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Basic question , but if I do a daily regression and get an alpha of 0.00004. Should the yearly alpha be : 0.00004 *100 *252 = 1.008% OR 0.00004 * 100 *365 = 1.46%. What is considered the yearly alpha if one wants to know the outperformance of a manager annually? Should one use 252 trading days or 365 days in a year?

Also in the case of fund that has a total performance of 25% in 500 days, should the formula to annualize be : 1.25^(252/500)-1 OR 1.25^(365/500)-1 ?

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You should use 252 trading days.

To annualize returns, multiply the average daily return by 252.

To annualize volatility, multiply the daily volatility by sqrt(252).

You can also use log daily returns if you prefer.

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