MFIN1151 Lecture Notes - Portfolio Diversification PDF

Topic 3. Portfolio Diversification Chapter 6 of the textbook Complete Version © McGraw Hill, LLC 1  Topics Covered Part One: Review of asset allocation problem with a risk- free asset and ONE risky...

Topic 3. Portfolio Diversification Chapter 6 of the textbook Complete Version © McGraw Hill, LLC 1  Topics Covered Part One: Review of asset allocation problem with a risk- free asset and ONE risky asset Part Two: Asset allocation problem with TWO risky assets Part Three: Asset allocation problem with a risk-free asset and TWO risky assets Part Four: Efficient frontier with Many risky assets Part Five: Asset allocation problem with a risk-free asset and Many risky assets Reading: textbook Chapter 6 © McGraw Hill, LLC 2  1. Review of asset allocation problem with a risk-free asset and ONE risky asset © McGraw Hill, LLC 3  Sharpe Ratio Sharpe (Reward-to-Volatility) Ratio: ratio of portfolio risk premium to standard deviation Portfolio Risk Premium 𝐸 𝑟𝑝 − 𝑟𝑓 𝑆𝑃 = = Standard Deviation of Excess Returns 𝜎𝑃 𝐸 𝑟𝑝 = Expected Return of the portfolio 𝑟𝑓 = Risk Free rate of return 𝜎𝑃 = Standard Deviation of portfolio excess return (in excess of risk-free rate) © McGraw Hill, LLC 4  Capital allocation line (CAL) Capital allocation line (CAL) is the line depicts the risk-return combinations available by choosing different values of y, that is, by varying capital allocation. If investors can borrow at the risk-free rate, the complete portfolio has the same Sharpe ratio, which is equal to the Sharpe ratio of the risky asset, regardless the value of weight y. The asset allocation line has the same slope everywhere. © McGraw Hill, LLC 5  Capital allocation line (CAL) If investors borrows at a rate higher than the risk-free rate, the borrowing cost increases, leading to a lower expected return on the complete portfolio. Consequently, the Sharpe ratio (slope) decreases, and the CAL becomes kinked, with a flatter slope on the leveraged portion. © McGraw Hill, LLC 6  2. Asset allocation problem with TWO risky assets © McGraw Hill, LLC 7  Diversification with two risky assets Example: We have two risky stocks. The return of Umbrella stock is 50% in rainy scenario and -25% in sunny scenario. The return of Resort stock is -20% in rainy scenario and 50% in sunny scenario. Assume the risk-free rate is 5%. What is the Sharpe ratio for each stock? If you invest 40% of weight in Umbrella stock and 60% of weight in Resort stock, what is the Sharpe ratio of your complete portfolio with two risky assets? © McGraw Hill, LLC 8  Diversification with two risky assets  Holding only Umbrella stock: Expected Return = 0.5 × 50% + 0.5 × (-25%) =12.5% Standard Deviation = 0.5 × (0.5 − 0.125)2 +0.5 × (−0.25 − 0.125)2 = 37.5% Sharpe Ratio = (12.5% - 5%) / 37.5% = 0.2  Holding only Resort stock: Expected Return = 0.5 × (-20%) + 0.5 × 50% = 15% Standard Deviation = 0.5 × (−0.2 − 0.15)2 +0.5 × (0.5 − 0.15)2 = 35% Sharpe Ratio = (15% - 5%) / 35% = 0.29 © McGraw Hill, LLC 9  Diversification with two risky assets  Portfolio with two risky assets (40% in Umbrella and 60% in Resort): Step 1: calculate the return of the complete portfolio under each scenario, using the weight of each stock in your