Time Value of Money Formulas
TVM formulas for present value, future value, and annuities.
Understand how money changes value over time with worked examples.
Future Value of a Lump Sum
The time value of money is one of the most important concepts in finance and economics. A dollar today is worth more than a dollar in the future because today's dollar can be invested to earn a return.
This principle drives all financial decision-making: loans, investments, retirement planning, and business valuations.
Present Value of a Lump Sum
Discounting is the reverse of compounding — it tells you what a future sum is worth in today's dollars. This is essential for comparing cash flows that occur at different times.
Future Value of an Ordinary Annuity
An annuity is a series of equal payments made at regular intervals. This formula tells you how much money you will accumulate by making regular payments over time.
Present Value of an Ordinary Annuity
Variables
| Symbol | Meaning |
|---|---|
| FV | Future value — the amount of money at a future date |
| PV | Present value — the current worth of future money |
| PMT | Payment amount per period (for annuities) |
| r | Interest rate per period (as a decimal) |
| n | Number of compounding periods |
Example 1: Future Value
You invest $10,000 at 6% annual interest compounded annually for 15 years. What will it be worth?
FV = PV × (1 + r)n = 10,000 × (1.06)15
FV = 10,000 × 2.3966
FV = $23,966 (your money nearly 2.4× in 15 years at 6%)
Example 2: Annuity Accumulation
You save $500 per month at 8% annual return (compounded monthly) for 30 years. How much will you have?
Monthly rate: r = 0.08/12 = 0.00667
Number of periods: n = 30 × 12 = 360
FV = 500 × [(1.00667)360 − 1] / 0.00667
FV = 500 × [10.9357 − 1] / 0.00667 = 500 × 9.9357 / 0.00667
FV = 500 × 1,489.6
FV ≈ $744,800 (from $180,000 in total contributions — the rest is interest earned)
Example 3: Present Value
You will receive $50,000 in 10 years. If the discount rate is 5%, what is it worth today?
PV = FV / (1 + r)n = 50,000 / (1.05)10
PV = 50,000 / 1.6289
PV ≈ $30,696 (you would need to invest $30,696 today at 5% to have $50,000 in 10 years)
When to Use These
TVM formulas are essential for virtually all financial decisions.
- Planning retirement savings and projecting account growth
- Comparing investment options with different time horizons
- Calculating loan payments and mortgage affordability
- Valuing businesses and real estate investments
- Deciding between a lump sum payment and an annuity (e.g., lottery winnings)