Consider a portfolio problem with n assets e g stocks over some fixed period The rate of return for asset i is defined to be the

Question

Consider a portfolio problem with n assets e g stocks over some fixed period The rate of return for asset i is defined to be the change in asset s price over the period For instance if the price changes from p to prew the rate of return is phew Pi P The rate of return for each asset i i 1 N is given by a random variable R with mean r E R Furthermore the covariance of R and R is equal to E E R r R r Denote the total budget for investment by B which we assume is equal to one for simplicity Let RT be the total return on investment In other words RT R I where I is the initial investment in asset i i 1 n A good investment strategy needs to balance the trade off between the total return and the risk measured by the variance of RT i e the higher the variance of Ry is the riskier the strategy is a Set up an appropriate convex optimization problem for maximizing the expected total return E R subject to a constraint that the variance of Rr does not exceed o2 b Set up an appropriate convex optimization problem for minimizing the variance of the total return subject to a constraint that the expected total return is at least Rmin

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