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Net present value formula and time-zero cash flow

Net present value is the sum of cash flows discounted to time zero. The time-zero cash flow enters directly because its discount factor is one.

Read the sum

Each term has an amount, a date measured in regular periods and a rate per period. A negative initial investment is included as CF₀ rather than subtracted a second time after it is already in the sequence.

NPV=CF0+∑t=1TCFt(1+r)tNPV=CF_0+\sum_{t=1}^{T}\frac{CF_t}{(1+r)^t}

Amounts and rate intervals must use the same timing convention.

Variables
SymbolMeaning
CF₀Cash flow at time zero
CF_tCash flow t regular periods later
rDiscount rate per regular period
TNumber of future periods

Check the zero-rate case

At a zero discount rate, every discount factor is one. For −1,000, 300, 400 and 500, NPV is therefore 200. This boundary case is a useful sign and timing check before applying a positive rate.

Interpret the sign with its assumptions

A positive or negative result belongs to the cash flows and rate supplied. For a decision comparison, estimates must be comparable in risk, timing and scope. NPV is not the same thing as the undiscounted sum unless the rate is zero.

Further references