Future Worth and Annual Worth Analysis

Future Worth and Annual Worth Analysis

$Learning Objectives$

Convert a cash-flow stream to an equivalent future worth at a stated terminal date.
Convert present worth to equivalent uniform annual worth over a stated life.
Decompose annual worth into capital recovery, operating cash flow, and salvage recovery.
Compare alternatives using AW when lives differ and the service assumptions support that comparison.
Verify that PW, FW, and AW give consistent decisions on the same economic basis.

$Future Worth$

Future worth is the equivalent value of all study cash flows at a selected future focal date, commonly the end of the study period.

Future Worth of Discrete Cash Flows

Compounds all signed cash flows to terminal period n.

FW_n=\sum_{t=0}^{n}CF_t(1+i)^{n-t}

Variables

Symbol	Description	Unit
$FW_n$	Future worth at terminal period n	-
$CF_t$	Net cash flow at period t	-
$i$	Effective rate per period	-
$n$	Terminal study period	-

$Annual Worth$

Annual worth is the uniform end-of-period series economically equivalent to a project's cash-flow stream over a stated analysis life at the MARR.

Annual Worth from Present Worth

Converts net PW to an equivalent uniform annual amount over n periods.

AW=PW(A/P,i,n)

Variables

Symbol	Description	Unit
$AW$	Equivalent uniform annual worth	-
$PW$	Net present worth	-
$i$	MARR per period	-
$n$	Number of annual-worth periods	-

Annual Worth of a Typical Asset

Separates first cost, annual net operating cash flow, and terminal salvage.

AW=-P(A/P,i,n)+A+S(A/F,i,n)

Variables

Symbol	Description	Unit
$P$	First cost magnitude at time zero	-
$A$	Uniform annual net operating cash flow, positive for net inflow	-
$S$	Positive terminal salvage value	-
$i$	MARR per year	-
$n$	Asset life in years	-

$Capital Recovery$

Capital recovery is the uniform annual amount that recovers the invested capital and the required return, accounting for any terminal salvage value.

Annual Capital-Recovery Cost

Annualized ownership cost before operating costs when salvage S is received at the end of life n.

CR=P(A/P,i,n)-S(A/F,i,n)

Variables

Symbol	Description	Unit
$CR$	Annual capital-recovery cost	-
$P$	First cost	-
$S$	Terminal salvage value	-
$i$	MARR	-
$n$	Asset life	-

$Interactive Equivalent-Worth Laboratory$

Compare two alternatives and inspect each one's PW, terminal FW, and AW. Equal-life alternatives should produce consistent PW/AW preferences; when lives differ, the laboratory deliberately prevents raw own-life PW from being treated as a valid ranking without a common study basis.

MARR9.00%

Alternative A

First costPHPAnnual net cash flowPHP/yearTerminal salvagePHP

Analysis life7 years

Alternative B

First costPHPAnnual net cash flowPHP/yearTerminal salvagePHP

Analysis life10 years

Alternative	Life	PW at time 0	FW at own terminal year	AW
A	7 years	₱1,248,471	₱2,282,254	₱248,059
B	10 years	₱2,854,873	₱6,758,524	₱444,847

Annual-worth preference

Alternative B

This unequal-life AW comparison assumes continuing service and a defensible replacement/repeatability basis for each alternative.

Equivalent-method cross-check

Raw own-life PW not ranked

Present worths shown over different terminal lives are descriptive values, not a valid mutually exclusive ranking unless a fixed common study period or explicit repeatability model is imposed.

A decision sign

Nonnegative AW

For an independent revenue project, nonnegative AW clears the economic screen at the stated MARR.

B decision sign

Nonnegative AW

Equivalent worth methods preserve accept/reject sign when the same cash flow, rate, and life are used.

Do not use a least-common-multiple life mechanically unless repeated replacement with comparable cost, performance, and salvage is a reasonable model. If repeatability is not defensible, define a fixed study period and model terminal values explicitly.

MARR10.00%

Common study life8 years

Alternative A

First costPHPAnnual net cash flowPHP/yearSalvagePHP

Alternative B

First costPHPAnnual net cash flowPHP/yearSalvagePHP

PW — Alternative A

₱1,741,625

Meets the MARR screen.

PW — Alternative B

₱1,968,658

Meets the MARR screen.

Preferred on this common basis

Alternative B

PW(B) − PW(A) = ₱227,033.

For unequal lives, do not force a common-life comparison without stating the repeatability assumption. Annual worth or a fixed study period is often clearer.

$Why Annual Worth Helps with Unequal Lives$

AW expresses each alternative as an equivalent annual consequence over its own analysis life. It is especially useful when the service is ongoing and each alternative can reasonably be replaced by a comparable successor. If that replacement assumption is not defensible, use a fixed study period with explicit terminal assumptions instead.

$Annual-Worth Decision Rules$

Independent revenue project: accept economically if $AW\ge0$ .
Mutually exclusive revenue alternatives: choose the largest AW on a consistent basis.
Cost-only alternatives providing equal service: choose the smallest equivalent annual cost.

$PW, FW, and AW Are Equivalent Methods$

For the same cash flows, MARR, and study basis, PW, FW, and AW are transformations of one another. They should not produce conflicting accept/reject decisions. A disagreement usually signals inconsistent timing, rate, sign convention, or study horizon.

$Automatic Repeatability Is an Assumption$

Do not assume an asset can be replaced forever at identical real cost, performance, and salvage simply because a textbook least-common-multiple comparison is convenient. State replacement assumptions explicitly.

$Equivalent-Worth Selection Workflow$

STEP-BY-STEP

Establish the common economic assumptions and service requirement.
Build each alternative's signed cash-flow model.
Select PW, FW, or AW as the reporting basis.
Convert all cash flows using the same MARR.
Apply the proper decision rule for revenue or cost alternatives.
Cross-check one alternative using another equivalent-worth basis when practical.
State any replacement or terminal-value assumption supporting unequal-life comparison.

Key Takeaways

FW moves all cash flows to a future focal date; AW converts them to a uniform series.
PW, FW, and AW are economically equivalent when based on the same cash flows, rate, and horizon.
Capital recovery annualizes first cost net of salvage recovery.
AW is useful for unequal lives only when its service/replacement assumptions are appropriate.
Revenue alternatives are maximized; cost-only equivalent annual cost is minimized.
Conflicting equivalent-worth decisions are a diagnostic signal of inconsistent modeling.