Optimal Combination of Risky Assets
? Introduce the risk free asset
? The highest trade-off line is the one connecting point F and T
? Portfolio T is referred as the optimal combination of risky assets
?r1?rf????r2?rf??12?1?2 w1?22?r1?rf??2??r2?rf??1??r1?rf?r2?rf??12?1?222w1?69.2% and w2?30.8% ? rT?12.2% and ?T?14.6%
? Portfolios along the line FT are now efficient portfolios
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The Risk-Return Trade-Off Curve0.160.14Expected Return0.120.100.080.060.040.020.0000.050.1T F 0.150.20.25Stansard Deviation
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Achieving a Target Expected Return
? Expected return is 10%
? wrT??1?w?rf?w?12.2%??1?w??6%?10%
? w?0.65
? Weight in riskless asset: 35%
Weight in risky asset 1: 0.65?69.2%?45% Weight in risky asset 2: 0.65?30.8%?20%
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Portfolio of Many Risky Assets
? Portfolio with N risky assets
Minimize ????wiwj?ij?i?j
i?1j?12pNNs.t. rP??wiri
i?1N?wi?1
Ni?1
? In order to solve problems with more than two securities requires tools such as quadratic programming
? The reduction in standard deviation of the portfolio depends on the correlation coefficients among security returns
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