# Time Value of Money > A dollar today is worth more than a dollar tomorrow. Learn present and future value, discounting, annuities and NPV, the maths behind bonds, valuations and options. Source: https://learn.tradelabsai.com/math/time-value-of-money/ Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03 Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice. Cite as: TradeLabs Learn, "Time Value of Money", https://learn.tradelabsai.com/math/time-value-of-money/ The time value of money is the idea that money available now is worth more than the same amount in the future, because money today can be invested to earn a return and because the future is uncertain. This simple principle underlies bond pricing, stock valuation, option pricing, loan payments and the cost of carry in futures. Learning to move cash flows between the present and the future with discounting is one of the most useful skills in finance. ## Future value and present value ``` future value FV = PV × (1 + r)^n present value PV = FV / (1 + r)^n ``` - **r:** interest or discount rate per period - **n:** number of periods **Example: Discounting a future payment** You will receive $10,000 in 5 years. With a discount rate of 6%, its present value is 10,000 / 1.06^5 ≈ $7,473. At 10%, it is about $6,209. The higher the rate, the less a future payment is worth today. See [Interest Rates](https://learn.tradelabsai.com/macro/interest-rates/). ## Net present value (NPV) NPV adds up the present values of all cash flows, including the initial cost: ``` NPV = Σ CF_t / (1 + r)^t - initial investment ``` A positive NPV means an investment earns more than the discount rate. Companies use NPV to choose projects; investors use the same logic to value businesses. See [Capital Allocation and Management](https://learn.tradelabsai.com/fundamentals/capital-allocation/) and [DCF Valuation](https://learn.tradelabsai.com/fundamentals/dcf-valuation/). ## Internal rate of return (IRR) The IRR is the discount rate that makes NPV equal to zero. For bonds, the IRR is the yield to maturity. See [Yield to Maturity](https://learn.tradelabsai.com/bonds-credit/yield-to-maturity/). ## Annuities and perpetuities | Cash flow pattern | Present value formula | Example | |---|---|---| | Annuity (fixed payment for n periods) | PMT × [1 minus (1 + r)^(minus n)] / r | Loan payments, bond coupons | | Perpetuity (fixed payment forever) | PMT / r | Some preferred shares | | Growing perpetuity | PMT_1 / (r minus g) | Terminal value in DCF. See [Terminal Value](https://learn.tradelabsai.com/fundamentals/terminal-value/) | **Example: Valuing a bond with TVM** A 3 year bond pays a $50 coupon each year and $1,000 at maturity. With a 6% yield: PV = 50 / 1.06 + 50 / 1.06² + 1,050 / 1.06³ ≈ 47.17 + 44.50 + 881.60 ≈ $973.27. The bond trades below par because its coupon (5%) is below the market yield (6%). See [How Bonds Work](https://learn.tradelabsai.com/bonds-credit/how-bonds-work/). ## Compounding frequency | Frequency | Effective annual rate for a 6% nominal rate | |---|---| | Annual | 6.00% | | Semiannual | 6.09% | | Monthly | 6.17% | | Continuous | 6.18% | ``` effective annual rate = (1 + r / m)^m - 1 continuous: e^r - 1 ``` Options and futures models usually use continuous compounding. See [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/). ## Where traders use TVM | Application | Lesson | |---|---| | Bond prices and yields | [How Bonds Work](https://learn.tradelabsai.com/bonds-credit/how-bonds-work/) | | Stock valuation (DCF) | [DCF Valuation](https://learn.tradelabsai.com/fundamentals/dcf-valuation/) | | Futures fair value and cost of carry | [Spot vs Futures](https://learn.tradelabsai.com/futures/spot-vs-futures/) | | Option pricing (discounting the strike) | [Put-Call Parity](https://learn.tradelabsai.com/options/put-call-parity/) | | Forward exchange rates | [FX Forwards and Forward Points](https://learn.tradelabsai.com/forex/fx-forwards-and-forward-points/) | | Discount rates and interest rate sensitivity | [WACC and Cost of Equity](https://learn.tradelabsai.com/fundamentals/wacc-and-cost-of-equity/) | ## Common mistakes - **Mismatching periods and rates,** such as using an annual rate with monthly periods. - **Forgetting inflation:** real vs nominal rates. See [Inflation](https://learn.tradelabsai.com/macro/inflation/). - **Ignoring risk:** riskier cash flows deserve higher discount rates. ## Discount rates and risk The discount rate should reflect both the time value of money and the risk of the cash flows. Government bond yields are used for nearly certain cash flows. Riskier cash flows, such as a young company's future profits, are discounted at higher rates that include a risk premium. Using the wrong rate is one of the most common sources of valuation error. See [WACC and Cost of Equity](https://learn.tradelabsai.com/fundamentals/wacc-and-cost-of-equity/). ## Real vs nominal Discount nominal cash flows with nominal rates and inflation adjusted cash flows with real rates. Mixing them, such as discounting nominal cash flows with a real rate, overstates value. See [Inflation](https://learn.tradelabsai.com/macro/inflation/). ## Frequently asked questions ### What is the time value of money? The principle that money today is worth more than the same amount in the future because it can earn a return and because the future is uncertain. ### What is present value? The current worth of a future cash flow, found by discounting it at an appropriate rate. ### What is NPV? Net present value: the sum of the present values of all future cash flows minus the initial investment. Next, learn the matrix maths behind portfolios in [Linear Algebra for Traders](https://learn.tradelabsai.com/math/linear-algebra-for-traders/). ## Continue learning - Next lesson: [Linear Algebra for Traders](https://learn.tradelabsai.com/math/linear-algebra-for-traders/) - Previous lesson: [Compounding and Geometric vs Arithmetic Returns](https://learn.tradelabsai.com/math/compounding/) - Related: [Compounding and Geometric vs Arithmetic Returns](https://learn.tradelabsai.com/math/compounding/): Compounding means returns earn returns over time. Learn the formulas, why losses hurt more than gains help, volatility drag and how it shapes position sizing. - Related: [DCF Valuation](https://learn.tradelabsai.com/fundamentals/dcf-valuation/): A DCF values a company by forecasting free cash flows and discounting them to today. Learn the steps, a worked example, sensitivity analysis and common mistakes. - Related: [How Bonds Work](https://learn.tradelabsai.com/bonds-credit/how-bonds-work/): A bond is a loan that pays interest and returns principal at maturity. Learn coupons, price and yield, why prices fall when rates rise and the main bond risks. - Related: [WACC and Cost of Equity](https://learn.tradelabsai.com/fundamentals/wacc-and-cost-of-equity/): WACC blends the cost of equity and the after tax cost of debt into a discount rate. Learn CAPM, beta, the equity risk premium, a worked example and common pitfalls. - Related: [Interest Rates](https://learn.tradelabsai.com/macro/interest-rates/): Interest rates are the price of money and a key driver of asset prices. Learn policy vs market rates, real rates and how rates move stocks, bonds and currencies.