\documentclass[10pt,a4paper]{article} % Packages \usepackage{fancyhdr} % For header and footer \usepackage{multicol} % Allows multicols in tables \usepackage{tabularx} % Intelligent column widths \usepackage{tabulary} % Used in header and footer \usepackage{hhline} % Border under tables \usepackage{graphicx} % For images \usepackage{xcolor} % For hex colours %\usepackage[utf8x]{inputenc} % For unicode character support \usepackage[T1]{fontenc} % Without this we get weird character replacements \usepackage{colortbl} % For coloured tables \usepackage{setspace} % For line height \usepackage{lastpage} % Needed for total page number \usepackage{seqsplit} % Splits long words. %\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely. \usepackage[normalem]{ulem} % For underlining links % Most of the following are not required for the majority % of cheat sheets but are needed for some symbol support. \usepackage{amsmath} % Symbols \usepackage{MnSymbol} % Symbols \usepackage{wasysym} % Symbols %\usepackage[english,german,french,spanish,italian]{babel} % Languages % Document Info \author{Natalie Moore (NatalieMoore)} \pdfinfo{ /Title (time-value-of-money.pdf) /Creator (Cheatography) /Author (Natalie Moore (NatalieMoore)) /Subject (Time value of money Cheat Sheet) } % Lengths and widths \addtolength{\textwidth}{6cm} \addtolength{\textheight}{-1cm} \addtolength{\hoffset}{-3cm} \addtolength{\voffset}{-2cm} \setlength{\tabcolsep}{0.2cm} % Space between columns \setlength{\headsep}{-12pt} % Reduce space between header and content \setlength{\headheight}{85pt} % If less, LaTeX automatically increases it \renewcommand{\footrulewidth}{0pt} % Remove footer line \renewcommand{\headrulewidth}{0pt} % Remove header line \renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit % This two commands together give roughly % the right line height in the tables \renewcommand{\arraystretch}{1.3} \onehalfspacing % Commands \newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour \newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols \newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns \newcommand{\tn}{\tabularnewline} % Required as custom column type in use % Font and Colours \definecolor{HeadBackground}{HTML}{333333} \definecolor{FootBackground}{HTML}{666666} \definecolor{TextColor}{HTML}{333333} \definecolor{DarkBackground}{HTML}{A35555} \definecolor{LightBackground}{HTML}{F9F4F4} \renewcommand{\familydefault}{\sfdefault} \color{TextColor} % Header and Footer \pagestyle{fancy} \fancyhead{} % Set header to blank \fancyfoot{} % Set footer to blank \fancyhead[L]{ \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{C} \SetRowColor{DarkBackground} \vspace{-7pt} {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}} } \end{tabulary} \columnbreak \begin{tabulary}{11cm}{L} \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Time value of money Cheat Sheet}}}} \\ \normalsize{by \textcolor{DarkBackground}{Natalie Moore (NatalieMoore)} via \textcolor{DarkBackground}{\uline{cheatography.com/19119/cs/11141/}}} \end{tabulary} \end{multicols}} \fancyfoot[L]{ \footnotesize \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{LL} \SetRowColor{FootBackground} \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}} \\ \vspace{-2pt}Natalie Moore (NatalieMoore) \\ \uline{cheatography.com/nataliemoore} \\ \uline{\seqsplit{www}.clipto.com/transcribe-audio-video-to-text-free?via=natalie} \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\ \vspace{-2pt}Published 19th March, 2017.\\ Updated 19th March, 2017.\\ Page {\thepage} of \pageref{LastPage}. \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}} \\ \SetRowColor{white} \vspace{-5pt} %\includegraphics[width=48px,height=48px]{dave.jpeg} Measure your website readability!\\ www.readability-score.com \end{tabulary} \end{multicols}} \begin{document} \raggedright \raggedcolumns % Set font size to small. Switch to any value % from this page to resize cheat sheet text: % www.emerson.emory.edu/services/latex/latex_169.html \footnotesize % Small font. \begin{multicols*}{3} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Variable key}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Where:}}} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} FV & = Future value of an investment \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} PV & = Present value of an investment (the lump sum) \tn % Row Count 4 (+ 2) % Row 3 \SetRowColor{white} r & = Return or interest rate per period (typically 1 year) \tn % Row Count 6 (+ 2) % Row 4 \SetRowColor{LightBackground} n & = Number of periods (typically years) that the lump sum is invested \tn % Row Count 8 (+ 2) % Row 5 \SetRowColor{white} PMT & = Payment amount \tn % Row Count 9 (+ 1) % Row 6 \SetRowColor{LightBackground} CFn & = Cash flow steam number \tn % Row Count 10 (+ 1) % Row 7 \SetRowColor{white} m & = \# of times per year r compounds \tn % Row Count 11 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Equation guide}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Future value of a lump sum:}}} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{FV = PV x (1 + r)\textasciicircum{}n\textasciicircum{}} \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} - & Future-value