A dollar today is not the same as a dollar next year
This topic builds directly on compound interest from the previous topic, and it introduces the single most important idea in all of finance: the time value of money. The idea is simple to state: a dollar in your hand right now is worth more than a dollar you are promised a year from now. Why? Because the dollar you have today can be invested immediately — put in a savings account, used to buy a property, lent out at interest — and start growing. The promised future dollar cannot start growing until you actually receive it. Since money has the ability to earn a return over time, its value depends on *when* you receive it, not just *how much* it is. Present value (PV) and future value (FV) are the two tools finance professionals use to put cash flows that happen at different times onto a level, comparable footing.
Future value: growing a known amount forward in time
Future value (FV) answers the question: 'If I have a certain amount of money today, and it earns a certain rate of return, how much will it be worth at some point in the future?' This is exactly the compound interest formula from the previous topic, just renamed with its finance-standard variable names.
Worked Example 1 — Future value. You deposit $20,000 today into an investment earning 7% per year, compounded annually, and you don't touch it for 5 years. FV = PV x (1 + r)^n = $20,000 x (1.07)^5. First calculate (1.07)^5 = 1.402551732. Then: $20,000 x 1.402551732 = $28,051.03. In 5 years, today's $20,000 will have grown into $28,051.03, assuming the 7% return holds every year.
Future Value of a Single Sum
FV = PV x (1 + r)^n
- FV
- — Future value — what the money will be worth later
- PV
- — Present value — the amount of money you have today
- r
- — Interest rate (rate of return) per period, as a decimal
- n
- — Number of periods between today and the future date
Take what you have today, and grow it forward period by period at the given rate. This is identical to the compound interest formula — future value IS compound interest, just applied to an investment question instead of a loan question.
Present value: the reverse — discounting a future amount back to today
Present value (PV) answers the opposite question: 'If I am promised a certain amount of money at some point in the future, what is that promise worth in today's dollars?' To find present value, you work the future value formula backward, using a discount rate (the same concept as an interest rate, but called a discount rate when you're moving money backward in time instead of forward). This process is called discounting.
Worked Example 2 — Present value. Suppose you know you will need $50,000 in 8 years to make a down payment on a property, and you can currently invest money at 6% per year, compounded annually. How much do you need to invest today? PV = FV / (1 + r)^n = $50,000 / (1.06)^8. First, (1.06)^8 = 1.593848. Then: $50,000 / 1.593848 = $31,370.62. If you invest $31,370.62 today at 6% compounded annually, it will grow into exactly $50,000 in 8 years. You can double-check this: $31,370.62 x (1.06)^8 = $50,000.00 (rounding aside) — present value and future value are simply two directions of the exact same relationship.
Present Value of a Single Sum
PV = FV / (1 + r)^n
- PV
- — Present value — what the future amount is worth today
- FV
- — Future value — the known amount to be received later
- r
- — Discount rate per period, as a decimal
- n
- — Number of periods between today and the future date
Take the future amount and shrink it back to today by dividing by (1 + r) once for every period between now and then. The higher the discount rate, or the further away the future payment is, the smaller its present value becomes.
Worked example: A commercial real estate investor expects to sell a property for $3,000,000 in exactly 10 years and uses a discount rate of 8% to reflect the risk and opportunity cost of waiting. PV = $3,000,000 / (1.08)^10 = $3,000,000 / 2.158925 = $1,389,580.46. In other words, the right to receive $3,000,000 in 10 years is worth about $1.39 million today at an 8% discount rate — this single calculation is the mathematical backbone of most real estate valuation and investment analysis.
Four things that always move present value and future value
- A higher interest/discount rate (r) increases future value, but decreases present value — because a higher rate means faster growth forward, and heavier discounting backward.
- More time (larger n) increases future value further and decreases present value further, since the effect of compounding (or discounting) has more periods to work.
- A larger starting present value produces a proportionally larger future value, and vice versa — the relationship is directly proportional (a doubled PV doubles the FV, all else equal).
- Present value and future value of the same cash flow, at the same rate and time, are always mathematically consistent: if you take a PV and grow it forward with the FV formula, you get back exactly the FV you started with — and vice versa.
A shortcut worth knowing: the Rule of 72
The 'Rule of 72' is a fast mental-math estimate for how many years it takes an investment to double at a given annual compound rate: divide 72 by the interest rate (as a whole number, not a decimal). At 8%, doubling takes about 72 / 8 = 9 years (the precise answer is about 9.01 years). At 6%, doubling takes about 72 / 6 = 12 years (the precise answer is about 11.90 years). It's an approximation, not an exact formula, but it's genuinely useful for quick sanity checks in your head before you reach for a calculator.
Module Check
A buyer expects to sell a small commercial property for $100,000 in exactly 6 years. Using a discount rate of 9% per year, what is that future sale price worth in today's dollars (present value)?