Present and Future Value
Also known as: time value of money, PV, FV, discounting
Grade 9-12
View on concept mapThe concept that money has different values at different points in time. 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.
Definition
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.
💡 Intuition
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.
🎯 Core Idea
Money today is worth more than the same amount in the future because of its earning potential. Discounting converts future cash flows to present-day equivalents, enabling fair comparison of money received at different times.
Example
- 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
Formula
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.
Notation
PV = present value, FV = future value, r = discount rate (or interest rate) per period, t = number of periods, NPV = net present value.
🌟 Why It 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.
💭 Hint When Stuck
Ask: am I moving money forward in time (use FV = PV*(1+r)^t) or backward (use PV = FV/(1+r)^t)? Draw a timeline to clarify.
Formal View
Related Concepts
🚧 Common Stuck Point
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.
⚠️ 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.
Go Deeper
Frequently Asked Questions
What is Present and Future Value in Math?
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.
Why is Present and Future Value important?
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 usually 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 Present and Future Value?
Before studying Present and Future Value, you should understand: compound interest.
Prerequisites
Next Steps
Cross-Subject Connections
How Present and Future Value Connects to Other Ideas
To understand present and future value, you should first be comfortable with compound interest. Once you have a solid grasp of present and future value, you can move on to annuities.