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

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 and 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.

Annuities and perpetuities#

Compounding frequency#

effective annual rate = (1 + r / m)^m - 1
continuous: e^r - 1

Options and futures models usually use continuous compounding. See Black-Scholes Model.

Where traders use TVM#

Common mistakes#

• Mismatching periods and rates, such as using an annual rate with monthly periods.
• Forgetting inflation: real vs nominal rates. See 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.

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.

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.