Stretching the covariance of a daily change to the year

Question

I'm diving into financial mathematics and have calculated a matrix that gives me the daily change of four given securities:

The start of it looks like this:

My Tutorial is now calculating the covariance of a stock over a year as $250\ Cov(d_1, d_2)$, where $d_x$ is the daily change of a stock.

250 is the number of trading days on my stock exchange.

So, for example, the iROB and DAX have a daily Covariance of $Cov(iROB, DAX) = -0.000173$ and therefore a yearly one of $250\ Cov(iROB, DAX) = -0.043201$.

This feels to me like comparing apples and oranges. Why can I scale the covariance up by a factor of days in order to get the yearly Covariance?

Answer 1

This is derived under assumption of Wiener process, more simply put if you have random independent identically normally distributed daily increments then covariance (and variance as well) simply adds up as time passes.

Compare this with:

Suppose: $$ Z=X+X' \quad X,X'\sim \mathcal{N}_{0,1} $$ Then: $$ Z \sim \mathcal{N}_{0,2} $$ Which you can easily verify using linearity of expectation as: $$\mathbb{E}[Z^2]=\mathbb{E}[(X+X')(X+X')]=\mathbb{E}[X^2]+\mathbb{E}[{X'}^2]+\mathbb{E}[XX']$$ And $\mathbb{E}[XX']=0$ as increments are independent.