portfolio (e.g., 0.4 for Umbrella and 0.6 for Resort) Step 2: calculate the expected return of the complete portfolio 𝐸 𝑟 , using the probability of each scenario (e.g., 0.5 for raining season and 0.5 for sunny season) Step 3: calculate the variance 𝜎 2 and the standard deviation 𝜎 of the complete portfolio, using the probability of each scenario (e.g., 0.5 for raining season and 0.5 for sunny season) Step 4: calculate the Sharpe ratio, using the expected return and the standard deviation calculated form step 2 and step 3. © McGraw Hill, LLC 10  Diversification with two risky assets  The portfolio with two risky assets (40% in Umbrella and 60% in Resort): Step 1: Return for rainy season = 0.4 × 50% + 0.6 × (-20%) = 8% Return for sunny season = 0.4 × (-25%) + 0.6 × 50% = 20% Step 2: Expected Return = 0.5 × 8% + 0.5 × 20% =14% Step 3: Standard Deviation = 0.5 × (0.08 − 0.14)2 +0.5 × (0.2 − 0.14)2 = 6% Step 4: Sharpe Ratio = (14% - 5%) / 6% = 1.5  Holding a complete portfolio with 40% in Umbrella and 60% in Resort has a better Sharpe ratio than investing 100% in Umbrella or 100% in Resort, mainly because the complete portfolio has a much smaller standard deviation.  Why is holding both two risky stocks better than holding individual stock? © McGraw Hill, LLC 11  Covariance and Correlation  The portfolio with two risky assets (40% in Umbrella and 60% in Resort): Suppose that the rate of return and standard deviation on two risky asset, A and B, are (𝑟𝐴 , 𝜎𝐴 ) and (𝑟𝐵 , 𝜎𝐵 ) respectively. Assume there exist several scenarios, and each scenario has a probability 𝑝 to happen.  Covariance: 𝐶𝑜𝑣(𝑟𝐴 , 𝑟𝐵 ) =σ𝑠 𝑝(𝑠)[𝑟𝐴 (𝑠) − 𝐸(𝑟𝐴 )][ 𝑟𝐵 (𝑠) − 𝐸(𝑟𝐵 )] The covariance between the two returns of Umbrella stock and Resort stock: 𝐶𝑜𝑣 𝑟𝐴 , 𝑟𝐵 =0.5 × (50% - 12.5%) × (-20% - 15%) + 0.5 × (-25% - 12.5%)× (50% -15%) = -0.13125 𝐶𝑜𝑣(𝑟𝐴 ,𝑟𝐵 )  Correlation Coefficient: 𝜌𝐴𝐵 = and 𝐶𝑜𝑣 𝑟𝐴 , 𝑟𝐵 = 𝜌𝐴𝐵 𝜎𝐴 𝜎𝐵 𝜎𝐴 𝜎𝐵 𝐶𝑜𝑣(𝑟𝐴 ,𝑟𝐵 ) −0.13125 𝜌= = = −1 𝜎𝐴 𝜎𝐵 0.375×0.35 © McGraw Hill, LLC 12  Covariance and Correlation  Correlation coefficient 𝜌 ranges from −1 to +1.  A correlation of +1 indicates perfect positive correlation. With a correlation of +1, you could predict 100% of the variability of one asset’s return. E.g., if stock X increases by 5%, then stock Y will also increase by 5%.  A correlation of −1 indicates perfect negative correlation, meaning that one asset’s return varies perfectly inversely with the other’s return. With a correlation of −1, you could also predict 100% of the variability of one. E.g., if stock X increases by 5%, then stock Y will also decrease by 5%.  A correlation of 0 indicates that the returns on the two assets are unrelated. © McGraw Hill, LLC 13  Covariance and Correlation For the returns of two risky assets, we normally have −1< 𝜌 < 1, because it is very rare for two assets’ returns to move in the exactly same or opposite direction. As long as the correlation coefficient 𝜌 is not equal to +1, we can achieve some degree of diversification. A lower correlation improves diversification, and a higher correlation reduces the effect of diversification. When the correlation coefficient 𝜌 is negative, diversification reduces risk significantly. © McGraw Hill, LLC 14  Market Risk and Firm-Specific Risk Market risk (systematic risk or non-diversifiable risk) It is the risk that remains after diversification and is attributable to risk sources common to the whole economy. E.g., business cycle such as economic recession. It affects all security returns and affects them in the same direction. It leads to positive correlation among securities. Firm-specific risk (non-systematic risk or diversifiable risk) It is the risk that can be eliminated by diversification. E.g., weather, oil price rise, management success, marketing achievement It only affects some stock returns but not the others and affects different stock returns in different direction. It leads to zero or negative correlation among securities. Diversification can only eliminate firm-specific risk, but not market risk. © McGraw Hill, LLC 15  Diversification and Risk When n increases (i.e., the number of different stocks in the portfolio increases), the firm-specific risk (i.e., diversifiable risk) can be reduced to 0, but the market risk won’t be reduced. © McGraw Hill, LLC 16  Weights on each risky asset How do the weights on each risky asset affect the portfolio’s return and risk? Return and standard deviation on risky assets A and B are 𝑟𝐴 , 𝜎𝐴 , 𝑟𝐵 , 𝜎𝐵 respectively. The risky portfolio P includes two risky assets A and B. How do the weights 𝑤𝐴 and 𝑤𝐵 affect the portfolio’s 𝐸 𝑟𝑃 and 𝜎𝑃 ? © McGraw Hill, LLC 17  Rules of two-risky-assets portfolios Three calculation rules of two-risky-assets portfolios Rule 1: The rate of return on a portfolio is the weighted average of returns on the component securities. The weights are the portfolio proportions. 𝑟𝑃 = 𝑤𝐴 𝑟𝐴 + 𝑤𝐵 𝑟𝐵 Rule 2: The expected rate of return on a portfolio is the weighted average of expected returns on the component securities. The weights are the portfolio proportions. 𝐸 𝑟𝑃 = 𝑤𝐴 𝐸 𝑟𝐴 + 𝑤𝐵 𝐸 𝑟𝐵 Rule 3: The variance of the rate of return on a two-risky-asset portfolio is the sum of the contributions of the component security variances 𝜎𝐴2 and 𝜎𝐵2 plus a term that involves the correlation coefficient 𝜌𝐴𝐵 (or covariance) between the returns on the component securities. The standard deviation is the square root of variance. 𝜎𝑃2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝐴 𝑤𝐵 𝐶𝑜𝑣 𝑟𝐴 , 𝑟𝐵 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝐴 𝑤𝐵 𝜎𝐴 𝜎𝐵 𝜌𝐴𝐵 𝜎𝑃 = 𝜎𝑃2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝐴 𝑤𝐵 𝜎𝐴 𝜎𝐵 𝜌𝐴𝐵 © McGraw Hill, LLC 18  Rules of two-risky-assets portfolios When the correlation coefficient between the component securities is lower (smaller or more negative), there will be a greater tendency for returns on the two securities to offset each other. Hence, diversification will reduce portfolio variance, standard deviation and portfolio risk, and increases portfolio’s Sharpe ratio. Special cases for Rule 3: If 𝜌𝐴𝐵 = 1, then 𝜎𝑃2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝐴 𝑤𝐵 𝜎𝐴 𝜎𝐵 = (𝑤𝐴 𝜎𝐴 + 𝑤𝐵 𝜎𝐵 )2 𝜎𝑃 = (𝑤𝐴 𝜎𝐴 + 𝑤𝐵 𝜎𝐵 )2 = |𝑤𝐴 𝜎𝐴 + 𝑤𝐵 𝜎𝐵 | If 𝜌𝐴𝐵 = -1, then 𝜎𝑃2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 − 2𝑤𝐴 𝑤𝐵 𝜎𝐴 𝜎𝐵 = (𝑤𝐴 𝜎𝐴 − 𝑤𝐵 𝜎𝐵 )2 𝜎𝑃 = (𝑤𝐴 𝜎𝐴 − 𝑤𝐵 𝜎𝐵 )2 = |𝑤𝐴 𝜎𝐴 − 𝑤𝐵 𝜎𝐵 | Since (𝑎𝑥 + 𝑏𝑦)2 = 𝑎2 𝑥 2 + 𝑏2 𝑦 2 + 2𝑎𝑏𝑥𝑦 and (𝑎𝑥 − 𝑏𝑦)2 = 𝑎2 𝑥 2 + 𝑏2 𝑦 2 − 2𝑎𝑏𝑥𝑦 © McGraw Hill, LLC 19  Investment opportunity set Investment opportunity