How do I annualise a covariance matrix?

Question

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 1

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:

1. $X^a_t$ 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 $\left(\text{cov}\left(X^a_t, X^b_t\right)=\sigma_{a,b} \forall t\right)$, $\forall a \neq b$;
2. $X^a_{t_1}$ 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:

1. The log-returns to be normally distributed;
2. The log-returns to have the same distribution accross time $t$;
3. The log-returns to have the same distribution accross assets $a$;
4. 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;
5. The log returns to be independent. They just need to be uncorrelated $\forall t_1 \neq t_2, \forall a, b$.