% !TeX spellcheck = en_US % !TeX encoding = UTF-8 \documentclass[10pt, a4paper]{article} % ----- packages ----- \usepackage{amsmath} % AMS mathematical facilities for LaTeX \usepackage{enumitem} % Control layout of itemize, enumerate, description \usepackage{fancyhdr} % Extensive control of page headers and footers in LaTeX2 \usepackage{geometry} % Flexible and complete interface to document dimensions \usepackage{graphicx} % Enhanced support for graphics \usepackage{hyperref} % Extensive support for hypertext in LaTeX \usepackage{parskip} % Layout with zero \parindent, non-zero \parskip \usepackage{titlesec} % Select alternative section titles \usepackage{multirow} % Create tabular cells spanning multiple rows % ----- pdf metadata ----- \hypersetup{ pdftitle={Financial Mathematics Cheat Sheet}, pdfsubject={Financial Mathematics Cheat Sheet - marcelomijas - CC-BY-4.0}, pdfauthor={Marcelo Moreno Porras}, pdfkeywords={latex, economics, cheatsheet, financial-mathematics} } % ----- custom commands ----- % simple interest formulas \newcommand{\Sif}{$C_{n} = C_{0} \cdot (1 + i \cdot n)$} \newcommand{\SifRateim}{$i_{m} = \dfrac{i}{m}$} \newcommand{\SifSolveCo}{$C_{0} = \dfrac{C_{n}}{1 + i \cdot n}$} \newcommand{\SifSolvei}{$i = \dfrac{\frac{C_{n}}{C_{0}} - 1}{n}$} \newcommand{\SifSolven}{$n = \dfrac{\frac{C_{n}}{C_{0}} - 1}{i}$} % compound interest formulas \newcommand{\Cif}{$C_{n} = C_{0} \cdot (1 + i)^{n}$} \newcommand{\CifRateim}{$i_{m} = (1 + i)^{1 / m} - 1$} \newcommand{\CifRatei}{$i = (i_{m} + 1)^{m} - 1$} \newcommand{\CifSolveCo}{$C_{0} = \dfrac{C_{n}}{(1 + i)^{n}}$} \newcommand{\CifSolvei}{$i = \left(\dfrac{C_{n}}{C_{0}}\right)^{1 / n} - 1$} \newcommand{\CifSolven}{$n = \dfrac{\log{\left(\frac{C_{n}}{C_{0}}\right)}}{\log{(1 + i)}}$} % simple discount formulas \newcommand{\Sdf}{$C_{0} = C_{n} (1 - d \cdot n)$} \newcommand{\SdfRatedm}{$d_{m} = \dfrac{d}{m}$} \newcommand{\Sdfr}{\textbf{Rational} \quad $C_0 = \dfrac{C_{n}}{1 + i \cdot n}$} % compound discount formula \newcommand{\Cdf}{$C_{0} = C_{n} \cdot (1 - d)^{n}$} % temporal unitary rent formulas \newcommand{\TurfPoa}{$a_{n \rceil i} = \dfrac{1 - (1 + i)^{-n}}{i}$} \newcommand{\TurfPos}{$S_{n \rceil i} = a_{n \rceil i} \cdot (1 + i)^{n}$} \newcommand{\TurfPra}{$\ddot{a}_{n \rceil i} = (1 + i) \cdot a_{n \rceil i}$} \newcommand{\TurfPrs}{$\ddot{S}_{n \rceil i} = (1 + i) \cdot S_{n \rceil i}$} % perpetual unitary rent formulas \newcommand{\PurfPoa}{$a_{\infty \rceil i} = \dfrac{1}{i}$} \newcommand{\PurfPra}{$\ddot{a}_{\infty \rceil i} = (1 + i) \cdot a_{\infty \rceil i}$} % temporal geometric rent formulas \newcommand{\TgrfPoA}{$A(C;q)_{n \rceil i} = \begin{cases} C \cdot \dfrac{1 - \left( \dfrac{q}{1 + i} \right)^n}{1 + i - q} & \mathrm{si} \quad q \neq 1 + i \\ C \cdot \dfrac{n}{1 + i} & \mathrm{si} \quad q = 1 + i \end{cases}$} \newcommand{\TgrfPoS}{$S(C;q)_{n \rceil i} = A(C;q)_{n \rceil i} \cdot (1 + i)^n$} \newcommand{\TgrfPrA}{$\ddot{A}(C;q)_{n \rceil i} = (1 + i) \cdot A(C;q)_{n \rceil i}$} \newcommand{\TgrfPrS}{$\ddot{S}(C;q)_{n \rceil i} = (1 + i) \cdot S(C;q)_{n \rceil i}$} % perpetual geometric rent formulas \newcommand{\PgrfPoA}{$A(C;q)_{\infty \rceil i} = \begin{cases} C \cdot \dfrac{1}{1 + i - q} & \mathrm{si} \quad q < 1 + i \\ \mathrm{Infinito} & \mathrm{si} \quad q \geq 1 + i \end{cases}$} \newcommand{\PgrfPrA}{$\ddot{A}(C;q)_{\infty \rceil i} = (1 + i) \cdot A(C;q)_{\infty \rceil i}$} % constant \newcommand{\Const}{Constante} % loan french type \newcommand{\Lfta}{$a = \dfrac{C_{0}}{a_{n \rceil i}}$} \newcommand{\LftA}{$A_{s} = A_{1} \cdot (1 + i)^{s - 1}$} \newcommand{\LftC}{$C_{0} = A_{1} \cdot S_{n \rceil i}$} \newcommand{\LftCRec}{$C_{s} = C_{s - 1} \cdot (1 + i) - a$} \newcommand{\LftCPro}{$C_{s} = a \cdot a_{n - s \rceil i}$} \newcommand{\LftCRet}{$C_{s} = C_{0} \cdot (1 + i)^{s} - a \cdot S_{s \rceil i}$} % loan american type \newcommand{\Latas}{$a_{s} = I_{s} = C_{0} \cdot i_{s} \quad \mathrm{si} \quad s \neq n$} \newcommand{\Latan}{$a_{n} = I_{n} + C_{0} = C_{0} \cdot i_{s} + C_{0} \quad \mathrm{si} \quad s = n$} \newcommand{\LatAs}{$A_{s} = 0 \quad \mathrm{si} \quad s \neq n$} \newcommand{\LatAn}{$A_{n} = C_{0} \quad \mathrm{si} \quad s = n$} \newcommand{\LatCs}{$C_{s} = C_{0} \quad \mathrm{si} \quad s \neq n$} \newcommand{\LatCn}{$C_{n} = 0 \quad \mathrm{si} \quad s = n$} % loan italian type \newcommand{\Lita}{$a_{s + 1} = a_{s} - i \cdot A$} \newcommand{\LitI}{$I_{s + 1} = I_{s} - i \cdot A$} \newcommand{\LitA}{$A = \dfrac{C_{0}}{n}$} \newcommand{\LitCRec}{$C_{s} = C_{s - 1} - A$} \newcommand{\LitCPro}{$C_{s} = C_{n - s} \cdot A$} \newcommand{\LitCRet}{$C_{s} = C_{0} - s \cdot A$} % loan geometrical type \newcommand{\Lgta}{$a = \dfrac{C_{0}}{A(1;q)_{n \rceil i}}$} \newcommand{\LgtCRec}{$C_{s} = C_{s - 1} \cdot (1 + i) - a \cdot q^{s - 1}$} \newcommand{\LgtCPro}{$C_{s} = A(a \cdot q^{s};q)_{n - s \rceil i}$} \newcommand{\LgtCRet}{$C_{s} = C_{0} \cdot (1 + i)^{s} - S(a;q)_{s \rceil i}$} % vertical text \newcommand{\vtext}[1]{ \rotatebox[origin=c]{90}{#1} } % ----- page customization ----- \geometry{margin=1cm} % margins config \pagenumbering{gobble} % remove page numeration \setlength{\parskip}{0cm} % paragraph spacing % title spacing \titlespacing{\section}{0pt}{2ex}{1ex} % ----- footer ----- \pagestyle{fancy} \renewcommand{\headrulewidth}{0pt} \cfoot{\href{https://github.com/marcelomijas/financial-math-cheatsheet}{\normalfont \footnotesize FM-25.01-EN - github.com/marcelomijas/financial-math-cheatsheet - CC-BY-4.0 license}} \setlength{\footskip}{12pt} % ----- document ----- \begin{document} \begin{center} \textbf{\LARGE \href{https://github.com/marcelomijas/financial-math-cheatsheet}{Financial Mathematics Cheat Sheet}} {\footnotesize By Marcelo Moreno Porras - Universidad Rey Juan Carlos} \end{center} \section*{Capitalization and