Ms. Sowns a portfolio containing some special securities that she believes will perform well compared to the market as a whole over the next several months. However, Ms. S realizes that her portfolio has a high beta relative to the market as a whole and so is exposed to a significant degree of market risk If the general market declines, her portfolio will also decline, even if her special securities do well. Let P represent the value of Ms. S's portfolio. In terms of the present value Po and portfolio return rate T p for the period from now to time T in the future, (vrite Po as PO. Tp as rP) Let M represent the market portfolio with value equal the Standard and Poors 500 stock index. In terms of the present value Mo and market return rate rM for the period from now to time T (2) M (T) = 1 The beta of portfolio P with respect to the market is defined as (3) bet ap-var(nr) Ms. S decides to hedge against market risk. She can change the beta of her portfolio by investing now in futures A future F is available, with dollar value equal 50 times the Standard and Poors index (write Mo as MO, rAr as rM) Ignoring interest so the discount factor is one, at time T the new portfolio has value (4) y(T) = P(T) + 50(M(T)-Ma)h where the second term represents the profit or loss from the investment in hfutures. Witing the return rate for the new portflio y as ry and replacing P(T) and M(T) with the expressions from (1) and (2), equation (4) becomes