Time Value of Money Tables and Formulas
The time value of money (TVM) is the principle that a dollar today is worth more than a dollar in the future because of its earning potential. TVM is the foundation of DCF analysis, bond pricing, loan amortization, and virtually every valuation method in finance.
Core TVM Formulas
| Concept | Formula | What It Answers |
|---|---|---|
| Future Value (FV) | FV = PV × (1 + r)n | What is $X worth in n years? |
| Present Value (PV) | PV = FV / (1 + r)n | What is a future $X worth today? |
| FV of Annuity | FV = PMT × [((1 + r)n − 1) / r] | What do regular payments grow to? |
| PV of Annuity | PV = PMT × [(1 − (1 + r)−n) / r] | What are regular payments worth today? |
| PV of Perpetuity | PV = PMT / r | What are infinite payments worth today? |
| PV of Growing Perpetuity | PV = PMT / (r − g) | What are growing infinite payments worth? |
Where PV = present value, FV = future value, PMT = periodic payment, r = discount rate per period, n = number of periods, g = growth rate.
Present Value of $1 Table
This table shows what $1 received in the future is worth today at various discount rates. Multiply any future cash flow by the appropriate factor to get its present value.
| Year | 3% | 5% | 7% | 10% | 12% | 15% |
|---|---|---|---|---|---|---|
| 1 | 0.9709 | 0.9524 | 0.9346 | 0.9091 | 0.8929 | 0.8696 |
| 2 | 0.9426 | 0.9070 | 0.8734 | 0.8264 | 0.7972 | 0.7561 |
| 3 | 0.9151 | 0.8638 | 0.8163 | 0.7513 | 0.7118 | 0.6575 |
| 5 | 0.8626 | 0.7835 | 0.7130 | 0.6209 | 0.5674 | 0.4972 |
| 10 | 0.7441 | 0.6139 | 0.5083 | 0.3855 | 0.3220 | 0.2472 |
| 15 | 0.6419 | 0.4810 | 0.3624 | 0.2394 | 0.1827 | 0.1229 |
| 20 | 0.5537 | 0.3769 | 0.2584 | 0.1486 | 0.1037 | 0.0611 |
| 30 | 0.4120 | 0.2314 | 0.1314 | 0.0573 | 0.0334 | 0.0151 |
Future Value of $1 Table
This table shows what $1 invested today grows to at various rates over time. It is the inverse of the PV table. Use it to project investment growth or the impact of compound interest.
| Year | 3% | 5% | 7% | 10% | 12% | 15% |
|---|---|---|---|---|---|---|
| 1 | 1.0300 | 1.0500 | 1.0700 | 1.1000 | 1.1200 | 1.1500 |
| 2 | 1.0609 | 1.1025 | 1.1449 | 1.2100 | 1.2544 | 1.3225 |
| 3 | 1.0927 | 1.1576 | 1.2250 | 1.3310 | 1.4049 | 1.5209 |
| 5 | 1.1593 | 1.2763 | 1.4026 | 1.6105 | 1.7623 | 2.0114 |
| 10 | 1.3439 | 1.6289 | 1.9672 | 2.5937 | 3.1058 | 4.0456 |
| 20 | 1.8061 | 2.6533 | 3.8697 | 6.7275 | 9.6463 | 16.3665 |
| 30 | 2.4273 | 4.3219 | 7.6123 | 17.4494 | 29.9599 | 66.2118 |
Present Value of Annuity Table
Shows the present value of receiving $1 per period for n periods. Essential for valuing bond coupon streams, lease payments, and loan calculations.
| Periods | 3% | 5% | 7% | 10% | 12% | 15% |
|---|---|---|---|---|---|---|
| 1 | 0.9709 | 0.9524 | 0.9346 | 0.9091 | 0.8929 | 0.8696 |
| 3 | 2.8286 | 2.7232 | 2.6243 | 2.4869 | 2.4018 | 2.2832 |
| 5 | 4.5797 | 4.3295 | 4.1002 | 3.7908 | 3.6048 | 3.3522 |
| 10 | 8.5302 | 7.7217 | 7.0236 | 6.1446 | 5.6502 | 5.0188 |
| 15 | 11.9379 | 10.3797 | 9.1079 | 7.6061 | 6.8109 | 5.8474 |
| 20 | 14.8775 | 12.4622 | 10.5940 | 8.5136 | 7.4694 | 6.2593 |
| 30 | 19.6004 | 15.3725 | 12.4090 | 9.4269 | 8.0552 | 6.5660 |
Annuity Due vs. Ordinary Annuity
| Feature | Ordinary Annuity | Annuity Due |
|---|---|---|
| Payment timing | End of each period | Beginning of each period |
| PV adjustment | Standard formula | Multiply ordinary PV × (1 + r) |
| FV adjustment | Standard formula | Multiply ordinary FV × (1 + r) |
| Common example | Bond coupons, loan payments | Rent, lease payments, insurance premiums |
| Value comparison | Lower (payments come later) | Higher (payments come earlier) |
Practical Applications
| Application | TVM Concept Used | Example |
|---|---|---|
| DCF Valuation | PV of future cash flows | Discount projected FCF to today |
| Bond Pricing | PV of annuity + PV of lump sum | Discount coupons + par value |
| Mortgage Payment | PV of annuity (solve for PMT) | Calculate monthly payment |
| Retirement Planning | FV of annuity | How much will monthly savings grow to? |
| WACC Discounting | PV at risk-adjusted rate | Discount enterprise value cash flows |
| Stock Valuation (DDM) | PV of growing perpetuity | Gordon Growth Model for dividend stocks |
Rule of 72
A quick mental math shortcut: divide 72 by the annual interest rate to estimate how many years it takes for money to double.
At 6%, money doubles in ~12 years. At 10%, ~7.2 years. At 12%, ~6 years. This is surprisingly accurate for rates between 2% and 15%.
When building a DCF model, the discount rate has an outsized impact on your valuation. A 1-2% change in WACC can swing the result by 20-30%. Always run a sensitivity table with different discount rates and growth assumptions to show the range of outcomes.
Key Takeaways
- TVM is the foundation of all valuation: DCF, bond pricing, loan math, and retirement planning.
- PV discounts future money to today; FV compounds today’s money into the future.
- Annuity formulas handle equal periodic payments; perpetuity formulas handle infinite payments.
- An annuity due (payments at the start) is always worth more than an ordinary annuity (payments at the end).
- The Rule of 72 is a fast way to estimate doubling time: 72 / rate.
Frequently Asked Questions
What is the time value of money in simple terms?
Money available today is worth more than the same amount in the future because you can invest it and earn a return. A dollar today can grow through compound interest, so it has greater purchasing power than a dollar received a year from now.
What is the difference between present value and future value?
Present value tells you what a future amount is worth today (discounting). Future value tells you what a current amount will grow to over time (compounding). They are inverse operations using the same formula: FV = PV × (1 + r)n.
When do you use the annuity formula vs. the perpetuity formula?
Use the annuity formula when payments occur for a fixed number of periods (bond coupons, loan payments). Use the perpetuity formula when payments continue forever — such as in the Gordon Growth Model for dividend valuation or for valuing preferred stock.
How does the discount rate affect present value?
Higher discount rates reduce present value. A $1,000 cash flow in 10 years is worth $614 at 5% but only $386 at 10%. This is why the choice of WACC or required return rate is so critical in valuation models.
What is the Rule of 72 used for?
The Rule of 72 estimates how long it takes for an investment to double in value. Divide 72 by the annual return rate. At 8% annual return, your money doubles in approximately 9 years (72 / 8 = 9).