Present and Future Value Formula

The Formula

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

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} = \1628.89
- Present value of \1000 received in 10 years: PV = \frac{1000}{(1.05)^{10}} = \613.91

Notation

PV = present value, FV = future value, r = discount rate (or interest rate) per period, t = number of periods, NPV = net present value.

What This Formula Means

The concept that money has different values at different points in time. Future value (FV) calculates what a present amount will grow to; present value (PV) 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)^t; PV = \frac{FV}{(1+r)^t}; NPV = \sum_{t=0}^{n}\frac{C_t}{(1+r)^t} where C_t is cash flow at time t

Worked Examples

Example 1

easy
Find the future value of \5{,}000 invested at 4\% annual interest compounded annually for 6$ years.

Solution

  1. 1
    The future value formula for compound interest is FV = PV \cdot (1 + r)^n.
  2. 2
    Substitute: FV = 5000(1.04)^6.
  3. 3
    (1.04)^6 \approx 1.2653, so FV \approx 5000 \times 1.2653 = \6{,}326.60$.

Answer

\$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)^n assumes interest is reinvested (compounded). The interest earned is \1{,}326.60 — more than simple interest (\1{,}200) because of compounding.

Example 2

medium
How much should you invest today at 5\% annual interest compounded quarterly to have \20{,}000 in 8$ years?

Common Mistakes

  • Comparing dollar amounts from different time periods without discounting: \1000 in 20 years is NOT the same as \1000 today. Always convert to the same point in time.
  • Confusing the discount rate with the inflation rate: they're related but different. The discount rate reflects opportunity cost; the real discount rate adjusts for inflation.
  • Using NPV incorrectly: a positive NPV means the investment earns MORE than the discount rate, making it worthwhile. An NPV of zero means it earns exactly the discount rate—not that it's worthless.

Why This Formula Matters

Present and future value are the foundation of all financial decision-making: valuing bonds, comparing investment options, pricing loans, evaluating business projects (NPV analysis), and retirement planning. Any time you compare cash flows at different times, you need these concepts.

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 (FV) calculates what a present amount will grow to; present value (PV) 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?

PV = present value, FV = future value, r = discount rate (or interest rate) per period, t = number of periods, NPV = net present value.

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

Present and future value are the foundation of all financial decision-making: valuing bonds, comparing investment options, pricing loans, evaluating business projects (NPV analysis), and retirement planning. Any time you compare cash flows at different times, you need these concepts.

What do students get wrong about Present and Future Value?

The discount rate r is the opportunity cost—what you could earn elsewhere. A higher discount rate makes future money worth LESS today, because your alternative investment would grow faster.

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

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

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