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I assume that risk it measured here in volatility. Then a portfolio with 100*$w$ percent invested in A and 100*$(1-w)$ percent invested in B has the annual variance $$v = w^2 0.25^2 + 2* 0.5 w(1-w) 0.25*0.5 + (1-w)^20.5^2.$$ Searching for the portfolio with the samllest variance is equivalent to searching for the smallest volatility. To get the minimum ...

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Portfolio beta is a linear combination of each asset's beta times the weight of the asset in the portfolio. Thus in general we have $$\beta = \sum_{i=1}^N w_i \beta_i$$ where $w_i$ is the weight of asset $i$ and $\beta_i$ its beta. If we assume that for stocks the betas are positive then $\beta$ above is positive for positive weights. If you have positive ...

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Some use the acronym SAMURAI: a benchmark should be S - Specified in Advance A - Appropriate M - Measurable U - Unambiguous R - Reflective of Current Investment Thinking A - Accountable I - Investable In principle it is the responsibility of the decision maker (the board of the pension fund) to set the benchmark, otherwise if the choice is left to ...

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See http://www.princeton.edu/~yacine/leverage.pdf The leverage effect refers to the observed tendency of an asset’s volatility to be negatively correlated with the asset’s returns. Typically, rising asset prices are accompanied by declining volatility, and vice versa. The term “leverage” refers to one possible economic interpretation of this ...

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yes u is the unit vector of all ones

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