Three‐point bounds on the effective conductivity &sgr;eof isotropic two‐phase composites, that improve upon the well‐known two‐point Hashin–Shtrikman bounds [J. Appl. Phys.23, 779 (1962)], depend upon a key microstructural parameter &zgr;2. A highly accurate approximation for &sgr;edeveloped by Torquato [J. Appl. Phys.58, 3790 (1985)] also depends upon &zgr;2. This paper reports a new and accurate algorithm to compute the three‐point parameter &zgr;2for dispersions of hard spheres by Monte Carlo simulation. Data are reported up to values of the sphere volume fraction &fgr;2near random close‐packing and are used to assess the accuracy of previous analytical calculations of &zgr;2. A major finding is that the exact expansion of &zgr;2through second order in &fgr;2provides excellent agreement with the simulation data for the range 0≤&fgr;2≤0.5, i.e., this low‐volume‐fraction expansion is virtually exact, even in the high‐density region. For &fgr;2>0.5, this simple quadratic formula is still more accurate than other more sophisticated calculations of &zgr;2. The linear term of the quadratic formula is the dominant one. Using our simulation data for &zgr;2, we compute three‐point bounds on the conductivity &sgr;eand Torquato’s approximation for &sgr;e.