Present and Future Value Formula

Present and future value is the concept that money has different values at different points in time.

The Formula

FV=PV⋅(1+r)tFV = PV \cdot (1 + r)^t
PV=FV(1+r)tPV = \frac{FV}{(1 + r)^t}
Net Present Value: NPV=∑t=0nCt(1+r)tNPV = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^t} where CtC_t is the cash flow at time tt.

When to use: Would you rather have \$100 today or \$100 in five years? Today, obviously—because you could invest the \$100 and have MORE than \$100 in five years. Present value answers: 'How much would I need TODAY to have \$X in the future?' Future value answers: 'If I invest \$X today, what will it become?' Discounting is the reverse of compounding—it shrinks future money back to today's value.

Quick Example

At 5% annual interest:
- Future value of $1000 in 10 years: FV=1000(1.05)10=$1628.89FV = 1000(1.05)^{10} = \$1628.89
- Present value of $1000 received in 10 years: PV=1000(1.05)10=$613.91PV = \frac{1000}{(1.05)^{10}} = \$613.91

Notation

PVPV = present value, FVFV = future value, rr = discount rate (or interest rate) per period, tt = number of periods, NPVNPV = net present value.

What This Formula Means

The concept that money has different values at different points in time. Future value (FVFV) calculates what a present amount will grow to; present value (PVPV) calculates what a future amount is worth today, using discounting.

Would you rather have \$100 today or \$100 in five years? Today, obviously—because you could invest the \$100 and have MORE than \$100 in five years. Present value answers: 'How much would I need TODAY to have \$X in the future?' Future value answers: 'If I invest \$X today, what will it become?' Discounting is the reverse of compounding—it shrinks future money back to today's value.

Formal View

FV=PV(1+r)tFV = PV(1+r)^t; PV=FV(1+r)tPV = \frac{FV}{(1+r)^t}; NPV=∑t=0nCt(1+r)tNPV = \sum_{t=0}^{n}\frac{C_t}{(1+r)^t} where CtC_t is cash flow at time tt

Worked Examples

Example 1

easy
Find the future value of $5,000\$5{,}000 invested at 4%4\% annual interest compounded annually for 66 years.

Answer

$6,326.60\$6{,}326.60

First step

1
The future value formula for compound interest is FV=PV⋅(1+r)nFV = PV \cdot (1 + r)^n.

Full solution

  1. 2
    Substitute: FV=5000(1.04)6FV = 5000(1.04)^6.
  2. 3
    (1.04)6≈1.2653(1.04)^6 \approx 1.2653, so FV≈5000×1.2653=$6,326.60FV \approx 5000 \times 1.2653 = \$6{,}326.60.
Future value tells you how much a present sum will be worth after earning compound interest. The formula FV=PV(1+r)nFV = PV(1+r)^n assumes interest is reinvested (compounded). The interest earned is $1,326.60\$1{,}326.60 — more than simple interest ($1,200\$1{,}200) because of compounding.

Example 2

medium
How much should you invest today at 5%5\% annual interest compounded quarterly to have $20,000\$20{,}000 in 88 years?

Example 3

medium
How much should you invest today at 6% annual compounding to have $5000 in 4 years? Use 1.064=1.262476961.06^4 = 1.26247696.

Common Mistakes

  • Discounting when you meant to grow (or vice versa) - dividing by (1+r)t(1+r)^t moves money BACK to today, multiplying moves it forward.
  • Comparing future and present amounts directly without converting - bring both to the same point in time before judging which is larger.
  • Using the wrong sign or direction in NPV - the initial cost at t=0t=0 is divided by (1+r)0=1(1+r)^0=1 and is usually negative.

Why This Formula Matters

It is the foundation of every investment and financing decision — you cannot fairly compare \$100 today against \$120 in two years without putting both at the same point in time, and discounting (NPV) is how businesses evaluate whether a project is worth funding. Recognizing it by "Am I moving one sum of money forward (growing) or backward (discounting) along the timeline?" — rather than by familiar numbers — is what lets a student tell it apart from compound interest and annuities and net present value (npv) in a mixed problem set.

Frequently Asked Questions

What is the Present and Future Value formula?

The concept that money has different values at different points in time. Future value (FVFV) calculates what a present amount will grow to; present value (PVPV) calculates what a future amount is worth today, using discounting.

How do you use the Present and Future Value formula?

Would you rather have \$100 today or \$100 in five years? Today, obviously—because you could invest the \$100 and have MORE than \$100 in five years. Present value answers: 'How much would I need TODAY to have \$X in the future?' Future value answers: 'If I invest \$X today, what will it become?' Discounting is the reverse of compounding—it shrinks future money back to today's value.

What do the symbols mean in the Present and Future Value formula?

PVPV = present value, FVFV = future value, rr = discount rate (or interest rate) per period, tt = number of periods, NPVNPV = net present value.

Why is the Present and Future Value formula important in Math?

It is the foundation of every investment and financing decision — you cannot fairly compare \$100 today against \$120 in two years without putting both at the same point in time, and discounting (NPV) is how businesses evaluate whether a project is worth funding. Recognizing it by "Am I moving one sum of money forward (growing) or backward (discounting) along the timeline?" — rather than by familiar numbers — is what lets a student tell it apart from compound interest and annuities and net present value (npv) in a mixed problem set.

What do students get wrong about Present and Future Value?

The procedure for present and future value is the easy part; the trap is discounting when you meant to grow (or vice versa). Asking "Am I moving one sum of money forward (growing) or backward (discounting) along the timeline?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Present and Future Value formula?

Before studying the Present and Future Value formula, you should understand: compound interest.

← Back to Present and Future Value See All Examples Practice Problems