Abstract: We present a comprehensive framework for comparing the merits of alternative portfolio insurance strategies in realistic contexts. Our findings add generality to previous results comparing option based and constant proportionality portfolio insurance strategies (OBPI and CPPI). The optimal OBPI and CPPI payoffs are determined by maximising expected utilities, with various degrees of risk sensitivity and over several investment horizons, using a general, two-parameter HARA utility. We consider two cases: either defined payoffs are purchased at fair prices or, as is typical in the implementation of portfolio insurance strategies, replicated discretely. The price dynamics of risky assets are modelled with either a geometric Brownian process or a time-changed geometric Brownian. Our results confirm the superiority of CPPI over OBPI in all cases. The effects of discrete replication and discontinuous price processes are examined by simulation and compared to the purchase at fair price of the theoretically optimal CPPI payoff when the underlying process is geometric Brownian.