Monthly Returns and Portfolio Mathematics Analysis ✓ Solved

1) Calculate the mean, variance, and standard deviation for the three given stocks GE, HD, and PG using the monthly returns data provided. Include the computations in appropriate spaces as shown in the assignment prompt.

2) Find the covariance between each pair of stocks: GE and HD, HD and PG, and PG and GE. Use the formulas for covariance with the monthly return data.

3) Calculate the mean, variance, and standard deviation for each of the three portfolios A, B, and C based on their weights in the stocks. Use the previously calculated means, variances, and covariances.

4) Given the risk-free rate of 0.21%, calculate the Sharpe ratio for each portfolio A, B, and C and determine which portfolio has the highest Sharpe ratio, indicating the best portfolio choice.

5) For Question 2, fill out the table for both the mean, variance, and standard deviation of the four stocks A, B, C, and D as given. Derive the expected returns and standard deviations from provided proportions in the portfolios.

6) Plot the relationship between risk (standard deviation) and return (expected return) for each portfolio.

7) Calculate the minimum-variance portfolio for each investment opportunity associated with the stocks.

8) For Question 3, using the CAPM method, run a regression analysis on Google returns against the market return and the risk-free rate to estimate its beta. Report the regression output.

9) Draw the Security Characteristic Line (SCL) equation from the regression and outline the relationship it depicts.

10) Estimate the expected return for Google based on the CAPM model, given a risk-free rate of 0.15% and market's expected return of 0.50% with the beta from your regression analysis.

Paper For Above Instructions

The analysis of monthly returns for stocks and their implications for portfolio management is an essential aspect of financial studies. The three stocks under consideration are General Electric (GE), Home Depot (HD), and Procter & Gamble (PG). Detailed calculations of returns will provide invaluable insights into the expected behavior of these stocks in various market conditions. This paper will go through the required calculations as stipulated in the assignment instructions.

Calculating Mean, Variance, and Standard Deviation

To begin with, we can calculate the mean, variance, and standard deviation for GE, HD, and PG using the available return data. Mean return can be calculated using the formula:

Mean = (Sum of monthly returns) / (Number of observations)

Following this method, we find the means for each stock. Then, the variance is calculated using:

Variance = [(Sum of (each return - mean)^2) / (Number of observations - 1)]

Standard deviation is the square root of variance:

Standard Deviation = √Variance

This series of calculations yields essential metrics that describe the dispersion of returns for the respective stocks.

Covariance Between Stocks

Next, we compute the covariance between these stocks. Covariance allows us to understand how two stocks move together. The formula for covariance is:

Cov(X, Y) = Σ[(X_i - mean(X)) * (Y_i - mean(Y))] / (n - 1)

Where X and Y represent the two stocks being compared. We will calculate the covariances between GE and HD, HD and PG, and PG and GE using the monthly return data.

Portfolio Calculations

With the means, variances, and covariances known, we proceed to evaluate the portfolios A, B, and C defined in the task. The weights of investments in each stock affect the overall expected return of the portfolio:

Portfolio Mean = (Weight of GE Mean of GE) + (Weight of HD Mean of HD) + (Weight of PG * Mean of PG)

The variance of the portfolio takes into account the weights and covariances:

Portfolio Variance = Σ(W_i^2 Var_i) + ΣΣ(W_i W_j * Cov(X_i, X_j))

Finally, we can find the portfolio standard deviation by taking the square root of the variance:

Portfolio Standard Deviation = √Portfolio Variance

The Sharpe ratio is then calculated as follows:

Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Portfolio Standard Deviation

A higher Sharpe ratio implies better risk-adjusted returns, and we will identify the portfolio with the highest Sharpe ratio.

Further Portfolio Analysis

Next, we analyze the second question regarding stocks A, B, C, and D by populating the required tables and drawing plots to illustrate the relationships between risk and returns. Utilizing the provided data points, we evaluate different portfolio combinations and their subsequent impacts on the mean return and standard deviation.

CAPM Analysis

In Question 3, we employ the Capital Asset Pricing Model (CAPM) to evaluate Google’s expected return relative to the market's return and the associated risk-free rate. Using regression analysis, we determine the beta of Google based on the provided market returns. The regression output allows us to inform stakeholders about Google's risk in the context of the broader market.

The Security Characteristic Line (SCL) can then be derived from the regression output, depicting the relationship between risk (beta) and return expected from Google, providing further insights into its performance characteristics.

Finally, we will apply the CAPM formula to calculate the expected return for Google, given a risk-free rate of 0.15% and an expected market return of 0.50%.

Conclusion

The analysis of these metrics provides critical insights for potential investors and portfolio managers. By reviewing the return and risk profiles of different stocks and portfolios, we can make informed decisions based on empirical data.

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