### Question Description

This is the third part of my finance 310 project that needs to be done using previous information from part 1 and part 2.

**FINANCE 310: INVESTMENTS**

### Fall 2019

PROJECT PART 3

**The Point of the Project**

You are a portfolio manager, and you are trying to put together a portfolio that is designed to beat the market (represented here by the S&P 500 index). To do this you will first pick ten stocks, and then you will figure out how much of each of them to buy, using monthly data from the last five years to make your decisions. You have 100 million dollars to play with and you will pick stocks before the start of trading on August 8^{th}.

You will decide if you have beaten the S&P 500 by looking at the performance of your portfolio over the period August 8^{th}–November 8^{th} of this year (this is the ‘Evaluation Period’). To do this you will compare the risk-adjusted returns of your portfolio with the risk-adjusted return of the S&P. The project has three parts.

**Project Part III**

We are now going to find the optimal portfolio of risky stocks using the CAPM. We will also evaluate your performance. You need to answer the following questions:

- (2 points) Find the optimal portfolio weights using CAPM inputs. Calculate the Sharpe Ratio. If your portfolio is extreme consider imposing short sale constraints.
- (2 points) Compare your optimal portfolio weights using CAPM to the historical weights (Project II). Which set of weights seems more reasonable? Which one would you select? Why?
- (2 points) You believe the CAPM is not the perfect model, but useful as a baseline. Continue using the CAPM for eight of your stocks, however, for the remaining two, substitute in your own estimates for expected returns. Give at least one reason why you think these inputs are reasonable. How does your optimal portfolio compare to the ‘pure’ CAPM portfolio in Question 1.
- (2 points) Re-estimate the betas for all the stocks using two years of monthly data. Are these better estimates of betas? What does your optimal portfolio look like with the 2-year betas?
- (3 points) Select an optimal portfolio.
**Explain**why you select it over the alternatives. - (2 points) Assume you invested in your optimal portfolio at the start of class and held it for almost three months (August 8, 2019 to November 8, 2019). Calculate the all the performance measures from class for your optimal portfolio: (i) Sharpe measure, (ii) M-squared, (iii) Treynor measure, (iv) alpha and (v) the appraisal ratio.
- (2 points) Did your portfolio in #6 beat the market? Why? Which performance measure is the most appropriate? Why?
- (3 points) Your client asks you
the portfolio weights you are recommending make sense. Provide a short and intuitive answer to his question. Use at most 100 words.*why*

P.(2 points) Presentation (i.e., see the next paragraph)

Make a title page that includes your name and section number and anything else you deem helpful. You should hand in a **hard-copy** at the beginning of class. Include answers to all the questions and any supporting material you think is helpful. Make it presentable to a “client,” or a prospective employer. It should be user friendly, concise, well-written, neat and convincing. Print only the relevant parts of the spreadsheet. The answers should be brief, but convincing. Make it easy to find your answers. Include supporting material in the appendix. Practice good printing etiquette. Write well. The assignment should be maximum **eight** pages (including any appendix, but excluding any cover page). Longer is not always better.

**DETAILED INSTRUCTIONS**

Try to follow these instructions at closely as possible.

Note that for Questions 1-5 below you should use historical data from August 8^{th}, 2014 to the end of trading on August 7^{th}, 2019. You want to use this data to find your optimal portfolio on August 8th, 2019.

**Find the optimal portfolio weights using CAPM inputs. Calculate the Sharpe Ratio. Does imposing short sales help?****Compare your optimal portfolio weights using CAPM to the historical weights (Project II). Which set of weights seems more reasonable? Which one would you select? Why?****You believe the CAPM is not the perfect model, but useful as a baseline. Continue using the CAPM for eight of your stocks, however, for the remaining two, substitute in your own estimates for expected returns. Give at least one reason why you think these inputs are reasonable. How does your optimal portfolio compare to the ‘pure’ CAPM portfolio in Question 2.****Re-estimate the betas for all the stocks using two years of monthly data. Why are they different? Are these better estimates of betas? What does your optimal portfolio look like with the 2-year betas?****Select an optimal portfolio. Explain why you select it over the alternatives.****Assume you invested in your optimal portfolio at the start of class and held it for almost three months (August 8, 2019–November 8, 2019). Calculate all the performance measures from class for your optimal portfolio: (i) Sharpe measure, (ii) M-squared, (iii) Treynor measure, (iv) alpha and (v) appraisal ratio.****Did your portfolio in #6 beat the market? Why? Which performance measure is the most appropriate? Why?****Your client asks you***why*the portfolio weights you are recommending make sense. Provide a short and intuitive answer to his question. Use at most 100 words.

