Future and Annual Worth Analysis

Learning Objectives

  • Understand the Future Worth (FW) and Annual Worth (AW) methods for evaluating alternatives.
  • Explain the major advantage of the Annual Worth method regarding lifespans.
  • Calculate Capital Recovery (CR) for a physical asset.
  • Compare investment alternatives utilizing equivalence metrics.

In addition to Present Worth (PW) Analysis, two other primary equivalence methods for evaluating engineering alternatives are Future Worth (FW) and Annual Worth (AW) analysis. These methods offer different perspectives but, when applied correctly, will always lead to the exact same investment decision as Present Worth Analysis.

Future Worth (FW) Analysis

Future Worth (FW) Method

Compares alternatives based on their equivalent value at a future date (usually the end of the project life). It is particularly useful when the primary goal is to maximize future wealth, such as in retirement planning, long-term investments, or assessing the terminal value of a portfolio.

Application of Future Worth

Future Worth is especially useful for individuals or corporations planning for a specific target date. For instance, if an engineer is depositing \5,000annuallyintoaretirementaccountearningannually into a retirement account earning7%$, a Future Worth analysis immediately yields the total portfolio value at retirement age. It translates current decisions directly into final, compounded outcomes.

Future Worth Evaluation Criteria

Lifespan Equalization Rule

Just like with Present Worth Analysis, when comparing mutually exclusive alternatives using Future Worth, you must explicitly evaluate them over the same time period (Least Common Multiple or a defined Study Period). Failing to equalize the lifespans will skew the future compounding and invalidate the comparison.

Annual Worth (AW) Analysis

Annual Worth (AW) Method

Converts all cash flows (inflows and outflows) into a series of equal annual equivalent amounts spread uniformly over the life of the project.

Equivalent Uniform Annual Cost (EUAC)

When dealing primarily with costs (and negligible revenues), this method is historically referred to as Equivalent Uniform Annual Cost (EUAC). EUAC is critical for "leasing vs. buying" decisions or evaluating competing machinery. By converting a massive lump-sum purchase into an equivalent annual "lease" payment, a company can easily compare it against actual annual lease options.

The Major AW Advantage

The absolute most significant advantage of the Annual Worth method is that you do not need to use the Least Common Multiple (LCM) of lives when comparing mutually exclusive alternatives with different lifespans.

Because the AW method mathematically assumes continuous replacement of the asset with an identical asset at the end of its life, the AW of one life cycle of an alternative is exactly equal to the AW of any number of identical, continuous life cycles. Therefore, you only need to calculate the AW for one single life cycle of each alternative.

Annual Worth Evaluation Criteria

Capital Recovery (CR)

Capital Recovery Concept

A foundational component of any AW analysis for physical assets is Capital Recovery (CR). It represents the equivalent uniform annual cost of owning an asset, accounting for the initial capital investment, the time value of money, and the final salvage value at the end of its useful life.

Understanding Depreciation and Return

Capital Recovery is conceptually made of two parts: the actual loss in value of the asset over time (depreciation), plus the return on the invested capital that the company lost out on by buying the equipment instead of investing the money elsewhere (interest).

Capital Recovery Formula

Calculates the equivalent uniform annual cost of owning an asset.

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

Variables

SymbolDescriptionUnit
CRCRCapital Recovery cost-
PPInitial purchase price or first cost (present worth)-
SSSalvage value (future worth at end of life nn)-
nnExpected life in years-
iiInterest rate (MARR)-

Alternative Capital Recovery Formulation

Alternatively, using the standard relationship (A/F,i,n)=(A/P,i,n)−i(A/F, i, n) = (A/P, i, n) - i, the formula can be expressed to calculate the loss in value plus interest on the salvage:

Capital Recovery (Alternative Formula)

An alternative calculation of the equivalent uniform annual cost of owning an asset.

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

Variables

SymbolDescriptionUnit
CRCRCapital Recovery cost-
PPInitial purchase price or first cost (present worth)-
SSSalvage value (future worth at end of life nn)-
nnExpected life in years-
iiInterest rate (MARR)-

Total Annual Worth Concept

The total Annual Worth is the Capital Recovery cost plus any equivalent uniform annual revenues minus any equivalent uniform annual operating expenses (AOC):

Total Annual Worth Formula

Calculates the Total Annual Worth.

AWtotal=CR+Arevenues−AexpensesAW_{total} = CR + A_{revenues} - A_{expenses}

Variables

SymbolDescriptionUnit
AWtotalAW_{total}Total Annual Worth-
CRCRCapital Recovery cost-
ArevenuesA_{revenues}Equivalent uniform annual revenues-
AexpensesA_{expenses}Equivalent uniform annual operating expenses-

Interactive Equivalence Visualizer

Interactive Simulation

Use the simulator below to see how a single set of cash flows (Initial Cost, Annual Benefit, and Salvage Value) can be equivalently expressed as a Total Present Worth, Total Annual Worth, or Total Future Worth. Notice that regardless of the metric chosen, if one is positive (indicating profitability at the MARR), all three will be mathematically positive. Explore how Future Worth and Annual Worth vary with interest rate and time.

Equivalence Visualizer

Initial Cost (P)10,000 $
Annual Benefit (A)3,000 $ / yr
Salvage Value (S)2,000 $
Life (n)5 yrs
Interest Rate (MARR)10 %

Total PW

$2,614

Total AW

$690

Total FW

$4,210

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Note that PW, AW, and FW all lead to the same accept/reject decision (positive vs negative).
Key Takeaways
  • Future Value Perspective: Evaluates equivalent project value at the end of the analysis period instead of the beginning.
  • Equivalence Rules: Like PW, mutually exclusive options must absolutely be evaluated over identical time horizons for FW.
  • Decision Rule: A project is justified if its FW≥0FW \ge 0 at the MARR.
  • The AW Advantage: Converts all lump sums into an equivalent uniform annual series. There is absolutely no need to find the LCM of lives when comparing mutually exclusive options.
  • Capital Recovery (CR): The equivalent uniform annual cost of owning an asset, combining the amortization of the initial investment and the return on the eventual salvage value over the asset's life.
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