On some difference-delay equations arising in a problem of capital deposits

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Introduction. We consider a real life problem: a. person has made a deposit of D0 dollars in bank B, which calculates interest on this deposit at in=100•in% after each n+l-st quarter and the interest is compounded at the end of each consecutive year since the deposit date, which means that the interest is capitalised yearly. In the case discussed the basic time unit is a quarter but the conversion period - the time interval at the end of which the interest is compounded - is four quarters (for the terminology see [3]). We ask the following question: what is the balance of the person after n time units, i.e. after n quarters? Such a question is important to people planning various sorts of investments or making arrangements with life insurance institutions. We cannot find an answer to this question in available literature. In particular, it is not to be found in recent books devoted to the subject [see 1-6]. We want to find general formulas that would allow us to express this balance by the other given quantities. Such formulas allow us to solve inverse problems consisting in finding the initial deposit Z)q , the interest rate i or the length of the period after which a capital reaches a given level. In this paper we give explicit formulas for the above-mentioned balance. They can be applied to the problems with any number of payments. At the end of this paper we give some examples of application of these formulas to solving the mentioned inverse problems. The obtained formulas make solving such problems easy. A straightforward application of backward recurrence formulas derived from formula (2), although possible, is quite troublesome.ct