International Journal of Mathematics and Mathematical Sciences

Volume 2016, Article ID 2152189, 14 pages

http://dx.doi.org/10.1155/2016/2152189

## Loan Transactions with Random Dates for the First and Last Periodic Instalments

Departamento de Economía y Empresa, Universidad de Almería, La Cañada de San Urbano s/n, 04120 Almería, Spain

Received 9 March 2016; Accepted 11 July 2016

Academic Editor: Chin-Chia Wu

Copyright © 2016 María del Carmen Valls Martínez and Salvador Cruz Rambaud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Usually, loan transactions contracted in practice are nonrandom; that is to say, all amounts received (principal) and paid (period instalments) by the borrower are previously agreed with the lender, as well as their respective dates. In this paper, two new alternative loan models are introduced, depending on whether the borrower survives or not to fulfil all repayment obligations. In this way, either the initial or the final date of repayments can be subject to this contingency. Additionally, the different parameters of such random transactions are determined, as well as several measures of profitability/cost for the lender/borrower, respectively. These transactions can be attractive for both the lender and the borrower, which therefore make them worthy of consideration and subsequent implementation for the benefit of both parties.

#### 1. Introduction

It is well known that the current economic and financial situation brought about by the crisis in the construction sector has favoured the design and availability of new banking products [1, pp. 53–60] and, more specifically, mortgage loans [2]. In effect, on the one hand, there exists an increasing concern about the purchasing power of future generations. This means that those who contract a mortgage loan [3] are in favour of any repayment agreement which is finalised in the case of their premature death and does not transfer to their descendants. Additionally, this situation can be of particular interest to banking institutions because it can stimulate the provision of certain mortgage loans (generally agreed to be repaid over a long-term period) by persons of advanced age. Thus, the first objective of this paper is to present a novel kind of loan which can be attractive for elderly people because the outstanding principal is cancelled when they die and consequently it does not suppose an economic liability for future generations. This takes into account the fact that the current situation of an uncertain labour market in all modern economies mainly affects young people.

On the other hand, the opposite scenario is also possible, promoting the design of loans which adapt to the possible financial difficulties of the contracting parties themselves. This paper therefore gives consideration to two possible situations:(1)Where premature death of the contract holder causes the mortgage repayment obligations to pass to those who inherit the mortgaged property (inverse mortgage). In this case, the property itself is a potential guarantee that the outstanding amount of mortgage repayment can be met.(2)Where those who inherit a mortgaged property cannot, or do not wish to, make an immediate sale to pay off the outstanding mortgage but do not have sufficient liquidity to face up to their inherited financial obligations.Observe that the first situation means that the initial payments are assured and the final payments are random, whilst the second situation leads to random initial payments but the final ones are assured [4]. Once we have explained the reasons for introducing these two novel random transactions, we then proceed, in the paragraphs which follow, to specify their financial conditions and main characteristics.

In traditional loan transactions, the amounts repaid by the borrower to amortize the principal and the corresponding periodicity of repayment are established at the beginning of the loan [5]. However, some loans are possible in which the dates for the initial and final instalment payments are random, where the transaction duration would be subject to the occurrence of an eventuality. There follows an example of each one of these possible innovative transactions:(I)Loans where the payment of the first instalment is fixed and the payment of the last one is random. In this case, the amounts which amortize the received principal start to be repaid on an agreed date following the granting of the mortgage loan. Nevertheless, the duration depends on the survival period of the borrower [6]. This would be the case of a person who requests a loan to be amortized with the revenues coming from a life insurance policy, for example. Thus, if the borrower dies, the periodic payments would terminate. More generally, this situation occurs when a person of a certain age requests a loan with the intention of not leaving the debt to his/her heirs in a will. Nowadays and in practice, this is similar to mortgage insurance [7] required by the financial institution in case of death or total incapacity of the borrower. In fact, they are two different financial transactions where the risk is assumed by the insurance company. Nevertheless, the transaction presented here differs in that the risk is assumed by the lender.(II)Loans where the date of the first payment is random and the final one is fixed. In this case, the instalments which amortize the received principal start to be paid at an uncertain moment after the agreed instant at which the principal is provided by the lender, but the final instant is explicit in the mortgage contract. This would be the case of a person who requests a loan to be amortized by his/her heirs when he/she dies. As the instant of death of the contract holder is not previously known, although the probability factor can be calculated, the date of the initial instalment will be random. In practice, this is similar to the so-called inverse mortgage, in which the borrower offers his/her house as a guarantee [8] and, as a consequence, he/she receives a single amount (the loan principal). Starting from the date of the borrower’s death, this amount will be repaid, together with the accrued interest, by his/her heirs. However, currently, in an inverse mortgage, in order to obtain a fiscal advantage from the inheritance, the principal must be amortized with a single payment, unless at this moment the heirs agree with the lender on another loan transaction which replaces the first one. What is proposed here consists of a single loan transaction in which the lender assumes the risk of a repayment by the heirs, depending on the moment of death of the contracting party.All these innovative loan agreements are independent of the principal repayment methods [9] which are common within the loan amortization models (French method, constant principal repaid method, American method, etc.). In effect, the contracting party of a mortgage loan [10–12] can use any of the traditional methods to repay the loan, among which we can cite the French method (equal payments for all periods) which is the most usual, the constant principal repaid method, and the American method [13]. Moreover, these methods can be combined with other financial procedures, such as interest-only periods and fixed or variable interest rates [14].

