What present value means
A dollar today is worth more than a dollar in ten years, because today’s dollar can be invested and earn a return in the meantime. Present value answers the reverse question: how much would you need to invest today, at a given rate, to end up with a future amount? It is the foundation of pricing bonds, valuing businesses, comparing a lump sum with a series of payments and choosing between settlement offers.
The formulas
With r the rate per period and n the number of periods:
- Present value of a future sum: PV = FV ÷ (1 + r)^n
- Present value of regular payments received at the end of each period: PV = payment × (1 − (1 + r)^−n) ÷ r
- Discount factor: 1 ÷ (1 + r)^n, the value today of $1 received at the end
For monthly compounding, r is the annual rate ÷ 12 and n is the number of years × 12.
Worked example
You will receive $10,000 in 10 years and money can earn 5% a year.
- Discount factor = 1 ÷ 1.05^10 = 1 ÷ 1.6289 = 0.6139.
- Present value = 10,000 × 0.6139 = $6,139.13.
- With monthly compounding instead, the same sum is worth $6,071.61 today, because money compounds faster.
Now add a payment of $1,000 at the end of each year for those 10 years. The payments are worth 1,000 × (1 − 1.05^−10) ÷ 0.05 = $7,721.73 today, so the whole package is worth $13,860.87, against $20,000 received in total over the years.
How the rate changes the answer
The present value of $10,000 in 10 years:
| Discount rate | Present value |
|---|---|
| 2% | $8,203 |
| 3% | $7,441 |
| 5% | $6,139 |
| 7% | $5,083 |
| 10% | $3,855 |
Choosing a discount rate
- Guaranteed money (a legal settlement, a Treasury bond): use a Treasury yield of the same maturity.
- Risky cash flows (a business, rental income): use the return you could expect on an investment of similar risk, often 6 to 10% or more.
- Purchasing power: use expected inflation to express a future price in today’s dollars.
A higher rate always gives a lower present value. For a project with an upfront cost, use the net present value calculator, which subtracts the investment.