set is the set of all available combinations of risk and return offered by portfolios formed with different proportions in risky assets. Example: conduct the asset allocation with two risky assets. One risky asset is a stock, which has an expected return of 10%, denoted as 𝐸 𝑟𝑆 , and a standard deviation of 19%, denoted as 𝜎𝑆. The other risky asset is a bond, which has an expected return of 5%, denoted as 𝐸 𝑟𝐵 , and a standard deviation of 8%, denoted as 𝜎𝐵. Assume the correlation coefficient between the two risky assets is 0.2, denoted as 𝜌𝑆𝐵. Construct a portfolio that consists of these two risky assets and show the investment opportunity set of different proportions in each risky asset. © McGraw Hill, LLC 20  Investment opportunity set A B C D 1 Stock Bond 2 𝐸 𝑟𝑆 𝐸 𝑟𝐵 𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 3 10% 5% 19% 8% 0.2 4 Portfolio Weights Expected Return 𝐄 𝒓𝑷 Std Dev 𝝈𝑷 5 𝑊𝑆 = 1 − 𝑊𝐵 𝑊𝐵 Col A*$A$3 + Col B*$B$3 (Rule 2) Rule 3 6 -0.2 1.2 4% 9.59% 7 -0.1 1.1 4.5% 8.62% 100% weight in bond 8 0 1 5% 8% Minimum-variance portfolio 9 0.092 0.908 5.46% 7.804% 10 0.1 0.9 5.5% 7.81% Example on next slide 11 0.2 0.8 6% 8.07% 12 0.3 0.7 6.5% 8.75% 13 0.4 0.6 7% 9.77% 14 0.5 0.5 7.5% 11.02% 15 0.6 0.4 8% 12.44% 16 0.7 0.3 8.5% 13.98% 17 0.8 0.2 9% 15.6% 18 0.9 0.1 9.5% 17.28% 100% weight in stock 19 1 0 10% 19% 20 1.1 -0.1 10.5% 20.75% 21 1.2 -0.2 11% 22.53% © McGraw Hill, LLC 21  Investment opportunity set Investment opportunity set is the set of all available combinations of risk and return offered by portfolios formed with different proportions in risky assets. The spreadsheet above shows a part of the investment opportunity set of the two-risky-assets portfolio, a stock (S) and a bond (B), as there are infinite divisions between every two numbers. Negative weights mean short position. (Spreadsheet 6.5 in the textbook chapter 6) Calculation example for row 10 (marked by the light blue color) Rule 2: 𝐸 𝑟𝑃 = 𝑤𝑆 𝐸 𝑟𝑆 + 𝑤𝐵 𝐸 𝑟𝐵 Rule 3: 𝜎𝑃2 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝐶𝑜𝑣 𝑟𝑆 , 𝑟𝐵 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 𝐸 𝑟𝑃 = 0.1×10% + 0.9×5% = 5.5% 𝜎𝑃 = 0.12 × 19%2 + 0.92 × 8%2 + 2 × 0.1 × 0.9 × 19% × 8% × 0.2 = 7.81% © McGraw Hill, LLC 22  Investment opportunity set How to graph the investment opportunity set with the portfolio expected return and standard deviation corresponding to each allocation? Step 1: Calculate possible combinations of 𝐸 𝑟𝑃 and 𝜎𝑃 by changing the weights in the Excel. See the associated Excel sheet (Note: Excel sheet operation will not be tested in the exams). Step 2: Create a diagram with X-axis as 𝜎𝑃 and the Y-axis as 𝐸 𝑟𝑃 by inserting a scatter plot chart in Excel. Step 3: Connect all the dots together. The curve or the line is the graph of investment opportunity set. © McGraw Hill, LLC 23  Investment opportunity set © McGraw Hill, LLC 24  Investment opportunity set Bonds portfolio (row 8, 0% in stock, 100% in bond): 𝐸 𝑟𝑃 = 5% and 𝜎𝑃 = 8% Points to the lower right of Bonds portfolio (row 7, -10% in stock, 110% in bond): 𝐸 𝑟𝑃 = 4.5% and 𝜎𝑃 = 8.62% Stocks portfolio (row 19, 100% in stock, 0% in bond): 𝐸 𝑟𝑃 = 10% and 𝜎𝑃 = 19% Points to the upper right of Stocks portfolio (row 20, 110% in stock, -10% in bond): 𝐸 𝑟𝑃 = 10.5% and 𝜎𝑃 = 20.75% © McGraw Hill, LLC 25  Minimum variance portfolio The risk-minimizing proportions are 9.2% in stock and 90.8% in bond. With these proportions, the portfolio standard deviation will be 7.804%, and the portfolio’s expected return will be 5.46%. This is called minimum variance portfolio. The weights are calculated using the formulas below. 