This paper investigated a new framework for the competitive analysis of the Bahncard problem. In contrast to the earlier approach we introduce the interest rate
and the risk tolerance
into the model, in which the traveller can develop the optimal trading strategies based on his risk preference. Set
. We prove that the Bahncard problem with the interest rate is
is the critical point. Then we further design a t-tolerance strategy and present a surprisingly flexible competitive ratio of
is the optimal competitive ratio for the Bahncard problem with the interest rate and
is the percentage of discount.