Present Value Equation

Present value answers one question: what is a future amount worth today? Reverse the compounding formula $FV = PV(1+r)^n$ and you get

$PV = \frac{FV}{(1+r)^n} = FV(1+r)^{-n}$

• $FV$ — the amount received later
• $r$ — the discount rate per period, as a decimal
• $n$ — the number of periods until it arrives

The factor $(1+r)^{-n}$ is the discount factor; it is always between 0 and 1 for $r > 0$, so present value is always below face value. The rate $r$ is a choice, not a fact: it represents what the money could otherwise earn, and the answer moves with it. Doubling $r$ does not halve $PV$ — the relationship is exponential, so long horizons are punished far harder than short ones.

When the amounts differ from period to period, discount each one on its own date and add:

$PV = \sum_{t=1}^{n} \frac{C_t}{(1+r)^t}$

Net present value subtracts what you pay up front:

$NPV = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+r)^t}$

$NPV > 0$ means the discounted inflows exceed the outlay at that rate; $NPV = 0$ defines the internal rate of return.

If the cash flows are all equal, the sum collapses to the annuity form:

$PV = C \cdot \frac{1 - (1+r)^{-n}}{r}$

and if they continue forever, to the perpetuity $PV = C/r$. Two more rearrangements are worth remembering: $FV = PV(1+r)^n$ recovers the future amount, and

$r = \left(\frac{FV}{PV}\right)^{1/n} - 1$

gives the rate implied by a pair of values.

• Percent left unconverted: $r = 0.06$, not $6$.
• Off-by-one on $t$: a payment one year out is discounted once, so $t = 1$. Only a payment received today has $t = 0$.
• Mixing periods: quarterly cash flows need a quarterly rate ($r_{\text{annual}}/4$) and $n$ counted in quarters.
• Averaging uneven cash flows first: discounting the average is not the average of the discounted values, because $(1+r)^{-t}$ is non-linear.
• Using the annuity shortcut on unequal payments: that formula assumes every $C_t$ is identical.
• Forgetting the sign of $C_0$: in an NPV, the initial outlay is negative. Adding it instead flips the decision.