2 2 𝜎𝐵 −𝐶𝑜𝑣(𝑟𝑆 ,𝑟𝐵 ) 𝜎𝐵 −𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 𝑤𝑆(𝑚𝑖𝑛) = = 𝜎2+𝜎2 −2𝜎 𝜎𝑆2 +𝜎𝐵2 −2𝐶𝑜𝑣(𝑟 ,𝑟 ) 𝑆 𝐵 𝑆 𝐵 𝑆 𝜎𝐵 𝜌𝑆𝐵 2 2 𝜎𝐵 −0 𝜎𝐵 When 𝜌𝑆𝐵 = 0, then 𝑤𝐴(𝑚𝑖𝑛) = 𝜎2+𝜎2 −0 = 𝜎2+𝜎2 𝑆 𝐵 𝑆 𝐵 𝑤𝐵(𝑚𝑖𝑛) = 1 − 𝑤𝑆(𝑚𝑖𝑛) Question: Is this risk-minimizing portfolio (with 9.2% in stock and 90.8% in bond) preferable to others (e.g., the Stock portfolio with 100% in stock and 0% in bond)? © McGraw Hill, LLC 26  Minimum variance portfolio Question: Is this risk-minimizing portfolio (with 9.2% in stock and 90.8% in bond) preferable to others (e.g., the Stock portfolio with 100% in stock and 0% in bond)? It depends on investor’s risk aversion, i.e., how the investor trades off risk against return, because the risk-minimizing portfolio has a lower risk but also has a lower expected return compared to other stock portfolios. © McGraw Hill, LLC 27  The Mean-Variance Criterion Investors desire portfolios with high expected returns (mean) and low volatility (variance), i.e., lie to the northwest in the figure. If portfolio A has higher mean return and lower variance or standard deviation than portfolio B, i.e., 𝐸 𝑟𝐴 ≥ 𝐸 𝑟𝐵 and 𝜎𝐴 ≤ 𝜎𝐵 , then portfolio A dominates portfolio B, and all investors prefer A over B. Graphically, portfolio A will lie to the northwest of B. Given a choice between portfolios A and B, all investors would choose A. Portfolios that lie below the minimum-variance portfolio (on the downward sloping portion) can be rejected as inefficient. Any portfolio on the downward-sloping portion of the curve is dominated by the portfolio that lies directly above it on the upward-sloping portion of the curve, because the portfolio lies directly above it has higher expected return and equal standard deviation. © McGraw Hill, LLC 28  The Mean-Variance Criterion The best choice does not exist among the portfolios on the upward sloping portion of the curve, if there are two risky assets. This is because portfolios with higher expected return also have greater risk, and investors with different level of risk aversion prefer different portfolio choice. The best choice exists among the portfolios on the upward sloping portion of the curve, if there are one risk-free asset and two risky assets. We will study the best choice on the upward sloping portion of the curve when we introduce a risk-free asset to the two-risky-asset portfolio in Part Three of this topic. © McGraw Hill, LLC 29  The Mean-Variance Criterion In the example above, we have assumed a correlation 𝜌𝑆𝐵 of 0.2 between stock and bond returns. If 𝜌𝑆𝐵 = 1 (perfect positive correlation between bonds and stocks), then Rule 2: 𝐸 𝑟𝑃 = 𝑤𝑆 𝐸 𝑟𝑆 + 𝑤𝐵 𝐸 𝑟𝐵 Rule 3: 𝜎𝑃2 = 𝑤𝑆2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝐶𝑜𝑣 𝑟𝑆 , 𝑟𝐵 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 𝜎𝑃2 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 = (𝑤𝑆 𝜎𝑆 + 𝑤𝐵 𝜎𝐵 )2 𝜎𝑃 = (𝑤𝑆 𝜎𝑆 + 𝑤𝐵 𝜎𝐵 )2 = |𝑤𝑆 𝜎𝑆 + 