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Chapter 8

Optimal Risky Portfolios. Chapter 8. Diversification and Portfolio Risk. Suppose your portfolio consists of one stock. What would be the sources of risk:

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Chapter 8

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  1. Optimal Risky Portfolios Chapter 8

  2. Diversification and Portfolio Risk • Suppose your portfolio consists of one stock. What would be the sources of risk: -Macroeconomic risks:Risks those come from conditions in the general economy (Business cycle, inflation, interest rates, exchange rates) -Firm specific risks (success in R&D, personnel changes).

  3. Diversification • Now consider the diversification strategy. You invest half of your funds on another stock. Because the firm-specific influences on the two stocks differ, diversification should decrease risk. • If we diversify into many more securities, we continue to decrease portfolio volatility. With a large number of stocks we cannot avoid risk together since all securities are affected by the common macroeconomic factors.

  4. Diversification • When all risk is firm-specific, diversification can decrease the risk to low levels. • When common sources of risk affect all firms, extensive diversification cannot eliminate risk.

  5. The risk that cannot be diversified is called as market risk, non-diversifiable risk. • The risk that can be eliminated is called “unique risk”, firm specific risk, non-systematic risk, diversifiable risk.

  6. Risk Reduction with Diversification St. Deviation Unique Risk Market Risk Number of Securities

  7. Portfolios of Two-Risky Assets • Suppose our portfolio consists of two risky assets; debt (D) and equity (E). • The rate of return on this portfolio; rp = wD rD + wE rE • The variance of the two-asset portfolio;

  8. Correlation Coefficients: Possible Values Range of values for D,E + 1.0 >r> -1.0 If r= 1.0, the securities would be perfectly positively correlated If r= - 1.0, the securities would be perfectly negatively correlated

  9. In the case of perfect positive correlation; • The standard deviation of the portfolio with perfect positive correlation is the weighted average of the component standard deviations.

  10. In all other cases, correlation coefficient less than 1, making the standard deviation less than the weighted average of the component standard deviations. • Portfolios of less than perfectly correlated assets always offer better risk return opportunities than the individual components on their own. • The lower the correlation between the assets, the greater the gain in efficiency.

  11. In the case of no correlation; • The weight of debt in the portfolio that minimizing the portfolio risk is;

  12. In the case of perfect negative correlation; • When ρ=-1, wD σD- wEσE=0 • The weight of debt in the portfolio that makes the std dev of the portfolio zero. When ρ=-1

  13. At any correlation coefficient rate, the weight of debt that gives the minimum variance;

  14. Portfolios of Two Risky Assets • Example 1: Suppose we will consider a portfolio comprised of two mutual funds, a bond portfolio (long term debt securities) denoted D, and a stock fund (equity securities), denoted E. Debt Equity Expected Return E(r) 8% 13% Standard Deviation (δ) 12% 20% Cov(rD, rE) 72 Corr. Coefficient, ρDE .30

  15. Portfolio Risk and Return E(rp)= 8wD+13 wE

  16. wmin(E)=1-.82=.18

  17. The return of the min variance portfolio; • Rp=8x0.82+13x0.18 = 8.9% • Standard deviation of the min-variance portfolio; =11.45%

  18. Correlation Effects • The relationship depends on correlation coefficient. • -1.0 << +1.0 • If r = +1.0, no risk reduction is possible. there is no benefit from diversification. • Portfolio Std Dev is simple the weighted average of the component assets standard deviations.

  19. If ρ=0 with no correlation between two asset, diversification is more effective and portfolio risk is lower. • Min. portfolio standard deviation when ρ=0 is 10.29% and again lower than the standard deviation of either asset. • wE=.265

  20. The smaller the correlation, the greater the risk reduction potential. r = -1.0, offers a perfect hedging opportunity and the maximum advantage of diversification. We can construct a zero-variance portfolio. wmin(E; ρ=-1)=1-.625= .375

  21. Portfolios with Different Correlations(Portfolio Opportunity Sets) E(r) 13%  = -1  = .3  = -1  = 1 %8 St. Dev 12% 20%

  22. When an investor wishes to select the optimal portfolio from the opportunity set, the best portfolio will depend on risk aversion. • Northeast=> provide higher rates of return but with greater risk. • Southwest=> investors with greater risk aversion will prefer these portfolios with lower expected return and lower risk. • The best trade-off depends on personal preference.

  23. Asset Allocation with Stocks, Bonds, and Bills • We saw the asset allocation between a risk-free asset and a risky portfolio. Now two risky assetsand a risk-free asset. • If we connect the portfolios which gives minimum risk at different levels of return we can have the min. variance frontier.

  24. Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.

  25. Alternative CALs CAL (B) CAL (P) E(r) M M P P CAL (A-Global minimum variance) B B A F  P P&F A&F M

  26. The first CAL is drawn from through the min-variance portfolio A, 82% in bonds and 18% in stocks. A’s expected return is 8.9% and standard deviation is 11.45%. (ρ=.30) • With a T-bill rate of 5%, the reward-to-variability ratio, slope CAL combining T-bills and the min-variance portfolio is;

  27. Now consider the CAL that was portfolio B instead of A. Portfolio B invest 70% in bonds and 30% in stocks, its expected return is 9.5% (risk premium is 4.5%) and its standard deviation is 11.70%. • Reward-to-variability ratio; • Portfolio B dominates A.

