Multiple Contracts: The Case of Periodic Balloon Payments – Constant Installments

I. INTRODUCTION

In a pioneering contribution De-Losso et al. (2013) introduced the idea of substituting a single contract by multiple contracts. Specifically, considering a loan of F units of capital, with a term of n periods, at the periodic rate i of compound interest, which has to be reimbursed by periodic constant payments, the usual practice is that the financial institution providing the loan requires the borrower to sign a single contract.

Alternatively, the financial institution may require the borrower to sign n contracts. One for each of the n payments of the single contract. The reason for this procedure is that, considering its cost of capital, the financial institution may have a reduction, in terms of present values, of the amount of taxes it must pay.

In the above-mentioned contribution, it was considered only the usual case of a loan that must be reimbursed by n periodic constant payments. In this paper it will be examined the case where, besides the n periodic payments, the borrower has also to pay a sequence of $\ell$ balloon payments, with periodicity m.

This may be very interesting in the case of Brazil. Since, by law, every employer must pay, to each registered employee, thirteen monthly payments per year. The thirteenth one being known as the "décimo terceiro salário" (the thirteenth wage), which is usually paid during the month of December. As will be shown, the financial institution is also better off if a single contract is substituted by multiple contracts in this new case.

II. THE CASE OF A SINGLE CONTRACT

Suppose that, besides the n constant equal payments p, the borrower has also to pay a sequence of $\ell$ balloon payments. Each one of them with the value P, with periodicity m, where $m=n/\ell$, and the first balloon payment at epoch m.

Denoting by $i_m$ the equivalent rate of compound interest relative to m periods, so that

the assumption of n constant payments with value p, and additional $\ell$ payments with value P, with periodicity equal to m, implies that the following equation must be satisfied:

Therefore, once the value P of the $\ell$ balloon payments has been established, the value p of the n periodic constant payments will be equal to:

Noticing that the $k^{th}$ payment, $p_k$, is such that

it follows that, denoting by $S_k$ the outstanding debt at epoch k, just after the $k^{th}$ payment $p_k$, by $J_k$ the parcel of interest, and by $A_k$ the corresponding parcel of amortization of $p_k$, we have:

with $S_0=F$ and $S_n=0$ and

As a simple numerical illustration of a single contract, consider the case where $F=$100,000.00$,$n = 12$,$\ell = 4$,$m = 3$,$P = $10,000.00$and$i = 1%$perperiod.Inthiscase,thesolutionofequation(3)impliesthat,besidesthe4balloonpaymentsof$ 10,000.00, 12 periodic payments of $ 5,584.66 will be necessary. Table 1 presents the evolution of debt.

III. THE CASE OF MULTIPLE CONTRACTS

Instead of a single contract, the financial institution providing the loan has the option of requiring the borrower to sign n individual contracts - one for each of the n payments that would be associated with the case of a single contract. With the value of the loan of the $k^{th}$ subcontract being the present value, at the same interest rate i, of the $k^{th}$ payment of the single contract.

That is, the value of the principal of the $k^{th}$ subcontract, denoted $F_k$, is:

In this case, the parcel of amortization associated with the $k^{th}$ payment, denoted as $A_k^{\prime }$ will be:

Namely, the parcel of amortization associated with the $k^{th}$ subcontract is exactly equal to the value of the loan of the $k^{th}$ subcontract.

On the other hand, from an accounting point of view, it follows that the parcel of interest associated with the $k^{th}$ subcontract, which will be denoted by $J_k^{\prime }$ wherein:

It should be especially noted that, although from the strict point of view, not taking into consideration the costs that may be associated with the bookkeeping and registration of the n subcontracts, the total of interest payments is the same in both cases. That is:

However, in terms of present values, and depending on the financial institution cost of capital, it is possible that the financial institution will be better off if it adopts the multiple contracts option. As will be illustrated in the case of our simple example.

Table 2 presents the values of the sequence of payments $p_k$, the sequence of the parcels of interest $J_k$ in the case of a single contract, and the sequence $J_k^{\prime }$ of the parcels of interest in the case of the adoption of the option of multiple contracts. As well as the sequence $F_k=A_k^{\prime }$ of principals and the sequence of amortization components of the subcontracts. Additionally, Table 2 presents also the sequence of differences, $d_k$, and the sequence of accumulated values of $d_k$, denoted as ${\mathrm{\Delta }}_k$, given by:

It is interesting to note that the sequence of the values of $d_k$ has more than one change of sign. However, adapting the proposition in Norstrom (1972), as the sequence of accumulated values of $d_k$, does not change sign, it follows that $d_k$ has a unique internal rate of return. Which is, in this case, null. Since present values of the interest sequences of both cases, single $V_s$ and multiple $V_m$, at interest rate $\rho$, that denotes the financial institution cost of capital, are $V_s(\rho )=V_m(\rho )=0$. With $\rho$ being relative to the same period as the financing rate i. With:

As previously noted, the first point that should be observed is that, although the sum of the parcels of interest are the same in the case of a single contract and in the case of multiple contracts, the timing and the values of their respective payments are not the same.

