# How to convert daily volatility to monthly?

Suppose I have a forecasted daily volatility for K days.

How can I get the forecasted monthly volatility from the daily ?

If you assume the underlying time series $x_t$ is not autocorrelated (which is a reasonable assumption for daily financial returns), then $$\text{Var}( x_{t+1} + \dotsc + x_{t+K} ) = \text{Var}(x_{t+1}) + \dotsc + \text{Var}(x_{t+K})$$ and you can substitute forecasts for the theoretical quantities: $$\widehat{\text{Var}}( x_{t+1} + \dotsc + x_{t+K} ) = \widehat{\text{Var}}(x_{t+1}) + \dotsc + \widehat{\text{Var}}(x_{t+K}).$$ You have your daily foreasts $\widehat{\text{Var}}(x_{t+1}), \dotsc, \widehat{\text{Var}}(x_{t+K})$ with $K$ around 22 for working days or 30 for calender days; this allows you to obtain the monthly forecast $\widehat{\text{Var}}( x_{t+1} + \dotsc + x_{t+K} )$.
Meanwhile, in presence of autocorrelation you would have $$\text{Var}( x_{t+1} + \dotsc + x_{t+K} ) = \text{Var}(x_{t+1}) + \dotsc + \text{Var}(x_{t+K}) + \sum_{i=1}^K \sum_{j=1}^K \text{Cov}(x_{t+i},x_{t+j})$$ and a corresponding expression for forecasts in places of theoretical quantities.