In this work we consider the shortest path problem with uncertainty in arc lengths and convex risk measure objective. We explore efficient implementations of sample average approximation (SAA) methods to solve shortest path problems when the conditional value at risk and entropic risk measures are used and there is correlation present in the uncertain arc lengths. Ourwork explores the use of different decomposition techniques to achieve an efficient implementation of SAA methods for these nonlinear convex integer optimization problems. A computational study shows the effect of geometry, uncertainty correlation and variance, and risk measure parameters on efficiency and accuracy of the methods developed.