Future Value Exercise: A lump sum cash flow of $$422\mathrm{.}41$ is available now $($t$=0).$ For this cash flow, the appropriate discount rate per period is $9\mathrm{.}0\%.$ What is the future value of this cash flow $10$ periods from now? We will calculate the future value of the single cash flow in three equivalent ways. First, we will calculate the future value using a time line, where each column corresponds to a period of calendar time. Second, we use a formula for the future value. Third, we use Excel$$s FV function for the future value. Procedures: $1.$ Inputs: Enter the inputs $($PV$,$ r and n$)$ in the range B$2$:B$4.2.$ Future Value using a Time Line: Create a time line from period $0$ to period $10.$ Enter the single cash flow in period $0.$ Calculate the period $10$ future value of each cash flow and sum the future values as follows. $$ Period: Enter $0,1,2,...,10.$ in the range B$7$:L$7.$ Cash Flows: Enter $=$B$2$ in cell B$8.$ Enter $$0\mathrm{.}00$ in cell C$8$ and copy it across to L$8.$ Future Value of Each Cash Flow $=($Cash Flow$)*(1+$ Discount Rate per Period$)^(($Number of Periods$)-($Current Period$)).$ o Enter $=$B$8*(1+$$B$$3)^($$B$$4-$B$7)$ in cell B$9$ and copy it across. The exponent term $($$B$$4-$B$7)$ causes the period $0$ cash flow to be compounded $10$ times into the future, the period $1$ cash flow to be compounded $9$ times into the future, the period $2$ cash flow to be compounded $8$ times into the future, etc. The $ signs in $B$$3$ and $B$$4$ lock the column and the row when copying. $$ Future Value $=$ Sum over all periods of the Future Value of Each Cash Flow. o Enter $=$SUM$($B$9$:L$9)$ in cell L$10.3.$ Future Value using the Formula: For a lump sum single cash flow, the formula is Future Value $=($Cash Flow$)(1+$ Discount Rate per Period$)^($Number of Periods$).$ Enter $=$B$2*(1+$B$3)^$B$4$ in cell L$13.4.$ Future Value using the FV Function: The Excel FV function can be used to calculate the future value of a single cash flow, the future value of an annuity, or the future value of a bond. For a single cash flow, the format is $=-$FV$($Discount Rate per Period, Number of Periods, $0,$ Single Cash Flow$).$ Enter $=-$FV$($B$3,$B$4,0,$B$2)$ in cell L$16.$ The Future Value of this Single Cash Flow is $$1,000\mathrm{.}00.$ Notice you get the same answer all three ways: using the time line, using the formula, or using the FV function! A B C D E F G H I J K L $1$ Inputs $2$ Single Cash Flow $$422\mathrm{.}413$ Discount Rate per Period $9\mathrm{.}0\%4$ Number of Periods $1056$ Future Value using a Time Line $7$ Period $0123456789108$ Cash Flows $$422\mathrm{.}41$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}009$ Future Value of Each Cash Flow $$1,000\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}00$ $$0\mathrm{.}0010$ Future Value $$1,000\mathrm{.}001112$ Future Value using the Formula $13$ Future Value $$1,000\mathrm{.}001415$ Future Value using the FV Function $16$ Future Value $$1,000\mathrm{.}00\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline & A & B & C & D & E & F & G & H & I & J & K & L $\backslash \backslash$

$\backslash$hline $1$ & Inputs & & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $2$ & Single Cash Flow & $\backslash$$$422\mathrm{.}41$ & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $3$ & Discount Rate per Period & $9\mathrm{.}0\backslash \%$ & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $4$ & Number of Periods & $10$ & & & & & & & & & & $\backslash \backslash$

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$\backslash$hline $6$ & Future Value using a Time Line & & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $7$ & Perrod & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10\backslash \backslash$

$\backslash$hline $8$ & Cash flows & $\backslash$$$422\mathrm{.}41$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00\backslash \backslash$

$\backslash$hline $9$ & Rubure Value of Each Cash Flow & $\backslash$$$1,000\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$ $90\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00$ & $\backslash$$$0\mathrm{.}00\backslash \backslash$

$\backslash$hline $10$ & Pubure Value & & & & & & & & & & & $\backslash$$$1,000\mathrm{.}00\backslash \backslash$

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$\backslash$hline $12$ & Future Value using the Formula & & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $13$ & Pubure Value & & & & & & & & & & & $\backslash$$$1,000\mathrm{.}00\backslash \backslash$

$\backslash$hline $14$ & & & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $15$ & Future Value using the FV Function & & & & & & & & & & & $\backslash \backslash$

$\backslash$hline $16$ & Pubure Value & & & & & & & & & & & $\backslash$$$1,000\mathrm{.}00\backslash \backslash$

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