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\author{Natalie Moore (NatalieMoore)}
\pdfinfo{
  /Title (time-value-of-money.pdf)
  /Creator (Cheatography)
  /Author (Natalie Moore (NatalieMoore))
  /Subject (Time value of money Cheat Sheet)
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        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\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/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Natalie Moore (NatalieMoore) \\
  \uline{cheatography.com/nataliemoore} \\
        \uline{\seqsplit{www}.jchmedia.com/}
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  \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}.
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  Measure your website readability!\\
  www.readability-score.com
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\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
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\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}}--}
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\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)
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Lump sum future value in excel}}  \tn
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\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}}--}
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\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}}--}
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\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}}-}
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\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}}--}
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\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}}-}
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\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}