Portfolio i’s return is described by: ri = rf +βi,m(rm −rf ) +βi,c(rc −rf ) + 0.5%, where rm is the return on the market index, rc is the return on the commodity index and rf is the risk-free rate.
(a) Construct a pure arbitrage trading using the market index, a commodity index, t-bills and portfolio i.
(b) If you are $1 million long and $1 million short in your arbitrage trade. What will be your profit 1 at the end of the period?
(c) Now, assume there is no arbitrage in the market (so the alpha of portfolio i should be 0, not 0.5%). What is the expected return on portfolio i if the risk-free rate is 2%, the beta on the market is 1, the risk premium on the market is 5%, the beta on the commodity index is 2, and the risk premium on the commodity index is 3%?
(d) Now assume portfolio i has some idiosyncratic risk (e.g., ri = rf +βi,m(rm −rf )+βi,c(rc −rf )+ 0.5% + i , where i is mean zero and has a variance of .025%). Can we construct a pure arbitrage strategy?