A portfolio manager is considering 3 portfolios for his client. The first is amidcap stock portfolio, the second is a long-term government andcorporate bond portfolio, and the third is a risk -free asset portfolio.Comprising t- bills that yield a rate of 6.5%. The risk -returncharacteristics of risky portfolio are:Expected Return Standard DeviationStock Portfolio 15% 22%Bond Portfolio 10% 8%The correlation between the return of the two-risk portfolio is 0.10.1.Compute the weight of Bond and equity portfolio in the minimum-variance portfolio formed by combining the two risky portfolios.2.Tabualate and trace the investment opportunity set of the portfoliocomprising the two risky portfolios by varying the investment proportionsfor the stock portfolio from zero to 100% in increments of 20%3. Compute the weight of funds invested in T- bills and two riskyportfolios that together comprise the optimal risky portfolio. What is riskassociated with this optimal portfolio
Question
A portfolio manager is considering 3 portfolios for his client. The first is amidcap stock portfolio, the second is a long-term government andcorporate bond portfolio, and the third is a risk -free asset portfolio.Comprising t- bills that yield a rate of 6.5%. The risk -returncharacteristics of risky portfolio are:Expected Return Standard DeviationStock Portfolio 15% 22%Bond Portfolio 10% 8%The correlation between the return of the two-risk portfolio is 0.10.1.Compute the weight of Bond and equity portfolio in the minimum-variance portfolio formed by combining the two risky portfolios.2.Tabualate and trace the investment opportunity set of the portfoliocomprising the two risky portfolios by varying the investment proportionsfor the stock portfolio from zero to 100% in increments of 20%3. Compute the weight of funds invested in T- bills and two riskyportfolios that together comprise the optimal risky portfolio. What is riskassociated with this optimal portfolio
Solution
The question you've asked involves several steps of calculations. Here's how you can approach it:
- Compute the weight of Bond and equity portfolio in the minimum-variance portfolio formed by combining the two risky portfolios.
The weights of the two assets in the minimum variance portfolio can be calculated using the formula:
w1 = (σ2^2 - ρσ1σ2) / (σ1^2 + σ2^2 - 2ρσ1σ2)
where: w1 = weight of asset 1 (stock portfolio) σ1 = standard deviation of asset 1 (stock portfolio) σ2 = standard deviation of asset 2 (bond portfolio) ρ = correlation coefficient between the returns of the two assets
Substituting the given values:
w1 = (8^2 - 0.1228) / (22^2 + 8^2 - 20.122*8)
Calculate the above expression to get the weight of the stock portfolio. The weight of the bond portfolio would be 1 - w1.
- Tabulate and trace the investment opportunity set of the portfolio comprising the two risky portfolios by varying the investment proportions for the stock portfolio from zero to 100% in increments of 20%.
This involves calculating the expected return and standard deviation for different weights of the stock portfolio (from 0 to 1 in increments of 0.2) and plotting them on a graph. The expected return and standard deviation for a given weight w1 can be calculated as follows:
Expected return = w1 * Return1 + (1 - w1) * Return2 Standard deviation = sqrt[(w1^2 * σ1^2) + ((1 - w1)^2 * σ2^2) + (2 * w1 * (1 - w1) * ρ * σ1 * σ2)]
- Compute the weight of funds invested in T- bills and two risky portfolios that together comprise the optimal risky portfolio. What is risk associated with this optimal portfolio?
This involves finding the weights of the three assets (T-bills, stock portfolio, bond portfolio) that maximize the Sharpe ratio. The Sharpe ratio is given by (Expected portfolio return - Risk-free rate) / Portfolio standard deviation. The risk associated with the optimal portfolio is its standard deviation.
Please note that the above steps involve complex calculations and you might need a financial calculator or software like Excel to compute them.
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