Because of inflation and alternative investments, money is more valuable now than in the future. As such, we must compare economic alternatives not simply by the sum of all of the cash flows, but rather by the net present worth. We may observe that the ratio between the present and future worth of a cash flow is given by the factor expressed in equation $[\bold 1]$. This and other factors may also be obtained from tables. $P$ is the present worth of the cash flow, it’s value in terms of todays dollars, $F$ is the future worth of the cash flow, that actual amount to be given in the future. The factor is given in terms of the discount rate, $r$, and the number of compounding periods until the future cash flow, $n$.
$$ [\bold 1]\>\>\>\>\>\>(P/F)=\frac 1{(1+r)^n} $$
Thus, by a summation, any series of cash flows may be resolved into an net present worth/value. Alternatively, one may analyze net present cost or net present benefit. Additionally, the inverse of the above factor may be used to give the net future worth.
We may use equation $[\bold 1]$ to derive further equations that give the present worth of common patterns of cash flows.
=NPV(discount, cashflow1, [cashflow2, ...]) gives the net present worth of an arbitrary list of cash flows starting at year 1. Sheets Docs: NPV=PMT(rate, number_of_periods, present_value, [future_value, end_or_beginning]) gives the equivalent annuity of a present worth Sheets Docs: PMT$$ [\bold 2]\>\>\>\>\>\>(P/A)=\frac{(1+r)^n-1}{r(1+r)^n} $$
$$ [\bold 3]\>\>\>\>\>\> (P/G)=\frac 1r\begin{pmatrix} \frac{(1+r)^n-1}{r(1+r)^n}-\frac n{(1+r)^n} \end{pmatrix} $$
Rather than using net present worth directly to compare alternatives, we may resolve an arbitrary series of cash flows into an equivalent series of constant annuities. This may be done by resolving the cash flows into a present worth, and then into an annuity using the inverse of equation $[\bold 2]$.
We may analyze Equivalent Uniform Annual Worth (EUAW), Benefit (EUAB), or Cost (EUAC).
In the case of a finite analysis period, equations $[\bold{1-3}]$ suffice for determining the EUAW. However, for an infinite analysis period, equations $[\bold{4-6}]$ hold as factors for the EUAW, $A_E$.
$$ [\bold 4]\>\>\>\>\>\> (\frac{A_E}P)=r $$
$$ [\bold 5]\>\>\>\>\>\> (\frac{A_E}A)=1 $$
$$ [\bold 6]\>\>\>\>\>\> (\frac{A_E}G)=\frac 1r $$
We may use plotting to compare alternatives across various discount rates. In this comparison, we may use consider present worth or equivalent annual cash flows.