Active Investing and Multi-factor Models

1 Task

You are a portfolio analyst determining an optimal active portfolio management strategy for your Portfolio Manager. You are required to construct an appropriately weighted active portfolio and allocate between this active portfolio and the market portfolio. You will then determine whether the Carhart 4-factor model significantly increases the explanatory power of your active portfolio’s returns in comparison to the SIM. Finally, you will prepare a 3-minute presentation video to your Portfolio Manager, where you will present an analysis and comparison of your results.

2 Context

• After evaluating various U.S. equities drawn from the S&P500, a number of stocks which exhibit significant alpha during the period from January 2014 to December 2018 have been identified. Your Portfolio Manager wants you to carry out an ex-post analysis which examines how an optimal active portfolio could have been formed from these alpha stocks which, when combined with an S&P 500 index ETF, would have maximized the combination’s Sharpe ratio.
• The portfolio manager decides to randomly allocate five unique stocks which exhibited alpha during the sample period to each portfolio analyst to undertake their analysis. Each portfolio analyst will use only these five unique allocated stocks to construct their active portfolio.

3 Allocated Stocks and Data Download

• You have been allocated 5 alpha stocks. (Note that these stocks are unique to you and each student will have a different combination of stocks). Your allocated stocks will be posted on Moodle with your Student ID in the Project 3 Stock Allocation Lookup.xls excel file in the Project 3 Assessments tab.
• Each stock in the spreadsheet is identified by its Factset identifier. Using these identifiers, for the period from January 2014 through December 2018, download the monthly returns for each allocated stock from FactSet (60 observations). All returns should be inclusive of dividends - in the FactSet dropdown box "Total Return" select “% Return."
• For the period from January 2014 through December 2018, download from FactSet the monthly returns for the S&P 500 Index (FactSet identifier: SP50) (60 observations). All returns should be total returns inclusive of dividends - FactSet dropdown box: “Total Return” select “% Return (Gross, Unhedged)”.

4 Single Index Model Regressions (Security Characteristics Line)

• Use a risk-free rate of 3.00% APR (i.e. fixed at 0.25% monthly) for this analysis.
• For each of the five stocks in your active portfolio, calculate monthly excess return: 𝑅𝑅𝑖𝑖𝑖𝑖 = 𝑟𝑟𝑖𝑖𝑖𝑖 − 𝑟𝑟𝑓𝑓 where 𝑟𝑟𝑖𝑖𝑖𝑖 is the return on stock 𝑖𝑖 for month 𝑡𝑡, and 𝑟𝑟𝑓𝑓 is the risk-free rate. (Note: make sure you use the fixed monthly risk-free rate above).
• For the S&P 500 index, calculate monthly excess return: 𝑅𝑅𝑀𝑀𝑀𝑀= 𝑟𝑟𝑀𝑀𝑀𝑀− 𝑟𝑟𝑓𝑓 where, 𝑟𝑟𝑀𝑀𝑀𝑀is the return on the S&P 500 for month 𝑡𝑡. Compute the annualised average excess return for the S&P 500 (simply by multiplying the monthly excess return by 12). This will be the proxy for the average annual market risk premium over the sample period.
• Estimate the Single Index Model alpha 𝜶𝜶𝒊𝒊 and beta 𝜷𝜷𝒊𝒊, (i.e. the Security Characteristic Line) for each stock in your active portfolio using the regression equation:
• Determine the residual variance (𝜎𝜎𝜀𝜀𝑖𝑖 2 ) (unsystematic risk) for each stock in your active portfolio, either by using the following equation:

Or by using the Excel regression tools.

• Annualize the alpha and the residual variance (unsystematic risk) by multiplying by 12. (Note: ensure your alphas are generated by regressing excess returns above the risk-free rate, not raw returns).

Construct the Active Portfolio

• Conduct confidence interval testing for each of your 5 stocks to confirm that the alpha is non-zero (significant) at the 90% confidence level.
• As your Portfolio Manager is concerned that past alpha may not be reflected in future alpha, you are required to risk-weight the alphas as follows: o Only include a stock in the active portfolio if its alpha is significant. Any stock that does not exhibit non-zero alpha at the 90% confidence level should be excluded from the active portfolio

o Risk-weight the alpha by multiplying it by 10% when calculating each stock’s reward-to-risk ratio. For example, if the ex-post alpha was 20% p.a. from the SIM regression, the numerator in your reward to risk-calculation for that stock will be 2% p.a.

