Why Covariance Scales with Observation Frequency Under Zero Serial Correlation
Summary
The document derives the familiar annualization rule for covariance of asset log returns: multiply the per-period covariance by the number of periods in a year. It uses the covariance-of-sums identity. When returns within each period may co-move but returns from different periods are uncorrelated, all cross-period covariance terms vanish, leaving the same-period covariance repeated once for each period.
The explanation extends to multiple assets when their variances and pairwise covariances are finite and constant over time, and all assets’ returns are uncorrelated across distinct periods. It emphasizes that normality and independence are not required; zero cross-period correlation is sufficient for the stated derivation. These are sufficient conditions, not necessary ones. If returns have serial correlation or time-varying covariance, simple frequency scaling may not apply. The note also observes that expected returns scale similarly only under an additional constant-mean assumption.
Key ideas
- Annual covariance equals per-period covariance multiplied by periods per year when cross-period return covariances are zero.
- The result follows by expanding the covariance of cumulative returns and eliminating cross-period terms.
- Returns need not be normally distributed or independent for the stated scaling argument.
- Constant finite variances and covariances and zero correlation across distinct periods are sufficient assumptions.
- Expected returns scale by frequency under the additional assumption of a constant per-period mean.
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Full text
# How do I annualise a covariance matrix?
# How do I annualise a covariance matrix?
It is very difficult to source a rigorous answer to the above question.
I know the answer is:
```
Ann. Covariance = covariance * frequency
```
Can anyone explain the mathematical idea behind this formula?
## Answer by FP0 (score 4)
https://quant.stackexchange.com/a/71655
This is a consequence of the covariance of linear combinations of random variables which are uncorrelated with respect to time.
See wikipedia: $$\text{cov}(aX+bY,cW+dV)=ac\,\text{cov}(X,W)+ad\,\text{cov}(X,V)+bc\,\text{cov}(Y,W)+bd\,\text{cov}(Y,V)$$
In the case of log-returns, imagine that you have 2 assets having respectively log-returns $X_t$ and $Y_t$. Assume that for each period $t$, $X_t$ and $Y_t$ can be correlated, but also that $X_{t_1}$ and $Y_{t_1}$ are not correlated with $X_{t_2}$ and $Y_{t_2}$, $\forall t_1 \neq t_2$.
With 2 observations, you then have: \begin{split} a&=b=c=d=1\\ X&\stackrel{\text{def}}{=}X_1\\ Y&\stackrel{\text{def}}{=}X_2\\ W&\stackrel{\text{def}}{=}Y_1\\ V&\stackrel{\text{def}}{=}Y_2 \end{split}
So, due to the absence of correlation of $X_t$ and $Y_t$ at different times, you have:
\begin{split} \text{cov}\left(X_1+X_2,Y_1+Y_2\right)&=\text{cov}\left(X_1,Y_1\right)+\text{cov}\left(X_2,Y_2\right) &=2\,\text{cov}\left(X_t,Y_t\right) \end{split}
and in general:
$$\text{cov}\left(\sum_{t=1}^{n}X_t,\sum_{t=1}^{n}Y_t\right)=n\,\text{cov}\left(X_t,Y_t\right)$$
This can be generalised for any number of periods and assets $a$ with log-returns $X^a_t$ as long as:
- $X^a_t$ have constant and finite variance $\left(\mathbb{V}\text{ar}\left(X^a_t\right)=\sigma^2_a \forall t\right)$, $\forall a$ and covariances $\left(\text{cov}\left(X^a_t, X^b_t\right)=\sigma_{a,b} \forall t\right)$, $\forall a \neq b$;
- $X^a_{t_1}$ and $X^b_{t_2}$ can be correlated during the same period ($t_1=t_2$), but are not correlated $\forall t_1 \neq t_2, \forall a, b$ (including $a=b$).
So with $n$ periods per year and with the assumptions listed above, the covariance of the log-returns of each pair of assets can simply be multiplied by the annual frequency of observations $n$, as your post suggests.
Please note that these conditions are sufficient. Even if these assumptions are quite common, you do not need:
- The log-returns to be normally distributed;
- The log-returns to have the same distribution accross time $t$;
- The log-returns to have the same distribution accross assets $a$;
- The log-returns to have a constant expected value $\left(\mathbb{E}\left(X^a_t\right)=\mu_a \forall t\right)$, $\forall a$, but this assumption allows you to get the yearly expected values by scaling the individual expected values by $n$, like the covariances;
- The log returns to be independent. They just need to be uncorrelated $\forall t_1 \neq t_2, \forall a, b$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.