Magnetic properties of core-shell cobalt ferrite nanoparticles 15 to 48nm prepared by a sol-gel route have been studied. It is shown that the coercivity follows non-monotonic size dependence varying as 1/d above the maximum (d is the particle size). Field cooled magnetization exhibited both horizontal (exchange bias) and vertical shifts. The exchange bias is understood as originating at the interface between a surface region with structural and spin disorder and a core ferrimagnetic region. The dependence of the exchange bias and vertical shifts on the particle sizes and cooling fields are found to have significant differences and the differences are explained in the light of recent results which suggest that both weakly and strongly pinned spins are present at the interface. It is suggested that the exchange bias is dominated by the weakly pinned spins while the vertical shift is affected by the strongly pinned ones.