By the end of this lesson, you can build a complete two-outcome model, calculate its expected one-year return and ending euro balance, and explain why neither is a promised result. You will then change a probability to see how much the answer depends on the assumptions.
A probability describes the chance of a defined outcome, on a scale from 0 to 1. A percentage is a number per hundred: 60% = 60 ÷ 100 = 0.60. A probability weight tells us how much an outcome contributes to an average. A weighted sum multiplies each value by its weight and adds the products.
A one-year simple return is the gain or loss over that year divided by the starting balance. Here it includes any investment income and is measured before fees, taxes and inflation, with no deposits or withdrawals. A +10% return changes €1,000 by €100; a −5% return changes it by −€50. Multiplying a percentage return by a euro balance requires its decimal form: 10% becomes 0.10.
A random variable is a number determined by an uncertain outcome. Call the one-year return R. Write E[R] for its expected value, the mean using probability weights. Call the ending balance W, measured in euros. These labels let us keep a percentage return separate from a money amount.
Illustrative model: start with €1,000 and hold for exactly one year. Assume only two possible returns: +10% with probability 60%, or −5% with probability 40%. The cases are mutually exclusive: they cannot both be the return for that same year. They are exhaustive: this model includes no other possible return. Check the weights: 0.60 + 0.40 = 1.
The probabilities and returns are chosen teaching inputs. We have not estimated them from a security or a historical sample. Real returns usually require a richer range of outcomes. A model that omits a severe loss cannot account for it merely by computing its mean correctly.
Multiply each return by its probability: the upside contributes 0.60 × 0.10 = 0.06, and the downside contributes 0.40 × (−0.05) = −0.02. Add them: E[R] = 0.06 − 0.02 = 0.04 = 4%. Keeping the negative sign is essential because a loss reduces the mean.
A simple average of the two quoted returns would be [10% + (−5%)] ÷ 2 = 2.5%. That assigns each case a 50% weight. It answers a different question from our stated 60% and 40% probabilities.
In the upside case, W = €1,000 × 1.10 = €1,100. In the downside case, W = €1,000 × 0.95 = €950. Weight those balances using the same probabilities: E[W] = 0.60 × €1,100 + 0.40 × €950 = €660 + €380 = €1,040. The expected gain is €1,040 − €1,000 = €40.
There is a second route to the same answer. For every case, W = 1,000 × (1 + R), with W in euros and R a decimal return. Therefore E[W] = 1,000 × (1 + E[R]) = 1,000 × 1.04 = €1,040. This works because the starting balance is fixed and the calculation only scales and adds. This property is called linearity of expectation. It does not require independent outcomes or repeated trials.
Under this model, the realized one-year return can only be +10% or −5%, and the ending balance can only be €1,100 or €950. Neither 4% nor €1,040 can occur. Expected value summarizes the possibilities before the outcome is known. It does not create an extra middle outcome.
The mean is positive, yet there is still a 40% probability of losing €50. A decision also depends on the range of losses, access to the money and the person's circumstances. Risk preference means how someone values uncertain gains and losses; it cannot be read from the mean alone.
The 4% is also not a compounded growth rate across years. Compounding multiplies successive growth factors, each equal to 1 plus that period's return. In a separate illustrative two-year path, +10% followed by −5% gives €1,000 × 1.10 × 0.95 = €1,045, a cumulative gain of 4.5%. Its geometric annual growth rate is the constant yearly rate giving that same two-year balance: √1.045 − 1 ≈ 2.2252%. That path is not the one-year distribution, and it does not tell us how likely a two-year sequence is. Multi-year analysis needs assumptions about how returns relate across time.
Keep the illustrative +10% and −5% outcomes, starting €1,000 balance and one-year horizon. Now assume the upside probability is 40%. What must the downside probability be? Calculate the expected return and ending balance. Can that expected return occur, and does its sign eliminate loss risk?
Answer: because the two cases remain exhaustive, the downside probability is 1 − 0.40 = 0.60. E[R] = 0.40 × 0.10 + 0.60 × (−0.05) = 0.04 − 0.03 = 0.01 = 1%. E[W] = 0.40 × €1,100 + 0.60 × €950 = €440 + €570 = €1,010. The expected gain falls to €10 because more weight is placed on the loss.
The realized return can still only be +10% or −5%, so 1% cannot occur in this model. Positive expectation still includes a loss, now with 60% probability. The calculation is exact for the assumptions; it supplies no evidence that those assumptions describe an actual investment.
General education, not investment advice. All balances, returns and probabilities below are illustrative assumptions, not measured market data, real forecasts or Quantustik performance claims. The exercise does not establish a suitable investment, guaranteed profit or an appropriate level of risk for any person.
No. The most likely return is the value with the greatest probability. In the first example that is +10%, while the expected return is 4%. They answer different questions.
Yes, for the complete, non-overlapping list used here. If they do not, check for missing cases, overlapping categories or incorrect inputs before interpreting an expectation.
No. Those are explicitly excluded from this illustration. A real application must state whether its outcomes include costs, taxes and inflation and use the same convention in every case.
Investor education only — not investment advice, and never a promise of profit. Every investment can lose value.