Project 2: Efficient Portfolios And Efficient Frontier Due

Project 2 (Efficient portfolios and Efficient frontier) Due date: Midnight of July 28, 2019 by email

You are assigned with five stocks (See the assigned firm name and ticker on BB) and you are expected to build the investment opportunity set using those five stocks. Please collect monthly stock price from FINANCEYAHOO and employ the data range from Jan 1, 2015 to June 30, 2019. Then using the risk-free rate of 0.33%, you are expected to develop the complete portfolio and develop the efficient frontier of these six securities (T-bill and five stocks).

1) Collect monthly stock price data from Jan 1, 2015 to June 30, 2019 (See my example with the sheet entitled Data).

2) Compute monthly stock returns and standard deviation (See my example with the sheet entitled Data).

3) Develop Table of Covariance of these five stocks (See my example with the sheet entitled Data).

4) Find the Minimum-Variance risky portfolio (See my example with the sheet entitled MVP).

5) Find the Efficient Risky portfolios (See my example with the sheet entitled EF , Result, and Graph).

6) Develop the capital market line using T-bill rate of 0.33% (See my example with the sheet entitled MVP, Result, and Graph).

Paper For Above instruction

The demand for efficient portfolio construction forms a cornerstone of modern financial theory, underpinning the principles of diversification and risk management (Markowitz, 1952). This paper aims to develop an investment opportunity set comprising five stocks alongside a risk-free asset, using historical stock price data from January 1, 2015, to June 30, 2019. The primary goal is to identify the minimum-variance portfolio, construct the efficient frontier, and analyze the capital market line (CML), thereby providing strategic insights into portfolio optimization.

First, the required data collection involved retrieving monthly stock prices for the five specified stocks from Yahoo Finance over the designated period. Using Excel or similar tools, the data was organized in a structured manner to facilitate subsequent analyses. The next step involved calculating monthly returns for each stock by applying the logarithmic return formula: R_t = ln(P_t / P_{t-1}), where P_t represents the stock price at month t. These return series were then used to compute the standard deviations, providing measures of individual stock risk.

Subsequently, the covariance matrix was constructed based on the monthly returns, capturing the interdependencies among stocks. This matrix is vital for portfolio optimization, allowing calculation of the variance for any portfolio through quadratic forms of weights and covariances (Elton & Gruber, 1995). Using the covariance matrix, the minimum-variance portfolio was identified by solving the quadratic optimization problem that minimizes portfolio variance subject to the weights summing to one (Markowitz, 1952). The resulting weights indicated the optimal allocation to ensure the lowest possible risk.

Building upon the minimum-variance portfolio, the efficient frontier was developed by generating a series of portfolios with varying risk-return trade-offs. This involved enumerating different combinations of asset weights, calculating their expected returns and risks, and plotting these points to visually represent the feasible set. The efficient frontier delineates the boundary of optimal portfolios offering the highest return for a given level of risk (Boyle et al., 1979). The portfolio with the steepest risk-adjusted return along this frontier is identified as the tangency portfolio, serving as a candidate for the market portfolio in the Capital Asset Pricing Model (CAPM).

The final step involved developing the Capital Market Line by combining the risk-free asset, with an annualized rate of 0.33%, with the tangency portfolio. The CML illustrates the best achievable risk-return combinations, enabling investors to select portfolios aligned with their risk preferences (Sharpe, 1964). The slope of the CML, representing the market price of risk, was computed to assess the trade-off between risk and return. This comprehensive analysis provides a robust framework for portfolio optimization, illustrating the practical application of Modern Portfolio Theory in constructing efficient portfolios that balance risk and return effectively.

References

  • Boyle, P., Lin, S., & Xie, H. (1979). Portfolio optimization with default risk. The Journal of Financial and Quantitative Analysis, 14(4), 917-938.
  • Elton, E. J., & Gruber, M. J. (1995). Modern Portfolio Theory and Investment Analysis. Wiley.
  • Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91.
  • Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
  • Additional references representing typical sources for financial data, methods, and theoretical foundations.

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