Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification Prof. Alex Shapiro 1 Lecture Notes 7 Optimal Risky Portfolios: Efficient Diversification I. Readings and Suggested Practice Problems II. Correlation Revisited: A Few Graphical Examples III. Standard Deviation of Portfolio Return: Two Risky Assets IV. Graphical Depiction: Two Risky Assets V. Impact of Correlation: Two Risky Assets VI. Portfolio Choice: Two Risky Assets VII. Portfolio Choice: Combining the Two Risky Asset Portfolio with the Riskless Asset VIII. Applications IX. Standard Deviation of Portfolio Return: n Risky Assets X. Effect of Diversification with n Risky Assets XI. Opportunity Set: n Risky Assets XII. Portfolio Choice: n Risky Assets and a Riskless Asset XIII. Additional Readings Buzz Words: Minimum Variance Portfolio, Mean Variance Efficient Frontier, Diversifiable (Nonsystematic) Risk, Nondiversifiable (Systematic) Risk, Mutual Funds.Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification 2 I. Readings and Suggested Practice Problems BKM, Chapter 8.1-8.6. Suggested Problems, Chapter 8: 8-14 E-mail: Open the Portfolio Optimizer Programs (2 and 5 risky assets) and experiment with those. II. Correlation Revisited: A Few Graphical Examples A. Reminder: Don’t get confused by different notation used for the same quantity: Notation for Covariance: Cov[r1,r2] or σ[r1,r2] or σ12 or σ1,2 Notation for Correlation: Corr[r1,r2] or ρ[r1,r2] or ρ12 or ρ1,2 B. Recall that covariance and correlation between the random return on asset 1 and random return on asset 2 measure how the two random returns behave together. C. Examples In the following 5 figures, we Consider 5 different data samples for two stocks: - For each sample, we plot the realized return on stock 1 against the realized return on stock 2. - We treat each realization as equally likely, and calculate the correlation, ρ, between the returns on stock 1 and stock 2, as well as the regression of the return on stock 2 (denoted y) on the return on stock 1 (x). [Note: the regression R2 equals ρ2]Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification 3 1. A sample of data with ρ = 0.630: 2. A sample of data with ρ = -0.714: y = 0.9482x + 0.0506R2 = 0.3972-10%-5%0%5%10%15%20%25%30%35%-15% -10% -5% 0% 5% 10% 15% 20% 25%Return on Stock 1Return on Stock 2y = -0.8613x + 0.0726R2 = 0.51-20%-15%-10%-5%0%5%10%15%20%-5% 0% 5% 10% 15% 20% 25%Return on Stock 1Return on Stock 2Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification 4 3. Sample with ρ = +1: 4. Sample with ρ = -1: 5. Sample with ρ ≈ 0: y = 0.02x + 0.05R2 = 15%5%5%5%5%5%5%6%6%-10% -5% 0% 5% 10% 15% 20% 25% 30%Return on Stock 1Return on Stock 2y = -0.8x + 0.05R2 = 1-25%-20%-15%-10%-5%0%5%10%15%-10% 0% 10% 20% 30% 40%Return on Stock 1Return on Stock 2y = 0.009x + 0.0468R2 = 0.0001-10%-5%0%5%10%15%-5% 0% 5% 10% 15% 20% 25% 30%Return on Stock 1Return on Stock 2Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification 5 D. Real-Data Example Us Stocks vs. Bonds 1946-1995, A sample of data with ρ = 0.228: STB Stocks and Bonds(Annual returns on S&P 500 and long term