Stock portfolio Q

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Suppose stock returns are determined by a general factor model. For each stock i, the return at time t is given by: r_i, t - r^f = alpha_i + beta_i F + epsilon_i, t, where F = F_t - E[F] is the deviation of the common factor from its mean at time t and epsilon_i, t is an idiosyncratic risk process with mean zero and variance sigma_i^2 of for each stock i. Consider a portfolio of two stocks, i and j. a: Write out the general form of r_p - r^f for a portfolio across stocks i and j (you may assume that i and j are the only stocks and weights w_i + w_j = 1). What is E[r_p] - r^f? beta_P? sigma_P^2? epsilon_P? b: Which variable represents the portfolio's idiosyncratic risk? Which variable indicates the portfolio's exposure to systemic risk? Now consider the no arbitrage condition: E[r_i] - r^f/beta_i = E[r_j] - r^f/beta_j for all stocks i, j. c: Does the no arbitrage condition guarantee that all stocks have the same return? Why? d: If the no arbitrage condition holds for stocks i and j, does it necessarily also hold for a portfolio across i and j? Use equations and/or graphs to explain your