Present worth method

The present worth method converts every cash flow in a project to its equivalent value at time zero (the “present”), then sums them to get a single number called the present worth (PW). The same idea is called net present value (NPV) in business-finance contexts; they’re the same calculation.

For a project with cash flows $C_0,C_1,C_2,\dots ,C_N$ at the end of periods $0,1,2,\dots ,N$, discounted at rate $i$ (usually the MARR):

$PW={\sum }_{t=0}^N\frac{C_t}{(1+i{)}^t}.$

Cash flows can be positive (revenues, salvage) or negative (capital cost, operating cost). Time zero is “now,” so $C_0$ is the initial outlay (usually negative, since you spend money upfront) and $C_t$ for $t\ge 1$ are future flows.

Decision rules:

• $PW>0$: project benefits exceed costs in present-value terms. Accept (it clears MARR).
• $PW=0$: project is exactly at break-even.
• $PW<0$: present-value costs exceed benefits. Reject.

For mutually exclusive alternatives, pick the one with the highest PW (most positive — could be the least-negative if all are negative-PW cost-only options).

In practice, the calculation usually doesn’t involve year-by-year discounting. Common cash-flow patterns get collapsed using interest factors:

• Single payments → $(P/F,i,N)$.
• Equal-payment annuities → $(P/A,i,N)$.
• Arithmetic gradients → $(P/G,i,N)$.

A worked example. A project requires $100,000 upfront, generates $25,000/year for 6 years, and has a $10,000 salvage value at year 6. At MARR = 10%:

• PW of initial cost: -\100{,}000$.
• PW of annuity: 25{,}000 \cdot (P/A, 10\%, 6) = 25{,}000 \cdot 4.355 = \108{,}885$.
• PW of salvage: 10{,}000 \cdot (P/F, 10\%, 6) = 10{,}000 \cdot 0.5645 = \5{,}645$.
• Total: -100{,}000 + 108{,}885 + 5{,}645 = \14{,}530$.

PW > 0, so accept. The project clears the 10% hurdle by about $14,500 in present-value terms.

Key assumptions for the PW method (and most comparison methods):

1. Costs and benefits are measurable.
2. Future cash flows are known with certainty (relaxed under risk and uncertainty analysis).
3. Inflation is ignored or handled explicitly via real/actual conversions (Inflation).
4. Taxes are ignored or handled explicitly (Corporate income tax, Capital tax factor).
5. Funds are available — the firm isn’t capital-rationed.

For comparison methods that work at the same level, see Future worth method (equivalent at the project endpoint), Annual worth method (equivalent annualised), Internal rate of return (rate at which PW = 0), External rate of return, and Payback period.