\documentclass[11pt,twoside]{article}\makeatletter

\IfFileExists{xcolor.sty}%
  {\RequirePackage{xcolor}}%
  {\RequirePackage{color}}
\usepackage{colortbl}
\usepackage{wrapfig}
\usepackage{ifxetex}
\ifxetex
  \usepackage{fontspec}
  \usepackage{xunicode}
  \catcode`⃥=\active \def⃥{\textbackslash}
  \catcode`❴=\active \def❴{\{}
  \catcode`❵=\active \def❵{\}}
  \def\textJapanese{\fontspec{Noto Sans CJK JP}}
  \def\textChinese{\fontspec{Noto Sans CJK SC}}
  \def\textKorean{\fontspec{Noto Sans CJK KR}}
  \setmonofont{DejaVu Sans Mono}
  
\else
  \IfFileExists{utf8x.def}%
   {\usepackage[utf8x]{inputenc}
      \PrerenderUnicode{–}
    }%
   {\usepackage[utf8]{inputenc}}
  \usepackage[english]{babel}
  \usepackage[T1]{fontenc}
  \usepackage{float}
  \usepackage[]{ucs}
  \uc@dclc{8421}{default}{\textbackslash }
  \uc@dclc{10100}{default}{\{}
  \uc@dclc{10101}{default}{\}}
  \uc@dclc{8491}{default}{\AA{}}
  \uc@dclc{8239}{default}{\,}
  \uc@dclc{20154}{default}{ }
  \uc@dclc{10148}{default}{>}
  \def\textschwa{\rotatebox{-90}{e}}
  \def\textJapanese{}
  \def\textChinese{}
  \IfFileExists{tipa.sty}{\usepackage{tipa}}{}
\fi
\def\exampleFont{\ttfamily\small}
\DeclareTextSymbol{\textpi}{OML}{25}
\usepackage{relsize}
\RequirePackage{array}
\def\@testpach{\@chclass
 \ifnum \@lastchclass=6 \@ne \@chnum \@ne \else
  \ifnum \@lastchclass=7 5 \else
   \ifnum \@lastchclass=8 \tw@ \else
    \ifnum \@lastchclass=9 \thr@@
   \else \z@
   \ifnum \@lastchclass = 10 \else
   \edef\@nextchar{\expandafter\string\@nextchar}%
   \@chnum
   \if \@nextchar c\z@ \else
    \if \@nextchar l\@ne \else
     \if \@nextchar r\tw@ \else
   \z@ \@chclass
   \if\@nextchar |\@ne \else
    \if \@nextchar !6 \else
     \if \@nextchar @7 \else
      \if \@nextchar (8 \else
       \if \@nextchar )9 \else
  10
  \@chnum
  \if \@nextchar m\thr@@\else
   \if \@nextchar p4 \else
    \if \@nextchar b5 \else
   \z@ \@chclass \z@ \@preamerr \z@ \fi \fi \fi \fi
   \fi \fi  \fi  \fi  \fi  \fi  \fi \fi \fi \fi \fi \fi}
\gdef\arraybackslash{\let\\=\@arraycr}
\def\@textsubscript#1{{\m@th\ensuremath{_{\mbox{\fontsize\sf@size\z@#1}}}}}
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\def\abbr{}
\def\corr{}
\def\expan{}
\def\gap{}
\def\orig{}
\def\reg{}
\def\ref{}
\def\sic{}
\def\persName{}\def\name{}
\def\placeName{}
\def\orgName{}
\def\textcal#1{{\fontspec{Lucida Calligraphy}#1}}
\def\textgothic#1{{\fontspec{Lucida Blackletter}#1}}
\def\textlarge#1{{\large #1}}
\def\textoverbar#1{\ensuremath{\overline{#1}}}
\def\textquoted#1{‘#1’}
\def\textsmall#1{{\small #1}}
\def\textsubscript#1{\@textsubscript{\selectfont#1}}
\def\textxi{\ensuremath{\xi}}
\def\titlem{\itshape}
\newenvironment{biblfree}{}{\ifvmode\par\fi }
\newenvironment{bibl}{}{}
\newenvironment{byline}{\vskip6pt\itshape\fontsize{16pt}{18pt}\selectfont}{\par }
\newenvironment{citbibl}{}{\ifvmode\par\fi }
\newenvironment{docAuthor}{\ifvmode\vskip4pt\fontsize{16pt}{18pt}\selectfont\fi\itshape}{\ifvmode\par\fi }
\newenvironment{docDate}{}{\ifvmode\par\fi }
\newenvironment{docImprint}{\vskip 6pt}{\ifvmode\par\fi }
\newenvironment{docTitle}{\vskip6pt\bfseries\fontsize{22pt}{25pt}\selectfont}{\par }
\newenvironment{msHead}{\vskip 6pt}{\par}
\newenvironment{msItem}{\vskip 6pt}{\par}
\newenvironment{rubric}{}{}
\newenvironment{titlePart}{}{\par }

