An investor must choose how much to consume and must allocate their wealth between stocks and a risk-free asset so as to maximize expected utility.
[1][2] Research has continued to extend and generalize the model to include factors like transaction costs and bankruptcy.
The utility function is of the constant relative risk aversion (CRRA) form: Consumption cannot be negative: ct ≥ 0,[2][3] while πt is unrestricted (that is borrowing or shorting stocks is allowed).
The optimal consumption and stock allocation depend on wealth and time as follows:[4]: 401 This expression is commonly referred to as Merton's fraction.
Because W and t do not appear on the right-hand side; a constant fraction of wealth is invested in stocks, no matter what the age or prosperity of the investor.