A derivation of conventional portfolios and a new linear utility method

Four known portfolios are derived using a new control theory approach. These are the mean-variance, the HARA utility, the log-optimal and the exponential utility portfolios. A single HJB equation is derived from the dynamics of the wealth logarithm. The value functions for different portfolios appear as solutions to this single equation, and thus unifying the derivation of the mean-variance and the other three portfolios. A new method for portfolio selection is proposed that uses the linear utility and penalizes the fractions of wealth allocated across the risky assets. Conventional portfolios appear as examples to this more general method.