Present Value and Future Value: The Time Value of Money
Understand why a dollar today is worth more than a dollar in the future, and learn to calculate present and future values to make sound financial decisions across time.
If someone offered you $10,000 today or $10,000 in 5 years, you'd take it today — obviously. But what if they offered $10,000 today or $14,000 in 5 years? Now it depends on what you could earn on the $10,000 in the meantime. This is the time value of money, and it's the foundation of every financial valuation: mortgages, pension payments, lottery lump-sum decisions, bond pricing, and business investment analysis all reduce to this one question: what is a future payment worth in today's dollars?
- Define present value (PV) and future value (FV) and explain the discount rate
- Calculate PV from a known FV, and FV from a known PV
- Apply time value of money reasoning to real decisions (lottery, loans, investments)
Compounding moves money forward in time (PV → FV). Discounting moves it backward (FV → PV). They are exact inverses.
The discount rate r represents the return you could earn on money invested today — your opportunity cost.
Present Value of $10,000 Received in the Future (Discount Rate = 6%)
| Years Away | PV Today | "Lost" to Time |
|---|---|---|
| 1 | $9,434 | $566 |
| 5 | $7,473 | $2,527 |
| 10 | $5,584 | $4,416 |
| 20 | $3,118 | $6,882 |
You invest $6,000 today at 5% annually for 7 years. What will it be worth?
\[ FV = 6{,}000 \times (1.05)^7 = 6{,}000 \times 1.4071 = \mathbf{\$8{,}442.60} \]You need $20,000 in 8 years for a down payment. If you can earn 4.5% annually, how much must you invest today?
\[ PV = \frac{20{,}000}{(1.045)^8} = \frac{20{,}000}{1.4221} = \mathbf{\$14{,}063} \]Invest $14,063 today and it grows to exactly $20,000 in 8 years at 4.5%.
You win a lottery jackpot: $1,000,000 paid over 20 equal annual payments of $50,000, or $620,000 cash today. Assuming a 5% discount rate, what is the present value of the annuity payments?
\[ PV_{\text{annuity}} = 50{,}000 \times \frac{1 - (1.05)^{-20}}{0.05} = 50{,}000 \times 12.462 = \mathbf{\$623{,}111} \]The annuity's PV ($623,111) is slightly higher than the lump sum ($620,000) at 5%. At higher discount rates, the lump sum wins; at lower rates, the annuity wins. This is the exact math lottery winners use to decide.
- What is the PV of $15,000 received in 6 years at a 7% discount rate?
- You invest $4,000 at 8% for 10 years. What is the FV?
- If FV = $25,000 in 5 years and PV = $18,000 today, what annual rate is implied?
▶ Show Answers
- \(PV = 15{,}000 / 1.07^6 = 15{,}000 / 1.5007 =\) $9,996.
- \(FV = 4{,}000 \times 1.08^{10} = 4{,}000 \times 2.1589 =\) $8,636.
- \(r = (25{,}000/18{,}000)^{1/5} - 1 = 1.3889^{0.2} - 1 \approx 0.0678 =\) 6.78%.
- Confusing which direction you're going: FV = PV × (1+r)^t (forward). PV = FV / (1+r)^t (backward). Write the direction first.
- Using the wrong discount rate: The discount rate should reflect your actual opportunity cost. Using too low a rate overvalues future payments.
- Ignoring inflation: These formulas give nominal values. Real PV/FV should use the inflation-adjusted (real) rate.
- FV = PV × (1 + r)^t — compound forward; PV = FV / (1 + r)^t — discount back.
- Money received sooner is worth more because it can earn returns in the meantime.
- Discount rate = your opportunity cost — the return you give up by not investing today.
- Lottery, pension, and settlement decisions all reduce to comparing present values.
Every business investment decision uses present value. When a company evaluates whether to buy new equipment, open a new location, or acquire another business, analysts calculate the Net Present Value (NPV) of all projected future cash flows. If NPV > 0, the investment creates value; if NPV < 0, it destroys value. Investment bankers use discounted cash flow (DCF) models — built entirely on PV/FV math — to value companies during mergers, acquisitions, and IPOs.
Calculator Connection
The Compound Interest Calculator solves FV from PV and also shows the growth curve year by year. The Annuity Calculator calculates the present or future value of a series of equal periodic payments.
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Present Value and Future Value: The Time Value of Money: Quiz
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