𝑤𝐵 𝜎𝐵 | The portfolio standard deviation is a weighted average of the component security standard deviations. Both the portfolio expected return and standard deviation are simple weighted averages. In this (and only this) case, diversification has no use. © McGraw Hill, LLC 30  The Mean-Variance Criterion The figure of investment opportunity set with perfect positive correlation (𝜌𝑆𝐵 = 1) is an upward sloping straight line through the component securities. No portfolio can be rejected as inefficient in this case, and the choice among portfolios depends on risk aversion. Perfect positive correlation (𝜌 = 1) is the only case that diversification generates no benefit. (See the Excel sheet for different correlation coefficients) © McGraw Hill, LLC 31  The Mean-Variance Criterion Whenever 𝜌 < 1, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities. When correlation is negative, there will be even greater diversification benefits. The more negatively correlated are the two assets, the greater the diversification benefits. With perfectly negative correlation (𝜌 = -1), diversification has the most benefits 𝜎𝑃2 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 = 𝑤𝑆2 𝜎𝑆2 + 𝑤𝐵2 𝜎𝐵2 − 2𝑤𝑆 𝑤𝐵 𝜎𝑆 𝜎𝐵 𝜎𝑃 = (𝑤𝑆 𝜎𝑆 − 𝑤𝐵 𝜎𝐵 )2 = |𝑤𝑆 𝜎𝑆 − 𝑤𝐵 𝜎𝐵 | © McGraw Hill, LLC 32  The Mean-Variance Criterion When the returns of two risky assets are perfectly negatively related to each other, diversification can reduce the standard deviation to zero. Minimum variance portfolio with perfect negative correlation (𝜌 = -1) 2 2 𝜎𝐵 −𝐶𝑜𝑣(𝑟𝑆 ,𝑟𝐵 ) 𝜎𝐵 −𝜎𝑆 𝜎𝐵 𝜌𝑆𝐵 𝑤𝑆(𝑚𝑖𝑛) = 2 2 −2𝐶𝑜𝑣(𝑟 ,𝑟 ) = 2 2 −2𝜎 𝜎 𝜌 = 0.2963, and 𝑤𝐵(𝑚𝑖𝑛) = 1 − 𝑤𝑆(𝑚𝑖𝑛) 𝜎𝑆 +𝜎𝐵 𝑆 𝐵 𝜎𝑆 +𝜎𝐵 𝑆 𝐵 𝑆𝐵 𝐸 𝑟𝑃 = 𝑤𝑆(𝑚𝑖𝑛) 𝐸 𝑟𝑆 + 𝑤𝐵(𝑚𝑖𝑛) 𝐸 𝑟𝐵 = 6.48 𝜎𝑃 = |𝑤𝑆(𝑚𝑖𝑛) 𝜎𝑆 − 𝑤𝐵(𝑚𝑖𝑛) 𝜎𝐵 | = 0 All the portfolio allocations with expected return less than 6.48% (i.e., all the points to the lower right of turning point or on the downward sloping straight line) can be eliminated. © McGraw Hill, LLC 33  3. Asset Allocation with a Risk-Free Asset and TWO Risky Assets © McGraw Hill, LLC 34  Optimal Risky Portfolio We now include a risk-free asset. When choosing their capital allocation between the risk-free asset F and the risky portfolio P that contains two risky assets A and B, investors want the risky portfolio that offers the highest reward- to-volatility or Sharpe ratio. When we add the risk-free asset (T-bills yielding 3%) to a risky portfolio constructed from stocks and bonds, the investment opportunity set is the straight line of CAL (capital allocation line) connecting the risk-free asset and the risky portfolio including two risky assets, stocks and bonds. © McGraw Hill, LLC 35  Optimal Risky Portfolio Portfolio A is better than MIN (minimum-variance portfolio) as the risky portfolio. CALA dominates CALMIN because it offers a higher slope (i.e., a higher Sharpe ratio). © McGraw Hill, LLC 36  Optimal Risky Portfolio We continue to move the CAL upward until it reaches the ultimate point of tangency with the investment opportunity set. The tangency portfolio is the optimal risky portfolio (denoted as portfolio O) to mix with risk-free asset, because it produces the steepest CAL with the highest feasible Sharpe ratio. © McGraw Hill, LLC 37  Optimal Risky Portfolio Portfolio weights between