Risk and return: the central trade-off

Every investment decision is a negotiation between two quantities: the return you hope to earn and the risk you must accept to earn it. Portfolio theory begins by making both measurable.

Return as an expected value

The expected return of an asset is the probability-weighted average of its possible outcomes — in practice we estimate it from history as the mean of past returns. If a stock returned 12 percent, then minus 4 percent, then 8 percent over three years, its average annual return is:

(12 + (-4) + 8) / 3 = 5.33 percent

Risk as dispersion

Return alone tells you nothing about reliability. Two assets can share a 5 percent average while one delivers it smoothly and the other lurches between plus 40 and minus 30. Portfolio theory defines risk as volatility — the standard deviation of returns, the typical distance of an outcome from the mean.

variance = average of (each return minus the mean) squared
standard deviation = square root of variance

A higher standard deviation means outcomes are more spread out, so any single year is less predictable.

Why this framing is powerful — and limited

Treating risk as standard deviation lets us do arithmetic on portfolios. But it quietly assumes that an upside surprise is as undesirable as a downside one, and that returns are roughly symmetric. Real markets have fat tails and crashes that a single number understates. Keep that caveat in mind: the elegance of the theory rests on an approximation of reality, not reality itself.