Annualising Volatility When Returns Are Serially Correlated

1. When is square-root-of-time scaling justified?

For equal-variance increments with zero cross-period covariance, the variance of their sum grows linearly with the number of periods. Taking its square root produces the familiar square-root scaling. Independence is sufficient but stronger than the covariance condition needed for that variance identity. If increments are correlated, additional covariance terms remain. Also specify whether the sum represents log returns, arithmetic increments or an approximation to compounded simple returns. A mathematically correct scaling rule for an additive object does not become an exact annual compounded-return risk forecast merely because the source observations were daily or monthly. The object and assumptions must travel together.

2. Write the omitted terms explicitly

For a weakly stationary additive sequence, let gamma-zero denote one-period variance and gamma-h the covariance h periods apart. The variance of a k-period sum is k times gamma-zero plus twice the sum of (k minus h) times gamma-h for h from one through k minus one. The weighting counts how many pairs lie at each lag. Positive covariance increases this sum's variance relative to the zero-covariance shortcut, while negative terms can reduce it. William Sharpe's discussion of time dependence explains why aggregation needs these terms. It does not establish stationarity or a particular dependence pattern for any supplied return history.

3. Reconcile a small synthetic horizon

Assume invented monthly additive increments have standard deviation two percent and lag-h correlation equal to 0.25 raised to h. One-period variance is 0.0004. For two months, the shortcut gives variance 0.0008 and volatility approximately 2.8284 percent. Including the single covariance pair gives variance 0.0010 and volatility approximately 3.1623 percent. For three months, the corresponding variance is 0.00165 and volatility approximately 4.0620 percent. These are properties of an assumed stationary covariance pattern, not estimates from a fitted market model. The table shows the horizon, omitted terms and resulting units so the arithmetic can be checked directly.

4. Annual frequency is another horizon, not extra evidence

Calling monthly volatility annualized selects a twelve-period horizon under the chosen convention. It does not add independent months to a short sample. The same synthetic dependence rule can be inserted into the finite-horizon covariance sum, but that extrapolation assumes the relationship persists through the year. Lo's Statistics of Sharpe Ratios public publisher summary warns that ordinary monthly-to-annual Sharpe scaling is not generally valid under serial correlation. Only that public summary was reviewed here; the full journal article is premium content. This guide's variance derivation and invented examples are independently calculated, rather than presented as figures extracted from that restricted article.

5. Estimating autocovariance creates another uncertainty problem

A sample autocovariance is not a known population parameter. Lag selection, finite coverage, changing volatility and overlapping observations affect its interpretation. Using many noisy lag estimates can make a horizon estimate unstable, while ignoring dependence can understate or overstate the chosen additive risk. Preserve the estimation window and all included lag terms rather than reporting only a corrected headline. Non-synchronous valuations or smoothed marks may create serial patterns unrelated to liquid executable returns. An annualization adjustment does not repair those upstream measurement issues. It also does not establish an appropriate confidence interval without assumptions about the underlying estimation and sampling process.

6. Tracking error and Sharpe scaling need their own series

For tracking error, apply dependence reasoning to aligned active returns, not automatically to portfolio returns alone. Portfolio and benchmark can each be serially dependent while their difference has another covariance structure. For a Sharpe ratio, the numerator's aggregation and the denominator's aggregation must refer to the same excess-return convention. Simple multiplication of a sample ratio by a calendar factor hides those choices. Keep raw period statistics and the assumed horizon transformation as separate outputs. A benchmark change, variable costs or irregular observation intervals can invalidate a previously chosen scaling contract even when the arithmetic function itself has not changed.

7. Questions before accepting an annual risk number

Which additive or compounded quantity is being measured? What establishes the observation frequency, and are intervals equal? Were overlapping horizons introduced upstream? Which lag covariances are assumed zero, estimated or imposed synthetically? Is the report clear that annualization is a transformation rather than an observed year? Common failures include labeling a sum-of-increments variance as exact compounded risk, treating a larger calendar multiplier as more evidence and applying portfolio autocorrelation to active returns without checking. Preserve the one-period variance, dependence assumptions and horizon together. The result can then support a convention review without becoming a claim of future volatility or a calibrated investment model.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• William F. Sharpe: The Sharpe Ratio, time dependence and aggregation
• Andrew W. Lo: The Statistics of Sharpe Ratios, public publisher summary only; full article premium

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