Present Value (PV) Calculator

Present Worth (PW): Definition, Formulas, and When to Use It

Present Worth (PW)—also called present value (PV)—is the value today of money you expect to receive or pay in the future. PW converts each future cash flow into its equivalent amount at time zero using a discount rate that reflects the time value of money and risk. If you’re evaluating a project, PW tells you the current dollar value of the benefits and costs on a comparable, “now” basis. In engineering economics and corporate finance, PW is a foundation for go/no-go decisions, ranking alternatives, and comparing uneven cash-flow streams on an apples-to-apples basis.

Core Present Worth Formulas

1) Single Future Amount → Present Worth (P/F, i, n)

$ P = \dfrac{F}{(1+i)^{n}} \;\;\equiv\;\; P = F \,(P/F, i, n) $

Use when you know one future amount $F$ received at the end of $n$ periods (years, quarters, or months) and want its equivalent value today using discount rate $i$ per year. This appears in salvage values, balloon payments, or a single milestone payment.

2) Uniform Series (Ordinary Annuity) → Present Worth (P/A, i, n)

$ P = A \left[\dfrac{1-(1+i)^{-n}}{i}\right] \;\;\equiv\;\; P = A \,(P/A, i, n) $

Use when you have a constant end-of-period payment $A$ for $n$ periods—for example, annual savings from a process change or a level maintenance contract. This is the most common equivalence factor after the single-sum case.

3) Growing Annuity → Present Worth (payments grow at g)

$ P = \dfrac{A_1}{i-g}\left[1-\left(\dfrac{1+g}{1+i}\right)^{n}\right] \quad \text{for } i \neq g$

If payments start at $A_1$ in period 1 and grow at rate $g$ each period (e.g., energy savings increasing with price inflation), use the growing-annuity formula. When $i=g$, the limit is $P \approx A_1 \cdot n / (1+i)$.

4) Mixed or Irregular Cash-Flow Stream → Present Worth (General Form)

$ PW = \displaystyle \sum_{t=1}^{n} \dfrac{CF_t}{(1+i)^{t_y}} – C_0 $

When the amounts differ by period, discount each cash flow $CF_t$ individually and sum. Subtract the upfront cost $C_0$ at $t=0$. If your timing unit is quarters or months, convert the exponent to years $t_y$ with $t/4$ or $t/12$ so an annual rate $i$ is applied correctly.

What You Need Before Calculating Present Worth

Discount rate $i$: Your required return or opportunity cost per year. Reflects risk and the time value of money.
Timing convention: End-of-period (ordinary) vs. beginning-of-period (annuity due). The formulas above assume end-of-period.
Number of periods $n$: Count in the same unit as your cash-flow schedule (years, quarters, months).
Cash flows: Single future amount $F$, level payment $A$, first payment $A_1$ and growth $g$, or a list $\{CF_t\}$ for irregular streams.
Upfront cost $C_0$: Cash outflow at $t=0$. Enter as a positive number but treat economically as an outflow.

How to Calculate Present Worth (Step-by-Step)

Choose the model: single sum, uniform annuity, growing annuity, or irregular stream.
Align units: If your cash flows are monthly but your discount rate is annual, use $t_y=t/12$ in the exponent.
Apply the right factor: $P/F$ for a single sum, $P/A$ for a level series, growing-annuity formula when payments escalate.
Sum components: For mixed streams, discount each $CF_t$, add them, then subtract $C_0$.
Interpret: A positive PW (or NPV) means the project creates value above your required return; a negative PW means it does not.

Worked Examples

Example 1 — Single Future Amount (Salvage)

You expect to sell a machine for $F=\$8{,}000$ in 4 years. Your required return is $i=9\%$ annually. The present worth is:

$ P = \dfrac{8000}{(1+0.09)^{4}} = \dfrac{8000}{1.41158} \approx \$5{,}667 $

Today, that future salvage is worth about \$5.7k. If this is the only benefit and the machine costs more than that in present terms, it wouldn’t clear your hurdle.