risky assets A and B for the optimal risky portfolio (i.e., tangent portfolio): WA is the weight for stock, WB and is the weight for bond. The risky portfolio constructed using the following weights produces the tangent portfolio, which has the highest possible Sharpe ratio (i.e., the steepest CAL) 2 [𝐸(𝑟𝐴 )−𝑟𝑓 ]𝜎𝐵 −[𝐸(𝑟𝐵 )−𝑟𝑓 ]𝐶𝑜𝑣(𝑟𝐴 ,𝑟𝐵 ) 𝑤𝐴 = [𝐸(𝑟 2 2 𝐴 )−𝑟𝑓 ]𝜎𝐵 +[𝐸(𝑟𝐵 )−𝑟𝑓 ]𝜎𝐴 −[𝐸(𝑟𝐴 )−𝑟𝑓 +𝐸(𝑟𝐵 )−𝑟𝑓 ]𝐶𝑜𝑣(𝑟𝐴 ,𝑟𝐵 ) 𝑤𝐵 = 1 − 𝑤𝐴 The portfolio has the following expected return and variance 𝐸 𝑟𝑃 = 𝑤𝐴 𝐸 𝑟𝐴 + 𝑤𝐵 𝐸 𝑟𝐵 𝜎𝑃2 = (𝑤𝐴 𝜎𝐴 )2 +(𝑤𝐵 𝜎𝐵 )2 +2 𝑤𝐴 𝜎𝐴 𝑤𝐵 𝜎𝐵 𝜌𝐴𝐵 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝐴 𝑤𝐵 𝐶𝑜𝑣(𝑟𝐴 , 𝑟𝐵 ) © McGraw Hill, LLC 38  Steps in constructing complete portfolio 1. Specify the characteristics of all securities (expected returns, variances, covariance) 2. Figure out how to establish the optimal risky portfolio O on the investment opportunity set (i.e., tangent portfolio on the upward sloping curve) by using the weight on each risky asset A and B (i.e., WA and WB). 3. Allocate funds between the optimal risky portfolio and the risk-free asset to construct complete portfolio. Any point on the tangent CAL can be taken as the allocation between the optimal risky portfolio and risk-free asset (i.e., y depends on investors' risk attitude A). An investor with a higher risk tolerance will put more weight on the optimal risky portfolio. © McGraw Hill, LLC 39  Separation property of investment decision Portfolio choice can be separated into two independent tasks. Task 1 (risky portfolio selection): Determine the optimal risky portfolio O containing two risky assets A and B (optimal for every investor regardless his/her risk attitude) on the upward sloping curve of investment opportunity set, i.e., determine the asset allocation between the two risky assets (Asset A and Asset B, or stock and bond) 2 [𝐸(𝑟𝐴 )−𝑟𝑓 ]𝜎𝐵 −[𝐸(𝑟𝐵 )−𝑟𝑓 ]𝐶𝑜𝑣(𝑟𝐴 ,𝑟𝐵 ) 𝑤𝐴 = 2 +[𝐸(𝑟 )−𝑟 ]𝜎2 −[𝐸(𝑟 )−𝑟 +𝐸(𝑟 )−𝑟 ]𝐶𝑜𝑣(𝑟 ,𝑟 ) [𝐸(𝑟𝐴 )−𝑟𝑓 ]𝜎𝐵 𝐵 𝑓 𝐴 𝐴 𝑓 𝐵 𝑓 𝐴 𝐵 𝑤𝐵 = 1 − 𝑤𝐴 Task 2 (asset allocation between risk-free asset and optimal risky portfolio): Construct the complete portfolio from risk-free asset F and this optimal risky portfolio O on the Capital Allocation line (dependent on investors' risk attitude) 𝐸(𝑟𝑂 )−𝑟𝑓 𝑦= 2 𝐴𝜎𝑂 in the optimal risky portfolio (from Topic 3), also denoted as 𝑤𝑂 𝑤𝑅𝑖𝑠𝑘−𝑓𝑟𝑒𝑒 = 1 − 𝑦 in the risk-free asset, also denoted as 1- 𝑤𝑂 © McGraw Hill, LLC 40  Ultimate weights Example: Suppose that the optimal tangent risky portfolio consists of 56.8% bond fund and 43.2% stock fund. Assume that an investor places 55% of her wealth in the tangent risky portfolio O and 45% in T-bills (risk-free asset). What will be the overall asset allocations of her complete portfolio? Weight in risk-free asset 45% Weight in bond fund 0.568×55%=31.24% Weight in stock fund 0.432×55%=23.67% Total 100% © McGraw Hill, LLC 41  4. Efficient Frontier with Many Risky Assets © McGraw Hill, LLC 42  How does additional risky assets improve investment opportunities? Say we have three risky assets, stock A, B, and C. Points A, B, and C represent the expected returns and standard deviations of three stocks. The curve passing through A and B shows the investment opportunity set of portfolios formed from those two stocks. The point