Foundations of Finance Optimal Risky Portfolios Efficient Diversification Prof Alex Shapiro 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 1 Foundations of Finance Optimal Risky Portfolios Efficient Diversification 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 r1 r2 or 12 or 1 2 Notation for Correlation Corr r1 r2 or r1 r2 or 12 or 1 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 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 2 2 Foundations of Finance Optimal Risky Portfolios Efficient Diversification 1 A sample of data with 0 630 35 y 0 9482x 0 0506 2 R 0 3972 30 Return on Stock 2 25 20 15 10 5 0 15 10 5 5 0 5 10 15 20 25 10 Return on Stock 1 A sample of data with 0 714 20 15 10 Return on Stock 2 2 5 0 5 0 5 10 15 5 10 15 y 0 8613x 0 0726 2 R 0 51 20 Return on Stock 1 3 20 25 Foundations of Finance Optimal Risky Portfolios Efficient Diversification Sample with 1 6 y 0 02x 0 05 R2 1 6 5 Return on Stock 2 3 5 5 5 5 5 5 10 5 0 5 10 15 20 25 30 Return on Stock 1 4 Sample with 1 15 Return on Stock 2 10 5 10 0 0 5 10 20 30 40 10 15 y 0 8x 0 05 2 R 1 20 25 Return on Stock 1 Sample with 0 15 10 y 0 009x 0 0468 Return on Stock 2 5 R2 0 0001 5 0 5 0 5 10 15 5 10 Return on Stock 1 4 20 25 30 Foundations of Finance Optimal Risky Portfolios Efficient Diversification D Real Data Example Us Stocks vs Bonds 1946 1995 A sample of data with 0 228 N Mean Std Dev Std Err Mean 50 13 16 16 57 2 34 Corr Stocks Bonds 50 5 83 10 54 1 49 0 228 50 4 84 3 18 0 45 50 4 43 3 82 0 54 50 8 31 17 20 2 43 60 y 0 3592x 0 1106 R2 0 0522 50 40 30 Return on S P 500 STB Stocks and Bonds Annual returns on S P 500 and long term US govt bonds Raw Data Excess over T bill S P500 LT Gov t T bill Inflation S P500 LT Gov t 1946 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 …
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