Suppose there is a risk-free asset in zero net supply and the risky asset returns have a statistical factor structure R˜i = ai +b

i F˜ + ˜εi , wherethe ε˜i have zero means and are independent of each other and of F˜. Assume there is no labor income and there is a representative investor with CARA utility.

Let α denote the risk aversion of the representative investor. Let π denote the vector of market weights. Denote initial market wealth by w0 and end-of-period market wealth by w˜ m = w0R˜ m. Let δi denote the APT pricing error defined in Section