This paper applies the subsets of the change in first-degree stochastic dominance (FSD) shifts to the insurance model for the risk-averse agent. An individual faced with the change in the distribution of the random damage will demand more insurance to protect his initial wealth. This widespread belief depends on the preference of the decision-maker and the restrictions on the change in the cumulative distribution functions (CDFs) or the probability density functions (PDs). This paper shows that the risk-averse decision makers buy a greater insurance coverage for the subset of FSD shifts including a left-side relatively weak first-degree stochastic dominance shift. This general comparative static statement can be also investigated in other problems such as the standard portfolio model or the competitive firm model under uncertainty which has a linear payoff.