\newcolumntype{L}[1]{){\raggedright\arraybackslash}p{#1}}
\newcolumntype{C}[1]{){\centering\arraybackslash}p{#1}}
\newcolumntype{R}[1]{){\raggedleft\arraybackslash}p{#1}}
\newcolumntype{P}[1]{){\arraybackslash}p{#1}}
\newcolumntype{B}[1]{){\arraybackslash}b{#1}}
\newcolumntype{M}[1]{){\arraybackslash}m{#1}}
\definecolor{label}{gray}{0.75}
\def\unusedattribute#1{\sout{\textcolor{label}{#1}}}
\DeclareRobustCommand*{\xref}{\hyper@normalise\xref@}
\def\xref@#1#2{\hyper@linkurl{#2}{#1}}
\begingroup
\catcode`\_=\active
\gdef_#1{\ensuremath{\sb{\mathrm{#1}}}}
\endgroup
\mathcode`\_=\string"8000
\catcode`\_=12\relax

\usepackage[a4paper,twoside,lmargin=1in,rmargin=1in,tmargin=1in,bmargin=1in,marginparwidth=0.75in]{geometry}
\usepackage{framed}

\definecolor{shadecolor}{gray}{0.95}
\usepackage{longtable}
\usepackage[normalem]{ulem}
\usepackage{fancyvrb}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{marginnote}

\renewcommand{\@cite}[1]{#1}


\renewcommand*{\marginfont}{\itshape\footnotesize}

\def\Gin@extensions{.pdf,.png,.jpg,.mps,.tif}

  \pagestyle{fancy}

\usepackage[pdftitle={Scan to know paper details and author's profile A Multiple Contracts Version of the SACRE},
 pdfauthor={}]{hyperref}
\hyperbaseurl{}

	 \paperwidth210mm
	 \paperheight297mm
              
\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{3}
\clubpenalty=8000
\emergencystretch 3em
\hbadness=4000
\hyphenpenalty=400
\pretolerance=750
\tolerance=2000
\vbadness=4000
\widowpenalty=10000

\renewcommand\section{\@startsection {section}{1}{\z@}%
     {-1.75ex \@plus -0.5ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
     {-1.75ex\@plus -0.5ex \@minus- .2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large}}
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
     {-1.5ex\@plus -0.35ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\large}}
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
     {-1ex \@plus-0.35ex \@minus -0.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\normalsize}}
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
     {1.5ex \@plus1ex \@minus .2ex}%
     {-1em}%
     {\reset@font\normalsize\bfseries}}


\def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{1.0em plus 1pt}
 \@tempdima 1.5em \begingroup
 \parindent \z@ \rightskip \@pnumwidth 
 \parfillskip -\@pnumwidth 
 \bfseries \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}\par
 \endgroup}
\def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\def\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\@ifundefined{c@section}{\newcounter{section}}{}
\@ifundefined{c@chapter}{\newcounter{chapter}}{}
\newif\if@mainmatter 
\@mainmattertrue
\def\chaptername{Chapter}
\def\frontmatter{%
  \pagenumbering{roman}
  \def\thechapter{\@roman\c@chapter}
  \def\theHchapter{\roman{chapter}}
  \def\thesection{\@roman\c@section}
  \def\theHsection{\roman{section}}
  \def\@chapapp{}%
}
\def\mainmatter{%
  \cleardoublepage
  \def\thechapter{\@arabic\c@chapter}
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \pagenumbering{arabic}
  \setcounter{secnumdepth}{6}
  \def\@chapapp{\chaptername}%
  \def\theHchapter{\arabic{chapter}}
  \def\thesection{\@arabic\c@section}
  \def\theHsection{\arabic{section}}
}
\def\backmatter{%
  \cleardoublepage
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \setcounter{secnumdepth}{2}
  \def\@chapapp{\appendixname}%
  \def\thechapter{\@Alph\c@chapter}
  \def\theHchapter{\Alph{chapter}}
  \appendix
}
\newenvironment{bibitemlist}[1]{%
   \list{\@biblabel{\@arabic\c@enumiv}}%
       {\settowidth\labelwidth{\@biblabel{#1}}%
        \leftmargin\labelwidth
        \advance\leftmargin\labelsep
        \@openbib@code
        \usecounter{enumiv}%
        \let\p@enumiv\@empty
        \renewcommand\theenumiv{\@arabic\c@enumiv}%
	}%
  \sloppy
  \clubpenalty4000
  \@clubpenalty \clubpenalty
  \widowpenalty4000%
  \sfcode`\.\@m}%
  {\def\@noitemerr
    {\@latex@warning{Empty `bibitemlist' environment}}%
    \endlist}

\def\tableofcontents{\section*{\contentsname}\@starttoc{toc}}
\parskip0pt
\parindent1em
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\newenvironment{reflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemsep}{0pt}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\itshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{sansreflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\itemsep}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\upshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{specHead}[2]%
 {\vspace{20pt}\hrule\vspace{10pt}%
  \phantomsection\label{#1}\markright{#2}%

