Ridge regression is a classical statistical technique that attempts to address the bias-variance trade-off in the design of linear regression models. A reformulation of ridge regression in dual variables permits a non-linear form of ridge regression via the well-known 'kernel trick'. Unfortunately, unlike support vector regression models, the resulting kernel expansion is typically fully dense. In this paper, we introduce a reduced rank kernel ridge regression (RRKRR) algorithm, capable of generating an optimally sparse kernel expansion that is functionally identical to that resulting from conventional kernel ridge regression (KRR). The proposed method is demonstrated to out-perform an alternative sparse kernel ridge regression algorithm on the Motorcycle and Boston Housing benchmarks.