variance or stdev is additive - an illustration

Imagine annual (log) return is controlled by a noisegen, whose mean is a constant value M and variance is another constant value sigma^2

Since we hit the noisegen once a year, over 5 years we get 5 random "numbers", all with the same M and sigma. Each number is the realized annual (log) return. The cumulative end-to-end return is the sum of the 5 independent random variables. This sum is a random variable with a variance, which is additive. Assumption is iid i.e. repeated noisegen hits.
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In a different scenario, suppose we hit the noisegen once only and multiply the same output number by 5 in a "projected 5Y return". Now std is additive.

In both cases, the mean of the 5Y end-to-end return is 5*M