Unequal-Life Projects With Replacement Chains

Why Project Life Matters

Two projects can provide the same service for different lengths of time. Comparing their initial NPVs alone may be misleading: a project that lasts two years may need to be purchased again while a four-year alternative is still operating. A fair comparison must put the alternatives on a consistent time basis and account for the cash flows of any replacements.

Two standard approaches are the replacement-chain method, which repeats each project until both alternatives cover a common horizon, and the equivalent annual value method, which converts each project’s NPV into a constant annual amount. For cost-only alternatives, the corresponding amount is an equivalent annual cost (EAC), and the lower EAC is preferred when the alternatives provide equivalent service.

Definition: For a project with life $n$, NPV $N$, and effective annual discount rate $i$, its equivalent annual value (EAV) is the constant year-end amount $E$ whose present value over $n$ years equals $N$. Thus $N=E a_{\overline{n}|}$, where $a_{\overline{n}|}=(1-v^n)/i$ and $v=1/(1+i)$, for $i\ne0$. Therefore $EAV=N/a_{\overline{n}|}$. For a cost-only project with present-value cost $C$, its EAC is $C/a_{\overline{n}|}$.

Use a consistent perspective. If project cash flows are expressed as benefits less costs, a higher EAV is preferred. If they are expressed as costs to provide the same service, a lower EAC is preferred. The discount rate must be appropriate for the projects’ risk and cash-flow basis, as in the earlier material on risk-matched discount rates for project NPV. Cash flows and the discount rate must also be in consistent nominal or real terms.

Equivalent Annual Value and Its Justification

The formula follows directly from the definition of an annuity-immediate: a constant amount $E$ paid at each year-end for $n$ years has present value $E a_{\overline{n}|}$. To make that present value equal to the project’s NPV, set the two present values equal and solve for $E$:

N=E\sum_{t=1}^{n}v^t =E a_{\overline{n}|} \quad\Longrightarrow\quad E=\frac{N}{a_{\overline{n}|}}.

This is an annualization, not a forecast that the project will actually earn the same amount each year. It expresses the project’s NPV as an equivalent level annual amount over its own life. An equivalent annual comparison is useful when projects offer comparable service and can be repeated on comparable terms. It does not make dissimilar projects interchangeable or remove the need to examine their cash-flow assumptions.

Worked Example: Comparing Cost-Only Assets Over a Common Horizon

A hypothetical insurer is comparing two systems that provide the same processing service. System A costs $10,000 initially, lasts two years, and costs $2,000 at each year-end to operate. System B costs $13,000 initially, lasts three years, and costs $1,200 at each year-end to operate. Assume a 10% effective annual discount rate, no taxes, no disposal proceeds, and that each system can be replaced immediately at the end of its life for the same initial price and with the same operating costs and service capacity. Which has the lower cost over a six-year horizon?

Method. First find each life-cycle present-value cost and EAC. Then repeat each system to cover six years. The key assumption is that each replacement is an identical copy on identical terms.

At 10%, the relevant factors, rounded to six decimals, are $a_{\overline{2}|}=1.735537$, $a_{\overline{3}|}=2.486852$, $a_{\overline{6}|}=4.355261$, $v^2=0.826446$, and $v^3=0.751315$.

System A. Its two-year present-value cost is its initial cost plus the present value of two year-end operating costs:

C_A=10{,}000+2{,}000(1.735537) =10{,}000+3{,}471.074 =\$13{,}471.07.

The EAC is $13{,}471.07/1.735537=\$7{,}761.90$ per year. Over six years, three two-year cycles are needed. The common-horizon present-value cost is:

C_{A,6}=13{,}471.074(1+v^2+v^4) =13{,}471.074(1+0.826446+0.683013) \approx \$33{,}805.12.

As a check, $\$7{,}761.90(4.355261)\approx \$33{,}805.12$, allowing for rounding in the displayed factors.

System B. Its three-year present-value cost is:

C_B=13{,}000+1{,}200(2.486852) =13{,}000+2{,}984.222 =\$15{,}984.22.

Its EAC is $15{,}984.222/2.486852=\$6{,}427.49$ per year. Two three-year cycles cover the six-year horizon:

C_{B,6}=15{,}984.222(1+v^3) =15{,}984.222(1+0.751315) \approx \$27{,}993.41.

The check using the annual amount gives $\$6{,}427.49(4.355261)\approx \$27{,}993.41$. System B has the lower EAC and the lower six-year present-value cost, so it is preferred on the stated cost basis. The method avoids comparing only the first purchase prices or ignoring the replacement needed for System A.

