The time value of money · RiskAtoms Academy

Know these first

L1.1.02 Money, credit and interest

Math Atoms

∑ Percentages

By the end you will be able to

1Explain why money today is worth more than the same amount of money tomorrow.
2Name the three main reasons: opportunity cost, inflation and risk.
3Move an amount one year forward or back at an interest rate, so you can compare payments made on different dates.
4Find the interest rate at which an amount today and an amount in a year are worth the same.

Why it matters

Would you rather have £100 today or £100 in a year? Almost everyone says today. Now make it £100 today or £105 in a year. Suddenly it is a real choice, and finance gives you a way to make it.

Core explanation

Money has a date on it

A pound, a dollar or a yen is not just an amount. It is an amount on a date. £100 today and £100 next year are different things, in the same way that a train ticket for today and one for next month are different tickets. This idea is called the time value of money, and almost everything else in finance is built on it.

There are three main reasons why money sooner beats money later.

Opportunity cost

Money you have today can be put to work. In a savings account at 4%, £100 becomes £104 in a year. Waiting means giving that up.

Inflation

Prices tend to rise. If they rise 3% this year, £100 next year buys what about £97 buys today.

Risk

A promise to pay later might not be kept. The payer could go bust, change their mind or simply be late.

The first reason, the interest you give up, is called the opportunity cost. It is the one we can put a number on most easily, so it drives the maths. Inflation gets its own lesson (L1.3.07), and risk runs through the whole course. In practice, all three are folded into a single number: the interest rate you demand for waiting.

Comparing amounts on different dates. You cannot compare £100 today with £105 next year directly, any more than you can compare 100 euros with 105 dollars without an exchange rate. The interest rate is the exchange rate between today and next year. It lets you move an amount forward to a later date, or back to today, so both sit on the same date. Then you can compare them.

A cash flow timeline

Year 0 · today

× (1 + r) →

← ÷ (1 + r)

Year 1

Year 0 is today. Each payment sits at the date it is paid. To compare two payments, move one of them along the line until both sit at the same date.

Moving money forward gives its future value (FV). Moving it back gives its present value (PV). Drawing payments on a cash flow timeline like the one above is the first step in nearly every valuation you will ever do.

Today or in a year?

Decide, then see why

You can earn 4% a year. Take £1,000 today, or £1,000 in one year?
You can earn 5% a year. Take €500 today, or €530 in one year?
You can earn 3% a year. Take $200 today, or $205 in one year?

The formal version

Now let's write the one-year move as a formula. This notation, PV, FV and r, will follow you through the rest of the course, so it is worth building slowly.

The two moves, for reference

\mathrm{FV} = \mathrm{PV} \times (1 + r) \qquad\qquad \mathrm{PV} = \frac{\mathrm{FV}}{1 + r}

$r$ is the interest rate for one year, written as a decimal: 4% is 0.04. Lesson L1.3.02 extends this to many years, where interest starts to earn interest of its own.

The break-even rate. Two offers, an amount today and an amount in a year, are worth the same when the rate turns one exactly into the other. Rearranging $\mathrm{FV} = \mathrm{PV}(1+r)$ gives that rate:

r_{\text{break-even}} = \frac{\mathrm{FV}}{\mathrm{PV}} - 1

If you can earn more than this rate, take the money today. If you can earn less, wait.

Try it

Start with £100 today against £105 in a year, at 4%. Then slide the interest rate up past 5% and watch the verdict flip.

Try itToday or in a year?

Open the compound interest calculator →

Offer A: paid today£100

Offer B: paid in one year£105

Interest rate you could earn4.00%

Compare in one year

A, grown for a year£104.00

£100 × (1 + 0.0400)

B, as paid£105.00

Already in one year

Compare today

A, as paid£100.00

Already today

B, brought back to today£100.96

£105 ÷ (1 + 0.0400)

Wait: the later offer is worth £0.96 more today.

Break-even rate: 5.00%

Worked example

Paid now or paid later?

