An optimal approach for solving serial replacement problems with uncertain rewards and a risk-averse decision maker was previously introduced by Brown. In this note, we extend Brown's model to allow rewards to be correlated across replacements. Correlation may be useful in modeling the effects of the external environment on the rewards. However, total enumeration appears to be the onty method to find optimal solutions when rewards are correlated. Thus, we develop a heuristic and compare it to several existing solution approaches. Computational results indicate that, on the average, our heuristic outperforms the best of the other approaches by about 8 percent when there is positive correlation and 11 percent when there is negative correlation.