• Using the Treynor-Black model, appropriately weight your active portfolio.

6 Optimally Combine the Active Portfolio and the Market Portfolio

• Assume an ETF that tracks the S&P500 is an appropriate proxy for the market portfolio.
• Derive the sample variance (𝜎𝜎𝑀𝑀 2 ) of the S&P500 over the sample period and annualise it.
• Using the active portfolio weightings derived in 5 above, calculate the annualised active portfolio alpha 𝜶𝜶𝑨𝑨, beta 𝜷𝜷𝑨𝑨, and residual variance (𝝈𝝈𝜺𝜺𝑨𝑨 𝟐𝟐 ). Calculate the reward-to-risk ratio for the active portfolio (as a whole).
• Based on the reward-to-risk ratio of the market portfolio over the sample period. appropriately weight between the active portfolio and the market (passive) portfolio using the Treynor-Black model.
• Calculate the annualised information ratio for each stock in your portfolio, as well as the information ratio for the active portfolio as a whole.
• Calculate the potential improvement in your Sharpe ratio compared to a stand-alone investment in the market (passive) portfolio as a result of this strategy. Verify how this result relates to the information ratio of the active portfolio (as a whole).

7 Active Portfolio Returns

• Using the active portfolio weightings derived in 5 above, as well as the monthly returns downloaded in 3, calculate the monthly weighted active portfolio returns for the sample period.
• Estimate the Single Index Model alpha 𝜶𝜶𝑨𝑨 and beta 𝜷𝜷𝑨𝑨, for your weighted active portfolio (note this is the active portfolio Security Characteristic Line) using the regression equation:
• Annualise the weighted active portfolio alpha 𝛼𝛼𝐴𝐴, and determine if the alpha and beta 𝛽𝛽𝐴𝐴 are significant, as well as deriving the R2 for the model over the sample period. Verify that the alpha and beta for the active portfolio (𝛼𝛼𝐴𝐴 and 𝛽𝛽𝐴𝐴) are the same as your answer to 6 above. (Note: remember that your alpha in 6 was risk-weighted to be only 10% of the actual alpha, so the alpha derived from this portfolio regression should be exactly 10x the active portfolio alpha calculated in 6).

8 Multi-factor Model

• Open the excel file “Fama French Carhart Factors.xls”. This file contains the monthly risk premiums (stated as decimals/percentages) for the Carhart 4-factor model over the sample period. (Note: check that this is in a compatible format to your existing data in terms of decimals).
• Using the Fama-French and Carhart monthly factor portfolio data provided (SMB, HML and WML) undertake a multi-factor regression analysis for each stock in your active portfolio. Derive the alpha (including its significance level) and R2 for these 4-factor models for each stock in your active portfolio.
• Using the Fama-French and Carhart monthly factor portfolio data provided, undertake a multi-factor regression analysis for your weighted active portfolio (using the active portfolio weightings derived in 5 above, as well as the monthly returns downloaded in 3). Use the regression equation:
• Derive the alpha (including its significance level) and R2 for this 4-factor model regression of your weighted active portfolio (Note: this is one single regression).

Part 9

Prepare a report for your Portfolio Manager which discusses your results, as well as the limitations of your analysis. Some suggestions of what could be covered in your report include:

A brief discussion of the alpha of each stock in your active portfolio and why you believe it exists

o How you constructed the active portfolio and allocated between it and the S&P500 ETF

o A brief comment on the improvement in your Sharpe ratio arising from the addition of the active portfolio, compared to holding the S&P500 ETF alone

o An examination of the differences in the results derived from the individual stock regressions compared to the active portfolio regression, referencing such statistical measures as the R2 and the alpha, and any other statistical measures you believe are relevant

o A discussion of what happens to your R2 and alpha (including its significance), and any other statistical measures you believe are relevant, when the additional Carhart factors are added to the model, as well as an explanation for why you believe this occurs

o A discussion of the differences between your multi-factor model individual stock results and your multi-factor model portfolio result

o Consideration of any limitations or shortcomings you believe exist in your models, addressing issues such as the risk weighting methodology, allocation methodologies, concentration issues and the unconstrained nature of the outcomes