  \pdfbookmark[2]{#2}{#1}%
  \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}%
  }{}
      \def\TheFullDate{1970-01-01 (revised: 01 January 1970)}
\def\TheID{\makeatother }
\def\TheDate{1970-01-01}
\title{Scan to know paper details and author's profile A Multiple Contracts Version of the SACRE}
\author{}\makeatletter 
\makeatletter
\newcommand*{\cleartoleftpage}{%
  \clearpage
    \if@twoside
    \ifodd\c@page
      \hbox{}\newpage
      \if@twocolumn
        \hbox{}\newpage
      \fi
    \fi
  \fi
}
\makeatother
\makeatletter
\thispagestyle{empty}
\markright{\@title}\markboth{\@title}{\@author}
\renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2\p@
\belowdisplayshortskip 4\p@ plus2\p@ minus2\p@
\def\@listi{\leftmargin\leftmargini
               \topsep 2\p@ plus1\p@ minus1\p@
               \parsep 2\p@ plus\p@ minus\p@
               \itemsep 1pt}
}
\makeatother
\fvset{frame=single,numberblanklines=false,xleftmargin=5mm,xrightmargin=5mm}
\fancyhf{} 
\setlength{\headheight}{14pt}
\fancyhead[LE]{\bfseries\leftmark} 
\fancyhead[RO]{\bfseries\rightmark} 
\fancyfoot[RO]{}
\fancyfoot[CO]{\thepage}
\fancyfoot[LO]{\TheID}
\fancyfoot[LE]{}
\fancyfoot[CE]{\thepage}
\fancyfoot[RE]{\TheID}
\hypersetup{citebordercolor=0.75 0.75 0.75,linkbordercolor=0.75 0.75 0.75,urlbordercolor=0.75 0.75 0.75,bookmarksnumbered=true}
\fancypagestyle{plain}{\fancyhead{}\renewcommand{\headrulewidth}{0pt}}

\date{}
\usepackage{authblk}

\providecommand{\keywords}[1]
{
\footnotesize
  \textbf{\textit{Index terms---}} #1
}

\usepackage{graphicx,xcolor}
\definecolor{GJBlue}{HTML}{273B81}
\definecolor{GJLightBlue}{HTML}{0A9DD9}
\definecolor{GJMediumGrey}{HTML}{6D6E70}
\definecolor{GJLightGrey}{HTML}{929497} 

\renewenvironment{abstract}{%
   \setlength{\parindent}{0pt}\raggedright
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
   \textcolor{GJBlue}{\large\bfseries\abstractname\space}
}{%   
   \vskip8pt
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
}

\usepackage[absolute,overlay]{textpos}

\makeatother 
      \usepackage{lineno}
      \linenumbers
      
\begin{document}

             \affil[1]{  }

\renewcommand\Authands{ and }

\date{\small \em Received: 1 January 1970 Accepted: 1 January 1970 Published: 1 January 1970}

\maketitle


\begin{abstract}
        




\end{abstract}


\keywords{}

\begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) 
\textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark}
\end{textblock*}

\begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) 
\textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}}
\end{textblock*}


