# Calculating Variance of a random variable with time dependence

I am currently trying to undertstand the concept of Value at Risk, which attempts to calculate the a value such that the potential loss on a portfolio is bounded by a number with 99% probability.

The variable of interest is $$\triangle{y}$$ which represents the change in the interest rate. This variable has mean 0, and annual variance $\sigma^2$. However, as we wish to calculate the VaR over a specific time interval, we use the standardize variable: $$x=\frac{\triangle y}{\sigma\sqrt{\triangle t}}$$

where the denominator contains this term $\sqrt{\triangle t}$. This term represents the time interval we are considering (presumably relative to a year, where the change in t represents the fraction of a year we are considering as the interval). What is the intuition/derivation behind this? Many thanks!

Imagine that $Y$ is a sum of independent increments on time intervals $\Delta t$ over the course of one year. I.e., we are assuming that $$Y = \sum_i \Delta y_i .$$ Now, in 1 year, we have $\frac{1}{\Delta t}$ time intervals, such that the summation is for $i = 1, ... , \frac{1}{\Delta t}$. However, we are given that the variance of $Y$ is $\sigma^2$. Thus, a question is what is the variance $\sigma_i^2$ of each $\Delta y_i$? Assuming they are all equal, and noting that the variance of a sum is the sum of the variances for independent variables (i.e. the independent increments), then we have that $$\sigma^2 = \sum_i \sigma_i^2 = \frac{1}{\Delta t} \sigma_i^2$$ which yields $$\sigma_i^2 = \sigma^2 \Delta t ,$$ or equivalently $$\sigma_i = \sigma \sqrt{\Delta t} .$$ In summary, the standard deviation of an increment over a duration $\Delta t$ is $\sigma_i = \sigma \sqrt{\Delta t}$.
Now, let $\Delta y = \Delta y_1$, i.e. the increment over just the time $\Delta t$ as you specified. By dividing $\Delta y$ by its standard deviation, we get a random variable $x$ with unit variance. You can now use Q function or whatnot to get probabilities and such. Note that this is all a bit hand-wavy, and you'll want to fully understand the (almost surely wrong) probabilistic assumptions you're making before you trust this analysis with your money.