Finally, observe that the issue considered in this paper is included in a wide variety of mortgage loans labelled, in general, “flexible loans” which the credit institutions have recently started to offer (see [15, pp. 173–189] and [16, pp. 829–853]). In addition to the wide range of offers described in the previous paragraph, there exist other possibilities of mortgage loan amortization which have not yet been defined and which can prove very interesting, due to their flexibility, depending on the economic situation of the borrower.

This paper is organized as follows. After describing in the Introduction the context of our research, in Sections 2 and 3, we will analyse the two innovative kinds of loan transaction which have been proposed. Finally, Section 4 summarizes and concludes the paper.

#### 2. Amortization of a Loan Transaction Where the First Repayment Instalment Is Fixed and the Last Is Random

Let us consider a loan transaction in which the borrower receives the principal at instant 0 to be repaid by means of periodic amounts , on specified dates (). If the last payment is subject to a possible contingency, the borrower would have to pay a series of amounts greater than those corresponding to a similar transaction which is not subject to uncertainty. Thus, we propose the following equation of financial random equivalence at instant 0, by using the exponential discount function with variable discount rate according to the corresponding period: where represents an additional payment to the lender to cover the element of risk in the transaction.

When the contingency is the death of the borrower, obviously, is the risk that the borrower dies and consequently the debt payment disappears to the detriment of the lender. Thus, by considering the survival of the borrower, one has (see [17]) being the probability that a person aged reaches the age and being the number of persons who survive beyond the age .

By considering (1) and (2) and defining , with being the probability of survival at instant , one haswhich coincides with Gil Luezas and Gil Peláez [18] or Gil Peláez [19], by considering the exponential discount function.

Observe that the subscript denotes the instant at which the corresponding payment is due, calculated from the start of the financial transaction. Therefore, is the number of living persons of the same age as the borrower when he/she contracted the loan. Thus, for instance, if the borrower is 60 at the formal contract date, is the number of persons of the same generation who reach such age. Analogously, is the number of persons of this generation who reach 68.

The outstanding principal at an intermediate instant , denoted by , is the (outstanding) balance still to be repaid by the borrower to pay off the loan. It can be calculated using three different methods [20, pp. 249–256].

*(I) Prospective Method (according to the Future Periodic Instalments to Be Paid, i.e., from Instant ** to the Last Payment Due **). *One hasfrom which simple algebra shows that

*(II) Retrospective Method (according to the Periodic Amounts Already Paid by the Borrower from the Commencement to Instant **)*. Starting from (4), one hasthat is to say,which can be simplified toand, multiplying and dividing by , finally one has

*(III) Recursive Method (according to the Outstanding Principal Calculated at a Former Date)*. Starting from (8) for period , this can be simplified using standard mathematical procedures, resulting inAs previously indicated, if the loan amortization is subject to a contingency, the borrower would have to pay the amounts involved in the equation of financial random equivalence (3). However, if the transaction is not subject to this contingency, the corresponding amounts, , would verify the following familiar equation of financial equivalence at the start of the transaction:By comparing (3) and (11), it can quite be simply observed that . Evidently, the difference between the two payments is exclusively due to the risk whereby this difference will be called the* risk quota* which will be denoted by . Therefore,By considering (10) and (12), the additional amount that the borrower has to pay to the lender in each period due to assumed risk can be simplified as follows:Observe that , called the* saving quota*, is the part of the payment corresponding to a riskless transaction and, consequently, it is allotted to pay the interest of period* s* () and the amortization of a part of the principal ():These random transactions can be agreed with constant or variable interest rates. Moreover, the amount of the periodic payment () can be chosen and then the problem is to determine the principal (). Alternatively, once the principal is fixed, one can determine the periodic instalments ; finally, once and the saving quota () derived from (11) are fixed, the amount of the risk quota () corresponding to each period can be determined, giving rise to a variable payment .

*Example 1. *Assume that, in 2013, a person aged 50 requests a loan of 50,000. In order to determine the risk, the financial entity applies the probability of survival corresponding to a person with the same age and sex (male) as the borrower (see tables PERM/F-2000P). According to this information, one can determine the periodic payments and the amortization schedule (observe that when the loan contract comes into force, the probability of survival is 1 because we are dealing with probabilities conditioned to the survival of the borrower at that moment). By applying the aforementioned survival tables, this person is expected to survive for a further 44 years, that is to say, to the age of 94. By considering that the periodic amounts will be constant and that the transaction has been agreed with an annual variable interest rate of 9% for the first 5 years and that it will be updated every 5 years with an increase of 0.2%, the different parameters of the loan transaction can be observed in Table 1.