Find minimum variance portfolio from the investment opportunity set drawn in (3). Specify the weights for stocks 1 and 2 for this portfolio. 10 6. Download risk-free rates (for example, 1-month Treasury yields) for the same period and same frequency with the two stocks. Calculate mean risk-free rate in your sample. By using this mean risk-free rate, draw the capital allocation line (CAL) and your new efficient frontier on the plane of expected returns and standard deviations. Find the tangency portfolio from the investment opportunity set drawn in (3). Specify the weights for stocks 1 and 2 for this portfolio.
First, we need to calculate the returns and variances of different portfolios consisting of stocks 1 and 2 from the investment opportunity set. We will consider various weight combinations (w1 and w2) for the two stocks to form these portfolios. The formula for the portfolio return (Rp) and portfolio variance (σ^2p) for a two-asset portfolio is as follows:
Rp = w1 * R1 + w2 * R2
σ^2p = (w1^2 * σ1^2) + (w2^2 * σ2^2) + 2 * w1 * w2 * ρ12 * σ1 * σ2
Where:
Rp is the portfolio return.
σ^2p is the portfolio variance.
R1 and R2 are the returns of stocks 1 and 2, respectively.
σ1 and σ2 are the standard deviations (volatility) of stocks 1 and 2, respectively.
ρ12 is the correlation coefficient between stocks 1 and 2.
Step 2: Find the Minimum Variance Portfolio
The minimum variance portfolio is the portfolio with the lowest portfolio variance. To find it, we need to calculate the portfolio variances for various weight combinations (w1 and w2) and identify the one with the smallest variance. The weights for stocks 1 and 2 for this portfolio will be the values that minimize the portfolio variance.
Step 3: Calculate Mean Risk-Free Rate
To construct the Capital Allocation Line (CAL) and the efficient frontier, we need the risk-free rate. You mentioned using 1-month Treasury yields. Find historical data for these yields for the same period as the stock returns data. Calculate the mean risk-free rate over this period.
Step 4: Construct the CAL
The Capital Allocation Line (CAL) is a linear relationship between expected portfolio returns and portfolio standard deviations. It is given by the formula:
E(Rp) = Rf + [(E(Rm) – Rf) / σm] * σp
Where:
E(Rp) is the expected portfolio return.
Rf is the risk-free rate.
E(Rm) is the expected return of the market portfolio.
σm is the standard deviation of the market portfolio.
σp is the standard deviation of the portfolio.
Step 5: Construct the Efficient Frontier
The efficient frontier is the set of portfolios that offer the highest expected returns for a given level of risk (standard deviation). To construct it, we will vary the portfolio weights (w1 and w2) for the two stocks and calculate the expected returns and standard deviations of each portfolio.
Step 6: Find the Tangency Portfolio
The tangency portfolio is the portfolio on the efficient frontier that is tangent to the CAL. It represents the optimal combination of risky assets (stocks 1 and 2) and the risk-free asset. To find it, locate the portfolio with the highest Sharpe ratio on the efficient frontier.
Once you have identified the tangency portfolio, you can specify the weights for stocks 1 and 2 in this portfolio.
By following these steps, you will be able to find the minimum variance portfolio, construct the CAL and efficient frontier, and identify the tangency portfolio based on the given data and risk-free rates.
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