The time value of money

A thousand euro today beats a thousand euro in a year. Not because you are impatient but because money in hand lets you do something. Save it, invest it, pay off a debt. Money that only arrives in a year does no work this year.

That simple idea is the foundation of all financial mathematics. Without it there is no way to compare an offer of "€100,000 now" with "€130,000 in five years".

Three reasons time changes value

Opportunity. Money you hold can earn. While you wait for it, you go without that return, and that return is the price of waiting.

Inflation. Prices rise over time, so the same sum buys less in a year. Money does not lose its number, it loses its purchasing power.

Risk. A promise of future payment may not be kept. The further away the payment, the greater the uncertainty, and the bigger the discount you demand for it today.

How it is calculated

Moving forward gives the future value:

$$FV = PV \cdot (1 + r)^n$$

Moving back to today is discounting, that is the present value:

$$PV = \frac{FV}{(1 + r)^n}$$

• $PV$: the value today
• $FV$: the value $n$ periods from now
• $r$: the rate of return for one period
• $n$: the number of periods

They are the same formula, only reversed. One moves money forward through time, the other back.

A worked example

Someone owes you €10,000 and offers two options: pay today, or €12,000 in four years.

Suppose you can earn 5% a year on your money. What is that €12,000 worth today?

$$PV = \frac{12{,}000}{(1.05)^4} = 9{,}872\ \euro$$

The answer is surprising: the future offer is actually worse today. The extra thousand sounds tempting, but over four years your own €10,000 would have grown to €12,155.

The break-even is the rate of return at which the two offers are equal: here about 4.7%. Earn more than that and you should take the money now.

Where you will use it

The time value of money sits behind every decision where payments fall at different times: pricing bonds, judging an investment project by its net present value, working out an annuity instalment, comparing a lease with a loan, and deciding whether to take severance as a lump sum or in instalments.

The procedure is always the same: bring every amount to one point in time, and only then compare. Adding up payments from different years is like adding euro to koruna.

Try the conversion in both directions in the discounting calculator; to judge several payments at once, use investment appraisal.