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.

FWn=∑t=0nCFt(1+i)n−tFW_n=\sum_{t=0}^{n}CF_t(1+i)^{n-t}

Variables

SymbolDescriptionUnit
FWnFW_nFuture worth at terminal period n-
CFtCF_tNet cash flow at period t-
iiEffective rate per period-
nnTerminal 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.

Equivalent-Worth Focal-Date Map

MethodFocal valueTransformation for a common study basisTypical comparison
Present worthTime zeroDiscount each dated flowMaximize net PW; minimize present cost-only value
Future worthSelected terminal dateCompound each dated flow to that dateSame preference as PW on the same basis
Annual worthUniform end-of-period seriesConvert equivalent PW or FW to AWMaximize AW; minimize equivalent annual cost

The three methods are equivalent transformations when the cash flows, MARR, dates, and analysis basis are identical.

Annual Worth from Present Worth

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

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

Variables

SymbolDescriptionUnit
AWAWEquivalent uniform annual worth-
PWPWNet present worth-
iiMARR per period-
nnNumber 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)AW=-P(A/P,i,n)+A+S(A/F,i,n)

Variables

SymbolDescriptionUnit
PPFirst cost magnitude at time zero-
AAUniform annual net operating cash flow, positive for net inflow-
SSPositive terminal salvage value-
iiMARR per year-
nnAsset 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)CR=P(A/P,i,n)-S(A/F,i,n)

Variables

SymbolDescriptionUnit
CRCRAnnual capital-recovery cost-
PPFirst cost-
SSTerminal salvage value-
iiMARR-
nnAsset 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.

Equivalent-Worth and Unequal-Life Laboratory

Concept and model scope

Compute PW, terminal FW, and AW from the same cash-flow model for two alternatives. When lives differ, annual-worth comparison is meaningful only under an explicit continuing-service or replacement assumption.

MARR9.00%
Alternative A
Analysis life7 years
Alternative B
Analysis life10 years
AlternativeLifePW at time 0FW at own terminal yearAW
A7 years₱1,248,471₱2,282,254₱248,059
B10 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.

Present-Worth Alternative Comparator

Concept and model scope

Discount every cash flow to time zero using one MARR and compare alternatives on the same study basis.

MARR10.00%
Common study life8 years
Alternative A
Alternative B
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.

Unequal-Life Comparison Basis

Service and replacement assumptionSuitable basisRequired disclosure
Alternatives provide repeatable service and successor assumptions are defensibleAW or equivalent annual costState the repeatability and cost/performance assumptions
Decision has a defined planning horizonFixed study periodEstimate terminal value, remaining service, or replacement at the horizon
Service requirements differ or replacement is not comparableReframe the decision before rankingDefine equivalent service or explain the non-economic constraint

Annual-Worth Decision Rules

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

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

Select PW, FW, or AW on one common cash-flow basis and guard unequal-life comparisons with explicit service assumptions.

For identical cash flows, MARR, timing, and study basis, PW, FW, and AW are equivalent transformations and must imply the same preference. Use AW for unequal lives only when service and replacement assumptions are defensible; otherwise define a fixed study period and terminal values.

Equivalent-Worth Method SelectionSelect PW, FW, or AW on one common cash-flow basis and guard unequal-life comparisons with explicit service assumptions.. For identical cash flows, MARR, timing, and study basis, PW, FW, and AW are equivalent transformations and must imply the same preference. Use AW for unequal lives only when service and replacement assumptions are defensible; otherwise define a fixed study period and terminal values.. Start → Define service need and alternatives; Define service need and alternatives → Build dated, signed cash flows and MARR; Build dated, signed cash flows and MARR → Useful lives differ?; Useful lives differ? — No → Requested focal-value method?; Useful lives differ? — Yes → Comparable repeat service defensible?; Comparable repeat service defensible? — Yes → Use AW or equivalent annual cost; Comparable repeat service defensible? — No → Set fixed study period and terminal values; Use AW or equivalent annual cost → Requested focal-value method?; Set fixed study period and terminal values → Requested focal-value method?; Requested focal-value method? — Present → Discount to time zero: PW; Requested focal-value method? — Future → Compound to terminal date: FW; Requested focal-value method? — Annual → Convert to uniform annual amount: AW; Discount to time zero: PW → Equivalent methods agree on preference?; Compound to terminal date: FW → Equivalent methods agree on preference?; Convert to uniform annual amount: AW → Equivalent methods agree on preference?; Equivalent methods agree on preference? — Yes → Apply revenue or cost-only decision rule; Equivalent methods agree on preference? — No → Audit dates, MARR, signs, and horizon; Audit dates, MARR, signs, and horizon → Build dated, signed cash flows and MARR; Apply revenue or cost-only decision rule → Report comparison basis

Start → Define service need and alternatives; Define service need and alternatives → Build dated, signed cash flows and MARR; Build dated, signed cash flows and MARR → Useful lives differ?; Useful lives differ? — No → Requested focal-value method?; Useful lives differ? — Yes → Comparable repeat service defensible?; Comparable repeat service defensible? — Yes → Use AW or equivalent annual cost; Comparable repeat service defensible? — No → Set fixed study period and terminal values; Use AW or equivalent annual cost → Requested focal-value method?; Set fixed study period and terminal values → Requested focal-value method?; Requested focal-value method? — Present → Discount to time zero: PW; Requested focal-value method? — Future → Compound to terminal date: FW; Requested focal-value method? — Annual → Convert to uniform annual amount: AW; Discount to time zero: PW → Equivalent methods agree on preference?; Compound to terminal date: FW → Equivalent methods agree on preference?; Convert to uniform annual amount: AW → Equivalent methods agree on preference?; Equivalent methods agree on preference? — Yes → Apply revenue or cost-only decision rule; Equivalent methods agree on preference? — No → Audit dates, MARR, signs, and horizon; Audit dates, MARR, signs, and horizon → Build dated, signed cash flows and MARR; Apply revenue or cost-only decision rule → Report comparison basis

  • Start: terminator
  • Define service need and alternatives: process
  • Build dated, signed cash flows and MARR: process
  • Useful lives differ?: decision
  • Comparable repeat service defensible?: decision
  • Use AW or equivalent annual cost: process
  • Set fixed study period and terminal values: process
  • Requested focal-value method?: decision
  • Discount to time zero: PW: process
  • Compound to terminal date: FW: process
  • Convert to uniform annual amount: AW: process
  • Equivalent methods agree on preference?: decision
  • Audit dates, MARR, signs, and horizon: process
  • Apply revenue or cost-only decision rule: process
  • Report comparison basis: terminator
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.

References