Volatility

Volatility starts with a series of returns. Compute their mean, take each return's deviation from it, square the deviations, average them and take the square root. For a sample, the average uses n − 1 in the denominator, to correct for measuring deviations from the sample mean rather than the true one. The result is in the same units as the returns: a stock with an annual volatility of 25% has returns that, if roughly normal, land within 25 percentage points of their average in about two years out of three.

Because variance is the square of volatility, it is quoted in squared units, percent squared or a decimal squared. That is why analysts report volatility for interpretation even though portfolio calculations are carried out in variance and covariance.

Volatility is usually quoted per year so that assets can be compared. To scale from a shorter interval, multiply by the square root of the number of intervals in a year. The rule assumes returns are independent and identically distributed from one period to the next; when returns trend or revert toward a mean, the square-root rule misstates risk over longer horizons.

Volatility also changes over time. Markets alternate between calm stretches and turbulent ones, and periods of high volatility tend to cluster. A figure estimated over a quiet decade can badly understate risk going into a crisis, which is why practitioners compare several look-back windows and watch implied volatility from option markets.

Volatility is the most common single measure of an investment's total risk. It feeds the Sharpe ratio, which divides excess return by volatility, and it is the input portfolio optimizers use to trade risk off against expected return. Diversification works because the volatility of a portfolio is less than the weighted average of its holdings' volatilities whenever correlations are below one.

Its limits follow from the definition. It is symmetric, so an unexpected gain raises volatility as much as an unexpected loss. It ignores the shape of the tails, so two assets with equal volatility can carry very different chances of a severe loss. And it says nothing about credit, liquidity or operational risk, which can dominate an investment's outcome.

Candidates should compute sample and population standard deviations, annualize them correctly, combine them into portfolio volatility using correlations, and interpret the result. They should also recognize that variance cannot be compared with returns directly because of its squared units, and that downside measures such as semideviation are related but distinct statistics.

In derivatives, volatility is an input to option pricing. Higher expected volatility raises the value of both calls and puts, because it widens the range of outcomes in which the option pays off, while the holder's loss remains capped at the premium paid.