A process for securely binding the adjoining fibers of a fibrous material in the absence of the use of phenol-formaldehyde reaction product is provided. A curable binding composition comprising an intermediate reaction product capable of undergoing free-radical polymerization as well as an esterification crosslinking reaction is provided. The intermediate reaction product is formed by the reaction of (i) at least one anhydride having at least one unsaturated double bond capable of undergoing free-radical polymerization, and (ii) at least one polyol crosslinker. When applied to a fibrous material, the binding composition comprising the intermediate reaction product is cured in the presence of a free-radical initiator and a polyol crosslinker wherein the double bonds present in the intermediate reaction product undergo free-radical polymerization and hydroxyl and carboxylic acid end groups undergo crosslinking to form a cured binder. In a preferred embodiment the securely bound fibrous product is fiberglass building insulation that well withstands water even at elevated temperatures.