An Alternative Multiple Contracts Version of SACRE

Taking into consideration that the SACRE (system of increasing amortization in real terms), as originally instituted by Caixa Econômica Federal, is not financially consistent, an exact procedure, denoted as SACRE*, was proposed in de Faro and Lachtermacher (2022). An alternative version (SACRE-F) of this system was proposed by Forger (2010). The present paper submits a multiple contract version of SACRE-F. It is shown that, taking into due account the financial institution cost of capital, it is always better to implement the multiple contracts approach instead of the single contract option. An comparison between the both alternatives are detailed.


Introduction
In 1996, Caixa Econômica Federal (CEF), which is the foremost house-financing institution in Brazil with a history of innovation, introduced a new debt amortization scheme named "Sistema de Amortizações Reais Crescentes" -SACRE (system of increasing amortizations in real terms).
As the CEF original version is not financially consistent, two distinct procedures have been suggested to remedy this deficiency.
The first one was developed by Forger (2010) and was based on the well-established principle which states that the present value of the sequence of the corresponding payments has to equal the value of the principal F being financed, taking into account the rate of interest.
The second one named SACRE*, however, proposed by de Faro and Lachtermacher (2022), was devised in such a way as to maintain the original CEF proposal as close as possible.
Given that de Faro and Lachtermacher (2023) has already proposed a multiple contracts version of the SACRE*, we will present a multiple contracts version of the Forger (2010) proposal, namely SACRE-F.

The Single Contract Version of the SACRE-F Procedure
Denoting by F the loan amount, and by i the periodic interest rate being charged, suppose that in the case where a single contract is considered, it has been stipulated that the debt has to be reimbursed by n periodic payments with the k-th payment being denoted as ̂, since will denote the k-th payment in the case of implementation of SACRE*.
Like the case of the original CEF scheme, the number n of the periodic payments is divided into subperiods, each one with m constant payments.
Fundamentally, it is supposed that: Furthermore, as also proposed by Forger (2010), it is convenient to decompose the index k, 1 ≤ k ≤ n, that identifies the k-th payment ̂, according to ℓ and m, as follows: ( ) 1 k p m q = −  + , with 1 p  and 1 qm  With this proviso, we can simplify the notation in such a way that: ̂=̂( −1)× + =̂′ , for any , 1 ≤ ≤ ℓ, and any , 1 ≤ ≤ With this notation, it follows that expression (1') can be rewritten as: or, after some simple manipulations Additionally, Forger (2010) also proposed that, from any subperiod to the next, there is a linear increase on the values of the constant payments. It was then derived that: In summary, the sequence of payments when implementing the Forger version of SACRE is: Before proceeding, it is interesting to point out, as shown in Forger (2010), that expression (4) encompasses two very particular and important cases. For instance, when ℓ = p = 1 and m = n, we have the case of constant payments. While, if ℓ = n, p = k and m = q = 1, we have the case of constant amortization.
With regard to the evolution of the outstanding debt at time k , also following Forger (2010), we have 1, 2, , kn = and 0 SF = : with the parcels of amortization and of interest, which comprises the payment ̂ being respectively given as: and

A numerical Example
Fixing the number of payments n = 12, with ℓ = 4 and m = 3, we present in Table 1 the corresponding sequences of ̂, ̂ and of ̂. Additionally, to make a numerical comparison with the case of SACRE*, Table 1 also presents the corresponding values of the sequence of payments , the sequence of the values of the outstanding debt , and also the sequence of the parcels of interest . In both cases, = 12,000 units of capital and = 1% per period. As it should be expected, both the SACRE-F and the SACRE* are confirmed to be financially consistent. That is, both procedures lead to full amortization of the debt.
On the other hand, although 1 >̂1, we have that ∑− 1 > ∑ −1 . Therefore, the debtor must pay more interest in the case of SACRE-F.
Moreover, this appears to be a general result. Since, as illustrated in Figure 1, whereas F equals to 1,200,000 units of capital, to magnify the numerical differences, and n = 240 months (20 years), we have that: where ρ, supposed to be relative to the same period as the interest rate i, denotes the financial institution cost of capital.
It should be noted that, in Figure 1, the interest rate i is monthly, while the opportunity cost, identified as ρ a , is expressed in annual terms.  Tables 2 and 3, which refer to the cases where i = 0.5% monthly and i =1% monthly, respectively, we have the percentual increase of the fiscal gain = [ 1 ( ) 2 ( ) − 1] × 100 where ρ a expresses the annual value of the opportunity cost, and where a n expresses the length of the contract in years. The results presented, which are even greater if the interest rate i is increased, confirm that the financing institution, when writing a single contract, should always choose the SACRE* version.