A small supplier in Singapore is owed money by a customer. The customer offers two options: S$10,000 now, or S$10,450 in one year. The supplier's bank pays 4% a year on deposits. Rates are annual and in Singapore dollars. Try each step before revealing it.

Which offer should the supplier take?

Step 1 of 5

1
Put both offers on a timeline
S$10,000 sits at year 0. S$10,450 sits at year 1. They are on different dates, so we move one of them.

The answer depends on the rate, and the right rate depends on the risk. That is why so much of risk management comes down to choosing the rate at which to discount.

Regulatory lens · looking ahead

No rules apply yet. Accounting and prudential rules ask firms to put a value today on amounts due later, for example loan loss allowances under IFRS 9 and insurance liabilities under IFRS 17 and Solvency II. They return in Level 4 (L4.6 and L4.7).

Sparks

SparkWhen money tomorrow was worth more

For years, some central banks set interest rates below zero. The European Central Bank's deposit rate was negative from June 2014 to July 2022, and the Bank of Japan kept a negative policy rate from 2016 to March 2024. Banks were in effect paying to keep money at the central bank. Holding large amounts of cash is costly and unsafe, so a small negative rate can persist for a while. The time value of money still works; the rate in the formula simply has a minus sign.

SparkThe marshmallow test

In experiments at Stanford from the late 1960s, psychologist Walter Mischel offered young children one treat now or two if they could wait a few minutes alone with the first. Some waited; many did not. Later studies found the link between waiting and success later in life weaker than first reported, but the test is a neat picture of what an interest rate measures: the price of patience.

Key takeaways

Money has a date on it: the same amount is worth more the sooner it arrives.
Three reasons: the interest you give up (opportunity cost), inflation and risk.
The interest rate is the exchange rate between dates. Forward one year: multiply by (1 + r). Back to today: divide by (1 + r).
Only compare amounts that sit on the same date. A timeline makes this easy to see.
The break-even rate FV ÷ PV − 1 tells you when waiting is worth it.

Reaction

Check what stuck

Question 1 · objective 2

Which of these is NOT a reason that money today is worth more than the same amount in a year?

AYou could invest it and earn interest in the meantimeBPrices may rise, so the money buys less laterCA promised future payment might not arriveDBanknotes wear out over time

Question 2 · objective 3

You deposit £250 for one year at 6% interest. How much do you have at the end of the year, in £?

Question 3 · objective 3

At 8% a year, what is $1,080 paid in one year worth today, in $?

Question 4 · objective 1

You can earn 5% a year. Which is worth more: €2,000 today or €2,090 in one year?

A€2,000 todayB€2,090 in one yearCThey are worth the same

Question 5 · objective 4

You are offered £400 today or £422 in one year. At what interest rate, in %, would the two offers be worth the same?

Glossary

Time value of money: The idea that an amount of money is worth more the sooner you receive it.
Present value (PV): What an amount of money is worth today.
Future value (FV): What an amount of money is worth at a later date, after interest.
Interest rate (r): The yearly return on money, as a share of the amount; 4% is r = 0.04.
Opportunity cost: What you give up by choosing one option: here, the interest you could have earned in the meantime.
Inflation: A general rise in prices, so that the same amount of money buys less over time.
Cash flow timeline: A line marked in periods (year 0 is today) with each payment placed at the date it is paid.

Deep Dives & Origins

Deep Dives

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill), chapter 2, “How to calculate present values”. Beginner.
Irving Fisher, The Theory of Interest (Macmillan, 1930), part I. Intermediate, historical.

Origins

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill).
Fisher, The Theory of Interest (Macmillan, 1930).
European Central Bank, key ECB interest rates (historical deposit facility rates).
Bank of Japan, statement on monetary policy, 19 March 2024.
Mischel, Shoda and Rodriguez, “Delay of gratification in children”, Science 244 (1989).
Watts, Duncan and Quan, “Revisiting the marshmallow test”, Psychological Science 29 (2018).

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Regulators and supervisors

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Simple and compound interest