The time value of money from first principles

Everyone has heard that a franc today is worth more than a franc tomorrow. Very few people can say precisely why, and the imprecision matters, because the reason determines what number you should use to convert between the two. Get the reason wrong and every valuation you build afterwards inherits the error, silently and at scale.

There are three distinct reasons, and they stack. First, capital is productive: a franc today can be put to work — lent, invested, used to buy a machine — and will come back as more than a franc. Second, consumption now is preferred to consumption later, so people demand compensation for postponing it. Third, a franc promised in the future may not arrive, so the promise must be discounted for the chance of default or disappointment. The first two together produce what we call the risk-free rate. The third produces the risk premium on top of it. This lesson deals with the machinery that converts between time periods; the sibling lesson on the discount rate deals with how to choose the number you feed into that machinery.

Compounding is the forward operation

If you invest CHF 1,000 at 4% a year, after one year you have CHF 1,040. In the second year you earn 4% not only on the original CHF 1,000 but also on the CHF 40 of interest, giving CHF 1,081.60. After three years, CHF 1,124.86.

The general form is a future value equal to the present amount multiplied by (1 + r) raised to the number of periods. The exponent is the entire story. Linear intuition badly underestimates it: at 4% the sum does not take 25 years to double, it takes roughly 18. The rule of 72 — divide 72 by the percentage rate for an approximate doubling time — gives 18 years here, and the exact answer is about 17.7.

Two things follow immediately. Small differences in rate produce large differences in outcome over long horizons, which is why fee drag of one percentage point a year is not a rounding error over an investing lifetime. And the growth is back-loaded: most of the absolute gain in a long compounding series happens in its final third, which is precisely when impatient owners have usually sold.

Discounting is the same operation run backwards

Valuation asks the reverse question. Not "what does CHF 1,000 become", but "what is a future CHF 1,000 worth now". You divide instead of multiply.

CHF 1,000 due in three years, at 4%, is worth 1,000 ÷ 1.04³ = 1,000 ÷ 1.124864 = CHF 889.

That figure — 1 ÷ (1 + r)^n — is called the discount factor. It is the price today of one franc delivered on a specific future date, and it is the only building block you need. Every discounted cash flow model, however elaborate, is a list of expected future amounts each multiplied by its own discount factor and then added up. The complexity in real models lives entirely in forecasting the amounts and choosing the rate; the discounting itself is arithmetic a schoolchild can do.

Note the asymmetry that the exponent creates. At a 2% rate, CHF 1,000 due in ten years is worth CHF 820.35. Raise the rate to 3% and it falls to CHF 744.09 — about 9.3% less. Do the same on a thirty-year claim: CHF 552.07 at 2%, CHF 411.99 at 3%, a fall of about 25%. The same one-percentage-point change in rates does roughly two and three-quarter times as much proportional damage to the thirty-year claim. This sensitivity of price to rates, scaling with the timing of cash flows, is what practitioners call duration. It explains why long-dated bonds move violently when rates shift, and equally why the valuations of companies whose profits sit far in the future — early-stage growth businesses, pre-revenue biotech, infrastructure with long build phases — reprice hardest when the rate environment changes. Nothing about those businesses changed. The discount factors did.

Annuities and perpetuities: the shortcuts worth knowing

Two patterns of cash flow recur often enough to deserve closed-form solutions.

An annuity pays a fixed amount each period for a fixed number of periods — a pension, a lease, a mortgage. A perpetuity pays a fixed amount forever, which no real contract quite does, but which is an excellent approximation for the distant tail of a durable business.

Take CHF 10,000 a year at a 4% discount rate. As a perpetuity it is worth 10,000 ÷ 0.04 = CHF 250,000. As a twenty-year annuity it is worth CHF 135,903.

Sit with that comparison. Twenty years of payments — a very long time by the standards of any forecast anyone can honestly make — accounts for only about 54% of the value of the infinite stream. The remaining 46% sits in years 21 onward, a period about which nobody knows anything specific. This is not an artefact of the example; it is the structural reason terminal value dominates DCF models, which the sibling lesson on terminal value takes up in detail. The honest response is not to pretend the tail away, but to recognise that a large fraction of any long-lived asset's value rests on the claim "this business will still be earning something decades from now" — and to test that claim rather than the third decimal place of next year's margin.

If the payment grows at a constant rate g, the perpetuity denominator becomes (r − g). The formula only makes sense when g is strictly below r, and it becomes numerically unstable as the two converge. That instability is the mathematics warning you that assuming permanent growth close to your discount rate is not a mildly aggressive assumption but a structurally broken one.

Frequency, effective rates and comparing like with like

A quoted rate is meaningless without its compounding convention. A nominal 6% compounded monthly means 0.5% applied twelve times: 1.005¹² = 1.06168, an effective annual rate of about 6.17%. Compounded continuously, the same nominal 6% gives about 6.18%.

