An inequality for variances of the discounted rewards

We consider the following two definitions of discounting: (i) multiplicative coefficient in front of the rewards , and (ii) probability that the process has not been stopped if the stopping time has an exponential distribution independent of the process. It is well known that the expected total discounted rewards corresponding to these definitions are the same. In this note we show that , the variance of the total discounted rewards is smaller for the first definition than for the second definition.