# Stretching the covariance of a daily change to the year

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?

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.
• Thanks, can you elaborate on what's $Z, X, X'$ and $\mathcal{N}_{0,1}$? – shredding Feb 20 at 10:06
• $Z,X,X'$ are random variables - they represent 2-day change, 1-day change, 1-day change respectively. $\mathcal{N}_{0,1}$ is standard normal distribution (cf. wiki Normal distribution) – Radost Feb 20 at 10:12