1. Money has a time stamp

Suppose someone offers you a choice: receive €100 today, or receive €100 in exactly one year. Most people choose €100 today. Not because they distrust the person making the offer, and not because they need the money urgently. They choose it because €100 today can be put to work immediately. Deposited in a bank at even a modest interest rate, it grows into something more than €100 by next year. Waiting receives the same nominal amount while giving up that growth.

This is the time value of money: the idea that the timing of a cash flow matters, not just its size. A euro received today and a euro received in one year are not economically equivalent, even though they carry the same label.

Three forces explain why.

Opportunity cost. Money available today can be invested immediately. If you deposit €100 at a rate of 5% per year, it becomes €105 in twelve months. Choosing to receive €100 in one year instead means giving up that extra €5. The opportunity forgone is a real cost, even if no cash changes hands.

Inflation. In most economies, prices rise over time. A euro today buys more goods and services than a euro in the future, because by the time you receive it, prices will be higher. Even setting aside investment returns, inflation erodes the purchasing power of any deferred payment.

Uncertainty. A payment promised in the future carries more risk than cash in hand. The counterparty might default, circumstances might change, or the payment might arrive late. Future cash flows are less certain than present ones, and that additional uncertainty has a cost.

Together, these three forces mean that deferring a cash flow has a price. That price is expressed as an interest rate: the compensation a lender demands for parting with money today, or equivalently, the return a borrower must offer to attract funds.

2. Future value and present value

2.1 Three examples

Three simple transactions illustrate how time and money interact.

These three examples share the same logic. There is a cash flow today and a cash flow in the future, and the interest rate connects them.

2.2 The two sides of the same coin

The two directions of this relationship have standard names.

The future value ($\text{FV}$) of a cash flow is what it grows into at a future date, given an interest rate. In the deposit example, the future value of €100 today at 10% for one year is €110.

The present value ($\text{PV}$) of a future cash flow is what that future amount is worth today. In the zero coupon bond example, the present value of €100 to be received in one year at 10% is €90.91.

These are two sides of the same calculation, run in opposite directions. Future value asks: if I invest now, what do I end up with? Present value asks: what would I need to invest today to end up with a specific amount in the future?

We call the process of converting a future cash flow back into a present value discounting. The interest rate used to do this is the discount rate.

3. Compounding: how interest accumulates

The examples above assumed a single year. When the time horizon extends beyond one period, you need a rule for how interest accumulates. That rule is the compounding convention, and different markets use different ones.

3.1 Simple interest

With simple interest, you calculate interest only on the original principal. The interest earned does not itself earn further interest. If you deposit €100 for two years at 5% per year with simple interest, you earn €5 in year one and €5 in year two, ending with €110. No matter how much has accumulated, you always calculate the interest for each period on the same starting amount.

Simple interest is the standard for short term money market instruments, typically those with maturities under one year. The future value $\text{FV}$ is calculated as follows:

where $P$ is the principal (€100), $r$ is the annual rate (5%), and $t$ is time in years.

3.2 Compound interest

With compound interest, each period’s interest is added to the principal, so that in the next period you earn interest on a larger amount. If you deposit €100 for two years at 5% with annual compounding:

• After year one: €100 × 1.05 = €105
• After year two: €105 × 1.05 = €110.25

The extra €0.25 compared to simple interest comes from the interest earned on the first year’s interest. Over long periods, this compounding effect becomes substantial. The general formula, assuming annual compounding, is:

Rearranging gives the present value $\text{PV}$ of a future amount $\text{FV}$:

This is the foundation of bond pricing. When you encounter a multi-year instrument that promises a series of future cash flows, you discount each one back to today using this formula and sum the results.

3.3 Continuous compounding

In financial theory and in most derivative pricing models, a third convention applies: continuous compounding. Instead of crediting interest once per year or once per quarter, interest accrues continuously, in infinitely small increments.

Continuous compounding is the mathematical limit of compound interest as the number of compounding periods grows without bound. The resulting formula is:

where $e$ is the mathematical constant approximately equal to 2.718. The corresponding present value formula is:

Continuous compounding is used throughout derivatives pricing because it simplifies the mathematics of how prices and rates evolve over time. When you see the expression $e^{-rt}$ in a pricing formula, it is discounting a future cash flow back to its present value. While continuous compounding is a theoretical concept, daily compounding is often used in practice as a close approximation.

