We prove a KPZ-type formula which relates the a.s. Hausdorff dimension of a set associated with the Schramm-Loewner evolution (SLE) or the conformal loop ensemble (CLE) to the a.s. Hausdorff dimension of a time set for a two-dimensional correlated Brownian motion. As applications of our formula we get new short proofs of the Hausdorff dimension of a number of special sets for SLE. Based on a joint work with Ewain Gwynne and Jason Miller.