Time value of money

The time value of money is the principle that a dollar today is worth more than a dollar tomorrow. If you have $1,000 today you can invest it and have more than $1,000 a year from now; if instead someone promises you $1,000 a year from now, you’ve given up the chance to earn that return in the interim. Money has a time dimension, and any honest comparison of cash flows occurring at different times has to account for it.

The mechanism is interest — the rate at which money grows over time. Given an interest rate $i$ per period, the relationship between a present amount $P$ and the future amount $F$ after $N$ periods of compounding is:

$F=P(1+i{)}^N$

Equivalently, the present-day equivalent of a future amount is

$P=\frac{F}{(1+i{)}^N}.$

Discounting backward is the inverse of compounding forward. The ratio $(1+i{)}^N$ is what makes $1 today equivalent to $$(1+i{)}^N$ in $N$ years.

The big payoff: cash flows at different times can be compared once you’ve moved them all to a common point in time using these formulas. This is the basis of the Present worth method, Future worth method, Annual worth method, Internal rate of return, and every other engineering-economic comparison technique. You cannot add cash flows at different times directly — you have to convert first.

The underlying reasoning has three components:

1. Opportunity cost. Money you hold could earn a return elsewhere. Forgoing that return is a real cost (see Opportunity cost).
2. Inflation. Prices generally rise, so a future dollar buys less than today’s dollar even if no interest were involved (see Inflation).
3. Risk. A future cash flow might not materialise. Even if you trust the promise, you’d want compensation for the risk.

The “interest rate” used in time-value calculations bundles these together. For a low-risk corporate context, the rate is close to the cost of capital plus inflation; for a high-risk venture, the rate is much higher to compensate for the chance of loss. This is the conceptual content of the MARR used to discount project cash flows.

For the simplest realisation — single payment compounding — see Compound interest. For the broader family of conversion factors, see Compound interest factor.