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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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If you make the assumption that your returns are iid normally distributed $R_i \sim \mathcal{N}(0, 1)$, then with the second definition, and using @ZRH provided formula... $$E[DD] = \sqrt{\int_{-\infty}^{c}x^2\frac{1}{\sqrt{2 \pi}}exp(-\frac{x^2}{2}) dx }$$ So expanding this out (integration by parts) you get; $$E[DD] = \sqrt{ \left [ - x \frac{1}{\sqrt{2 ...


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