This paper derives some equivalent conditions for tail equivalence of a distribution G and the convolution G*H, where G belongs to the exponential distribution class and H is another distribution. This generalizes some existing sufficient conditions and gives further insight into closure properties of the exponential distribution class. If G also is O-subexponential, then the new conditions are satisfied. The obtained results are applied to investigating asymptotic behavior for the finite-time ruin probability in a discrete-time risk model with both insurance and financial risks, where the distributions of the insurance risk or the product of the two risks may not belong to the convolution equivalence distribution class.