Time Value of Money Explained: PV, FV & Annuities
What Time Value of Money Means
Money available now is worth more than the same amount received later, because money in hand can be invested today and start earning a return immediately — money that hasn't arrived yet can't.
Present Value and Future Value, Plainly
Future value (FV)is what an amount grows into after earning a return over some period of time. If you know what you're starting with and how it will grow, FV tells you what you'll end up with.
Present value (PV)runs the same idea backward. If you know what you want to end up with at some future date, PV tells you what that goal is worth right now — or, put differently, how much you'd need today to reach it.
Neither number is more "correct" than the other — they're two views of the same relationship, and which one you solve for just depends on which side of the timeline you already know.
This shows up constantly outside a finance textbook, too. A loan payment schedule is a present-value problem in disguise (a lender wants to know what a stream of future payments is worth today). A retirement savings target is a future-value problem (what will a recurring contribution grow into by a future date). Once you can spot which direction a question is really asking, choosing the right formula becomes far less confusing than it first looks.
The Core Formula
FV = PV × (1 + r)^n
PV is the amount you're starting with. r is the interest rate per period, written as a decimal (5% becomes 0.05). n is the number of periods the money grows for. Multiply those together in this shape, and you get what the starting amount becomes after all that growth compounds on top of itself.
Worked Example: Growing a Starting Amount
Suppose you set aside $600 today in an account earning 4% annually, and leave it untouched for 3 years. What does it grow to?
Notice the growth isn't just $600 × 4% × 3 years — that would only be $672 under simple, non-compounding interest. The extra 92 cents comes from compounding: each year's interest earns its own interest the following year, not just the original $600.
Reversing It: Solving for Present Value
Now flip the question around. Say you want to have $2,400 saved in 3 years for a study-abroad trip, and you expect a 5% annual return. How much do you need to set aside today?
Depositing $2,073.21 today, at 5% for 3 years, grows to exactly $2,400 — the same relationship as the first example, just solved from the opposite direction.
Why Compounding Frequency Matters
The formula above assumes interest compounds once per period. In practice, many accounts compound more often — monthly or daily rather than annually — and more frequent compounding produces a slightly higher future value for the same stated annual rate, since interest starts earning its own interest sooner.
Take $800 at a 6% annual rate over 2 years. Compounded once a year, it grows to $898.88. Compounded monthly instead (splitting the 6% into a monthly rate and running 24 compounding periods), it grows to $901.73 — a modest but real difference from compounding frequency alone, with nothing else about the deal changed.
The gap between the two grows larger as the rate rises, the timeline lengthens, or the compounding periods get more frequent still (daily rather than monthly, for instance). Over a short 2-year stretch at a moderate rate, the difference is small enough to round away in casual conversation. Over a multi-decade investment horizon, the same effect compounds into a genuinely material gap — which is exactly why loan and investment disclosures are required to state a compounding basis, not just a headline annual rate.
What Is an Annuity? Ordinary vs. Due
An annuity is simply a series of equal payments made at regular intervals — a car payment, a recurring deposit, a subscription fee. The timing of each payment within its period matters for the math, and finance separates that timing into two types:
| Type | Payment Timing | Typical Example |
|---|---|---|
| Ordinary Annuity | End of each period | A monthly loan payment |
| Annuity Due | Beginning of each period | Rent paid at the start of the month |
Because an annuity-due payment sits invested for one extra period compared to the same payment under an ordinary annuity, its future value comes out slightly higher for otherwise identical numbers.
Worked Example: A Simple Annuity
Say you save $75 at the end of each month in an account earning 4% annually (compounded monthly), for 24 months, starting from zero.
You contributed $1,800 total over the 24 months (75 × 24). The remaining $70.72 is compounding growth on the earlier deposits — proportionally small over just 2 years, but the same mechanism that becomes far more significant over longer horizons.
Financial Calculator Quick Reference
Most dedicated financial calculators (and TVM worksheets in spreadsheet software) organize every problem around the same five variables. Enter any four, solve for the fifth:
| Key | What It Represents |
|---|---|
| N | Number of periods |
| I/Y | Interest rate per period (as a percentage) |
| PV | Present value (starting or current amount) |
| PMT | Recurring payment per period (0 if none) |
| FV | Future value (ending amount) |
One habit worth building early: most calculators expect cash you pay out to be entered as a negative number and cash you receive as positive. Mixing up the signs is the single most common reason a TVM answer comes back looking wildly wrong.
Run Your Own Numbers Instantly
Skip the manual algebra and solve for PV, FV, or an annuity directly.
Frequently Asked Questions
What's the difference between present value and future value?
Present value is what a future amount of money is worth today, discounted back by an interest rate. Future value is the reverse — what an amount available today will grow to by some point in the future, given a rate of return. They describe the same relationship from opposite ends.
Why is a dollar today worth more than a dollar in the future?
Because a dollar in hand today can be invested immediately and start earning a return, while a dollar promised for later can't do anything until it arrives. That earning potential is exactly what an interest rate measures, and it's the entire reason present and future amounts aren't directly comparable without adjusting for time first.
What's the difference between an ordinary annuity and an annuity due?
In an ordinary annuity, each payment lands at the end of its period. In an annuity due, each payment lands at the beginning. Because an annuity-due payment sits in the account for one extra period, it accumulates slightly more growth than the same payment schedule structured as an ordinary annuity.
Does the time value of money account for inflation?
Not directly — the standard PV/FV formulas use whatever interest rate you plug in, and that rate can represent an investment return, a discount rate, or something else entirely depending on the problem. If you want a result that reflects purchasing power rather than raw dollar growth, you'd use a rate adjusted for expected inflation, but that's a separate input choice, not something built into the formula itself.
How do I calculate TVM on a financial calculator?
Enter any four of the five standard TVM variables (N, I/Y, PV, PMT, FV) and solve for the fifth. Most calculators require you to enter cash outflows as negative numbers and inflows as positive — an easy step to forget, and usually the reason a calculated answer comes out with an unexpected sign.
What's the Rule of 72 and how does it relate to TVM?
The Rule of 72 is a quick mental shortcut for estimating how long it takes an amount to double at a given annual rate — divide 72 by the interest rate (as a whole number) to get the approximate number of years. It's a fast approximation built on the same compounding idea behind the full FV formula, useful for a ballpark answer when you don't need exact precision.
Why do textbooks sometimes call the interest rate a 'discount rate' instead?
Same number, different direction of use. When you're moving a value forward in time (PV to FV), it's usually called an interest or growth rate. When you're moving a value backward (FV to PV), the identical rate is often called a discount rate, since it's being used to discount a future amount down to its present worth.
Can PV and FV formulas handle negative growth?
Yes, mathematically — a negative rate simply shrinks a value over time instead of growing it, which can model a depreciating asset or a declining cash flow. The formula structure doesn't change; only the sign and size of the rate does.