This Part Of The Project Deals With Portfolio Returnrisk Calculations

This part of the project deals with portfolio return/risk calculations. To complete the project, you will hand in a set of answers to the questions listed below along with any supporting calculations and graphs. This project should be neat and well organized so that I can easily find your answers to each of the questions.

1. Data: This project makes use of annual data for two risky securities: the S&P 500 Index and Gold. Annual values for each of these securities during the 29-year period are provided in a spreadsheet named GroupProject1Data.xls. The spreadsheet is available on the class web page. You will also need an estimate of the annual risk-free rate. To get this rate, you should take the most recent annual rate on U.S. Government Securities (note: select the Treasury Security you feel is most relevant for a one-year investment horizon). These rates can be found on the web. You should specify your estimate of the annual Risk-Free rate, the date used to identify this interest rate, and the U.S. Treasury category used to identify this rate.

2. Return Calculations: Calculate annual returns for each of the two securities for each of the 28 years from 1976 through 2003. Calculate the average annual return, the standard deviation of annual returns, and the correlation between the returns of the two securities during this period and fill in the table provided. Attach the spreadsheet showing all relevant calculations as Exhibit 1.

3. Capital Allocation Lines: Assume that the mean return, standard deviation, and correlation estimates provide a reasonable forecast of expected returns and risks for the coming year. Plot the two risky securities on an expected return – standard deviation graph with the risk-free security. Label all three securities and draw the Capital Allocation Line (CAL) for each risky security. Attach the graph as Exhibit 2.

4. Risky Portfolios: Calculate the expected returns and standard deviations of portfolios combining the two risky securities with weights varying from 0% to 100% in 5% increments (resulting in 21 portfolios). Attach the relevant calculations as Exhibit 3.

5. Opportunity Set and the Optimal Risky Portfolio: Plot all 21 portfolios and the risk-free security on the expected return – standard deviation graph. Label the S&P 500, Gold, and the risk-free security. Identify and label the Minimum Variance Portfolio and the Optimal Risky Portfolio, and draw the CAL for the optimal portfolio. Attach the graph as Exhibit 4. Determine the portfolio weights in the Optimal Risky Portfolio, the standard deviation of the Minimum Variance Portfolio, and answer related questions.

6. Capital Allocation using the Optimal Risky Portfolio: Choose a target annual return between 2% and 10%. Calculate the portfolio weights in the risky and risk-free assets to achieve this return, calculate the portfolio's standard deviation, and illustrate this point on the graph from Exhibit 4. Fill in the table accordingly.

7. Sensitivity Analysis: (a) Repeat questions 4 and 5 assuming the correlation between the two risky securities is 0.30, using the same expected return and standard deviation forecasts. Attach the calculations and graph (Exhibits 5 and 6). (b) Repeat question 6 using the optimal risky portfolio from part (a). Fill in the table and compare the results. (c) Discuss how changes in correlation affect diversification and risk reduction based on your findings.

Paper For Above instruction

In this paper, we will explore the process of portfolio return and risk calculations, emphasizing the importance of diversification, correlation, and efficient asset allocation. The analysis begins with selecting two risky assets—namely, the S&P 500 Index and Gold—and calculating their historical returns, volatility, and correlation over a specified period (1976–2003). Using this data, we will estimate expected returns, construct the Capital Allocation Line (CAL), and determine optimal portfolio weights that maximize Sharpe ratio. Additionally, the sensitivity of the portfolio to changes in correlation coefficients will be assessed, demonstrating the impact of asset correlation on diversification benefits.

Initial data collection involves extracting annual returns for the S&P 500 and Gold, alongside the risk-free rate. Calculations of mean returns, standard deviations, and correlation coefficients enable us to forecast risk and return profiles of these assets. Graphically, plotting these securities on an expected return versus standard deviation graph illustrates their relative risk-return trade-offs and facilitates the visualization of the efficient frontier. The CAL emanating from the risk-free asset intersects this frontier, indicating the best possible combinations of the risky assets for given levels of risk.

Constructing portfolios with varying weights allocates different proportions to the S&P 500 and Gold, generating a set of possible risk-return combinations. These portfolios are plotted to determine the minimum variance portfolio—characterized by the lowest possible risk—and the optimal risky portfolio, which maximizes the Sharpe ratio. Critical points, such as the minimum variance and optimal risky portfolios, are identified and labeled on the graph. The optimal portfolio's weights are calculated using mean-variance optimization techniques, considering the expected returns, volatilities, and correlations.

Next, applying the Capital Asset Pricing Model (CAPM), we analyze the relationship between expected return and beta for each security. The Market Model further decomposes total risk into systematic (market-related) and firm-specific components, providing insights into the nature of risk. Regression analysis yields estimates of alpha and beta, while risk decomposition clarifies the proportion of risk attributable to market movements versus idiosyncratic factors.

The Security Market Line (SML) is graphed using the CAPM's insight that expected returns are linearly related to beta, with securities above the line considered undervalued and those below overvalued. The analysis discusses whether the international ETFs are properly priced according to CAPM assumptions and offers investment recommendations based on the pricing anomalies identified.

Finally, the sensitivity analysis explores how adjusting the correlation between assets influences the diversification benefits and optimal portfolio composition. When the correlation increases from the original estimate to 0.30, the portfolio's risk declines less with diversification, affecting the efficient frontier and the optimal weights. Re-evaluating the target returns under these changed assumptions illustrates the robustness of portfolio decisions under different correlation scenarios and reinforces the critical role of asset correlation in portfolio management.

References