Several sources of uncertainty affect the assessment of existing buildings, including uncertainties associated with the material properties and the displacement capacity of the elements. In engineering practice, Monte Carlo simulations of the nonlinear seismic response of structures lead often to excessive computational costs and therefore rarely carried out. This paper proposes a simple logic tree approach, where a moment-matching technique is proposed to define the optimal sampling points and combination weights to apply to its branches. As a more refined method, a novel application to structural engineering of a recently proposed Point Estimate Method (PEM), which aims at reducing the required number of simulations further, is tested. Both methods are applied to a historical stone masonry building, which is modelled by an equivalent frame approach. The methods are benchmarked against the results of a Monte Carlo simulation and other approximate methods applied in the literature (FOSM, response surface method), which highlights the good accuracy of such methods for estimating the performance uncertainty of the tested building. Moreover, the effect of the different sources of uncertainty on the modelled performance of the building are discussed, identifying the displacement capacity as a major source of uncertainty, whose effect can be compared in terms of order of magnitude to the record-to-record variability.