  28. P=> tangency portfolio is the optimal risky portfolio to mix with T-bills. The expected return & risk can be read from the graph. • E(rp)=11% σp=14.2%

  29. To construct optimal risky portfolios from more than two risky assets we need a spreadsheet or a computer program. We will see here a portfolio construction problem with only two risky assets and a risk free asset. We can derive a formula for the weights of each asset in the optimal portfolio.

  30. The objective is to find the weights wD and wE that result in the highest slope of the CAL=> the weights that result in the risky portfolio with the highest reward-to-variability ratio. Therefore the objective is to maximize the slope of the CAL for any portfolio P.

  31. For the portfolio with two risky assets, the expected return and standard deviation of portfolio P are E(rp) = wDE(rD)+ wEE(rE) = 8 wD+ 13 wE

  32. When we maximize the objective function Sp, we have to satisfy the constraint that the portfolio weights sum to 1, wD+wE=1 subject to

  33. Optimal Risky Portfolio; wE=1-.40=.60

  34. E(rp)=(.40 x 8)+ (.6 x 13)= 11% • The CAL of this portfolio has a slope of; (reward-variability ratio of portfolio P)

  35. Optimal Complete Portfolio • A=4=> coefficient of risk-aversion • 74.39% invest in portfolio P • 1-.7439=.2561 (25.61% invest in T-bills) • Portfolio P → 40% Bonds → ywD=.4 x .7439=.2976 =>29.76% →60% C.stock → ywE=.6 x .7439=.4463 =>44.63% => % invest in portfolio P

  36. The steps to find the complete portfolio; 1. Specify the return characteristics of all securities ( expected return, variances, covariances) 2. Establish the risky portfolio a. Calculate optimal risky portfolio, p( equation 8.7) b. Calculate the properties of portfolio P using the weights determined pg.239 In step a and equation (8.1) and (8.2) 3. Allocate funds between the risky portfolio and the risk-free asset. a. Calculate the fraction of the complete portfolio allocated to portfolio P and T-bills. b. Calculate the share of the complete portfolio invested in each asset and T-bills.

  37. The Markowitz Portfolio Selection Model Harry Markowitz (1952) Model of portfolio selection 1990=> Nobel Prize for economics We can generalize the portfolio construction problem to the case of many risky assets and a risk free asset. As in the two risky assets example the portfolio construction problem has 3 parts. 1. Identify the risk-return combinations available from the set of risky assets. 2. Identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL. 3. Choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio.

  38. 1. Detemining the risk-return opportunities available to the investor • The optimal combinations result in lowest level of risk for a given return. (Min variance frontier of risky assets) • The optimal trade-off is described as the efficient frontier. • These portfolios are dominant. • All the portfolios lie on the min-variance frontier from the global min-variance portfolio and upward provide the best risk-return combinations and candidates for the optimal portfolio. This portfolio is the efficient frontier of risky assets. • The bottom part of the min-variance frontier is inefficient.

  39. Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.

  40. 2. Identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL. • Now our portfolio optimization problem involves the risk free asset. We search for CAL with the highest reward-to-variability ratio. • The CAL that is supported by the optimal portfolio, P, is tangent to the efficient frontier. The CAL dominates all alternative feasible lines. • Portfolio P is the optimal risky portfolio. • The optimal combination becomes linear. • A single combination of risky and riskless assets will dominate.

  41. The opportunity set of debt and equity funds with the optimal CAL and the optimal risky portfolio CAL E(r) B Q P A rf F St. Dev

  42. 3. Choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio. • Individual investor chooses the appropriate mix between the optimal risky portfolio P and T-bills

  43. Determination of the optimal overall portfolio CAL E(r) B Q P A rf F St. Dev

  44. Capital Allocation and the Separation Property • Portfolio P is the point at which CAL tangent to the efficient frontier. Portfolio P maximize the reward-to-variability ratio, the slope of the line from F to portfolio on the efficient frontier. Portfolio P is the optimal risky portfolio that a portfolio manager will offer to all clients regardeless of their degree of risk aversion.

  45. Seperation Property • The degree of risk aversion (A) comes into play only in the selection of the desired point along the CAL. More risk averse investor will invest more in rf and less in optimal risky portfolio than will a less risk averse investor. However both will use portfolio P as their optimal risky portfolio. • This is called a separation property (J. Tobin)

  46. Portfolio Selection without Rf & Risk Aversion U’ U’’ U’’’ E(r) Efficient frontier of risky assets S P Q Less risk-averse investor More risk-averse investor St. Dev

  47. Efficient Frontier with Lending & Borrowing CAL E(r) B Q P A rf F St. Dev

  48. HOMEWORK • Problems 10, 14, and 31 (a,b,c,d,e).

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