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The performance of an optimal filter is lower bounded by the Bayesian Cramér-Rao Bound (BCRB). In some cases, this bound is tight (achieved by the optimal filter) asymptotically in information, i.e., high signal-to-noise ratio (SNR). However, for jump Markov linear Gaussian systems (JMLGS) the BCRB is not necessarily achieved for any SNR. In this paper, we derive a new bound which is tight for all SNRs. The bound evaluates the expected covariance of the optimal filter which is represented by one deterministic term and one stochastic term that is computed with Monte Carlo methods. The bound relates to and improves on a recently presented BCRB and an enumeration BCRB for JMLGS. We analyze their relations theoretically and illustrate them on a couple of examples.