The Multiple Contracts Alternative
Instead of a single contract, the financial institution has the option of requiring the borrower to write n subcontracts -one for each of the n payments that would be associated with the case of a single contract, with the principal 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 principal of the k-th subcontract, denoted by ̂, is: In this case, the parcel of amortization associated with the k-th payment, denoted by ̂′ , will be: Namely, the parcel of amortization associated with the k-th subcontract is exactly equal to the value of the corresponding principal.
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 ̂′ , wherein: From the strict accounting point of view, not taking into consideration the costs that may be associated with the bookkeeping and registration of the subcontracts, the total interest payment is the same in both cases. However, in terms of present values, and depending on the financial institution's opportunity cost, it is possible that the financial institution will be better off if adopts the option of multiple contracts.

A Simple Numerical Example
Considering the same simple numerical example in section 3, but now fixing F=120,000 units of capital, Table 4 presents the values of the sequence of payments ̂, the sequence of the parcels of interest, ̂ in the case of a   (1974), which internal rate of return is known to be unique, and, in this case, is equal to zero. Therefore, we are assured that: That is, at least in the case of our simple numerical example, the financial institution granting the loan will be better off if it adopts the multiple contracts option.

A General Analysis
In the previous section, focusing attention on the case of a contract with only 12 payments, it was verified that the sequence dk of differences of the interest payments has just one change of sign, thereby assuring us of the uniqueness of the corresponding internal rate of return, which was known to be zero.
However, when the number of payments is increased, it is possible to have instances where more than one change of sign can occur in the sequence dk .
This possibility is illustrated in Figure 2, which refers to the case where the contract has a term of 15 years, with ℓ = 15 and monthly payments, with the monthly interest rate i going from 0.5% up to 3%, and with the value of the loan amount still fixed at 120,000 units of capital as well.
While we have only one change of sign if the interest rate is 0.5%, 1.5%, 2.5% or 3%, we have 3 changes of sign if is 1% or 2%.
However, considering a classical result first stated by Norstrom (1972), which is based on the sequence of the accumulated values of the sequence dk , we can still guarantee the uniqueness of the corresponding internal rate of return, which value we already know to be zero. Therefore, in these instances we also have ̂> 0 as illustrated in Figure 3, where 0% < ρ ≤ 10% monthly, and with the monthly interest rate i going to 0.5% to 3%.    Therefore, as indicated by the results in Tables 5 to 8, the financial institution should always prefer the multiple contracts version of the SACRE-F scheme.

A multiple contracts paradox
In section 3, comparing the SACRE* with the SACRE-F for the case of single contracts, it was shown that the financing institution should always prefer the first. However, and thus leading to a paradox, in the case of multiple contracts this is not always true.
For instance, fixing F = 1,200,000 units of capital, to magnify the numerical differences, if the term of the contract is 15 years, and the financing rate is 0.5% per month, Table 9 presents the value of V3(ρ) as given in relation (13), and the value of V4(ρ), defined follows: As can be seen, and thus evidencing a paradox, in the case of multiple contracts, we have situations where the financing institution should prefer the SACRE-F scheme.
For instance, this occurs when either the opportunity cost is 25% in annual terms, or 30%. In both cases, we have ˆ0   , which means that, in the case of adopting multiple contracts, it is not always true that the financing institution is better off if it adopts the SACRE* scheme.
In Table 10, which refers to the case where na is 30 years, we also have situations where the financing institution would be better off if the multiple contracts version of the SACRE-F were applied.  Table 11 that the SACRE-F is always dominated. Table 11. Values of the numerical differences ̂′ when i = 1.0% p.m. and na = 15 years 3 ( ) 4 ( ) ̂′ On the other hand, whenever i is greater than 1% per month, it appears that the paradox vanishes establishing that the SACRE* version of multiple contracts is the better choice.

Conclusion
Given that the SACRE, as originally proposed by CEF, is not financially consistent, two variants have been proposed.
As shown here, the de Faro and Lachtermacher (2022) SACRE* variant, as analyzed in de Faro and Lachtermacher (2023), appears to be the dominant one, both in the case of a single contract and in the case of multiple contracts.
Notwithstanding, for unusually high values of the financing institution opportunity cost, a multiple contracts version of the SACRE-F proposal may be the dominant one.
Furthermore, if the SACRE-F is chosen to be implemented, the financing institution should always make use of the multiple contract's version.