Replacement Chains and the Common-Horizon Result

For lives $n_A$ and $n_B$, the smallest horizon that is an exact whole number of cycles for both is their least common multiple $H$. Repeat each project as many times as needed to reach $H$, discounting every cycle’s cash flows back to time zero. For a level service that can be replicated, the replacement-chain comparison represents the decision to keep providing that service through the entire common horizon.

The relationship between the two methods can be shown directly. Suppose a project with life $n$ has NPV $N$, and it is repeated identically $m$ times, so $H=mn$. Each cycle begins $n$ years after the preceding one. The present value of the chain is therefore:

N_{\text{chain}} =N(1+v^n+v^{2n}+\cdots+v^{(m-1)n}).

The $H$-year annuity factor can be grouped into $m$ blocks of $n$ year-end payments. Each block is discounted by the appropriate number of years:

a_{\overline{H}|} =a_{\overline{n}|}(1+v^n+v^{2n}+\cdots+v^{(m-1)n}).

Since $N=EAV\cdot a_{\overline{n}|}$, substituting into the chain formula gives $N_{\text{chain}}=EAV\cdot a_{\overline{H}|}$. Thus, for alternatives repeated over the same horizon, comparing EAVs gives the same ranking as comparing replacement-chain NPVs. For cost-only alternatives the same identity applies to costs: compare EACs or common-horizon present-value costs.

Choose a consistent basis.
Use the same effective annual discount rate and the same nominal or real cash-flow basis for all alternatives.

Find each project’s life-cycle NPV or cost.
Include all relevant cash flows over that project’s life, including any terminal cash flow at its end.

Annualize or extend.
Divide NPV or present-value cost by the project’s annuity factor, or repeat each project to the least common multiple of the lives.

Check the interpretation.
Prefer the higher EAV for net benefits, or the lower EAC for equivalent-service cost alternatives. Confirm the replacement assumptions before treating the ranking as decisive.

Worked Example: Ranking Unequal-Life Projects by EAV

A hypothetical company is choosing between two mutually exclusive service platforms. Platform A requires $100,000 at time zero and produces net operating benefits of $60,000 at each year-end for two years. Platform B requires $120,000 at time zero and produces net operating benefits of $50,000 at each year-end for three years. Assume a 10% effective annual discount rate, no taxes, and that each platform can be repeated with the same investment and operating benefits. Compare their EAVs and their replacement chains over six years.

Method. Compute each project’s NPV from its stated cash flows, annualize that NPV over its own life, and then use the six-year annuity factor to check the chain comparison.

For Platform A, the present value of benefits is $60{,}000a_{\overline{2}|}=60{,}000(1.735537)=\$104{,}132.23$. Therefore:

NPV_A=104{,}132.23-100{,}000=\$4{,}132.23, \qquad EAV_A=\frac{4{,}132.23}{1.735537} \approx \$2{,}380.95.

For Platform B, the present value of benefits is $50{,}000a_{\overline{3}|}=50{,}000(2.486852)=\$124{,}342.60$. Thus:

NPV_B=124{,}342.60-120{,}000=\$4{,}342.60, \qquad EAV_B=\frac{4{,}342.60}{2.486852} \approx \$1{,}746.22.

Although Platform B has the larger NPV over its own life, Platform A has the larger EAV. The replacement-chain check uses $a_{\overline{6}|}=4.355261$:

NPV_{A,6}\approx 2{,}380.95(4.355261)=\$10{,}369.67, \qquad NPV_{B,6}\approx 1{,}746.22(4.355261)=\$7{,}605.24.

Platform A has the higher six-year chain NPV, matching the EAV ranking. This conclusion depends on the stated repeatability: the future investments and annual benefits must remain as assumed, and each repeated platform must provide a comparable service. The method avoids treating the larger standalone NPV as automatically superior when the project lives differ.

When Repeated Replacement Is a Valid Assumption

A replacement chain is not justified merely because the arithmetic can be performed. Its interpretation depends on assumptions about future availability, service, and cash flows. In particular, the comparison should address the following points:

Comparable service. The alternatives must provide sufficiently equivalent output or service quality over the comparison horizon. If one option has materially different capacity, that difference must be valued or incorporated in the analysis.
Repeatability. A replacement must be available when required, and its relevant purchase, operating, and disposal cash flows must be known or reasonably represented by the repeated-cycle assumptions.
Consistent operating conditions. The operating benefits and costs, technology, and required service should not change in ways that make later cycles materially different unless those changes are included explicitly.
Consistent discounting. The same appropriate discount rate can be used for comparable cash flows in each cycle. If risk or financing assumptions change, the relevant cash flows and discounting must reflect that change.
Complete terminal treatment. Disposal proceeds, tax effects, working-capital recovery, and any other relevant terminal cash flows must be included at the correct dates. They cannot be omitted simply because a chain is being built.

