answer the queston
ERIC WANG
Chapter6solution.pptx
6. Efficient Diversification
Instructor: Seongcheol Paeng
7/17/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
Explain Covariance and Correlation in Asset Allocation with Two Risky Assets.
The scenario analysis in Spreadsheet 6.1 posits four possible scenarios for the economy: a severe recession, a mild recession, normal growth, and a boom. The last row of Spreadsheet 6.1 shows that the expected return of the stock fund is 10% and that of the bond fund is 5%. The variance is the probability-weighted average of the squared deviation of actual return from the expected return; the standard deviation is the square root of the variance. These values are computed in Spreadsheet 6.2. Suppose we form a portfolio with 40% in stocks and 60% in bonds. Notice that while the portfolio’s expected return is just the weighted average of the expected return of the two assets, the portfolio standard deviation is actually lower than that of either component fund. (6.1) The negative value for the covariance indicates that the two assets, on average, vary inversely; when one performs well, the other tends to perform poorly. (6.2) Correlation is a pure number and can range from −1 to +1. A correlation of −1 indicates that one asset’s return varies perfectly inversely with the other’s. The correlation coefficient of = −.49 in Equation 6.2 confirms the tendency of the returns on the stock and bond funds to vary inversely. In fact, a fraction of = .24 of the variance of stocks can be explained by the returns on bonds. Equation 6.2 shows that whenever the covariance is called for in a calculation we can replace it with the following expression using the correlation coefficient: (6.3)
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2. Explain the Mean-Variance Criterion.
These risk-return combinations are plotted in Figure 6.3. Investors desire portfolios that lie to the “northwest” in Figure 6.3. These are portfolios with high expected returns (toward the “north” of the figure) and low volatility (to the “west”). These preferences mean that we can compare portfolios using a mean-variance criterion in the following way: Portfolio A is said to dominate portfolio B if all investors prefer A over B. This will be the case if it has higher mean return and lower variance or standard deviation: and For example, the stock fund in Figure 6.3 dominates portfolio Z; the stock fund has higher expected return and lower volatility. Portfolios that lie below the minimum-variance portfolio in the figure can therefore be rejected out of hand as inefficient. Figure 6.4 shows the opportunity set with perfect positive correlation—a straight line through the component securities. No portfolio can be discarded as inefficient in this case, and the choice among portfolios depends only on risk aversion. Perfect positive correlation is the only case in which there is no benefit from diversification. Whenever ρ < 1, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities. Therefore, there are benefits to diversification whenever asset returns are less than perfectly positively correlated.
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Assignments
Problem Sets (Paraphrase with your own words.)
3. Explain optimal risky portfolio and draw the graph.
When we add the risk-free asset to a stock-plus-bond risky portfolio, the resulting opportunity set is the straight line that we called the CAL (capital allocation line). We now consider various CALs constructed from risk-free bills and a variety of possible risky portfolios, each formed by combining the stock and bond funds in alternative proportions. We start in Figure 6.5 with the opportunity set of risky assets constructed only from the bond and stock funds. The lowest-variance risky portfolio is labeled MIN (denoting the minimum-variance portfolio). is drawn through it and shows the risk-return tradeoff with various positions in T-bills and portfolio MIN. Spreadsheet 6.6 (see bottom panel of column E) shows that portfolio MIN’s expected return is 5.46% and its standard deviation (SD) is 7.80%. Portfolio A (row 10 in Spreadsheet 6.6) offers an expected return of 6% with an SD of 8.07%. (6.8) ; (6.9) optimal risky portfolio: The best combination of risky assets to be mixed with safe assets to form the complete portfolio.
. (6.10)
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3. Explain optimal risky portfolio and draw the graph.
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Problem Sets (Paraphrase with your own words.)
4. Figure 6.7 shows one possible choice for the preferred complete portfolio, C. The investor places 43% of wealth in portfolio O and 57% in Treasury bills. The rate of return and volatility of the portfolio are
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Problem Sets (Paraphrase with your own words.)