\begin{textblock*}{10cm}(1.05cm,3cm)
{{\textit{CrossRef DOI of original article:}} \underline{}}
\end{textblock*}\let\tabcellsep& 	 	 		 
\section[{I. INTRODUCTION}]{I. INTRODUCTION}\par
In 1996, the "Caixa Econômica Federal" (CEF), which is the main institution for housing financing in Brazil, introduced a debt amortization scheme named "Sistema de Amortizações Reais Crescentes" -SACRE (system of increasing amortizations in real terms).\par
In its original version, this very peculiar amortization system is not financially consistent. Namely, even if all contractual payments are dutifully made, a residual debt remains, which must be paid in full by the borrower, usually one month after the end of the term of the contract.\par
Given that de Faro and Lachtermacher (2022) proposed a financially consistent variant of the SACRE, the purpose of this paper is to formulate a multiple contracts version of this system. Similar to cases of the adoption of either the constant payments scheme or the constant amortization scheme of debt financing, which were considered in De-Losso et al  {\ref (2013)} and in de Faro (2022), it will be shown that the financial institution granting the loan, depending on its cost of capital, may derive substantial income tax reductions in terms of present values. 
\section[{II. THE CASE OF A SINGLE CONTRACT}]{II. THE CASE OF A SINGLE CONTRACT}\par
Denoting by F the loan amount, and by i the periodic rate of compound interest , suppose that, in the case where a single contract is considered, it is stipulated by the financing institution granting the loan that the debt must be repaid in n periodic payments, in accordance with the SACRE scheme.\par
Since the SACRE scheme is a combination of the constant payments scheme with the constant amortization scheme, the number n of payments is divided into â??" subperiods, each with m payments. The numbers n, â??" and m are integer numbers with n = â??" ? m , and with m constant payments in each of the first sub-periods. , for (2)? ? = 1 + ? ( ) × ? ? -1 -? ? ? = 1 , 2 ,..., ?\par
Therefore, using the presumed recurrence method to determine the debtor's balance, we have:\par
(\par
where .? = 1 + ? ( ) ? -1 [ ] / ?\par
This relationship, in view of the value of P 1 presented in Equation  {\ref 1}, can be rewritten as:\par
( At this point, as suggested in de Faro and Lachtermacher (2022), rather than being constant, the last m payments should decrease linearly in accordance with an arithmetic progression of ratio equal to , with , which is a procedure assured to be ? × ?â??" -1 ( ) / ? ? â??" -1 ( ) ? + 1 = ? â??" -1 ( ) ? × 1 + ? × ? ( ) / ?\par
financially consistent whenever the interest rate, i, is less than 10\% per month, and which is far above the current rates charged in the Brazilian house-financing system. Currently, the monthly rate is reflected at 1.5\%.\par
In summary, the sequence of the first payments will be as follows: ? -? With regard to the sequence of the parcels of amortization, it should be noted that, as shown in de Faro and Lachtermacher (2012, p. 243), and similar to the case of the constant payments scheme, the parcels of amortization, in each set of constant payments, follow a geometric sequence of ratio equal to 1+ i . Accordingly, we have:  
\section[{III. THE MULTIPLE CONTRACTS ALTERNATIVE}]{III. THE MULTIPLE CONTRACTS ALTERNATIVE}\par
Rather than engaging a single contract, the financial institution has the option of requiring the borrower to adhere to 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.\par
Namely, the principal of the k -th subcontract, denoted by , is:? ? (\textbf{15})\par
In this case, the parcel of amortization associated with the k -th payment, which will be denoted by , ? ? will be:(16)\par
Ergo, the parcel of amortization associated with the k -th subcontract is exactly equal to the value of the corresponding principal.\par
Conversely, from an accounting point of view, it follows that the parcel of interest associated with the k -th subcontract, which will be denoted by , is:? ? (\textbf{17})\par
From a 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 payments is the same in both cases. However, in terms of present values, and depending on the financial institution opportunity cost, it is possible that the financial institution will be better off if it adopts the multiple contracts option. 
\section[{A simple numerical example}]{A simple numerical example}\par
Before presenting a numerical illustration, it is appropriate to give due credit to the one who has introduced the idea of associating a specific contract with each of the payments of the main contract.\par
As far as we know, the concept was originally proposed by \hyperref[b6]{Sandrini (2007)}, in his Master's thesis for the Federal University of Paraná. However, an actual contract for each of the payments was not effectively proposed. The goal was to imply, specifically for the case of the constant payments scheme of debt amortization, the occurrence of what is named, in legal terms, anatocism -to wit, the charge of interest upon interest.\par
Later, De-Losso et al. (  {\ref 2013}) presented a formalization of the concept of multiple contracts. Focusing on the case of the constant payments scheme. Later de Faro (2022) extended the analysis to consider the Constant Amortization System. Now, as a numerical illustration, consider a loan of 12,000 units of capital, for the case of ? = 12 periodic payments, with , and , with the periodic rate of interest, i, being equal to 1\% per ? = 3 â??" = 4 period.\par
Table \hyperref[tab_0]{1} presents the sequence of the 12 payments, which is the same both in the case of a single contract, as well as in the 12 individual contracts.\par
Also, in Strictly from an accounting point of view, there is no gain if a single contract is substituted by multiple contracts since the sums of the corresponding parcels of interest are the same. Hence, Yet, depending on the opportunity cost of the financial institution, which will be denoted as , the ? financial institution may derive substantial financial gains in terms of income tax deductions.\par
In other words, it is possible that:(18)\par
where the interest rate is supposed to be relative to the same period of the interest rate i . ? Moreover, as the sequence of differences has only one change of sign, thus characterizing what is ? ? termed a conventional financing project, cf. de Faro (1974), whose internal rate of return is unique, and in this particular case null, it follows that ? = ? 1 ? ( ) -? 2 ? ( ) > 0 ð??"?? ? > 0 .\par
Figure  {\ref 1} outlines the evolution of Additionally, we also have the evolution of , ?, ð??"?? 0 ? ? ? 5\% . when the interest rate, i , is equal to 0.5\%, 1\%, 1.5\%, 2\% and 3\%. 
\section[{| |}]{| |}\par
A Multiple Contracts Version of the SACRE
\begin{quote}
Figure 1\end{quote}
\par
For instance, if per period, and if per period, we will have ? = 1\% ? = 2\% units of capital. Namely, the financing institution ? = ? 1 2\% ( ) -? 2 2\% ( ) = 
\section[{IV. GENERAL ANALYSIS}]{IV. GENERAL ANALYSIS}\par
In the previous section, focusing attention on the case of a contract with only 12 payments, it was verified that the sequence, , of differences of the interest payments yielded just one change of sign, ? ? thereby assuring us of the uniqueness of the corresponding internal rate of return, which was known to be zero.\par
However, when the number of payments is increased, it is possible to have instances wherein more than one change of sign can occur.\par
This possibility is illustrated in Figure  {\ref 2}, which refers to the case where a loan of 1,200,000 units of capital has a term of 15 years (180 months), with â??" = 15, monthly payments, and with the monthly interest rate, i, going from 0.5\% up to 3\%. 
\section[{22}]{22}\par
A Multiple Contracts Version of the SACRE
\begin{quote}
Figure 2\end{quote}
\par
Wherefore, for the cases where the monthly interest rate i assumes the values of 1\%, 2\% and 3\%, we have three changes of sign in the sequences of differences with only one change of sign in the other ? ? three cases.\par
However, considering a classical result first stated by \hyperref[b5]{Norstrom (1972)}, which is based on the sequence of the accumulated values of the sequence , we can still guarantee the uniqueness of the ? ? corresponding internal rate of return, and which we already know is null. Moreover, we are also assured that the difference of present values is positive whenever the opportunity cost is greater ? ? than zero.\par
Taking into consideration that in Brazil the monthly interest rates charged in house-financing contracts do not exceed 2\% per month, in real terms, Tables  {\ref 2}\hyperref[tab_4]{3}\hyperref[tab_5]{4}\hyperref[tab_6]{5}   The results presented in Tables 2 to 5 are self-evident. They illustrate a compelling support for the substitution of a single contract by multiple contracts.\par
For instance, if the interest rate i charged by the financial institution granting the loan is 0.5\% per month, the percentual value of can be as high as 47\% when its opportunity cost is 30\% annually, the ? contract has a 5-year term, and with a percentage fiscal gain over 248\%, if the contract is of 30 years, and =30\%per year. ? ? Furthermore, even though the fiscal gain decreases when the interest rate, i , being charged is increased, the percentage gain is no less than 35\% in every case.\par
Accordingly, one can conclude that the financial institution is well advised whenever it substitutes a single contract by multiple contracts, one for each of the payments of the single contract, whenever using our version of SACRE scheme.\par
A Multiple Contracts Version of the SACRE 
\section[{V. A COMPARISON WITH TWO ALTERNATIVE SYSTEMS OF AMORTIZATION}]{V. A COMPARISON WITH TWO ALTERNATIVE SYSTEMS OF AMORTIZATION}\par
Given that the financial institution granting the loan may have the option of choosing an alternative system of amortization, this section addresses two such possibilities, since both alternatives have also been considered in the Brazilian House-Financial program.\par
The first one is the system of constant payments. In this case, as shown in De-Losso et al. (  {\ref 2013}) and also in de Faro (2022), the present value of the sequence of interest payments, if multiple contracts are adopted, is equal to:\par
where and . ? = ? × ? / 1 -1 + ? ( )-? [ ] ? = ? + ? + ? × ? ( )\par
Tables 6 to 9 illustrate the percentage increase of the fiscal gain , ? ' = ?1 ? ? ( ) / ? 3 ? ? ( ) -1 [ ] × 100\par
wherein the financial institution adopts the multiple contracts version of the SACRE instead of the constant payments scheme. As indicated in the overwhelming majority of the cases, the financial institution should not choose the multiple contracts version of the SACRE. That is, if possible, the best option is to adopt the multiple contracts version of the constant payments scheme.\par
On the other hand, in the case of the system of constant amortization, the present value of the sequence of interest payments, where multiple contracts are adopted as shown in de Faro (2022), is equal to:\par
Tables 10 to 13 portray the percentage increase of the fiscal gain , when ? = ?   Similarly, it is clear that in the overwhelming majority of cases, the financial institution should opt for the multiple contracts version of the constant amortization scheme. 
\section[{VI. CONCLUSION}]{VI. CONCLUSION}\par
In similarity to the cases where either the constant payments system or the constant amortization system is adopted, a financial institution which implements our version of the SACRE, will be well advised if a multiple contract scheme, rather than a single contract, is implemented. 
\section[{| |}]{| |}\par
However, if the financial institution has the option of rather than adopting the SACRE, choosing either the constant payment system or the constant amortization one, in the vast majority of cases, SACRE is not the best option.\begin{figure}[htbp]
\noindent\textbf{111}\includegraphics[]{image-2.png}
\caption{\label{fig_0}â??" - 1 1 ? 1 =}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6}\includegraphics[]{image-3.png}
\caption{\label{fig_1}6 )}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}18A}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}A}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{figure7}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-8.png}
\caption{\label{figure8}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{}
\end{longtable} \par
 
