Ion selectivity in a simple cyclic peptide nanotube, composed of four cyclo[-(D-Ala-Glu-D-Ala-Gln)$_2-$] units, is investigated by calculating the PMF profiles of $Na^+$, $K^+$, and $Cl^-$ ions permeating through the peptide nanotube in water. The final PMF profiles of the ions obtained from the umbrella sampling (US) method show an excellent agreement with those from the thermodynamic integration (TI) method. The PMF profiles of $Na^+$ and $K^+$ display free energy wells while the PMF curve of $Cl^-$ features free energy barriers, indicating the selectivity of the cyclic peptide nanotube to cations. Decomposition of the total mean force into the contribution from each component in the system is also accomplished by using the TI method. The mean force decomposition profiles of $Na^+$ and $K^+$ demonstrate that the dehydration free energy barriers by water molecules near the channel entrance and inside the channel are completely compensated for by attractive electrostatic interactions between the cations and carbonyl oxygens in the nanotube. In the case of $Cl^-$, the dehydration free energy barriers are not eliminated by an interaction between the anion and the peptide nanotube, leading to the high free energy barriers in the PMF profile. Calculations of the coordination numbers of the ions with oxygen atoms pertaining to either water molecules or carbonyl groups in the peptide nanotube reveal that the stabilization of the cations in the midplane regions of the nanotube arises from the favorable interaction of the cations with the negatively charged carbonyl oxygens.