Currently I am preparing for quant interview and I encounter the following question in Heard on the street.
Question: If the standard deviation of continuously compounded annual stock returns is $10\%,$ what is the standard deviation of continuously compounded four-year stock returns?
Assuming continuously compounded returns follow an arithmetic Brownian motion, variance of returns grows linearly with the compounding period. This is because consecutive returns in a random walk are independent, and the variance of a sum of independent random variables is just the sum of variances. This means that the four-year $\sigma^2$ equals four times the one-year $\sigma^2.$ It follows that the four-year $\sigma$ is two times the one-year $\sigma.$ The answer is therefore $20\%.$
I have a few doubts on the solution.
- Why can we assume that the returns follow an arithmetic Brownian motion (ABM)? I think ABM satisfies the SDE $$dS_t = \mu dt+\sigma dW_t$$ where $S_t$ is stock returns and $W_t$ is Brownian motion.
- For second bolded sentence, how does it explain that the variance of returns grows linearly with the compounding period?