\begin{quote}
? ? ? ? ? = ? ? -? ?\end{quote}

\caption{\label{tab_0}Table 1 ,}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.041846153846153845\textwidth}P{0.2693846153846154\textwidth}P{0.17523076923076922\textwidth}P{0.17784615384615382\textwidth}P{0.18569230769230768\textwidth}}
k\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
1\tabcellsep 1,120.00\tabcellsep 120.00\tabcellsep 11.09\tabcellsep 108.91\\
2\tabcellsep 1,120.00\tabcellsep 110.00\tabcellsep 22.07\tabcellsep 87.93\\
3\tabcellsep 1,120.00\tabcellsep 99.90\tabcellsep 32.94\tabcellsep 66.96\\
4\tabcellsep 1,086.35\tabcellsep 89.70\tabcellsep 42.39\tabcellsep 47.31\\
5\tabcellsep 1,086.35\tabcellsep 79.73\tabcellsep 52.73\tabcellsep 27.01\\
6\tabcellsep 1,086.35\tabcellsep 69.67\tabcellsep 62.96\tabcellsep 6.71\\
7\tabcellsep 1,051.16\tabcellsep 59.50\tabcellsep 70.72\tabcellsep -11.22\\
8\tabcellsep 1,051.16\tabcellsep 49.58\tabcellsep 80.43\tabcellsep -30.85\\
9\tabcellsep 1,051.16\tabcellsep 39.57\tabcellsep 90.04\tabcellsep -50.48\\
10\tabcellsep 1,011.16\tabcellsep 29.45\tabcellsep 95.77\tabcellsep -66.32\\
11\tabcellsep 1,001.34\tabcellsep 19.63\tabcellsep 103.82\tabcellsep -84.18\\
12\tabcellsep 991.52\tabcellsep 9.82\tabcellsep 111.60\tabcellsep -101.78\\
\tabcellsep 12,776.55\tabcellsep 776.55\tabcellsep 776.55\tabcellsep 0.00\end{longtable} \par
 
