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Understand economics at your own pace: step-by-step guided courses, and exercises to practice.
Return and risk
Every hoped-for gain is paid for in accepted uncertainty.
R = (P₁ − P₀ + income) / P₀ × 100Problem / motivation
You buy a share for €100. A year later it is worth 108 and it has paid you a €3 dividend. Have you gained 8%, 3% or 11%? (Share? Dividend? These words are defined just below.)
Two words first, because the whole course rests on them. A SHARE is a small part of a company: whoever holds one owns a piece of the business, and sometimes receives a DIVIDEND — a portion of profits, paid in cash. A BOND is a loan cut into parts: whoever holds one has lent money (to a state, to a company) and collects a COUPON — the interest paid each year. Both can be bought and sold on: they are called SECURITIES (the course “Share or bond?” compares them in detail).
The return on an investment therefore does not boil down to the rise in its price: you have to add what the security paid out along the way. And a return never comes alone — the more it makes you hope, the more it makes you tremble. This course first measures the complete gain, then puts a real figure on risk: the uncertainty about what you will actually collect.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Let's take the opening again. The price moved: 108 − 100 = +€8. And the security paid €3. The complete gain adds the two: (P₁ − P₀) + income — P for Price, subscript 0 for the start, 1 for the end of the period, “income” for the payment collected along the way (dividend on a share, coupon on a bond). That is assumption 2 as a formula: here, 8 + 3 = €11.
Is €11 a lot? Impossible to say without knowing what was committed: €11 of gain on €100 does not taste the same as on €1,000. So we RELATE the gain to the starting capital — mind the word: “relate to”, in mathematics, means DIVIDE BY (nothing to do with “this investment pays”). Hence the ÷ P₀: 11 ÷ 100 = 0.11. That is assumption 1 as a formula.
That leaves making the result readable and comparable: × 100 turns 0.11 into 11% — a common language for every investment, whatever its size. And a return is always stated WITH its period (assumption 3): our 11% is “over one year”; the same 11% “over five years” would be an entirely different matter.
One last, decisive point: this formula computes a REALISED return — the prices in it are observed, the story is over. For tomorrow, nobody knows P₁: we then reason in terms of EXPECTED return, the average of possible scenarios (assumption 4) — defined and put in figures at the Interpretation step. Keep the distinction in mind: it is what separates measuring the past from betting on the future. Click each term:
Solving / calculation
Let's settle the opening for good: share bought at €100, sold at €108, €3 of dividend collected. 8%, 3% or 11%? Let's unroll the formula step by step, then handle the values yourself.
- Capital gain: the change in price alone
108 − 100+€8 (the “8%” counts only this) - We add the income paid (the dividend)
8 + 3€11 of complete gain - We relate it (= divide it) to the starting capital
11 / 1000.11 - As a percentage
0.11 × 100R = 11%
Answer: 11%. The two other answers in the opening were the pieces: 8% = the price alone, 3% = the dividend alone. Ignoring the income paid amputates performance here by 3 points out of 11 — a little over a quarter.
R = (P₁ − P₀ + income) / P₀ × 100Set the prices, the income paid and inflation: everything else follows. The gap between the first two lines is the weight of the income; the gap on the last line is the price of the “− inflation” shortcut.
Economic interpretation
A return is never judged on its own: against inflation on one side, against the risk taken on the other. Let's put a figure on each.
An investment at 5% when prices rise by 3% makes you only about 2% richer: real return ≈ nominal − inflation. The “≈” matters: the exact rule DIVIDES the multiplying coefficients (+5% = ×1.05, +3% = ×1.03 — the % ↔ coefficient bridge from the course “GDP growth”; 1.05 ÷ 1.03 → +1.94%), and the subtraction is only a reliable shortcut for small rates — the simulator above shows both (course “Do your savings beat inflation?”, and “Inflation and purchasing power” for the complete mechanics).
Investment A: +8% for certain. Investment B: one year in two +20%, one year in two −4%. Expected return of B = (20 + (−4)) ÷ 2 = +8% — the same as A! Same expectation, opposite journeys: B is VOLATILE, its results deviate strongly from their average, upwards as well as downwards. RISK is that dispersion — and it can be measured: A never departs from its 8% (0 points), B ranges from +20 to −4, a 24-point spread (financiers summarise it in a single number, the standard deviation — same idea, equipped). The expected return, for its part, remains an average of scenarios, never a promise: next year's realised return will depart from it.
Who would choose B at equal expectation? Nobody — which is precisely why, on the markets, volatile investments have to offer a HIGHER expectation to find a taker. That supplement is the RISK PREMIUM: the ~7% real return on shares from the poll, against bonds that pay less but are steadier, is its century-long version. (An expert's nuance: DIVERSIFYING — spreading across many securities — erases part of the dispersion without sacrificing the expectation, so that the premium ultimately only rewards the risk that cannot be diversified away. Above all, hold on to the idea: no high return without accepted risk.)
Limits / critiques
Exercises
Share bought at €250, sold at €262, dividend of €8. What is the total return (in %)?
Share bought at €120, sold at €105, dividend of €3. What is the total return (in %, sign included)?
An investment returns +14% one year in two, and −2% the other year. What is its expected return (in %)?
True or false: the expected return on an investment is the return it will pay for certain.
Same investment (250 → €262, dividend €8): what is the PRICE-ONLY return, without the dividend (in %)?
Your savings account pays 2% this year, inflation is 4%. Your purchasing power…
Investment A: +6% for certain. Investment B: +26% or −14%, on a coin toss. Which statement is correct?
True or false: an investment advertising “+10% over five years” returns 10% a year.