# Index Models, CAPM and Arbitrage Pricing Theory

Question 1

Index Models:

Download 61 months (October 2008 to October 2013) of monthly data for the S&P 500 index (symbol = ^GSPC). Download 61 months (October 2008 to October 2013) of Apple Inc. data and 61 months (October 2008 to October 2013) of Exxon Mobil Corporation data. Download 60 months (November 2008 to October 2013) of the 13 week T-bill rate (symbol = ^IRX). Be sure to use end-of-month data! Construct the following on a spreadsheet:

1. Calculate 60 months of returns for the S&P 500 index, Apple and Exxon. Use November 2008 to October 2013. Note this means you need price data for October 2008. On the answer sheet report the average monthly returns for the S&P 500 index, Apple and Exxon, as well as the average monthly risk-free rate.

2. Calculate excess returns for the S&P 500 index, Apple and Exxon. Note you must divide the annualized risk-free rate (^IRX) by 1200 to approximate the monthly rate in decimal form. On the answer sheet report the average monthly excess returns for the S&P 500 index, Apple and Exxon.

3. Regress excess Apple returns on the excess S&P 500 index returns and report, on the answer sheet, Î±, Î², the r-square and whether Î± and Î² are different from zero at the 5% level of significance. Briefly explain your inference.

4. Decompose total risk for Apple into systematic risk and firm-specific risk. That is, calculate total risk, systematic risk and firm-specific risk for Apple.

5. Regress excess Exxon returns on the excess S&P 500 index returns and report, on the answer sheet, Î±, Î², the r-square and whether Î±, Î² are different from zero at the 5% level of significance. Briefly explain your inference.

6. Decompose total risk for Exxon into systematic risk and firm-specific risk. That is, calculate total risk, systematic risk and firm-specific risk for Exxon.

7. Estimate the covariance and correlation of Apple and Exxon excess returns.

Question 2

Q2: CAPM and APT:

1. The expected rate of return on the market portfolio is 8% and the risk-free rate of return is 2%. The standard deviation of the market portfolio is 23%. What is the representative investor's average degree of risk aversion?

2. Stock A has a beta of 1.15 and a standard deviation of return of 35%. Stock B has a beta of 2.50 and a standard deviation of return of 48%. Assume that you form a portfolio that is 60% invested in Stock A and 40% invested in Stock B. Using the information in question 1, according to CAPM, what is the expected rate of return on your portfolio?

3. Using the information in questions 1 and 2, what is your best estimate of the correlation between stocks A and B?

4. Your forecasting model projects an expected return of 11% for Stock A and an expected return of 19% for Stock B. Using the information in questions 1 and 2 and your forecasted expected returns, what is your best estimate of the alpha of your portfolio when using CAPM to determine a fair level of expected return?

5. A different analyst uses a two-factor APT model to evaluate expected returns and risk. The risk premiums on the factor 1 and factor 2 portfolios are 4% and 6%, respectively, while the risk-free rate of return remains at 2%. According to this APT analyst, your portfolio formed in question 2 has a beta on factor 1 of 1.8 and a beta on factor 2 of 2.4. According to APT, what is the expected return on your portfolio if no arbitrage opportunities exist?

6. Now assume that your forecasting model of question 4 accurately projects the expected return of Stocks A and B and therefore your portfolio, and that the APT model of question 5 describes the fair rate of return for your portfolio. Do any arbitrage opportunities exist? If yes, would you invest long or short in your portfolio constructed in question 2?

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#### Solution Preview

Question 1

Index Models:

1. Calculate 60 months of returns for the S&P 500 index, Apple and Exxon. Use November 2008 to October 2013. Note this means you need price data for October 2008. On the answer sheet report the average monthly returns for the S&P 500 index, Apple and Exxon, as well as the average monthly risk-free rate.

Average Monthly Returns S&P 500 = 1.106%

Average Monthly Returns Apple = 2.984%

Average Monthly Returns Exxon = 0.431%

Average Monthly Risk Free Rate = 0.008% (This is the monthly return on a Risk Free ...

#### Solution Summary

The solution contains solved problems in Asset Pricing theories including CAPM, APT and single index models with detailed explanations and calculation steps. The solution is based on actual market data, making it easier to understand the application of the concepts.