discount} \begin{center} \renewcommand{\arraystretch}{2.5} \begin{tabular}{|c|cc|cc|} \hline & \multicolumn{2}{c|}{\textbf{Capitalization}} & \multicolumn{2}{c|}{\textbf{Discount}} \\ \hline \multirow{2}{*}{\vtext{\textbf{Simple}}} & \multirow{2}{*}{\Sif} & \multirow{2}{*}{\SifRateim} & \Sdf & \SdfRatedm \\ & & & \multicolumn{2}{c|}{\textbf{\Sdfr}} \\ \hline \multirow{2}{*}{\vtext{\textbf{Compound}}} & \multirow{2}{*}{\Cif} & \CifRateim & \multicolumn{2}{c|}{\multirow{2}{*}{\textbf{\Cdf}}} \\ & & \CifRatei & & \\ \hline \end{tabular} \end{center} \vspace*{0.5cm} Notes: $C_{n}$ capital in $t = n$, $C_{0}$ capital in $t = 0$, $n$ periods, $m$ subperiods, $i$ interest rate, $d$ discount rate. There is also the so-called \textbf{fractional capitalization}, $i_{m} = \frac{j(m)}{m}$, where $j(m)$ is the nominal interest rate payable per $m$. \section*{Clearing interest formulas} \begin{center} \renewcommand{\arraystretch}{2.4} \begin{tabular}{|c|c|} \hline \textbf{Simple interest formulas} & \textbf{Compound interest formulas} \\ \hline \Sif & \Cif \\ \hline \SifSolveCo & \CifSolveCo \\ \hline \SifSolvei & \CifSolvei \\ \hline \SifSolven & \CifSolven \\ \hline \end{tabular} \end{center} \section*{Annuities} \begin{center} \renewcommand{\arraystretch}{2.6} \begin{tabular}{|c|c|c|c|} \hline & & \textbf{Unitary} & \textbf{Variable in geometric progression} \\ \hline \multirow{4}{*}{\vtext{\textbf{Temporal}}} & \multirow{2}{*}{\textbf{Postpayable}} & \TurfPoa & \TgrfPoA \\ & & \TurfPos & \TgrfPoS \\ \cline{2-4} & \multirow{2}{*}{\textbf{Prepayable}} & \TurfPra & \TgrfPrA \\ & & \TurfPrs & \TgrfPrS \\ \hline \multirow{2}{*}{\vtext{\textbf{Perpetual}}} & \textbf{Postpayable} & \PurfPoa & \PgrfPoA \\ \cline{2-4} & \textbf{Prepayable} & \PurfPra & \PgrfPrA \\ \hline \end{tabular} \end{center} \vspace*{0.5cm} Notes: \begin{itemize}[leftmargin=*] \item[] $q =$ factor. \item[] \textbf{Present discounted value}, example, $V_{0} = C \cdot a_{n \rceil i}$ \item[] \textbf{Final capitalized value}, example, $V_{n} = C \cdot S_{n \rceil i}$ \end{itemize} \newpage \section*{Amortization table} \begin{center} \renewcommand{\arraystretch}{2} \scalebox{0.90}{ \begin{tabular}{|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{\textbf{Period}} & \textbf{Interest} & \textbf{Amortization} & \textbf{Interest} & \textbf{Amortization} & \multirow{2}{*}{\textbf{Outstanding capital}} & \multirow{2}{*}{\textbf{Amortized capital}} \\ & \textbf{rate} & \textbf{term} & \textbf{payment} & \textbf{payment} & & \\ \hline 0 & - & - & - & - & $C_{0}$ & - \\ \hline $t_{1}$ & $i_{1}$ & $a_{1}$ & $I_{1} = C_{0} \cdot i_{1}$ & $A_{1} = a_{1} - I_{1}$ & $C_{1} = C_{0} - A_{1}$ & $M_{1} = C_{0} - C_{1}$ \\ \hline $t_{2}$ & $i_{2}$ & $a_{2}$ & $I_{2} = C_{1} \cdot i_{2}$ & $A_{2} = a_{2} - I_{2}$ & $C_{2} = C_{1} - A_{2}$ & $M_{2} = C_{1} - C_{2}$ \\ \hline $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\ \hline $t_{s}$ & $i_{s}$ & $a_{s}$ & $I_{s} = C_{s - 1} \cdot i_{s}$ & $A_{s} = a_{s} - I_{s}$ & $C_{s} = C_{s - 1} - A_{s}$ & $M_{s} = C_{s - 1} - C_{s}$ \\ \hline $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\ \hline $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\ \hline $t_{n}$ & $i_{n}$ & $a_{n}$ & $I_{n} = C_{n - 1} \cdot i_{n}$ & $A_{n} = a_{n} - I_{n}$ & $C_{n} = C_{n - 1} - A_{n} = 0$ & $M_{n} = C_{0} - C_{n} = M_{n - 1} + A_{n} = C_{0}$ \\ \hline \end{tabular} } \end{center} \section*{Loans} \begin{center} \renewcommand{\arraystretch}{3} \scalebox{0.90}{ \begin{tabular}{|cc|c|c|c|c|} \hline & & \multirow{2}{*}{\textbf{French}} & \multirow{2}{*}{\textbf{American}} & \multirow{2}{*}{\textbf{Italian}} & \textbf{Geometric progression} \\ & & & & & \textbf{terms} \\ \hline \multicolumn{2}{|c|}{$i$} & \Const & - & - & - \\ \hline \multicolumn{2}{|c|}{\multirow{2}{*}{\textbf{$a$}}} & \Const; & \Latas & \multirow{2}{*}{\Lita} & \multirow{2}{*}{\Lgta} \\ & & \Lfta & \Latan & & \\ \hline \multicolumn{2}{|c|}{$I$} & - & - & \LitI & - \\ \hline \multicolumn{2}{|c|}{\multirow{2}{*}{\textbf{$A$}}} & \multirow{2}{*}{\LftA} & \LatAs & \Const; & \multirow{2}{*}{-} \\ & & & \LatAn & \LitA & \\ \hline \multicolumn{2}{|c|}{\multirow{2}{*}{\textbf{$C$}}} & \LftC & \LatCs & \multirow{2}{*}{-} & \multirow{2}{*}{-} \\ & & & \LatCn & & \\ \hline \vtext{\textbf{Recursive}} & \vtext{\textbf{method}} & \LftCRec & - & \LitCRec & \LgtCRec \\ \hline \vtext{\textbf{Prospective}} & \vtext{\textbf{method}} & \LftCPro & - & \LitCPro & \LgtCPro \\ \hline \vtext{\textbf{Retrospective}} & \vtext{\textbf{method}} & \LftCRet & - & \LitCRet & \LgtCRet \\ \hline \end{tabular} } \end{center} \newpage \section*{Graphical analysis of loans} Suppose a loan with principal $C_{0} = 10.000$ at an interest rate $i = 10\%$ ending at $n = 12$ in which an amount is paid in each period that will depend on the type of loan. \textbf{French loan}: constant interest rate, all amortization terms are constant, amortization payments vary in geometric progression of factor $(1 + i)$: \begin{center} \includegraphics[width=11cm]{../figures/french-en.pdf} \end{center} \textbf{American loan}: the debtor only pays interest at the end of each period except in the last, in which he also pays the nominal amount of the loan; the payments are only interest, the live capital does not vary until the last period. \begin{center} \includegraphics[width=11cm]{../figures/american-en.pdf} \end{center} \textbf{Italian loan}: constant amortization payments, amortization terms and interest payments decrease in arithmetic progression $(-i \cdot A)$. \begin{center} \includegraphics[width=11cm]{../figures/italian-en.pdf} \end{center} \textbf{Loan with geometric progression terms}: amortization terms vary in geometric progression of factor $q$ (in this case, $q = 1.05$). \begin{center} \includegraphics[width=11cm]{../figures/geometric-en.pdf} \end{center} \end{document}