The first all-electron ab initio study of Young's modulus and Poisson ratio for a number of closed single-walled nanotubes is presented. At the Hartree-Fock 6-31G* level, the results obtained compare well with experimental as well as previous theoretical studies, predicting a Young's modulus higher than 1 TPa. The calculated Young's modulus for a graphene layer is found to be smaller than for its (5,5)-nanotube counterpart.