Value service that continues forever
An infinite-life equivalent is useful for continuing service, endowments, or permanent infrastructure, but its assumptions are strong.
The replacement cost equals I and recurs at N, 2N, … forever, with zero salvage. Infinite-life formulas require i>0.
Pay first cost at time 0.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
What the model means
A uniform end-of-year amount A continuing forever has P=A/i for i>0. If the first cost I must be repeated every N years with no salvage, the future replacements have PW I/[(1+i)ᴺ−1]. The first purchase at year 0 is separate.
Build the calculation
Capitalized cost is the present equivalent positive magnitude of all costs. An equivalent permanent annual cost equals i times capitalized cost. Positive recurring flows at zero discount have unbounded infinite present value; do not force the finite annuity limit into a perpetuity.
Set up. Calculate. Explain.
Rebuild the starting scenario
Initial cost: 20000 $; Annual recurring cost: 2000 $/year; Effective annual rate: 10 %; Replacement every N years: 5 years. Calculate the capitalized cost.
Setup hint
Pay first cost at time 0.
Open the visual solution
- Pay first cost at time 0.
- Annual costs beginning year 1 capitalize to A/i = $20,000.00.
- Replacement costs form a geometric series at years N,2N,…; their PW is I/[(1+i)^N−1]. Equivalent perpetual annual cost is iCC.
The replacement cost equals I and recurs at N, 2N, … forever, with zero salvage. Infinite-life formulas require i>0.
Pay first cost at time 0.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Transfer the idea to a new scenario
An asset costs $30,000 now and every ten years forever, with zero salvage; annual operation is $2,500. At 8%, find capitalized cost.
Setup hint
Pay first cost at time 0.
Open the visual solution
- Pay first cost at time 0.
- Annual costs beginning year 1 capitalize to A/i = $31,250.00.
- Replacement costs form a geometric series at years N,2N,…; their PW is I/[(1+i)^N−1]. Equivalent perpetual annual cost is iCC.
The replacement cost equals I and recurs at N, 2N, … forever, with zero salvage. Infinite-life formulas require i>0.
Pay first cost at time 0.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
For $1,000/year forever beginning in year 1 at 5%, its PW is:
Can you defend your setup?
Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?
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