Scaling Volatility with Horizon

The standard deviation of continuously compounded annual stock returns is 10 percent. What is the standard deviation of continuously compounded four-year returns?

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#Log returns add

A continuously compounded return over four years is the sum of the four annual log returns,

r_{1:4} = r_1 + r_2 + r_3 + r_4. \tag{1}

Under the standard assumption that yearly log returns are independent and identically distributed, variances add with no covariance terms,

\Var(r_{1:4}) = \sum_{t=1}^{4}\Var(r_t) = 4\sigma^2. \tag{2}

#Square root of time

Standard deviation is the square root of variance, so it grows with the square root of the horizon rather than the horizon itself,

\sigma_4 = \sqrt{4}\,\sigma = 2\sigma = 2 \cdot 10\% = 20\%. \tag{3}

Continuously compounded returns add across years, so their variances add and the standard deviation scales with the square root of the horizon. Quadrupling the horizon only doubles the volatility, taking the annual 10 percent to 20 percent over four years.

#Why log and not simple returns

Simple returns compound multiplicatively, so they do not add across periods and this clean linear scaling of variance fails for them. That is exactly why log returns are the natural unit for converting risk between horizons, with volatility scaling as $\sqrt{T}$ and the annualized number recovered by dividing a horizon volatility by $\sqrt{T}$ .