Variance

Variance is built in three steps: measure each outcome's distance from the mean, square those distances so that positive and negative deviations do not cancel, then average the squares. For a random variable, the averaging is done with probabilities rather than counts, so variance equals the sum over outcomes of probability times squared deviation from the expected value. An algebraically identical shortcut is the expectation of the squared variable minus the square of its mean, which is often faster when a probability table is given.

Squaring has a cost: variance is expressed in squared units, so a return series measured in percent yields a variance in percent squared, which has no intuitive reading. Taking the square root restores the original units and produces standard deviation. Variance nevertheless remains the working quantity in theory because it adds cleanly across independent components, while standard deviations do not.

For a two-asset portfolio, variance equals the first weight squared times the first variance, plus the second weight squared times the second variance, plus twice the product of the two weights and the covariance between the assets. Covariance can be rewritten as correlation times the two standard deviations, which makes the diversification mechanism explicit: with correlation below one, the cross term is smaller than it would be under perfect co-movement, and portfolio standard deviation falls below the weighted average of the individual standard deviations.

As the number of holdings grows, covariance terms outnumber variance terms and come to dominate total risk. This is the analytical foundation of the distinction between systematic and diversifiable risk, and the reason mean-variance optimisation requires a full covariance matrix rather than a list of standalone volatilities.

Under the assumption of independent, identically distributed returns, variance scales linearly with the length of the measurement interval while standard deviation scales with its square root. Annualising a daily variance therefore means multiplying by the number of trading days, roughly 250, and annualising a daily standard deviation means multiplying by the square root of that figure. The assumption is convenient but imperfect: return autocorrelation and volatility clustering both distort simple scaling.

In practice, variance estimated from historical data feeds risk budgets, tracking-error limits, value-at-risk models and option pricing inputs. Because volatility is persistent rather than constant, practitioners often replace the equally weighted historical estimate with exponentially weighted or conditional models that give recent observations more influence.

A candidate should be able to compute both population and sample variance from raw data, identify from the context which divisor applies, and compute variance for a discrete random variable from a probability distribution. Equally important is the two-asset portfolio variance calculation in both covariance and correlation form, including recovering an implied covariance from a stated correlation.

Beyond computation, variance underpins several inference procedures: the chi-square test addresses a single population variance, the F-test compares two variances, and degrees of freedom in each depend on sample sizes minus one. Finally, recognise the measure's limitation. Variance penalises upside and downside deviations equally, which motivates downside-focused alternatives such as semivariance and target semideviation when return distributions are skewed.