The time value of money

Definition

Receiving 1 000 € today or 1 000 € in a year's time: the two amounts are written the same way, and yet they are not the same thing. The one available today can be invested, and will be worth more in a year as soon as the investment earns something. The reverse operation is possible — obtaining today a sum expected later is borrowing — and its price has the same origin: the return the lender gives up, below which the rate demanded will not go.

The time value of money is the principle that an amount has economic meaning only when associated with a date, and that two amounts bearing different dates can be compared only once brought back to a common date. As long as the alternative use earns a positive return, a sum available earlier is worth more than a sum available later.

Adding or subtracting two amounts of different dates directly is therefore an error of the same kind as adding lengths expressed in different units.

The origin of this gap does not lie in time itself: it lies in the existence of an alternative use. If no investment earned anything at all, the two sums would be worth the same. It is because available capital can be put to use — and because that use earns something — that waiting has a cost. The time value of money is thus opportunity cost applied to the calendar: what waiting costs is the return on the best use it makes you give up.

Discounting, the inverse operation of compounding

Compounding carries a sum into the future by multiplying it by $(1+i)^n$(1+i)n. The question asked here is the symmetric one: what sum today is equivalent to an amount $V_n$Vn​ available in $n$n periods? The one that, compounded, would give back exactly $V_n$Vn​. It is enough, then, to divide instead of multiplying.

$V_0 = \frac{V_n}{(1+i)^n}$V0​=(1+i)nVn​​

This operation is called discounting, $V_0$V0​ the present value of the flow, and the number $1/(1+i)^n$1/(1+i)n the discount factor — the one by which the future flow is multiplied. As long as $i$i is positive, this factor is less than 1 and decreases geometrically with maturity.

Compounding and discounting are therefore one and the same relation, read in two directions. Every property of the one has its counterpart in the other: as long as the rate stays positive, the discount factor falls all the faster the higher the rate, and a very distant flow has an arbitrarily small present value.

The discount rate is an opportunity cost

The rate $i$i used for discounting is not a property of the sum being discounted. It is the return on the best comparable alternative use to which the capital would be assigned were it not tied up in the operation under examination.

Three consequences follow, often poorly understood.

• The rate depends on who is doing the calculation. Two people with different options legitimately discount the same flow at different rates, and arrive at different present values. This is not an inconsistency: they are not giving up the same thing.
• There is no "true" rate to be read off the flow. The rate is a reasoned choice, justified by the alternative adopted as the reference. Change the alternative and the result changes.
• The present value is highly sensitive to that choice, all the more so the longer the horizon, since the factor $1+i$1+i is there raised to the power $n$n. An honest comparison therefore examines how the conclusion shifts when the rate varies.

Comparing a series of flows: net present value

Most decisions bear not on a single amount but on a series of staggered sums: an initial outflow, then inflows. Each is brought back to the reference date, and the present values — homogeneous from then on — add up. Counting outflows negatively, the sum of these present values is the net present value (NPV).

$\mathrm{NPV} = \sum_{t=0}^{n} \frac{F_t}{(1+i)^t}$NPV=t=0∑n​(1+i)tFt​​

where $F_t$Ft​ is the net flow of period $t$t, counted negatively if it is an outflow. The flow at date 0 is not discounted, its factor being 1.

Take an outflow of 1 000 € today against three inflows of 400 € at periods 1, 2 and 3. At a rate of 5 %:

$\mathrm{NPV} = -1000 + \frac{400}{1{,}05} + \frac{400}{1{,}05^2} + \frac{400}{1{,}05^3} = -1000 + 380{,}95 + 362{,}81 + 345{,}54 = 89{,}30$NPV=−1000+1,05400​+1,052400​+1,053400​=−1000+380,95+362,81+345,54=89,30

The NPV is positive: the operation earns 89,30 € more, in today's euros, than the reference investment at 5 %.

The same calculation at 15 % gives $-1000 + 347{,}826 + 302{,}457 + 263{,}006 \approx -86{,}71$−1000+347,826+302,457+263,006≈−86,71. The NPV is negative, and the same flows become disadvantageous. Nothing has changed except the alternative they are being compared with: NPV does not measure an intrinsic quality, but a gap relative to a reference.

It is indeed this gap, and not a gross balance, that decides. Note that the undiscounted total of the inflows, 1 200 €, exceeds the outflow of 1 000 € in both cases: that comparison settles nothing, since it adds up amounts of different dates.

