A dollar today can be invested and become more than a dollar later, so money at different dates can't be added directly. Compound Interest grew money forward. Present value runs the same calculation backward, and Infinite Geometric Series handles payments that go on for many years.
Key idea
At interest rate $r$ per period: $$PV = \frac{FV}{(1 + r)^t}.$$ An annuity paying $C$ at the end of each of $n$ periods is worth $C\,\dfrac{1 - (1 + r)^{-n}}{r}$; a perpetuity (forever) is worth $\dfrac{C}{r}$; a perpetuity that starts at $C$ and grows at rate $g < r$ is worth $\dfrac{C}{r - g}$.
Why it works
If you put $PV$ in the bank today, it grows to $PV(1 + r)^t$ in $t$ periods. The present value of $FV$ is the deposit that grows to exactly $FV$, so divide. A stream of payments is worth the sum of each payment's present value. For a perpetuity that is a geometric series with first term $\dfrac{C}{1 + r}$ and ratio $\dfrac{1}{1 + r}$: $$\frac{C/(1 + r)}{1 - 1/(1 + r)} = \frac{C}{r}.$$ An annuity is a perpetuity minus the same perpetuity started $n$ periods later, which gives the factor $1 - (1 + r)^{-n}$.
Worked example: One payment
What is \$1,210 received in 2 years worth today at 10%?
- $PV = \dfrac{1{,}210}{(1.1)^2} = \dfrac{1{,}210}{1.21} = 1{,}000$.
- Check forward: \$1,000 grows to \$1,100 after a year and \$1,210 after two.
Answer: \$1,000.
Worked example: An annuity and a perpetuity
At 10%, value (a) \$1,000 a year for 3 years and (b) \$500 a year forever, each payment at the end of the year.
- (a) $$\dfrac{1{,}000}{1.1} + \dfrac{1{,}000}{1.21} + \dfrac{1{,}000}{1.331} \approx 909.09 + 826.45 + 751.31 = 2{,}486.85$$
- Formula check: $1{,}000\cdot\dfrac{1 - 1.1^{-3}}{0.1} \approx 2{,}486.85$.
- (b) $\dfrac{500}{0.1} = 5{,}000$.
Answer: (a) about \$2,486.85; (b) \$5,000.
Worked example: A growing perpetuity
A payment of \$300 arrives next year and then grows 2% a year forever. Interest is 8%.
- $PV = \dfrac{300}{0.08 - 0.02} = \dfrac{300}{0.06} = 5{,}000$.
- The formula needs $g < r$; otherwise the payments grow faster than they are discounted and the sum has no finite value.
Answer: \$5,000.
Common mistake
Multiplying by $(1 + r)^t$ when you should divide, or discounting over the wrong number of periods. Multiplying gives a future value; a present value must be smaller than the amount (for $r > 0$). And a payment at the end of year 1 is discounted once, at the end of year 3 three times. Check: is your present value less than the payment itself? Then count the periods between today and the payment.
Try the compounding and bond calculators on Calculators & Links to see how the rate and the timing change a present value.