Make PW, FW, and AW tell the same story
Present, future, and annual worth are three equivalent representations of the same cash-flow sequence when date, rate, and horizon are consistent.
Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The independent project meets the MARR on these assumed cash flows.
Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
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
Use net cash flows: receipts minus disbursements at each date. PW discounts to time 0; FW accumulates to the chosen terminal date; AW converts PW into a uniform end-of-period amount over the study period.
Build the calculation
For independent projects without resource restrictions, a nonnegative net worth at MARR meets the economic criterion. Under constraints, portfolio choice can be more complicated. For mandatory equal-service cost alternatives, compare equivalent cost magnitudes and minimize rather than demand positive revenue.
Set up. Calculate. Explain.
Rebuild the starting scenario
Initial investment: 50000 $; Annual net benefit: 15000 $/year; MARR: 10 % per year; Number of years: 5 years. Calculate the net present worth.
Setup hint
Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
Open the visual solution
- Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
- PW = −I + A(P/A,i,N) = $6,861.80.
- FW = PW(F/P,i,N); AW = PW(A/P,i,N). Positive conversion factors preserve the sign and decision.
Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The independent project meets the MARR on these assumed cash flows.
Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
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
Invest $36,000 now and receive $9,000 net annually for six years. At MARR 12%, find PW.
Setup hint
Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
Open the visual solution
- Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
- PW = −I + A(P/A,i,N) = $1,002.67.
- FW = PW(F/P,i,N); AW = PW(A/P,i,N). Positive conversion factors preserve the sign and decision.
Period 0: −$36,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
The independent project meets the MARR on these assumed cash flows.
Draw the investment as negative at time 0 and net benefit as positive at years 1–N.
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.
If PW at MARR is positive, what follows for the SAME horizon and cash flows?
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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