E on the AB curve is one portfolio chosen from the combination of stock A and stock B. The curve passing through B and C shows the investment opportunity set of portfolios formed from those two stocks. The point F on the BC curve is one portfolio chosen from the combination of stock B and stock C. The curve that passes through point E and F represents the investment opportunity set of portfolios constructed from portfolios E and F. Portfolios E and F are constructed from stock A, B, and C, this curve shows the investment opportunity set of portfolios constructed from these three stocks. The curve EF extends the investment opportunity set to the desired northwest. © McGraw Hill, LLC 43  How does additional risky assets improve investment opportunities? We can continue to take points (each representing portfolios) from these three curves and combine them into another new portfolio, thus shifting the investment opportunity set further to the northwest. E.g., take point A and a point on the top of curve EF to form a new portfolio with an investment opportunity set further to the northwest. © McGraw Hill, LLC 44  How does additional risky assets improve investment opportunities? The boundary of all the curves will lie quite away from the individual stocks in the northwesterly direction Efficient frontier is the graph that connects all the northwestern-most portfolios, which is the best investment opportunity set for many risky assets. It represents the set of portfolios that offers the highest possible portfolio expected rate of return for each level of portfolio standard deviation. These are efficiently diversified portfolios. The tangent portfolio on the efficient frontier is the most optimal portfolio. © McGraw Hill, LLC 45  5. Asset allocation problem with a risk-free asset and Many risky assets © McGraw Hill, LLC 46  A risk-free asset and Many risky assets Investors choose the optimal risky portfolio (the one that is the tangent point and has the highest Sharpe Ratio), and hold a complete portfolio made of the risk- free asset and the optimal risky portfolio (on the steepest feasible CAL). © McGraw Hill, LLC 47  A risk-free asset and Many risky assets The asset allocation problem with a risk-free asset and many risky assets is similar to the asset allocation problem with a risk-free asset and two risky assets. Both problems can be decomposed into two tasks. Task 1 (risky portfolio selection, or asset allocation among many risky assets or two risky assets): Determine the optimal risky portfolio O containing either many risky portfolio or two risky assets A and B (optimal for every investor regardless his/her risk attitude). It is the tangent point on the upward sloping curve of efficient frontier or investment opportunity set and has the highest Sharpe Ratio. It determines the asset allocation among many risky assets or between the two risky assets (Asset A and Asset B, or stock and bond) Task 2 (asset allocation between risk-free asset and optimal risky portfolio): Construct the complete portfolio made of the risk-free asset F and this optimal risky portfolio O, i.e., choosing a point on the Capital Allocation line (the location on the CAL dependent on investors' risk attitude) © McGraw Hill, LLC 48