Let (Formula presented.) be a probability space, ξ be a random variable on (Formula presented.), (Formula presented.) be a sub-σ-algebra of (Formula presented.), and let (Formula presented.) be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of (Formula presented.) in terms of the moments of ξ. This allows us to find the optimal constant in the bounded compact approximation property of (Formula presented.), (Formula presented.).