From compound to continuous: where does $e^{rt}$ come from?

So far we have compounded once per year. But what happens if we compound more frequently? What if we compound every six months, every quarter, or even every day?

When you compound $m$ times per year, the rate per period becomes $r/m$ and the number of periods becomes $m \times t$. The future value formula generalises to:

Let us see what happens to €100 invested at 5% for one year as we increase $m$:

Notice two things. First, each step produces a slightly higher value than the one before, because you start earning interest on interest slightly sooner. Second, the increases get smaller and smaller. The values are converging toward a ceiling.

That ceiling is exactly what you reach when you let $m$ go to infinity. In mathematics, this limit is a famous result:

This is actually one way to define the number $e$. Set $r = 1$, and $e$ is the value that $(1 + 1/m)^m$ converges to as $m$ grows. It is approximately 2.71828.

Substituting back, the future value under continuous compounding becomes:

So the exponential function $e^{rt}$ is not an arbitrary mathematical choice. It is the natural outcome of compounding interest as frequently as possible. And its inverse, $e^{-rt}$, is simply the corresponding discount factor. It tells you the amount you would need to invest today in order to receive one unit of currency at time $t$.

4. Discount factors

4.1 What a discount factor is

A discount factor $\text{DF}(t)$ is the present value of one unit of currency (say, €1) to be received at time $t$. It captures, in a single number, how much a future payment is worth today.

If $\text{DF}(2) = 0.9070$, it means that €1 received in two years is worth €0.9070 today. A payment of €500,000 in two years is therefore worth $500{,}000 \times 0.9070 = €453{,}500$ today.

When interest rates are positive, the discount factor always lies between zero and one: a future euro is always worth less than a current euro. The further away the payment, the smaller the discount factor.

4.2 Three formulas, one concept

The three compounding conventions each produce a different formula for the discount factor:

Simple discounting (used for short dated instruments):

Compound discounting (standard for bonds):

Continuous discounting (standard in derivatives pricing):

These three formulas express the same economic idea in different mathematical languages. At short maturities, the three conventions produce very similar numbers. Over longer horizons, the differences become material, which is why it matters to know which convention a system or model applies.

The table below shows the discount factor under each convention for a discount rate of 5% at various maturities.

At one year, all three conventions give nearly the same result. At ten years, simple discounting implies that €1 is worth €0.667 today, while continuous discounting gives €0.607. The choice of convention moves the present value by more than 9% over a ten-year horizon.

4.3 Discount factors and zero coupon bonds

A discount factor is not only a mathematical construct. It has a direct market interpretation.

A zero coupon bond is a bond that pays no coupons during its life and returns only its face value at maturity. If a zero coupon bond with a face value of €1 maturing in $t$ years trades at a price of $\text{DF}(t)$, then $\text{DF}(t)$ is precisely the market’s discount factor for that maturity.

That price implies a rate called the zero rate for that maturity: the single discount rate that equates the present value of the bond’s only cash flow to its current market price. Zero rates extracted from market prices at different maturities form the zero curve, also called the zero-coupon yield curve or discount curve. Practitioners use the term “yield curve” loosely in conversation to refer to the same concept, but the chapter Yield curves clarifies the distinction between par curves, zero coupon curves, and forward curves as three separate constructs.

The zero-coupon yield curve is the fundamental input for pricing almost any fixed income or derivative instrument. You value a swap, for example, by discounting each of its future cash flows at the zero rate appropriate for that cash flow’s payment date, then summing the resulting present values. There is no single discount rate for the whole instrument: each cash flow has its own discount factor, drawn from the curve at the right maturity.

5. Discounting in practice

5.1 The value of any instrument

The value of any financial instrument is the sum of its future cash flows, each discounted to today:

where $C_i$ is the cash flow at time $t_i$ and $\text{DF}(t_i)$ is the discount factor for that date. This formula reappears throughout the chapters that follow, for bonds, loans, swaps, and FX forwards alike. The structure is always the same: identify the future cash flows, determine the discount factor for each payment date, multiply, and sum. What varies between instruments is the nature of the cash flows, whether fixed or floating, certain or contingent, in one currency or two. The chapter Net present value & basis point value applies the formula to a contract with cash flows in both directions.

