For the multi-objective time series search problem, Hasegawa and Itoh [Theoretical Computer Science, Vol.78, pp.58-66, 2018] presented the best possible online algorithm balanced price policy for any monotone function f:Rk→R. Specifically the competitive ratio with respect to the monotone function f(c1,...,ck)=(c1+…+ck)/k is referred to as the arithmetic mean component competitive ratio. Hasegawa and Itoh derived the explicit representation of the arithmetic mean component competitive ratio for k=2, but it has not been known for any integer k≥3. In this paper, we derive the explicit representations of the arithmetic mean component competitive ratio for k=3 and k=4, respectively. On the other hand, we show that it is computationally difficult to derive the explicit representation of the arithmetic mean component competitive ratio for arbitrary integer k in a way similar to the cases for k=2, 3, and 4.