a=0.05using the p-value approach.

(d) H_{0}: , H_{1}: , with a=0.05using the p-value approach.

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Q12 **(4 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 formula from lectures. Verify the value of the F-Statistic by performing the Wald Test in Eviews and using p-values. Paste your Eviews output below.

Verify your conclusion by performing the Wald test in Eviews and considering *p*-values. What is your conclusion?

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Q13 **(4 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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