Estimating Variance From Historical Returns

Why Estimate Variance From Historical Returns?

A finite history of returns gives an investor observations, not a complete description of every return the investment could produce. The sample mean summarizes the observed returns, and the sample variance measures how spread out those observations are around that mean. Under suitable assumptions, sample variance also serves as an estimate of the return variance in the underlying process.

The distinction between a historical sample and a specified probability distribution matters. If the possible returns and their probabilities are given, variance is calculated using those probabilities. If only a finite series of realized returns is available, the probabilities are not known; the sample variance uses the observations and, for the usual unbiased estimate, divides by \(n-1\), not \(n\).

The sample variance has squared return units. If returns are entered as percentage points, such as \(3\) for \(3\%\), the variance is in squared percentage points. If returns are entered as decimals, such as \(0.03\) for \(3\%\), the variance is in decimal-return squared. These are two ways of expressing the same result: converting percentage-point returns to decimals multiplies the variance by \(0.01^2=0.0001\).

Calculating the Sample Variance

For a historical series, first calculate the arithmetic mean of the observations. Then subtract that mean from each return, square each deviation, add the squared deviations, and divide by \(n-1\). The number \(n-1\) is the degrees-of-freedom adjustment used when estimating the variance of an underlying return process from a sample.

Why the Denominator Is \(n-1\)

The reason for the \(n-1\) denominator is not that a historical series literally contains only \(n-1\) returns. Rather, when the sample mean is estimated from the same observations, the deviations from that mean are constrained: they sum to zero. Under a standard sampling model, dividing by \(n-1\) corrects the downward bias that would result from dividing by \(n\).

Suppose the observed returns \(R_1,\ldots,R_n\) are independent and identically distributed, with population mean \(\mu\) and finite population variance \(\sigma^2\). These are assumptions for the unbiasedness result, not facts guaranteed by observing a return history. The sample mean is \(\bar{R}\). The identity below follows by writing \(R_t-\bar{R}=(R_t-\mu)-(\bar{R}-\mu)\), expanding the squares, and using \(\sum_{t=1}^n(R_t-\mu)=n(\bar{R}-\mu)\):

Taking expectations gives \(n\sigma^2\) for the first sum. Under independence and identical distribution, \(\operatorname{Var}(\bar{R})=\sigma^2/n\), and \(E[(\bar{R}-\mu)^2]=\sigma^2/n\). Thus the expected sum of squared deviations from the sample mean is \(n\sigma^2-n(\sigma^2/n)=(n-1)\sigma^2\). Dividing by \(n-1\) therefore gives \(E[s_R^2]=\sigma^2\). This is the justification for calling \(s_R^2\) an unbiased estimator of the population variance under these assumptions.

If returns are dependent, have changing distributions over time, or otherwise fail the assumptions, this unbiasedness result need not apply. The formula still describes the spread in the observed finite series, but its interpretation as an unbiased estimate of a stable population variance requires care.

Historical Estimates and Specified Distributions

Historical returns are a finite set of observed outcomes. In contrast, a specified probability distribution provides possible outcomes and their probabilities. For a discrete distribution, the variance is the probability-weighted average of squared deviations from the expected return. It is not the sample variance formula, because its probabilities describe the model rather than being estimated from a sample of observations.

Common Mistakes and Checks

Variance calculations are especially vulnerable to small arithmetic and interpretation errors. In an exam response, show the mean, deviations, sum of squares, and denominator; doing so makes it possible to identify the source of a discrepancy.

• Using \(n\) instead of \(n-1\) for sample variance. In the six-return example, the squared deviations sum to \(64\). Dividing by \(6\) gives \(10.6667\), which is not the usual sample variance estimate. The correction is \(64/(6-1)=12.8000\) squared percentage points.
• Using a wrong sum of squared deviations. The deviations for \(2,4,4,6,8,12\) about \(6\) are \(-4,-2,-2,0,2,6\), whose squares sum to \(64\), not \(60\). Rechecking each squared deviation prevents a plausible-looking but incorrect final value.
• Mixing percentage points and decimals. A variance of \(12.8\) squared percentage points is not \(12.8\) decimal-return squared. In decimal units it is \(0.00128\).
• Centering on the wrong value. Sample variance uses deviations from the sample mean. A supplied expected return may be the correct center for a probability-distribution variance, but it is not automatically the center for a historical sample variance.
• Overstating what the estimate establishes. A historical sample variance summarizes the observations and, under assumptions, estimates an underlying variance. It does not prove that future returns follow a stable distribution.