\caption{\label{tab_1}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.07727272727272727\textwidth}P{0.31939393939393934\textwidth}P{0.04893939393939394\textwidth}P{0.010303030303030303\textwidth}P{0.028333333333333332\textwidth}P{0.11075757575757576\textwidth}P{0.12621212121212122\textwidth}P{0.12878787878787878\textwidth}}
i =1\%p.m.\tabcellsep \tabcellsep \tabcellsep \tabcellsep ? ?\tabcellsep (\%)\tabcellsep \\
n (years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep \tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 6.9818\tabcellsep \multicolumn{2}{l}{14.0369 21.1427}\tabcellsep \tabcellsep 28.2780\tabcellsep 35.4240\tabcellsep 42.5637\\
10\tabcellsep \multicolumn{3}{l}{12.6403 25.8461 39.4650}\tabcellsep \tabcellsep 53.3526\tabcellsep 67.3789\tabcellsep 81.4325\\
15\tabcellsep \multicolumn{3}{l}{17.1122 35.2698 54.0650}\tabcellsep \tabcellsep 73.1370\tabcellsep 92.1981\tabcellsep 111.0376\\
20\tabcellsep \multicolumn{3}{l}{20.6713 42.6924 65.3157}\tabcellsep \tabcellsep 87.9659\tabcellsep 110.2641\tabcellsep 131.9929\\
25\tabcellsep \multicolumn{3}{l}{23.5050 48.4791 73.8287}\tabcellsep \tabcellsep 98.8507\tabcellsep 123.1814\tabcellsep 146.6734\\
30\tabcellsep \multicolumn{3}{l}{25.7789 52.9968 80.2719}\tabcellsep \multicolumn{2}{l}{106.8814}\tabcellsep 132.5477\tabcellsep 157.2110\end{longtable} \par
 
\caption{\label{tab_4}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.08\textwidth}P{0.59\textwidth}P{0.01\textwidth}P{0.017499999999999998\textwidth}P{0.02\textwidth}P{0.01\textwidth}P{0.12249999999999998\textwidth}}
i =1.5\%p.m.\tabcellsep \tabcellsep \tabcellsep ? ?\tabcellsep (\%)\tabcellsep \tabcellsep \\
n (years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep \multicolumn{5}{l}{6.4400 12.9049 19.3754 25.8347 32.2678}\tabcellsep 38.6619\\
10\tabcellsep \multicolumn{5}{l}{11.0264 22.3577 33.8719 45.4608 57.0333}\tabcellsep 68.5156\\
15\tabcellsep \multicolumn{5}{l}{14.2645 29.0323 44.0129 58.9738 73.7455}\tabcellsep 88.2143\\
20\tabcellsep \multicolumn{5}{l}{16.6191 33.8043 51.0807 68.1246 84.7455}\tabcellsep 100.8487\\
25\tabcellsep \multicolumn{5}{l}{18.3616 37.2551 56.0467 74.3817 92.1025}\tabcellsep 109.1625\\
30\tabcellsep \multicolumn{5}{l}{19.6818 39.8015 59.6129 78.7830 97.2111}\tabcellsep 114.8975\end{longtable} \par
 
\caption{\label{tab_5}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.07566765578635015\textwidth}P{0.5120178041543026\textwidth}P{0.1261127596439169\textwidth}P{0.09584569732937685\textwidth}P{0.020178041543026708\textwidth}P{0.010089020771513354\textwidth}P{0.010089020771513354\textwidth}}
i =2\%p.m.\tabcellsep \tabcellsep \tabcellsep ? ?\tabcellsep (\%)\tabcellsep \tabcellsep \\
n (years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 5.9649\tabcellsep 11.9192\tabcellsep \multicolumn{4}{l}{17.8476 23.7367 29.5750 35.3530}\\
10\tabcellsep 9.7461\tabcellsep \multicolumn{5}{l}{19.6393 29.5842 39.5008 49.3250 59.0074}\\
15\tabcellsep \multicolumn{6}{l}{12.1807 24.5843 37.0028 49.2816 61.3156 73.0401}\\
20\tabcellsep \multicolumn{6}{l}{13.8378 27.8861 41.8294 55.4706 68.7046 81.4871}\\
25\tabcellsep \multicolumn{6}{l}{15.0065 30.1631 45.0706 59.5276 73.4576 86.8487}\\
30\tabcellsep \multicolumn{6}{l}{15.8621 31.7896 47.3310 62.3081 76.6821 90.4695}\end{longtable} \par
 
\caption{\label{tab_6}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6} \par 
\begin{longtable}{P{0.08993174061433447\textwidth}P{0.12474402730375425\textwidth}P{0.11604095563139932\textwidth}P{0.12474402730375425\textwidth}P{0.17986348122866894\textwidth}P{0.0841296928327645\textwidth}P{0.13054607508532423\textwidth}}
i =0.5\%p.m.\tabcellsep \tabcellsep \tabcellsep ? ?\tabcellsep (\%)\tabcellsep \tabcellsep \\
n(years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 5.7794\tabcellsep 5.2285\tabcellsep 4,6985\tabcellsep 4.1891\tabcellsep 3.7003\tabcellsep 3.2316\\
10\tabcellsep 9.4904\tabcellsep 7.6829\tabcellsep 5.9746\tabcellsep 4.3761\tabcellsep 2.8930\tabcellsep 1.5266\\
15\tabcellsep 12.4358\tabcellsep 8.8406\tabcellsep 5.5845\tabcellsep 2.7002\tabcellsep 0.1883\tabcellsep -1.9736\\
20\tabcellsep 14.6772\tabcellsep 9.0254\tabcellsep 4.2075\tabcellsep 0.2399\tabcellsep -2.9578\tabcellsep -5.5094\\
25\tabcellsep 16.3350\tabcellsep 8.5602\tabcellsep 2.4188\tabcellsep \multicolumn{2}{l}{-2.2365 -5.7050}\tabcellsep -8.2918\\
30\tabcellsep 17.4969\tabcellsep 7.7185\tabcellsep 0.6241\tabcellsep \multicolumn{2}{l}{-4.3241 -7.7642}\tabcellsep -10.2021\end{longtable} \par
 
