Time Value of Money

Definition

The time value of money is the principle that a given amount of money is worth more today than the same amount in the future. This is not merely an assertion about inflation — it holds even in a world without price changes. A dollar today can be invested to earn a return, so receiving it now rather than later means forgoing that earning potential. The interest rate is the price that reconciles this trade-off: it tells you how much extra you must receive in the future to be willing to give up a dollar today.

This single principle underpins virtually every financial calculation: the present value of a bond’s future coupons, the cost-benefit analysis of a capital project, the price of a stock, the actuarial math behind pension liabilities, and the logic of compound interest on a savings account or a credit card debt.

Why it matters

Key takeaways

Future value (FV) = Present value × (1 + r)^n — a dollar invested at rate r for n periods grows by compounding.
Present value (PV) = Future value ÷ (1 + r)^n — discounting converts a future cash flow into its equivalent today. The higher the rate or the longer the wait, the less a future dollar is worth now.
The discount rate reflects the opportunity cost of capital: what you could earn by investing elsewhere at comparable risk. A 'risk-free' rate (e.g., Treasury bond yield) is the floor; riskier projects use higher rates.
Net present value (NPV): the sum of discounted future cash flows minus the initial investment. A positive NPV means the project creates value; negative NPV means it destroys value.
Perpetuity shortcut: PV of a constant cash flow C paid forever = C ÷ r. Useful for valuing steady-income assets.
Compounding frequency matters: $1,000 at 6% annual compounding grows differently than 6% compounded monthly. The effective annual rate (EAR) standardizes the comparison.

Present value and discounting

Read it as: $747 today and $1,000 in 5 years are economically equivalent at a 6% discount rate — they sit on the same indifference curve. Discounting converts the future amount to today’s value; compounding converts the present amount to a future value. The discount rate does the translating — raise it and the present value falls; lower it and future cash flows become nearly as valuable as present ones.

Compounding: the most powerful force in finance

How compounding works

Compounding means earning returns on your returns. At 10% annually, $1,000 grows to $1,100 in year 1, then to $1,210 in year 2 (10% on $1,100), not $1,200 — the $10 of extra gain is “interest on interest.” Over long periods, this exponential growth dwarfs simple-interest accumulation. $1,000 at 10% for 30 years compounded annually reaches $17,449; at simple interest it would reach only $4,000.

The Rule of 72

A useful approximation: divide 72 by the annual interest rate to find how many years it takes for an investment to double. At 6%, money doubles in about 12 years (72 ÷ 6). At 3%, it takes about 24 years. At 9%, about 8 years. This shortcut highlights how sensitive long-run wealth outcomes are to seemingly small differences in return.

Net present value and investment decisions

NPV is the central tool for capital allocation. A firm evaluating a new factory sums the present values of all expected future cash flows the factory will generate, then subtracts the upfront cost. If NPV > 0, build the factory — it generates more value than the capital cost. If NPV < 0, the capital is better deployed elsewhere. The discount rate used is the firm’s weighted average cost of capital (WACC) — the blended return required by its debt and equity holders.