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We develop an exchange rate target zone model with finite exit time and non-Gaussian tails. We show how the tails are a consequence of time-varying investor risk aversion, which generates mean-preserving spreads in the fundamental distribution. We solve explicitly for stationary and non-stationary exchange rate paths, and show how both depend continuously on the distance to the exit time and the target zone bands. This enables us to show how central bank intervention is endogenous to both the distance of the fundamental to the band and the underlying risk. We discuss how the feasibility of the target zone is shaped by the set horizon and the degree of underlying risk, and we determine a minimum time at which the required parity can be reached. We prove that increases in risk after a certain threshold can yield endogenous regime shifts where the ``honeymoon effects'' vanish and the target zone cannot be feasibly maintained. None of these results can be obtained by means of the standard Gaussian or affine models. Numerical simulations allow us to recover all the exchange rate densities established in the target zone literature. The generality of our framework has important policy implications for modern target zone arrangements.