Considering the period of i and $\rho$ one month, Table 3 shows, for our example, the present values of the interest sequences for several values of the cost of capital, in annual terms, ${\rho }_a$.

Therefore, in the case of our simple numerical example, we have , if . Consequently, the financial institution providing the loan should prefer to implement the multiple contracts option.

Although a more general analysis should consider the case where the periodic balloon payments P can assume any value, as long , otherwise p would either be null or negative, we are going to focus attention on the case where P = p. Since this is the case that better contemplates the Brazilian peculiarity of thirteen yearly wages of the employees.

In this case, it follows, from equation (2) (see appendix A for demonstration), that the value of the periodic and balloon payments, p, will be such that:

As a numerical illustration, consider the case where $F=$100,000.00$,i=1$4,363.31, plus the two payments of the same value at epoch 12 and epoch 24. Table 4 presents the evolution of the debt of a single contract of this case and Table 5 the corresponding multiple contracts.

Table 5: Multiple contracts – "thirteenth wage" Although the sequence

$d_k$ also has more than one change of sign, the sequence of its accumulated values ${\mathrm{\Delta }}_k$ has no change of sign. Therefore, the sequence of differences $d_k$ has a unique internal rate of return. Consequently, we are assured that for all Table 6 shows the values of $V_s(\rho )$ and $V_m(\rho )$

4.1. Reduction in the value of the installments

Given that, considering a contract with a term of n years, the effect of the "thirteenth wage" is to imply the value of the monthly payments to be reduced, as compared to the case of no "thirteenth wage", it is interesting to give a numerical illustration of the size of the reduction.

For instance, considering the constant installments methods, if the contract has a term of n = 20 years, with i = 1% per month, with and without the "thirteenth wage", it follows that the value of the monthly payment will be p = $1,101.09.Whileinthecaseofthe“thirteenthwage”,thevalueoftheresultingmonthlypaymentwillbe{p}^{\prime }=$1,020.61. That is, we will have a reduction of 7.31% in the value of the monthly payment.

However, as shown in Table 7, the resulting reduction decreases when the value of the financing interest rate i is increased. Furthermore, for every value of i, the resulting reduction does not change with the value of the term n.

A comprehensive analysis would have to consider different values of the periodicity m of the balloon payments. However, given that De-Losso et al. (2013), did not address the behavior of what can be defined as the fiscal gain $\delta$, given by:

We are going to focus on only two cases. The one with no balloon payments and the one with the "thirteenth wage".

5.1. The case of periodic payments only

In Tables 8 to 13, a comparison of single and multiple contracts values of $\delta$ are presented. In terms of the annual cost of capital value ${\rho }_a$, for different values of the monthly interest rate i, for contracts with term n ranging from 5 to 30 years.

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As shown, in all cases the fiscal gain is substantial. Which implies that the financial institution providing the loan should always choose the option of implementing multiple contracts.

5.2. The case of the "thirteenth wage"

Analogously, Tables 14 to 19 present the value of the fiscal gain $\delta$, in terms of the annual value ${\rho }_a$ for different values of the monthly interest rate i, for contracts with length n of 5 to 30 years.

Additionally, fixing the financing interest rate i at 1% per month, Figures 3 and 4 depict the behavior of the fiscal gain, respectively for the cases where the opportunity cost varies from 5% to 30% per year, and when the length of the contract varies from 5 to 30 years, for the case of thirteen wages.

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As shown, in all cases fiscal gain is very substantial. Therefore, given that implementing the policy of considering the "thirteenth wage" is beneficial to the borrower, the financial institution providing the loan, should also prefer to adopt the option of substituting a single contract by multiple contracts.

V. CONCLUSIONS

As shown in the other cases previously analyzed, cf. De-Losso et al. (2013), for the case of constant payments, de Faro (2021), for the case of periodic payments only, de Faro (2022), for the case of the constant amortization, de Faro and Lachtermacher (2023b), for the case of the an alternative version of the SACRE, and de Faro and Lachtermacher (2024), for the case of the German system of amortization, it always better for the financial institution providing the loan to implement the policy of multiple contracts.

Appendix A

since