Group Project

Portfolio Analysis

In this first part, we use the index model to choose an optimal portfolio of risky assets. One unfortunate truth about the statistics of stock markets is that data generally provides poor estimates for expected return and alpha. However, estimates of beta and firm-specific volatility tend to be much more accurate.

In this part of the project, we will make assumptions about alpha and expected return, but we will use the data to inform us about beta and volatility. We will then construct and examine optimal risky portfolios.

Assumptions:

You should make the following assumptions about annualized expected returns and the alphas of your four stocks for all of the exercises below. All other values will be calculated using data or the index model

Annual CAPM Alpha of Stock 1: 4%

Annual CAPM Alpha of Stock 2: -4%

Annual CAPM Alpha of Stock 3: 5%

Annual CAPM Alpha of Stock 4: 5%

Market Risk Premium: 6%

Value Factor Risk Premium: 4%

Size Factor Risk Premium: 2%

Tasks:

Calculate the estimated sharpe ratio of the market portfolio. This simply 6% (our assumed Market Risk Premium), divided by the annualized standard deviation of the excess market return over our five year period.

Using the index model, calculate the weights for the optimal risky portfolio which combines Stock 1, Stock 2, and a position in the Market portfolio. Call this "Portfolio A". Record the portfolio's weights, Sharpe Ratio, Market Beta, CAPM alpha and expected excess return.

To do this I would proceed using the following steps.

Create a spreadsheet which calculates the alpha, beta, and firm-specific risk for a portfolio with weights w1, w2, and wM = (1 - w1 - w2).

Use these values, along with the Market Risk Premium, to calculate the expected excess return and standard deviation for the portfolio. The ratio of these will be the sharpe ratio.

Use excel's solver to choose w1 and w2 to maximize the sharpe ratio.

Repeat the previous exercise but this time construct the optimal risky portfolio of Stock 3, Stock 4, and a position in the Market Portfolio.

Call this "Portfolio B". Record the portfolio's weights, Sharpe Ratio, Market Beta, CAPM alpha and expected excess return.

Repeat the exercise but now use all four stocks and the market portfolio. Call this optimal risky portfolio "Portfolio C". Record the portfolio's weights, Sharpe Ratio, Market Beta, CAPM alpha and expected excess return.

Portfolio C might represent a trading strategy implemented by a hedge fund. Our goal now is to analyze the riskiness of this strategy using a multi-factor model. We will use the Fama-French three factor model.

Calculate the betas of Portfolio C with respect to the market, value, and size factors.

Using the weights calculated for portfolio C, construct the 5-year time series of excess returns.

Regress this time series on the Market, Value, and Size Factors.

Use these betas, along with the assumed Risk Premia for Market, Value, and Size to calculate the predicted expected return from F-F 3-factor model. You have already calculated the Portfolio's expected excess return (E[RC]) in Part 4). The difference between this expected return is the F-F 3-factor alpha.

α_FF=E[R_C ]- β_MKT 6%- β_HML 4%-β_SMB 2%

Recall that the CAPM Alpha would be

α_CAPM=E[R_C ]- β_MKT 6

Part 2 write up:

You should report the

The recorded values for Portfolios A, B, and C, along with the market sharpe ratio.

The regression output for the F-F 3 factor regression calculated in 5)

The F-F 3 Factor alpha calculated in 6.

Answer the following questions (a few sentences for each).

Rank the Sharpe Ratios of Portfolio's A, B, and C. Explain the intuition behind this ranking (why is each ranked where it is).

Look at the weights for Stocks 1,2,3, and 4 in portfolio C. What has the largest positive weight and why? The largest negative weight?

Does Portfolio C have statistically significant exposure to the value and size factors? How does this exposure (or lack of exposure) effect the interpretation of the expected return of Portfolio C?

(Hint: How does the alpha of Portfolio C change when considering the Portfolio using the F-F 3-factor Model rather than the CAPM).