\caption{\label{tab_7}Table 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7} \par 
\begin{longtable}{P{0.40138888888888885\textwidth}P{0.4486111111111111\textwidth}}
Values of for ? '\tabcellsep ? = 1 . 0\% ? . ? .\end{longtable} \par
 
\caption{\label{tab_8}Table 7 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{10} \par 
\begin{longtable}{P{0.09086206896551724\textwidth}P{0.1143103448275862\textwidth}P{0.11724137931034483\textwidth}P{0.12017241379310345\textwidth}P{0.0029310344827586207\textwidth}P{0.20810344827586205\textwidth}P{0.06741379310344828\textwidth}P{0.1289655172413793\textwidth}}
i =0.5\%p.m.\tabcellsep \tabcellsep \tabcellsep ?\tabcellsep ?\tabcellsep (\%)\tabcellsep \\
n(years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep \tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 1.2310\tabcellsep 1.0836\tabcellsep 0.9439\tabcellsep \tabcellsep 0.8114\tabcellsep 0.6861\tabcellsep 0.5675\\
10\tabcellsep 1.1235\tabcellsep 0,8599\tabcellsep 0.6193\tabcellsep \tabcellsep 0.4016\tabcellsep 0.2061\tabcellsep 0.0315\\
15\tabcellsep 1.0147\tabcellsep 0.6658\tabcellsep 0.3650\tabcellsep \tabcellsep 0.1106\tabcellsep -0.1015\tabcellsep -0.2766\\
20\tabcellsep 0.8942\tabcellsep 0.5125\tabcellsep 0.1968\tabcellsep \tabcellsep \multicolumn{2}{l}{-0.0552 -0.2521}\tabcellsep -0.4043\\
25\tabcellsep 0.7927\tabcellsep 0.3987\tabcellsep 0.0889\tabcellsep \tabcellsep \multicolumn{2}{l}{-0.1445 -0.3168}\tabcellsep -0.4438\\
30\tabcellsep 0.7084\tabcellsep 0.3151\tabcellsep 0.0216\tabcellsep \tabcellsep \multicolumn{2}{l}{-0.1884 -0.3371}\tabcellsep -0.4436\end{longtable} \par
 
\caption{\label{tab_9}Table 10 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{11} \par 
\begin{longtable}{P{0.09343971631205673\textwidth}P{0.1175531914893617\textwidth}P{0.12056737588652483\textwidth}P{0.1296099290780142\textwidth}P{0.012056737588652482\textwidth}P{0.12056737588652483\textwidth}P{0.1476950354609929\textwidth}P{0.10851063829787233\textwidth}}
i =1.0\%p.m.\tabcellsep \tabcellsep \tabcellsep ? ?\tabcellsep (\%)\tabcellsep \tabcellsep \\
n(years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep \tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 2.5282\tabcellsep 2.2448\tabcellsep 1.9781\tabcellsep \tabcellsep 1.7273\tabcellsep 1.4915\tabcellsep 1.2700\\
10\tabcellsep 2.3441\tabcellsep 1.8686\tabcellsep 1.4429\tabcellsep \tabcellsep 1.0645\tabcellsep 0.7298\tabcellsep 0.4351\\
15\tabcellsep 2.1678\tabcellsep 1.5681\tabcellsep 1.0653\tabcellsep \tabcellsep 0.6496\tabcellsep 0.3087\tabcellsep 0.0302\\
20\tabcellsep 1.9540\tabcellsep 1.3198\tabcellsep \multicolumn{2}{l}{0.8130}\tabcellsep 0.4170\tabcellsep 0.1106\tabcellsep -0.1265\\
25\tabcellsep 1.7694\tabcellsep 1.1305\tabcellsep \multicolumn{2}{l}{0.6452}\tabcellsep 0.2849\tabcellsep 0.0184\tabcellsep -0.1807\\
30\tabcellsep 1.6121\tabcellsep 0.9843\tabcellsep \multicolumn{2}{l}{0.5301}\tabcellsep 0.2070\tabcellsep \multicolumn{2}{l}{-0.0246 -0.1942}\end{longtable} \par
 
