Time Value of Money (TVM)

The time value of money (TVM) is the principle that a sum of money in hand today is worth more than the same nominal sum received in the future. Money available now can earn Interest, be spent, or be kept safe against the chance that a promised payment never arrives. TVM is the basic arithmetic behind savings accounts, loans, bond prices, pensions, insurance and almost every valuation method used in finance.

The idea links two quantities: present value, what a future payment is worth today, and future value, what a sum invested today grows into. Moving money forward in time is called compounding; moving it back is called discounting. The rate used for both is the discount rate, which reflects the return available elsewhere, expected inflation and the risk of the payment. TVM underpins Net Present Value (NPV), the Internal Rate of Return (IRR) and the Dividend Discount Model (DDM).

Imagine a friend offers you a choice: a gift of 100 today or the same 100 a year from now. Almost everyone takes it today. You could put it in a bank and have a little more by next year, you could use it now, and a year is long enough for plans, prices or your friend's mood to change.

Turn the choice around and the same logic tells you how much a future promise is really worth. If a bank pays a few per cent a year, then 100 promised next year is worth a bit less than 100 today, because a smaller sum put in the bank now would grow to 100 by then. Finance makes this instinct exact, so payments on different dates can be compared fairly.

With compound interest, a sum grows by the factor $(1 + r)$ each period. The future value of a single amount is:

Reversing it gives the present value of a payment due in the future:

• $PV$ is the present value, the amount today.
• $FV$ is the future value, the amount at the end of the period.
• $r$ is the interest or discount rate per period, written as a decimal.
• $n$ is the number of periods.

Worked example: 1,000 invested at 5% a year for 3 years grows to $1{,}000 \times 1.05^{3} = 1{,}157.63$. Going the other way, 1,000 promised in 3 years is worth $1{,}000 \div 1.05^{3} = 863.84$ today at a 5% rate.

A level stream of payments, called an annuity, has its own shortcut. The present value of a payment $C$ made at the end of each of $n$ periods is:

For example, 100 a year for 5 years at 5% is worth $100 \times (1 - 1.05^{-5}) \div 0.05 = 432.95$ today. The same formula sets the instalments on a mortgage. If the payments go on forever, the value of this perpetuity collapses to $C \div r$.

The discount rate is the heart of every TVM calculation, and it has several parts. Economists usually describe it as a risk-free rate, such as the yield on short-dated government debt such as Treasury bills, plus a risk premium for the chance that the payment will be late, smaller or missing. In nominal terms the rate also compensates for expected inflation; a calculation done in real terms strips inflation out of both the cash flows and the rate.

The rate also reflects opportunity cost: the best return given up by tying money into one use instead of another. That is why companies discount projects at their cost of capital, investors use their required return, and governments use a social discount rate. In September 2026 the UK Treasury accepted a review recommendation to cut the standard rate in its Green Book appraisal guidance from 3.5% to 3.0% for the first 30 years of an appraisal (confirmed in the Treasury's October 2026 response), which raises the present value of benefits far in the future.

• Lending and saving. Banks quote borrowing costs as an Annual Percentage Rate (APR) and savings returns as an Annual Percentage Yield (APY); both are TVM conversions between periodic and annual rates.
• Bond pricing. The price of a bond is the present value of its coupons and principal, and its yield to maturity is the single rate that makes that present value equal the market price.
• Company valuation. Discounted cash-flow models value a business as the present value of its future cash flows; the DDM does the same with dividends.
• Pensions and insurance. Pension funds and insurers discount promised payments decades ahead to decide how much money they must hold today.
• Public projects. Cost-benefit appraisals of roads, energy and climate policy discount future costs and benefits so they can be weighed against spending now.

TVM results are only as good as the inputs. Small changes in the rate move present values sharply over long horizons: 1,000 due in 50 years is worth about 228 today at 3% but only about 87 at 5%. That sensitivity makes the choice of rate contested in climate economics, where a low rate gives heavy weight to future generations and a higher one gives them much less.

Standard TVM also assumes a constant rate and steady, exponential discounting. Research in behavioral economics finds that people often discount the near future far more steeply than the distant future, a pattern known as present bias or hyperbolic discounting, which helps explain under-saving for retirement. Finally, future cash flows are themselves estimates; precise discounting of guessed numbers can give a false sense of accuracy.

• Investor.gov (US SEC): Compound Interest Calculator
• HM Treasury: The Green Book, appraisal and evaluation in central government