(Bounds on returns) Consider a universe of just three securities. They have expected rates of return of 10%, 20%, and 10%, respectively. Two portfolios are known to lie on the minimum-variance set. They are defined by the portfolio weights w = .60 .20 .20 ⎤ ⎦, v= − .80 .20 .40 ⎤ ⎦ It is also known that the market portfolio is efficient. (a) Given this information, what are the minimum and maximum possible values for the expected rate of return on the market portfolio? (b) Now suppose you are told that w represents the minimum-variance portfolio. Does this change your answers to part (a)?

1. alwaysrain irrigation, inc. would like to determine capacity requirements for the next four years. currently, two production lines are in place for making bronze and plastic sprinklers. three types of sprinklers are available in both bronze and plastic: 90-degree nozzle sprinklers, 180-degree nozzle sprinklers, and 360-degree nozzle sprinklers. management has forecast demand for the next four years as follows: both production lines can produce all the different types of nozzles. the bronze machines needed for the bronze sprinklers require two operators and can produce up to 12,000 sprinklers. the plastic injection molding machine needed for the plastic sprinklers requires four operators and can produce up to 200,000 sprinklers. three bronze machines and only one injection molding machine are available. what are the capacity requirements for the next four years? (assume that there is no learning.)