\caption{\label{tab_10}Table 11 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{12} \par 
\begin{longtable}{P{0.09512635379061371\textwidth}P{0.11967509025270757\textwidth}P{0.12274368231046931\textwidth}P{0.12581227436823103\textwidth}P{0.14115523465703972\textwidth}P{0.12274368231046931\textwidth}P{0.12274368231046931\textwidth}}
i =1.5\%p.m.\tabcellsep \tabcellsep \tabcellsep ?\tabcellsep (\%) ?\tabcellsep \tabcellsep \\
n(years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 3.8908\tabcellsep 3.4788\tabcellsep 3.0936\tabcellsep 2.7336\tabcellsep 2.3972\tabcellsep 2.0829\\
10\tabcellsep 3.6425\tabcellsep 2.9811\tabcellsep 2.3967\tabcellsep 1.8829\tabcellsep 1.4330\tabcellsep 1.0398\\
15\tabcellsep 3,4074\tabcellsep 2.5917\tabcellsep 1.9182\tabcellsep 1.3674\tabcellsep 0.9186\tabcellsep 0.5529\\
20\tabcellsep 3.0991\tabcellsep 2.2443\tabcellsep 1.5721\tabcellsep 1.0511\tabcellsep 0.6482\tabcellsep 0.3350\\
25\tabcellsep 2.8242\tabcellsep 1.9654\tabcellsep 1.3226\tabcellsep 0.8473\tabcellsep 0.4941\tabcellsep 0.2278\\
30\tabcellsep 2.5846\tabcellsep 1.7397\tabcellsep 1.1366\tabcellsep 0.7078\tabcellsep 0.3981\tabcellsep 0.1689\end{longtable} \par
 
\caption{\label{tab_11}Table 12 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{13} \par 
\begin{longtable}{P{0.09547101449275362\textwidth}P{0.1201086956521739\textwidth}P{0.12318840579710146\textwidth}P{0.12626811594202897\textwidth}P{0.0030797101449275364\textwidth}P{0.13550724637681158\textwidth}P{0.12318840579710146\textwidth}P{0.12318840579710146\textwidth}}
i =2.0\%p.m.\tabcellsep \tabcellsep \tabcellsep ?\tabcellsep ?\tabcellsep (\%)\tabcellsep \\
n(years)\tabcellsep 5\%\tabcellsep 10\%\tabcellsep 15\%\tabcellsep \tabcellsep 20\%\tabcellsep 25\%\tabcellsep 30\%\\
5\tabcellsep 5.3187\tabcellsep 4.7826\tabcellsep 4.2842\tabcellsep \tabcellsep 3.8207\tabcellsep 3.3898\tabcellsep 2.9890\\
10\tabcellsep 5.0078\tabcellsep 4.1727\tabcellsep 3.4415\tabcellsep \tabcellsep 2.8038\tabcellsep 2.2487\tabcellsep 1.7659\\
15\tabcellsep 4.7111\tabcellsep 3.6897\tabcellsep 2.8547\tabcellsep \tabcellsep 2.1760\tabcellsep 1.6247\tabcellsep 1.1756\\
20\tabcellsep 4.3022\tabcellsep 3.2309\tabcellsep 2.3969\tabcellsep \tabcellsep 1.7531\tabcellsep 1.2553\tabcellsep 0.8670\\
25\tabcellsep 3.9287\tabcellsep 2.8489\tabcellsep \multicolumn{2}{l}{2.0482}\tabcellsep 1.4577\tabcellsep 1.0179\tabcellsep 0.6847\\
30\tabcellsep 3.5992\tabcellsep 2.5325\tabcellsep 1.7778\tabcellsep \tabcellsep 1.2421\tabcellsep 0.8542\tabcellsep 0.5657\end{longtable} \par
 
\caption{\label{tab_12}Table 13 :}\end{figure}
 			\label{foot_0}\footnote{\label{foot_0} © 2023 Great ] Britain Journals Press} 			\label{foot_1}\footnote{\label{foot_1} Compilation 1.0 © 2023 Great ] Britain Journals Press} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Norstrom ()]{b5}\label{b5} 	 		‘A Sufficient Condition for a Unique Non Negative Internal Rate of Return’.  		 			C Norstrom 		.  	 	 		\textit{Journal of Financial and Quantitative Analysis}  		1972. 3 p. .  	 
\bibitem[De Faro and Lachtermacher ()]{b2}\label{b2} 	 		 			C De Faro 		,  		 			G Lachtermacher 		.  		\textit{Introdução à Matemática Financeira, FGV/Saraiva},  		2022.  	 
\bibitem[De-Losso et al. ()]{b4}\label{b4} 	 		 			R De-Losso 		,  		 			B C Giovannetti 		,  		 			A S Rangel 		.  	 	 		\textit{Sistema de Amortização por Múltiplos Contratos: a Falácia do Sistema Francês},  		2013. 4 p. .  	 
\bibitem[De Faro and Lachtermacher ()]{b3}\label{b3} 	 		‘O SACRE no Regime de Juros Compostos’.  		 			C De Faro 		,  		 			G Lachtermacher 		.  	 	 		\textit{Estudos e Negócios Academics},  		2022. 4 p. .  	 
\bibitem[De Faro ()]{b0}\label{b0} 	 		‘On the Internal Rate of Return Criterion’.  		 			C De Faro 		.  	 	 		\textit{The Engineering Economist}  		1974. 19 p. .  	 	 (Nº) 
\bibitem[Sandrini ()]{b6}\label{b6} 	 		 			J C Sandrini 		.  		\textit{Sistemas de Amortização de Empréstimos e a Capitalização de Juros: Análise dos Impactos Financeiros e Patrimoniais},  		2007.  		 			Federal University of Paraná 		 	 	 (Master Thesis) 
\bibitem[De Faro (2022)]{b1}\label{b1} 	 		‘The Constant Amortization Scheme With Multiple Contracts’.  		 			C De Faro 		.  	 	 		\textit{Revista Brasileira de Economia}  		april-june, 2022. 76 p. .  	 
\end{bibitemlist}
 			 		 	 
\end{document}