5. Explain the Efficient Frontier of Risky Assets, Choosing the Optimal Risky Portfolio, and the Preferred Complete Portfolio and a Separation Property.
the Efficient Frontier of Risky Assets
Now we can continue to take other points (each representing portfolios) from these three curves and further combine them into new portfolios, thus shifting the opportunity set even farther to the northwest. You can see that this process would work even better with more stocks. Moreover, the boundary or “envelope” of all the curves thus developed will lie quite away from the individual stocks in the northwesterly direction, as shown in Figure 6.10. The analytical technique to derive the efficient set of risky assets was developed by Harry Markowitz in 1951 and ultimately earned him the Nobel Prize in Economics. We sketch his approach here. The graph that connects all the northwesternmost portfolios is called the efficient frontier of risky assets. It represents the set of portfolios that offers the highest possible expected rate of return for each level of portfolio standard deviation. These portfolios may be viewed as efficiently diversified. One such frontier is shown in Figure 6.10. The three ways to draw the efficient frontier are (1) maximize the risk premium for any level of SD; (2) minimize the SD for any level of risk premium; and (3) maximize the Sharpe ratio for any level of SD (or risk premium).
7/17/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
5. Explain the Efficient Frontier of Risky Assets, Choosing the Optimal Risky Portfolio, and the Preferred Complete Portfolio and a Separation Property.
Choosing the Optimal Risky Portfolio
The second step of the optimization plan involves the risk-free asset. Using the current risk-free rate, we search for the capital allocation line with the highest Sharpe ratio (the steepest slope), as shown in Figures 6.5 and 6.6. The CAL formed from the optimal risky portfolio (O) will be tangent to the efficient frontier of risky assets discussed above. This CAL dominates all feasible CALs.
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Assignments
Problem Sets (Paraphrase with your own words.)
5. Explain the Efficient Frontier of Risky Assets, Choosing the Optimal Risky Portfolio, and the Preferred Complete Portfolio and a Separation Property.
The Preferred Complete Portfolio and a Separation Property
Finally, in the third step, each investor chooses the appropriate mix between the optimal risky portfolio (O) and T-bills, exactly as in Figure 6.7. A portfolio manager will offer the same risky portfolio (O) to all clients, no matter what their degrees of risk aversion. Risk aversion comes into play only when clients select their desired point on the CAL. Regardless of risk aversion, all clients will use portfolio O as the optimal risky investment vehicle. This result is called a separation property, introduced by James Tobin (1958), the 1983 Nobel Laureate for Economics: Its name reflects the fact that portfolio choice can be separated into two independent tasks. The first task, to determine the optimal risky portfolio (O), is purely technical. Given the input data, the best risky portfolio is the same for all clients regardless of risk aversion. The second task, construction of the complete portfolio from bills and portfolio O, is personal and depends on risk aversion. Here the client is the decision maker. When different managers use different input data, they will develop different efficient frontiers and offer different “optimal” portfolios.
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Problem Sets (Paraphrase with your own words.)
6. Explain the Single Index Model.
+ + (6.11)
Let us use to denote the excess return on a security, that is, the rate of return in excess of the risk-free rate: = -
is the excess return on a broad market index (the S&P 500 is commonly used for this purpose), so variation in this term reflects the influence of economywide or macroeconomic events that generally affect all stocks to greater or lesser degrees. The security’s beta, , is the typical response of that particular stock’s excess return to changes in the market index’s excess return. The term in Equation 6.11 represents the impact of firm-specific or residual risk. The expected value of is zero, as the impact of unexpected events must average out to zero. The term in Equation 6.11 is not a risk measure. Instead, represents the expected return on the stock beyond any return induced by movements in the market index. This term is called the security alpha. A positive alpha is attractive to investors and suggests an underpriced security: Among securities with identical sensitivity (beta) to the market index, securities with higher alpha values will offer higher expected returns. Because the firm-specific component of the stock return is uncorrelated with the market return, we can write the variance of the excess return of the stock as
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Assignments
Problem Sets (Paraphrase with your own words.)
6. Explain the Single Index Model.
Variance () = Variance ( + + )
= Variance ( ) + Variance ()
= + ()
= Systematic risk + Firm-specific risk (6.12)
Therefore, the total variance of the rate of return of each security is a sum of two components:
1. The variance attributable to the uncertainty of the entire market. This variance depends on both the variance of , denoted , and the beta of the stock on .
2. The variance of the firm-specific return, , which is independent of market performance.
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