I´ve been trying to compute a volatility of invesment with currency hedging and I have a question. Let's take this example. We have our money in a fond copying the S&P500 index, which has 16% volatility, we also know that the current volatility of a dollar toward our currency is 5%. We want to know the volatility of the whole invesment.
Can I compute as following? If so, what is the reason for adding the two deviations instead of mulitplying them considering the volalitity of an index and a currency are mutualy independent. $$\sigma=\sqrt{(16^2)+(5^2)}$$
You are almost right except that there could be correlation between the portolio return ( devoid of exchange rate fluctuations) and the exchange returns. Let us say the variances are notated $\sigma_p^2$ and $\sigma_c^2$ and $\rho$ is the correlation coefficient. then the variance of the portfolio is given by
$$\sigma_T^2 = \sigma_p^2 + \sigma_c^2 + 2\rho\sigma_p\sigma_c$$. They are not mutiplicative in the pure sense but are governed by the above relationship where you split the returns owing to underlying asset and returns owing to underlying currrency. Thus the volatility is $\sigma_T$