The decision rule follows from reading the NPV as a gap: $\mathrm{NPV} > 0$NPV>0, the operation is preferable to the reference alternative; $\mathrm{NPV} = 0$NPV=0, the two are worth the same; $\mathrm{NPV} < 0$NPV<0, the alternative is preferable.

The internal rate of return

Since the NPV depends on the rate, the question can be reversed and one can look for the rate that brings it to zero. That rate is the internal rate of return (IRR): the value of $i$i for which the operation and its reference alternative are exactly equivalent.

In the previous example the IRR is about 9,70 %, a value obtained by numerical solution: the equation is polynomial of degree $n$n, with no general formula by radicals beyond degree 4, and unworkable by hand well before that. When a single future flow faces an initial amount, the IRR can instead be computed directly: $i = (V_n/V_0)^{1/n} - 1$i=(Vn​/V0​)1/n−1. It reads off directly: the operation is preferable to any alternative earning less than 9,70 %, and disadvantageous against any alternative earning more. This is the tipping point observed between the two calculations at 5 % and at 15 %.

Two limits are worth knowing. Bringing the NPV to zero amounts to solving $\sum_t F_t\,x^t = 0$∑t​Ft​xt=0, where $x = 1/(1+i)$x=1/(1+i); only the roots $x > 0$x>0 — that is, $i > -1$i>−1 — have a meaning, and the IRR follows from them by $i = 1/x - 1$i=1/x−1. As long as the flows change sign only once — an outflow, then inflows — such a root exists and is unique, the IRR being, moreover, able to be negative if the inflows total less than the outflow. When they change sign several times, the equation may admit several admissible roots, or none, and the IRR then ceases to be unambiguously defined. Moreover, being a rate, it says nothing about the size of the operation: a high IRR on a small sum may create less value than a modest IRR on a large one. To rank competing operations, the NPV, which is expressed as an amount, remains the relevant quantity.

The case of uncertain flows

The preceding formulas assume flows known in amount and in date. As soon as the future amount is uncertain, two distinct adjustments are possible: weighting the flows by their probability, and raising the discount rate by a risk premium above the return on a use deemed safe. Both adjustments are commonly practised — it is their justification, not their existence, that remains debated.

They are not of the same nature. The first is merely the computation of an expectation: it is well defined as soon as the probabilities are. But deciding on the expectation alone already presupposes indifference to risk — an assumption that the second adjustment precisely rejects, without for all that following from the principle: the principle says that flows must be brought back to a common date, it does not say by how much to raise the rate. The size of the appropriate premium, and even the legitimacy of making a single rate carry two things as different as the passage of time and exposure to risk, remain debated. The discounting of an uncertain flow is therefore to be treated as a matter of convention, whose result does not have the same status as a calculation on certain flows.

What the principle assumes

• A single unit. The flows must be expressed in the same currency and treated homogeneously with regard to inflation: flows in current units are discounted at a nominal rate, the one not corrected for inflation, and flows expressed in constant purchasing power at a real rate, the one obtained by $(1+i)/(1+\pi) - 1$(1+i)/(1+π)−1 — where $i$i here denotes the nominal rate and $\pi$π the inflation rate over the period — and not by simple subtraction. Mixing the two is the most frequent error in discounting calculations. (The word nominal here stands opposed to real; as noted in the previous principle, elsewhere it stands opposed to effective and then presupposes a given compounding frequency.)
• A rate suited to each maturity. Nothing requires the rate to be unique over the whole horizon. If it differs from one period to another, the factor becomes a product of successive factors, and writing it as a power is only the special case of a constant rate.
• Explicit dates. Discounting requires knowing the date of each flow, not just its amount: $n$n is as much a parameter of the calculation as $i$i.
• A genuinely available alternative. The NPV compares against a use assumed to be accessible for the same amounts and at the same dates. If the alternative is capped or unavailable, the rate adopted no longer measures what is actually given up.

One remark, finally. If $i$i lies between $-1$−1 and 0, the discount factor exceeds 1, and a future sum is worth more than the same sum today. The principle therefore does not assert that money loses value over time — it asserts that an amount without a date is incomplete, and that the direction of the gap is fixed by the sign of the rate.

Summary

An amount has meaning only when accompanied by its date: to compare sums staggered over time, they must be brought back to a common date by dividing them by $(1+i)^n$(1+i)n, at the rate of the best alternative use.