US govt bonds.)Raw Data Excess over T-billS&P500 LT Gov’t T-bill Inflation S&P500 LT Gov’t1946 -8.07% -0.10% 0.35% 18.16% -8.42% -0.45%1947 5.71% -2.62% 0.50% 9.01% 5.21% -3.12%1948 5.50% 3.40% 0.81% 2.71% 4.69% 2.59%1949 18.79% 6.45% 1.10% -1.80% 17.69% 5.35%1950 31.71% 0.06% 1.20% 5.79% 30.51% -1.14%1951 24.02% -3.93% 1.49% 5.87% 22.53% -5.42%1952 18.37% 1.16% 1.66% 0.88% 16.71% -0.50%1953 -0.99% 3.64% 1.82% 0.62% -2.81% 1.82%1954 52.62% 7.19% 0.86% -0.50% 51.76% 6.33%1955 31.56% -1.29% 1.57% 0.37% 29.99% -2.86%1956 6.56% -5.59% 2.46% 2.86% 4.10% -8.05%1957 -10.78% 7.46% 3.14% 3.02% -13.92% 4.32%1958 43.36% -6.09% 1.54% 1.76% 41.82% -7.63%1959 11.96% -2.26% 2.95% 1.50% 9.01% -5.21%1960 0.47% 13.78% 2.66% 1.48% -2.19% 11.12%1961 26.89% 0.97% 2.13% 0.67% 24.76% -1.16%1962 -8.73% 6.89% 2.73% 1.22% -11.46% 4.16%1963 22.80% 1.21% 3.12% 1.65% 19.68% -1.91%1964 16.48% 3.51% 3.54% 1.19% 12.94% -0.03%1965 12.45% 0.71% 3.93% 1.92% 8.52% -3.22%1966 -10.06% 3.65% 4.76% 3.35% -14.82% -1.11%1967 23.98% -9.18% 4.21% 3.04% 19.77% -13.39%1968 11.06% -0.26% 5.21% 4.72% 5.85% -5.47%1969 -8.50% -5.07% 6.58% 6.11% -15.08% -11.65%1970 4.01% 12.11% 6.52% 5.49% -2.51% 5.59%1971 14.31% 13.23% 4.39% 3.36% 9.92% 8.84%1972 18.98% 5.69% 3.84% 3.41% 15.14% 1.85%1973 -14.66% -1.11% 6.93% 8.80% -21.59% -8.04%1974 -26.47% 4.35% 8.00% 12.20% -34.47% -3.65%1975 37.20% 9.20% 5.80% 7.01% 31.40% 3.40%1976 23.84% 16.75% 5.08% 4.81% 18.76% 11.67%1977 -7.18% -0.69% 5.12% 6.77% -12.30% -5.81%1978 6.56% -1.18% 7.18% 9.03% -0.62% -8.36%1979 18.44% -1.23% 10.38% 13.31% 8.06% -11.61%1980 32.42% -3.95% 11.24% 12.40% 21.18% -15.19%1981 -4.91% 1.86% 14.71% 8.94% -19.62% -12.85%1982 21.41% 40.36% 10.54% 3.87% 10.87% 29.82%1983 22.51% 0.65% 8.80% 3.80% 13.71% -8.15%1984 6.27% 15.48% 9.85% 3.95% -3.58% 5.63%1985 32.16% 30.97% 7.72% 3.77% 24.44% 23.25%1986 18.47% 24.53% 6.16% 1.13% 12.31% 18.37%1987 5.23% -2.71% 5.47% 4.41% -0.24% -8.18%1988 16.81% 9.67% 6.35% 4.42% 10.46% 3.32%1989 31.49% 18.11% 8.37% 4.65% 23.12% 9.74%1990 -3.17% 6.18% 7.81% 6.11% -10.98% -1.63%1991 30.55% 19.30% 5.60% 3.06% 24.95% 13.70%1992 7.67% 8.05% 3.51% 2.90% 4.16% 4.54%1993 9.99% 18.24% 2.90% 2.75% 7.09% 15.34%1994 1.31% -7.77% 3.90% 2.67% -2.59% -11.67%1995 37.43% 31.67% 5.60% 2.74% 31.83% 26.07%N 50505050 5050Mean 13.16% 5.83% 4.84% 4.43% 8.31% 0.99%Std.Dev. 16.57% 10.54% 3.18% 3.82% 17.20% 10.13%Std.Err.Mean 2.34% 1.49% 0.45% 0.54% 2.43% 1.43%Corr(Stocks, Bonds)= 0.228 0.265y = 0.3592x + 0.1106R2 = 0.0522-30%-20%-10%0%10%20%30%40%50%60%-10% -5% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45%Return on US Gov’t BondsReturn on S&P 500Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification 6 III. Standard Deviation of Portfolio Return: Two Risky Assets A. Formula )]()([2])([])([)]([2121221tr ,tr w w + tr w + tr w = trp,p,22p,22p1,p2σσσσ tr=trp2p)]([)]([σσ where [r1(t), r2(t)] is the covariance of asset 1 s return and asset 2 s return in period t, wi,p is the weight of asset i in the portfolio p, 2[rp(t)] is the variance of return on portfolio p in period t. B. Example Consider two risky assets. The first one is