factor (FVF) table \tn % Row Count 3 (+ 1) % Row 3 \SetRowColor{white} - & Excel future value formula FV= \tn % Row Count 4 (+ 1) % Row 4 \SetRowColor{LightBackground} - & Compound interest. Formula for simple interest is PV + (n x (PV x r)) \tn % Row Count 6 (+ 2) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Future Value of an Ordinary Annuity}}} \tn % Row Count 7 (+ 1) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{FV = PMT x \{ {[} ( 1 + r )\textasciicircum{}n\textasciicircum{} - 1 {]} / r\}} \tn % Row Count 8 (+ 1) % Row 7 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Future Value of an Annuity Due}}} \tn % Row Count 9 (+ 1) % Row 8 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{FV (annuity due) = PMT x \{ {[} ( 1 + r)\textasciicircum{}n\textasciicircum{} -1 {]} / r \} x (1 + r)} \tn % Row Count 11 (+ 2) % Row 9 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Future Value of Cash Flow Streams}}} \tn % Row Count 12 (+ 1) % Row 10 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{FV = CF1 x (1 +r)\textasciicircum{}n-1\textasciicircum{} + CF2 x (1 + r)\textasciicircum{}n-2\textasciicircum{} + ... + CFn x (1 + r)\textasciicircum{}n-n\textasciicircum{}} \tn % Row Count 14 (+ 2) % Row 11 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Present value of a lump sum in future}}} \tn % Row Count 15 (+ 1) % Row 12 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{PV = FV / (1 + r)\textasciicircum{}n\textasciicircum{} = FV x {[} 1 / (1+ r)\textasciicircum{}n\textasciicircum{} {]}} \tn % Row Count 16 (+ 1) % Row 13 \SetRowColor{white} - & Present-value factor (FVF) table \tn % Row Count 17 (+ 1) % Row 14 \SetRowColor{LightBackground} - & Excel present value formula PV= \tn % Row Count 18 (+ 1) % Row 15 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Present Value of a Mixed Stream}}} \tn % Row Count 19 (+ 1) % Row 16 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{PV = {[}CF1 x 1 / (1 + r)\textasciicircum{}1\textasciicircum{}{]} + {[}CF2 x 1 / (1 + r)\textasciicircum{}1\textasciicircum{}{]} + ... + {[}CFn x 1 / (1 + r)\textasciicircum{}1\textasciicircum{}{]}} \tn % Row Count 21 (+ 2) % Row 17 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Present Value of an Ordinary Annuity}}} \tn % Row Count 22 (+ 1) % Row 18 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{PV = PMT/r x {[}1 - 1 / (1 + r)\textasciicircum{}n\textasciicircum{}{]}} \tn % Row Count 23 (+ 1) % Row 19 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Present Value of Annuity Due}}} \tn % Row Count 24 (+ 1) % Row 20 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{PV (annuity due) = PMT/r x {[}1 - 1 / (1 + r)\textasciicircum{}n\textasciicircum{}{]} x (1 + r)} \tn % Row Count 26 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Lump sum future value in excel}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/nataliemoore_1489232119_2017-03-11 21_34_36-Future Value Calculators - Excel.png}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Present Value of a Growing Perpetuity}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Most cash flows grow over time} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{This formula adjusts the present value of a perpetuity formula to account for expected growth in future cash flows} \tn % Row Count 4 (+ 3) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Calculate present value (PV) of a stream of cash flows growing forever (n = ∞) at the constant annual rate g} \tn % Row Count 7 (+ 3) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{PV = CF1 / r - g r \textgreater{} g}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Loan Amortization}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{A borrower makes equal periodic payments over time to fully repay a loan} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{E.g. home loan} \tn % Row Count 3 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Uses} \tn % Row Count 4 (+ 1) % Row 3 \SetRowColor{white} - & Total \$ of loan \tn % Row Count 5 (+ 1) % Row 4 \SetRowColor{LightBackground} - & Term of loan \tn % Row Count 6 (+ 1) % Row 5 \SetRowColor{white} - & Frequency of payments \tn % Row Count 7 (+ 1) % Row 6 \SetRowColor{LightBackground} - & Interest rate \tn % Row Count 8 (+ 1) % Row 7 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Finding a level stream of payments (over the term of the loan) with a present value calculated at the loan interest rate equal to the amount borrowed} \tn % Row Count 11 (+ 3) % Row 8 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Loan amortization schedule}} Used to determine loan amortisation payments and the allocation of each payment to interest and principal} \tn % Row Count 14 (+ 3) % Row 9 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Portion of payment representing interest declines over the repayment period, and the portion going to principal repayment increases}}} \tn % Row Count 17 (+ 3) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{PMT = PV / \{1 / r x {[} 1 - 1 / (1 + r)\textasciicircum{}n\textasciicircum{} {]} \}}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Deposits Needed to Accumulate a Future Sum}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Determine