Beta, as a measure of risk based on market prices of shares, has been widely debated and researched in the strong, semi-strong and weak markets. It has been proved that there is neither negative nor abnormal beta. Past studies rarely considered frontier and infant markets such as Dar es Salaam Stock Exchange (DSE) while studying beta and its behavior. By means of the corresponding closing share prices of 17 companies during a continuous 246-day trading period in 2018 extracted from DSE database, this study examines the trading frequency anomalies in infant markets by testing returns and sensitivity of shares and portfolios. Through computing the betas of DSE traded shares, this study has found many abnormalities. The shares showed infrequent trading like bonds. The prices were constant over a short period of time, and sometimes the shares were not traded at all. Due to this small volatility, the shares showed abnormal behavior which resulted in negative beta sometimes. We concluded that this could be due to two major reasons. Firstly, there is insufficient knowledge on the share market among the East African investors and the public, and secondly, the markets are rather young and the trading platforms and infrastructures are not so well-established. We, therefore, suggest the policy makers to optimize share trading in the region by considering the findings of this study.