This study noted that despite the numerous studies addressing the problem of stock return distributions, there has been relatively little research in Korea on the application of the stable Paretian distribution, and therefore focused on the application of this distribution. After defining the concept of the stable Paretian distribution and explaining the method for measuring the characteristic exponent, 50 listed firms were selected and their weekly return data were used to test whether the stable Paretian distribution holds for individual stock returns. The test results showed that depending on the stock, cases exhibited normal distributions, Cauchy distributions, or stable Paretian distributions, and the possibility of infinite variance of returns was raised. Furthermore, through the verification of additive stability, the possibility of infinite variance in returns that had been raised was explored using a derived simple statistical method.