the annual deposit necessary to accumulate a certain amount of money at some point in the future} \tn % Row Count 3 (+ 3) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{E.g. house deposit} \tn % Row Count 4 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Can be derived from the equation for fi nding the future value of an ordinary annuity} \tn % Row Count 6 (+ 2) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Can also be used to calc required deposit} \tn % Row Count 7 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{PMT = FV \{{[}( 1 + r)\textasciicircum{}n\textasciicircum{} - 1 {]} / r\}}} \newline \newline Once this is done substitute the known values of FV, r, and n into the righthand \newline side of the equation to find the annual deposit required.} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Stated Versus Effective Annual Interest Rates}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Make objective comparisons of loan costs or investment returns over different compounding periods} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Stated annual rate}} is the contractual annual rate charged by a lender or promised by a borrower} \tn % Row Count 4 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Effective annual rate (EAR)}} AKA the true annual return, is the annual rate of interest actually paid or earned} \tn % Row Count 7 (+ 3) % Row 3 \SetRowColor{white} - & Reflects the effect of compounding frequency \tn % Row Count 9 (+ 2) % Row 4 \SetRowColor{LightBackground} - & Stated annual rate does not \tn % Row Count 10 (+ 1) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Maximum effective annual rate for a stated annual rate occurs when interest compounds continuously} \tn % Row Count 12 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{EAR = ( 1 + r/m )\textasciicircum{}m\textasciicircum{} - 1}} \newline \newline Compounding continuously: {\bf{EAR (continuous compounding) = e\textasciicircum{}r\textasciicircum{} - 1}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Concept of future value}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Apply simple interest, or compound interest to a sum over a specified period of time.} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Interest might compound: annually, semiannual, quarterly, and even continuous compounding periods} \tn % Row Count 4 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Future value}} value of an investment made today measured at a specific future date using compound interest.} \tn % Row Count 7 (+ 3) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Compound interest}} is earned both on principal amount and on interest earned} \tn % Row Count 9 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Principal}} refers to amount of money on which interest is paid.} \tn % Row Count 11 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Important to understand}} \newline After 30 years @ 5\% a \$100 principle account has: \newline - Simple Interest: balance of \$250. \newline - Compound interest: balance of \$432.19 \newline \newline {\bf{FV = PV x (1 + r)\textasciicircum{}n\textasciicircum{}}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{The Power of Compound Interest}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/nataliemoore_1489354179_2017-03-13 07_29_17-William L Megginson, Scott B. Smart-Introduction to Corporate Finance, Abridged .png}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{Future Value of One Dollar} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Present value}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Used to determine what an investor is willing to pay today to receive a given cash flow at some point in future.} \tn % Row Count 3 (+ 3) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Calculating present value of a single future cash payment} \tn % Row Count 5 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Depends largely on investment opportunities of recipient and timing of future cash flow} \tn % Row Count 7 (+ 2) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Discounting}} describes process of calculating present values} \tn % Row Count 9 (+ 2) % Row 4 \SetRowColor{LightBackground} - & Determines present value of a future amount, assuming an opportunity to earn a return (r) \tn % Row Count 12 (+ 3) % Row 5 \SetRowColor{white} - & Determine PV that must be invested at r today to have FV, n from now \tn % Row Count 14 (+ 2) % Row 6 \SetRowColor{LightBackground} - & Determines present value of a future amount, assuming an opportunity to earn a given return (r) on money. \tn % Row Count 17 (+ 3) % Row 7 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{We lose opportunity to earn interest on money until we receive it} \tn % Row Count 19 (+ 2) % Row 8 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{To solve, inverse of compounding interest} \tn % Row Count 20 (+ 1) % Row 9 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{PV of future cash payment declines longer investors wait to receive} \tn % Row Count 22 (+ 2) % Row 10 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Present value declines as the return (discount) rises.