When a new investment opportunity of purchasing a new device occurs,the investors must decide whether or not and when to buy this device in an online fashion.That is,the online player must make an investment decision while neither future demand for orders nor future investment opportunities are known.This problem which generalizes the basic leasing problem has been introduced by Azar et al.,and then two special cases have been studied by Damaschke.In the so-called equal prices model a 2-competitive algorithm is devised and a 1.618 lower bound is given.Here we make use of an averaging technique and obtain a better tight lower bound of 2,in other words,this lower bound cannot be improved. Furthermore,another special case which only considers two-stage device replacement is studied in this paper.Accounting for the interest rate is an essential feature of any reasonable financial model.Therefore,we explore the two-stage model with and without the interest rate respectively.In addition,we introduce the risk-reward model to analyze this problem and improve the competitive ratio performance. When a new investment opportunity of purchasing a new device occurs, the investors must decide whether or not and when to buy this device in an online fashion.That is, the online player must make an investment decision while neither future demand for for nor nor future investment Opportunities are known. This problem which generalizes the basic leasing problem has been introduced by Azar et al., and then two special cases have been studied by Damaschke. In the so-called equal prices model a 2-competitive algorithm is devised and a 1.618 lower bound is given. Here we make use of an averaging technique and obtain a better tight lower bound of 2, in other words, this lower bound can not be improved. Further, another special case which only considers two-stage device replacement is studied in this paper.Accounting for the interest rate is an essential feature of any reasonable financial model.Therefore, we explore the two-stage model with and without the interest rate respectively. In addition, we i ntroduce the risk-reward model to analyze this problem and improve the competitive ratio performance.