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Minimization of the expected value of age of information (AoI) is a risk-neutral approach, and it thus cannot capture rare, yet critical, events with potentially large AoI. In order to capture the effect of these events, in this paper, the notion of conditional value-at-risk (CVaR) is proposed as an effective coherent risk measure that is suitable for minimization of AoI for real-time IoT status updates. In the considered monitoring system, an IoT device monitors a physical process and sends the status updates to a remote receiver with an updating cost. The optimal status update process is designed to jointly minimize the AoI at the receiver, the CVaR of the AoI at the receiver, and the energy cost. This stochastic optimization problem is formulated as an infinite horizon discounted risk-aware Markov decision process (MDP), which is computationally intractable due to the time inconsistency of the CVaR. By exploiting the special properties of coherent risk measures, the risk-aware MDP is reduced to a standard MDP with an augmented state space, for which we derive the optimal stationary policy using dynamic programming. In particular, the optimal history-dependent policy of the risk-aware MDP is shown to depend on the history only through the augmented system states and can be readily constructed using the optimal stationary policy of the augmented MDP. The proposed solution is shown to be computationally tractable and able to minimize the AoI in real-time IoT monitoring systems in a risk-aware manner.