} \tn % Row Count 24 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{E.g. value now of \$100 cash flow that will come at some future date is less than \$100 \newline \newline {\bf{PV = FV / (1 + r)\textasciicircum{}n\textasciicircum{} = FV x {[} 1 / (1+ r)\textasciicircum{}n\textasciicircum{} {]}}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{The Power of Discounting}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/nataliemoore_1489357892_2017-03-13 08_30_46-William L Megginson, Scott B. Smart-Introduction to Corporate Finance, Abridged .png}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4577 cm} x{3.29544 cm} x{0.82386 cm} } \SetRowColor{DarkBackground} \mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Special applications of time value}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{3}{x{5.377cm}}{Use the formulas to solve for other variables} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} - & Cash flow & CF or PMT \tn % Row Count 3 (+ 2) % Row 2 \SetRowColor{LightBackground} - & Interest / Discount rate & r \tn % Row Count 4 (+ 1) % Row 3 \SetRowColor{white} - & Number of periods & n \tn % Row Count 5 (+ 1) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{3}{x{5.377cm}}{Common applications and refinements} \tn % Row Count 6 (+ 1) % Row 5 \SetRowColor{white} - & Compounding more frequently than annually & \tn % Row Count 8 (+ 2) % Row 6 \SetRowColor{LightBackground} - & Stated versus effective annual interest rates & \tn % Row Count 10 (+ 2) % Row 7 \SetRowColor{white} - & Calculation of deposits needed to accumulate a future sum & \tn % Row Count 13 (+ 3) % Row 8 \SetRowColor{LightBackground} - & Loan amortisation & \tn % Row Count 14 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}---} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{x{3.83229 cm} p{1.14471 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Compounding More Frequently Than Annually}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Financial institutions compound interest semiannually, quarterly, monthly, weekly, daily, or even continuously.} \tn % Row Count 3 (+ 3) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{The more frequently interest compounds, the greater the amount of money that accumulates} \tn % Row Count 5 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Semiannual compounding}}} \tn % Row Count 6 (+ 1) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Compounds twice per year} \tn % Row Count 7 (+ 1) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Quarterly compounding}}} \tn % Row Count 8 (+ 1) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Compounds 4 times per year} \tn % Row Count 9 (+ 1) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{m values:}}} \tn % Row Count 10 (+ 1) % Row 7 \SetRowColor{white} Semiannual & 2 \tn % Row Count 11 (+ 1) % Row 8 \SetRowColor{LightBackground} Quarterly & 4 \tn % Row Count 12 (+ 1) % Row 9 \SetRowColor{white} Monthly & 12 \tn % Row Count 13 (+ 1) % Row 10 \SetRowColor{LightBackground} Weekly & 52 \tn % Row Count 14 (+ 1) % Row 11 \SetRowColor{white} Daily & 365 \tn % Row Count 15 (+ 1) % Row 12 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Continuous Compounding}}} \tn % Row Count 16 (+ 1) % Row 13 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{m = infinity} \tn % Row Count 17 (+ 1) % Row 14 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{e = irrational number \textasciitilde{}2.7183.\textasciicircum{}13\textasciicircum{}} \tn % Row Count 18 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{General equation: {\bf{FV = PV x (1 + r / m)\textasciicircum{}mxn\textasciicircum{}}} \newline \newline Continuous equation: {\bf{FV (continuous compounding) = PV x ( e\textasciicircum{}rxn\textasciicircum{} )}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Future Value of Cash Flow Streams}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Evaluate streams of cash flows in future periods.} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Two types:} \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Mixed stream}} = a series of unequal cash flows reflecting no particular pattern} \tn % Row Count 4 (+ 2) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Annuity}} = A stream of equal periodic cash flows} \tn % Row Count 6 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{More complicated than calc future or present value of a single cash flow, {\bf{same basic technique}}.} \tn % Row Count 8 (+ 2) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Shortcuts available to eval an annuity} \tn % Row Count 9 (+ 1) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{AKA terminal value} \tn % Row Count 10 (+ 1) % Row 7 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{FV of any stream of cash flows at EOY = sum of FV of individual cash flows in that stream, at EOY} \tn % Row Count 12 (+ 2) % Row 8 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Each cash flow earns interest, so future value of stream is greater than a simple sum of its cash flows} \tn % Row Count 15 (+ 3) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{FV = CF1 x (1 +r)\textasciicircum{}n-1\textasciicircum{} + CF2 x (1 + r)\textasciicircum{}n-2\textasciicircum{} + ... + CFn x (1 + r)\textasciicircum{}n-n\textasciicircum{}}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Future Value of an Ordinary Annuity}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Two basic types of annuity:} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Ordinary annuity}} = payments made into it at end of each period} \tn % Row Count 3 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{Annuity due}} = payments made into it at the beginning of each period (arrives 1 year sooner)} \tn % Row Count 5 (+ 2) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{So, future value of an annuity due always greater than ordinary annuity}}} \tn % Row Count 7 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Future value of an ordinary annuity can be calculated using same method as a mixed stream} \tn % Row Count 9 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{FV = PMT x \{ {[} ( 1 + r )\textasciicircum{}n\textasciicircum{} - 1 {]} / r\}}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Finding the Future Value of an Annuity Due}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Slight change to those for an ordinary annuity} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Payment made at beginning of period, instead of end} \tn % Row Count 3 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Earns interest for 1 period longer} \tn % Row Count 4 (+ 1) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Earns more money over the life of the investment} \tn % Row Count 5 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{FV (annuity due) = PMT x \{ {[} ( 1 + r)\textasciicircum{}n\textasciicircum{} -1 {]} / r \} x (1 + r)}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Present Value of Cash Flow Streams}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Present values of cash flow streams that occur over several years} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Might be used to:} \tn % Row Count 3 (+ 1) % Row 2 \SetRowColor{LightBackground} - & Value a company as a going concern \tn % Row Count 4 (+ 1) % Row 3 \SetRowColor{white} - & Value a share of stock with no definite maturity date \tn % Row Count 6 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{= sum of the present values of CFn} \tn % Row Count 7 (+ 1) % Row 5 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{{\bf{Perpetuity}}: A level or growing cash flow stream that continues forever} \tn % Row Count 9 (+ 2) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Same technique as a lump sum} \tn % Row Count 10 (+ 1) % Row 7 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Present Value of a Mixed Stream = Sum of present values of individual cash flows} \tn % Row Count 12 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Mixed stream: \newline {\bf{PV = {[}CF1 x 1 / (1 + r)\textasciicircum{}1\textasciicircum{}{]} + {[}CF2 x 1 / (1 + r)\textasciicircum{}1\textasciicircum{}{]} + ... + {[}CFn x 1 / (1 + r)\textasciicircum{}1\textasciicircum{}{]}}} \newline \newline Present value of an ordinary annuity} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Present Value of an Ordinary Annuity}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Similar to mixed stream} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Discount each payment and then add up each term} \tn % Row Count 2 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{PV = PMT/r x {[}1 - 1 / (1 + r)\textasciicircum{}n\textasciicircum{}{]}}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Present Value of Annuity Due}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Similar to mixed stream / ordinary annuity} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Discount each payment and then add up each term} \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Cash flow realised 1 period earlier} \tn % Row Count 3 (+ 1) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Annuity due has a larger present value than ordinary annuity} \tn % Row Count 5 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{PV (annuity due) = PMT/r x {[}1 - 1 / (1 + r)\textasciicircum{}n\textasciicircum{}{]} x (1 + r)}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{p{0.4977 cm} x{4.4793 cm} } \SetRowColor{DarkBackground} \mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Present Value of a Perpetuity}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Level or growing cash fl ow stream that continues forever} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Level = infinite life} \tn % Row Count 3 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{Simplest modern example = prefered stock} \tn % Row Count 4 (+ 1) % Row 3 \SetRowColor{white} \mymulticolumn{2}{x{5.377cm}}{Preferred shares promise investors a constant annual (or quarterly) dividend payment forever} \tn % Row Count 6 (+ 2) % Row 4 \SetRowColor{LightBackground} - & express the lifetime (n) of this security as infi nity (∞) \tn % Row Count 8 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}--} \SetRowColor{LightBackground} \mymulticolumn{2}{x{5.377cm}}{{\bf{PV = PMT x 1/r = PMT/r}}} \tn \hhline{>{\arrayrulecolor{DarkBackground}}--} \end{tabularx} \par\addvspace{1.3em} % That's all folks \end{multicols*} \end{document}