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As indicated by @AlexC and @amdopt, the formula is exact for log returns and approximate for discrete returns. Define the factor by which a price changes as $k$ so that price tomorrow $P_{t+1}$ is the price today times $k$ : $P_{t}*k$.Then the change in the price over a business year is $$\prod_{i \in [1, 252]}{k}$$ The log of the change is by properties of ...


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The returns (or rather alphas, i.e. returns relative to the benchmark) plotted are logarithmic returns, not the simple returns usually reported by investment managers. This makes them additive over time. The green line is a straight line with slope 18% (the expected annual alpha). The thin purple curve is $\sigma \sqrt{t}$ above and below the green line, ...


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