A completely algebraic formulation of the design of dynamical observers of the Luenberger type is available in the literature. This formulation places in evidence all the available degrees of freedom in the design of the observer. A natural question which arises is whether the design freedom can be utilized to achieve certain desirable overall system properties. Attention is turned in this paper to the possibility of realizing zero sensitivity to plant parameter variations of a quadratic performance index. It is established that the desired insensitivity is impossible to achieve for all initial states with the available design freedom.