These are modeling assumptions, not universal facts about real projects. If the future replacement price is known to change, if the service requirement ends before the common horizon, or if technology makes later cycles materially different, model those facts directly. A simple identical-cycle chain may then give a misleading ranking.

Worked Example: A Known Change in Replacement Cost

A hypothetical firm needs a system for exactly four years. System R costs $20,000 initially, lasts two years, and costs $2,000 to operate at each year-end. A replacement at the end of year two is available, but its known price will be $26,000; it then serves years three and four. System S costs $43,000 initially, lasts four years, and costs $1,000 at each year-end. Assume a 10% effective annual discount rate, no taxes, no disposal proceeds, and equivalent service. Which system has the lower four-year present-value cost?

Method. Discount the actual stated cash flows over the required four-year service period. Do not assume the second R purchase has the same price as the first.

At 10%, $v^2=0.826446$ and $a_{\overline{4}|}=3.169865$. System R’s operating costs have present value $2{,}000(3.169865)=\$6{,}339.73$. Its second purchase is made at time two, so its present value is $26{,}000(0.826446)=\$21{,}487.60$. Including the initial price:

C_R=20{,}000+21{,}487.60+6{,}339.73 =\$47{,}827.33.

System S’s operating costs have present value $1{,}000(3.169865)=\$3{,}169.87$, so:

C_S=43{,}000+3{,}169.87=\$46{,}169.87.

System S has the lower four-year present-value cost. To see the trap, incorrectly treating R’s replacement as another $20,000 purchase would give $20{,}000+20{,}000v^2+2{,}000a_{\overline{4}|}\approx \$42{,}868.66$, making R appear cheaper. That calculation contradicts the stated $26,000 replacement price. The correction is to use the actual known replacement cash flow, not an unsupported identical-cycle assumption.

Common Mistakes and Practical Interpretation

A frequent mistake is to compare project NPVs calculated over different lives as if they covered the same service period. NPV remains a valid measure for each project’s stated cash flows, but the decision between unequal-life alternatives may require a common service horizon or an explicit annual equivalent. Another trap is to annualize a project’s initial cost alone while leaving its operating costs, disposal proceeds, or other cash flows out of the life-cycle NPV. The annual equivalent must reflect the complete relevant cash-flow set.

A second mistake is to assume that the least common multiple is automatically the economically relevant horizon. It is a convenient exact horizon when repeated cycles are genuinely comparable; it is not a substitute for the actual required service period. If the firm needs the service for a fixed period that is not a whole number of cycles, model the partial final cycle, any resale value, and any other end effects explicitly.

Key takeaway: EAV and EAC put a project’s life-cycle NPV or cost on a level annual basis. A replacement chain gives the equivalent common-horizon comparison when repeated cycles are valid. Always state why the projects can be repeated on comparable terms before relying on that ranking.

Check Your Understanding

Use year-end cash flows and effective annual discounting unless a question states otherwise. Amounts are hypothetical.

A project has a two-year present-value cost of $17,355.37. At 10%, find its EAC to the nearest dollar. (Answer: $17{,}355.37/1.735537\approx \$10{,}000$ per year.)
A three-year project has NPV of $12,000 at 10%. Find its EAV to the nearest dollar. (Answer: $12{,}000/2.486852\approx \$4{,}825$ per year.)
Two cost-only assets provide the same service. Asset M has a two-year life and EAC of $9,000; Asset N has a four-year life and EAC of $8,000. At 10%, with identical replacement cycles and a four-year common horizon, which has the lower common-horizon present-value cost? (Answer: N; its cost is $8{,}000a_{\overline{4}|}=8{,}000(3.169865)\approx \$25{,}359$, versus M at $9{,}000(3.169865)\approx \$28{,}529$.)
A two-year asset’s first purchase price is $15,000, but a known replacement at the end of year two will cost $19,000. Explain whether a chain that repeats the $15,000 purchase price is appropriate. (Answer: No. Discount the stated $19,000 replacement cash flow at time two, or explicitly justify a different forecast; the identical-cycle chain assumption is contradicted by the known price.)
Project U has a four-year life and EAV of $3,000; Project V has a two-year life and EAV of $3,500. Both are repeatable on identical terms, provide equivalent service, and have the same risk-matched discount rate. Which has the higher six-year replacement-chain NPV? (Answer: V. Both use $a_{\overline{6}|}$, so U has $3{,}000a_{\overline{6}|}$ and V has $3{,}500a_{\overline{6}|}$; since the factor is positive, V’s chain NPV is higher.)

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