Present worth method

Convert every cash flow in a project to its equivalent value at time zero (the “present”), then sum them. The result is a single number, the present worth (PW). Business-finance contexts call the same calculation net present value (NPV).

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:

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

For mutually exclusive alternatives, pick the highest PW (most positive, which could be the least-negative if every option is a negative-PW cost-only choice).

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 → $(P/G,i,N)$.

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.

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, i.e. the firm isn’t capital-rationed.

The other equivalence-based comparison methods are Future worth method (equivalent at the project endpoint), Annual worth method (equivalent annualised), and Internal rate of return (the rate at which PW = 0).