Example 2 — Uniform Annual Savings (Level Annuity)

A process upgrade yields annual savings of $A=\$3{,}000$ for $n=6$ years. If $i=8\%$,

$ P = 3000 \left[\dfrac{1-(1+0.08)^{-6}}{0.08}\right] = 3000 \times 4.62288 \approx \$13{,}869 $

The stream of savings is worth about \$13.9k today. Compare that to the project’s upfront cost to judge value creation.

Example 3 — Growing Annuity (Costs Escalate with Inflation)

An annual service contract starts at $A_1=\$2{,}500$ next year and grows with inflation $g=3\%$ for $n=5$ years. Your discount rate is $i=9\%$.

$ P = \dfrac{2500}{0.09-0.03}\left[1-\left(\dfrac{1.03}{1.09}\right)^{5}\right] = \dfrac{2500}{0.06}\left[1-(0.94673)\right] \approx 41666.67 \times 0.05327 \approx \$2{,}221 $

The present worth of this growing cost stream is about \$2.22k. If an alternative contract has a lower PW, it’s the better deal all else equal.

Example 4 — Irregular Cash-Flow Project

Upfront cost $C_0=\$10{,}000$. Annual net cash flows: $CF_1=\$2{,}000$, $CF_2=\$3{,}500$, $CF_3=\$4{,}800$, and a terminal value of $CF_3 += \$1{,}500$. With $i=10\%$,

$ PW = \dfrac{2000}{1.10} + \dfrac{3500}{(1.10)^2} + \dfrac{(4800+1500)}{(1.10)^3} – 10000 \\ = 1818.18 + 2892.56 + 5469.57 – 10000 \approx 180.31 $

PW is slightly positive (~\$180). That means at a 10% required return, the project just clears your hurdle; a higher discount rate would likely turn it negative.

Choosing a Discount Rate (i)

The discount rate should reflect your opportunity cost and the risk of the cash flows. For core corporate projects many use a weighted average cost of capital (WACC) as a baseline. For riskier initiatives or cash flows tied to volatile markets, add a premium. When modeling real (inflation-adjusted) cash flows, use a real discount rate; for nominal cash flows, use a nominal rate. Because small changes in $i$ can move PW a lot, perform sensitivity checks (e.g., 8%, 10%, 12%) to understand how robust your decision is.

Common Pitfalls (and How to Avoid Them)

Unit mismatch: Monthly cash flows with an annual rate require converting the exponent with $t/12$, not dividing the rate by 12 in this PW framework.
Wrong timing: Assuming beginning-of-period payments when formulas are for end-of-period will overstate or understate PW.
Mixing nominal and real: Keep inflation treatment consistent across cash flows and discount rate.
Forgetting terminal value or working capital: Include expected sale proceeds and net working-capital recovery at the end.
Using an average rate for diverse risks: Consider risk-adjusted rates or scenario analysis for uncertain projects.

PW vs. NPV vs. Other Metrics

In engineering economy texts, present worth method and NPV are essentially the same concept: discount all cash flows to today and sum. PW/NPV answers “How much value does this create in today’s dollars?” Other metrics complement the picture:

IRR: The discount rate that makes PW = 0. Intuitive but can be misleading for non-standard cash flows or mutually exclusive options.
Payback: Time to recover the investment. Ignores time value beyond the cutoff and ignores benefits after payback.
Profitability Index (PI): PV of inflows divided by initial outlay; helpful when capital is constrained.

When in doubt, lean on PW/NPV for primary decisions, then use IRR, payback, and PI as supporting views.

Bottom Line

Present Worth distills future cash flows into today’s dollars so you can compare options fairly and decide with confidence. Choose the correct model (single sum, uniform series, growing series, or irregular stream), align your units, use a defensible discount rate, and test sensitivity. A positive PW means the investment clears your required return and creates value; a negative PW means it does not. With these principles and formulas, you can analyze equipment purchases, energy projects, service contracts, and investments with clarity and consistency.