5.2 Why the discount rate matters so much

A small change in the discount rate produces a meaningful change in present value, particularly for long dated cash flows. Discounting a bond’s coupons at 3% produces a higher value than discounting at 4%. When market interest rates rise, the present values of future cash flows fall, and the market value of existing fixed rate instruments declines accordingly.

6. When interest rates turn negative

6.1 The puzzle

Everything above assumes that interest rates are positive. A positive discount rate means that future cash flows are worth less than equivalent present cash flows, and the further away a payment, the less it is worth today.

What happens when interest rates are negative?

Mathematically, inserting a negative rate into the compound formula gives a discount factor greater than one. €1 received in one year would be worth more than €1 today. The usual intuition reverses: deferring a cash flow becomes advantageous, and receiving money sooner carries a penalty.

6.2 Historical context

Following the 2008 financial crisis, central banks in Europe and Japan cut interest rates to unprecedented levels in an attempt to stimulate growth and combat deflationary pressure.

The European Central Bank (ECB) lowered its deposit facility rate below zero for the first time in June 2014, initially to -0.10%. Over the following years, the rate fell further, reaching -0.50% by September 2019. For most of the period from 2014 to 2022, short term euro (EUR) interest rates were negative. Even longer-term interest rates, such as the German government bond (Bund) yields also turned negative: by 2019, a ten year Bund yielded approximately -0.70%, meaning investors were willing to pay the German government to hold their money for a decade.

The Bank of Japan (BoJ) introduced a negative deposit rate in January 2016, a policy it maintained for eight years before abandoning it in March 2024.

At the peak of the negative rate era, in early 2021, roughly 18 trillion United States dollars (USD) of global government bonds traded at negative yields.

6.3 What this means for the intuition

In a negative rate environment, the three forces described in section 1 no longer all point in the same direction.

Opportunity cost inverts: placing money in a bank earns a negative return. Parking €100 at -0.50% for one year returns €99.50. In nominal terms, receiving €100 in one year is preferable to depositing it today. Inflation and uncertainty still push in the traditional direction, but if inflation is near zero or actually negative (deflation), those forces can be weak enough that the overall effect reverses.

Negative rates also created practical complications. Some instruments, such as floating rate notes linked to EURIBOR (Euro Interbank Offered Rate), embedded floor clauses specifying that if the reference rate falls below zero, the clause treats it as zero for coupon purposes. Pricing those floors correctly requires a model that can accommodate negative rates, a challenge that generated considerable debate and revision among practitioners and academics throughout the 2010s.

The negative rate era ended as inflation returned sharply in 2022. The ECB raised its deposit facility rate from -0.50% in July 2022 to a peak of 4.00% by September 2023, one of the fastest tightening cycles in the institution’s history. Short term EUR rates are comfortably positive again as of early 2026. But the episode confirmed that the time value of money, while intuitive in a world of positive rates, is ultimately a mathematical relationship that the sign of the interest rate can invert.

7. Key takeaways

• The time value of money reflects three forces: opportunity cost, inflation, and uncertainty. Together they mean that a euro today is worth more than a euro in the future when interest rates are positive.
• Future value asks what a current cash flow grows into over time. Present value asks what a future cash flow is worth today. They are two sides of the same calculation, connected by the interest rate.
• Compounding conventions determine how interest accumulates. Simple interest is standard for short dated money market instruments, compound interest for bonds, and continuous compounding for derivatives models.
• A discount factor $\text{DF}(t)$ is the present value of €1 to be received at time $t$. Three equivalent formulas express it: $1/(1 + r \cdot t)$ for simple, $1/(1+r)^t$ for compound, and $e^{-rt}$ for continuous compounding.
• The three conventions give nearly identical results at short maturities but diverge meaningfully over longer horizons.
• A zero coupon bond is the market embodiment of a discount factor: its price equals the present value of a single future cash flow, and that price implies the zero rate for that maturity.
• You can value any financial instrument by identifying its future cash flows, applying the appropriate discount factor to each, and summing the results.
• When interest rates are negative, discount factors exceed one and the usual intuition inverts. The ECB and the Bank of Japan operated with negative rates for several years after 2014, creating real complications for pricing and risk management before the return of positive rates in 2022.