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**Unformatted text preview: **Optional Proof of the Security Market Line 1 The security market line states that for any asset with return R i , E [ R j ] = R f + β j E [ R M- R f ] where R f is the return on the riskfree asset, R M is the return on the market port- folio, and β i is the regression coefficient from a regression of R i on R M . That is, β j = Cov( R j , R M ) / Var( R M ). Suppose that there are I individuals in the economy. Let W i > 0 be the initial wealth of individual i . Let X ij be the proportion of W i invested in security j . The total wealth in the economy is W m = Σ I i =1 W i . Let ˜ W i denote the value of individual i ’s porfolio. That is, if i chooses a portfolio with return R p , ˜ W i = W i (1 + R p ). Each individual maximizes expected utility E [ u i ( ˜ W )]. We assume that the individual prefers more to less and is risk averse. This implies that u i > 0 and u i < 0. The individual solves max u i ( ˜ W ) subject to ˜ W = W i ˆ X j X ij R j + (1- X j X ij ) R f !...

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