How Serial Correlation Changes Multi-Period Return Variance

# How does autocorrelation bias annualizing variance?


# How does autocorrelation bias annualizing variance?












I read somewhere that autocorrelation prevents someone from annualizing variance. But how does it bias it? Let's say you have daily returns. If autocorrelation is high, should that overstate or understate annualized variance when you multiply by 252. What about if you have negative autocorrelation?

Thanks!

## Answer by Pleb (score 4, accepted)

https://quant.stackexchange.com/a/72184

In Andrew W. Lo's paper, The statistics of Sharpe Ratios (2002) he derives the variance of non-IID returns (returns that can exhibit serial correlation) under the assumption of (covariance) stationary returns with common variance (eq. 19):

\begin{align} \mathbb{V}ar\left(R_{t}(q)\right) &= \sum_{i=0}^{q-1} \sum_{j=0}^{q-1}\mathbb{C}ov(R_{t-i}, R_{t-j})\\ &= q\sigma^2 + 2\sigma^2 \sum_{k=1}^{q-1}(q-k)\rho_k, \end{align} where $\rho_k = \frac{\mathbb{C}ov(R_{t}, R_{t-k})}{\mathbb{V}ar\left(R_{t}\right)}$ is the k'th order autocorrelation (under stationarity) and $R_{t}(q)$ is the $q$'th period return defined by, $$ R_{t}(q) = R_t + R_{t-1} + \ldots + R_{t-q+1}. $$

With $\sigma^2 \geq 0$ we can observe the following from the above equation:

- Positive autocorrelations, $\rho_k > 0$, will upward bias the variance and hence also the volatility.

- For $\rho_k < 0$ the variance will be downward biased and so will the volatility.

- For $\rho_k=0$ the formula reduces to the scaled variance of the $q$'th-period return, $\mathbb{V}ar\left(R_{t}(q)\right) = q \sigma^2$ which implies that $Sd(R_t(q)) = \sqrt{q} \cdot \sigma$.

As a conclusive note, the author further states that (p. 41):

> [...] The reason is that positive serial correlation implies that the variance of multiperiod returns increases faster than holding-period q; hence, the variance of $R_t(q)$ is more than $q$ times the variance of $R_t$, yielding a larger denominator in the Sharpe ratio than the IID case. For returns with negative serial correlation, the opposite is true: The variance of $R_t(q)$ is less than $q$ times the variance of $R_t$, yielding a smaller denominator in the Sharpe ratio than the IID case.

Maybe this is the paper you are looking for? The paper contains some good examples on how serial correlation can affect Sharpe Ratios. It is worth a read.