# How do I annualise a covariance matrix?

It is very difficult to source a rigorous answer to the above question.

Ann. Covariance = covariance * frequency


Can anyone explain the mathematical idea behind this formula?

• That comes from the definition of a Weiner process (or Brownian motion), which you can find more about on the Chapter 14 (Wiener Processes and Itô’s Lemma) from Hull's Options, Futures, and Other Derivatives. There Hull discuss the square root rule for scaling volatility, which ultimately is the reason why you're multiplying variances (or covariances) by the time (or frequency as you say in your question). Jul 12, 2022 at 21:05
• $\sigma_{XY} = \sigma_X \sigma_Y \rho_{XY}$. Assuming $\rho$ is independent of time scale, you can annualize each volatility independently, which will each be annualized by $\sqrt{\mathrm{freq}}$ Aug 5, 2022 at 14:18

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 $$\forall t_1 \neq t_2, \forall a, b$$.