- To this we need the betas for each stock. We will estimate betas using the Index Model from class (see Lecture Notes: CAPM).
- Calculate excess returns for each stock and for the market. See the worksheet called ‘excess returns.’ Each column represents the difference between the asset returns and the risk-free rates. In other words, you want columns of numbers that represent excess returns. For example, if a stock’s raw return is in column B and the t-bill return is in column Z, then to get the excess return, the formula for row 1 should be =B1-$Z1. Do not forget the $ sign, since this ensures that you always subtract the t-bill (and not something else) when you drag the formula across cells.
- To get the betas we will estimate beta using a
**regression**. You will have to run an index model regression for each stock. This is quite easy in Excel. Simply choose regression from the “Data Analysis,” which can be found under tools. Your “Y” variable will be the cells containing the data on the individual stock’s excess returns, and the “X” variable will be the cells containing the data on the S&P’s excess returns. Allow excel to open a new worksheet for each regression (this is easier and is the default in excel; I have ten worksheets for this (one for each stock) in my example after ‘excess returns’). - The beta is now in B18. If you select labels it is the coefficient on the ‘S&P’. If you did not select labels it the coefficient of ‘X Variable 1’ (see also the Excel Example: Finding Beta).
- Enter your stocks’ betas into the worksheet ‘CAPM Inputs.’ Enter the historical standard deviations. This spreadsheet calculates the CAPM expected returns and covariances.
- Use the CAPM Inputs (returns, standard deviations and covariances) to get the optimal portfolio weights. Follow the instruction provided in Project Part II. The procedure is the same except that you have different inputs. When you copy your covariance matrix make sure to paste ‘value’ (paste special); sometimes the formula references change. Also make sure the diagonal of the CAPM Covariance Matrix is the historical variance (and not estimated using the CAPM covariance formula).
- If you get crazy weights implement the Troubleshooting tips discussed in Project Part II.

- Double check your covariance matrix and other inputs. Did you copy using paste special to make sure formulas did not change?
- Make sure the diagonals of the covariance are equal to the historical variance.
- Try copying in a fresh copy of the excel spreadsheet and start again.
- Manually input an equally weighted portfolio as your starting values.
- If none of this works then introduce additional constraints to place upper (and lower) bounds on your weights to prevent them from getting too crazy. Example: (i) weights can be no larger than 2 and (ii) no less than -1 (you can use other values). If you do this please explain it in your project.

Compare the weights obtained using CAPM inputs to those obtained using historical inputs. Present both sets of portfolio weights in your write up.

There are three main factors to consider when deciding which one to select: (I) the Sharpe Ratios, (II) the weights (see 3 above) and (III) the relative appeal of using the CAPM vs. historical inputs (i.e. CAPM is forward looking, but not an ideal model).

This is the exactly the same as in Question 1 except that you replace to CAPM expected returns with your own estimates. Remember that these are monthly returns!!

To justify your estimate use any available news story to come up with an estimate. Find at least one justification for each of the two stocks. The grading of this question is not so much about how convincing the estimate is, as how you answer the question.

Most likely the optimal portfolio is now going to change in the direction of your estimates. For example, if you are bearish on a stock the portfolio program is going to allocate less weight to that stock.

To estimate the betas follow the instructions in Question 1 part c, but instead of selecting all 60 months of data, select only the **most recent** 24 months as the inputs into the regression. To get the optimal portfolio weights use the new betas to estimate the covariance matrix and expected returns – and then follow the instructions in #1.

How do you pick the optimal portfolio? There are many alternative optimal portfolio weights that we have calculated: (1) historical, (2) historical w/short sales, (3) green investing, (4) CAPM with 5 year betas, (5) CAPM with 2 year betas, (6) CAPM w/your own estimates and (7) CAPM without short selling. All the optimal portfolios are found using historical data available on August 8^{th}, 2019. Optimal portfolios (1)-(3) are from Project #2.

Factors to consider when choosing your optimal portfolio are (1) Sharpe Ratios, (2) how reasonable are the weights, (3) how good are the inputs, etc. It is important to convincing discuss why you think the portfolio you select is superior to the other alternatives on August 8th (ex-ante).

You want to evaluate the performance of your portfolio during the ‘Evaluation Period’ (August 8, 2019–November 8, 2019). You want to:

- Download date from August 8, 2019–November 8, 2019 for all ten stocks, the S&P500 and T-bills.
- Calculate daily holding period returns. Remember that T-bills are an annual percent!! To convert the T-bill return into a monthly return divide by the number of trading days (251) times 100.
- Calculate daily portfolio returns on your optimal portfolio
- Calculate the all the performance measures for your portfolio

See Project Part I for detailed instruction on (i) and (ii). Daily returns on your optimal portfolio are a weighted average (using the optimal portfolio weights from 5 above) of the individual stock returns. To get the alpha and information ratio (appraisal ratio) you need to run a SCL regression (see #1) using the excess returns on your portfolio (from Y-variable) on the excess returns on the market portfolio (X-variable) from August 8, 2019–November 8, 2019.

Select an appropriate performance measure. Explain your choice based on the discussion in class. Based on this performance measure did you beat the market?

Assume your client has knowledge of the basics. Do not talk about why you initially picked your stocks. Convince me that you have understood the calculations that you have done. I am looking for **intuition** not mechanical explanations. Think of this as a two minute opportunity (an elevator spiel) to persuade your client that the portfolio you are recommending is sensible.