The hydration free energies of ions exhibit an approximately quadratic dependence on the ionic charge, as predicted by the Born model. We analyze this behavior using second-order perturbation theory. The average and the fluctuation of the electrostatic potential at charge sites appear as the first coefficients in a Taylor expansion of the free energy of charging. Combining the data from different charge states (e.g., charged and uncharged) allows calculation of free-energy profiles as a function of the ionic charge, The first two Taylor coefficients of the free-energy profiles can be computed accurately from equilibrium simulations, but they are affected by a strong system-size dependence, We apply corrections for these finite-size effects by using Ewald lattice summation and adding the self-interactions consistently. An analogous procedure is used for the reaction-field electrostatics. Results are presented for a model ion with methane-like Lennard-Jones parameters in simple point charge water, We find two very closely quadratic regimes with different parameters for positive and negative ions. We also studied the hydration free energy of potassium, calcium, fluoride, chloride, and bromide ions. We find negative ions to be solvated more strongly (as measured by hydration free energies) compared to positive ions of equal size, in agreement with experimental data. We ascribe this preference of negative ions to their strong interactions with water hydrogens, which can penetrate the ionic van der Waals shell without direct energetic penalty in the models used, In addition, we consistently find a positive electrostatic potential at the center of uncharged Lennard-Jones particles in water, which also favors negative ions. Regarding the effects of a finite system size, we show that even using only 16 water molecules it is possible to calculate accurately the hydration free energy of sodium, if self-interactions are considered.