The fixed core stochastic variational method is used to investigate positron binding to a model alkali atom with a continuously adjustable ionization potential. Positron binding is possible for model atoms which have ionization potential (IP) ranging from 0.1767 to 0.479 Hartree (corresponding to dipole polarizabilities ranging from 209 to 23.5 a03). Results of the model indicate that positron binding is likely for gold, but not for potassium, rubidium or caesium. The annihilation rate was largest (1.97 × 109 s-1) when the IP is smallest and smallest (0.07 × 109 s-1) when the IP is largest. The presence of a positronium (Ps) cluster configuration in the wavefunction is found to be important for an accurate estimate of the annihilation rate even in circumstances when the positron and electron are located large distances apart.