The gap looks trivial at 6%. It is not trivial at consumer-credit rates, and it is not trivial when you are comparing a Swiss mortgage quoted one way against a bond yield quoted another. Before comparing two rates, convert both to the same effective annual basis. Swiss lenders and platforms are required to disclose an effective annual rate for consumer credit for exactly this reason: nominal rates with different conventions are not comparable numbers, and quoting them side by side is a standard way of making an expensive product look cheap.

Real, nominal, and the currency you are actually paid in

Interest rates come in two flavours. A nominal rate is the number printed on the contract. A real rate is what remains after inflation — the change in what the money can actually buy.

The relationship is multiplicative, not additive. A 3% nominal return with 1% inflation gives a real return of 1.03 ÷ 1.01 − 1 = 1.98%, not exactly 2%. At low inflation the subtraction shortcut is fine. At high inflation it is badly wrong, and mixing the two is a classic modelling error: discounting real cash flows at a nominal rate systematically undervalues a business, and discounting nominal cash flows at a real rate systematically overvalues one. Pick a convention and hold it through the entire model.

For Swiss-resident investors there is a further discipline. Discount rates are currency-specific, because they embed that currency's inflation expectations. Swiss franc interest rates have for decades sat structurally below those of most other developed currencies, reflecting persistently low Swiss inflation and demand for the franc as a store of value. That is not free money: it is the arithmetic counterpart of a currency that has tended to appreciate over long periods. Discounting franc cash flows at a dollar or euro rate — or the reverse — imports a currency view into your valuation without your noticing. If you value a US business in dollars, discount at a dollar rate and convert the result at the exchange rate, rather than mixing conventions inside the model.

Switzerland also supplied the clearest modern demonstration that nothing in the mathematics requires a positive rate. The Swiss National Bank held its policy rate below zero for several years from January 2015 — at −0.75% on sight deposits for most of that period — before returning to positive territory in 2022. When r is negative, the discount factor exceeds one and a future franc is worth more than a franc today. Depositors paid to store money and long-dated bonds traded at prices implying a guaranteed nominal loss if held to maturity. The framework did not break. It simply reported that, under those conditions, safe future money was genuinely scarcer than present money.

Where this actually bites in valuation

Three practical consequences carry through the rest of this course.

Timing is a variable you can be wrong about independently of amount. A project that delivers the same total cash two years later is worth measurably less, and that is a real economic difference, not a modelling artefact.

Consistency beats precision. The discount rate, the cash flow definition, the currency and the inflation convention must all match. Most valuation errors in practice are mismatches, not bad forecasts.

And sensitivity is structural. Because value is a sum of exponentially weighted terms, long-duration assets will always be more volatile in price than short-duration ones for reasons entirely unrelated to business quality. Knowing this in advance is the difference between reading a 30% drawdown as new information and reading it as a change in the discount rate.

What to take away

• The time value of money has three separate sources — the productivity of capital, preference for present consumption, and the risk that promised cash fails to arrive — and only the third is a risk premium.
• Discounting is compounding reversed: the discount factor 1 ÷ (1 + r)^n is the price today of one franc on a given future date, and every DCF is just a weighted sum of such factors.
• Duration is arithmetic, not sentiment: a one-point rise in the rate cut a ten-year CHF 1,000 claim by about 9%, and a thirty-year claim by about 25%, which is why long-horizon businesses reprice hardest when rates move.
• Twenty years of a CHF 10,000 annuity at 4% is worth CHF 135,903 against CHF 250,000 for the perpetual version, so roughly 46% of the value of a long-lived asset sits beyond any horizon you can honestly forecast.
• Rates are currency-specific and convention-specific: convert to effective annual terms before comparing, keep real and nominal separate, and remember the Swiss experience of sub-zero policy rates from 2015 to 2022 as proof that the framework tolerates negative r.

Sources

• Irving Fisher, The Theory of Interest (1930) — the classical decomposition of interest into time preference and the productivity of capital.
• Eugen von Böhm-Bawerk, Capital and Interest — the earlier Austrian treatment of why present goods command a premium over future goods.
• Frederick Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856 (1938) — the original definition of duration.
• Zvi Bodie, Alex Kane and Alan Marcus, Investments — standard treatment of present value, effective rates and bond price sensitivity.
• Jonathan Berk and Peter DeMarzo, Corporate Finance — annuity and perpetuity formulas with worked derivations.
• Swiss National Bank — monetary policy assessments and interest rate statistics, including the negative interest rate period introduced in January 2015.
• Swiss Federal Statistical Office — the Swiss consumer price index, the reference series for converting nominal Swiss returns into real terms.
• Bank for International Settlements — research and statistics on long-term interest rates across currencies.