Annualize Multi-Year Volatility

This question is about scaling the volatility of a stock's returns from an annual horizon to a multi-year horizon in a simple continuous-compounding setting. The candidate is given an annualized volatility and asked to infer the volatility over a longer period, assuming a standard model with independent increments for returns. It checks whether the candidate can connect the everyday "annual vol" quote used on trading floors to the risk over a multi-year holding period, and whether they understand how risk aggregates through time rather than just memorizing a number.

The reasoning leans on thinking of returns as a sum of independent shocks and understanding how variance and standard deviation behave under such summation. It uses the square-root-of-time rule, which is ubiquitous in option pricing, risk management, and portfolio construction. Interviewers watch for comfort with time scaling, awareness of the underlying independence assumption, and the ability to manipulate standard deviations without confusing them with variances or expected returns.

What it tests

For processes where returns are modeled as independent increments—such as Brownian motion or arithmetic random walks—the total variance over a time interval is the sum of the variances of each subinterval. This means that if the variance over one period is $\sigma^2$, then over $T$ periods it is $T\sigma^2$. The key reason is that independence ensures no covariance terms: the variance of a sum of independent variables is just the sum of their variances. Volatility, being the standard deviation, thus scales with the square root of time: $\sqrt{T}\sigma$. This scaling law is fundamental in quantitative finance for aggregating risk over different time horizons, and it arises directly from the additive property of variance under independence.