<p><a href="https://en.wikipedia.org/wiki/Variance#Sum_of_uncorrelated_variables_.28Bienaym.C3.A9_formula.29" rel="nofollow">The variances of the sum of independent random variables are additive</a>. Therefore, if X and Y are independent, Var(X+Y)= Var(X) + Var(Y). The standard deviation is defined as $\sigma_z = \sqrt{Var(z)}$. Therefore, if X and Y are independent, $\sigma_{X+Y} = \sqrt{Var(X+Y)} = \sqrt{Var(X) + Var(Y)}$. This gives the result above. </p>

CC BY-SA 3.0
2016-05-02T15:57:36.890Z
2016-05-02T15:57:36.890Z
BKay
<p>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&amp;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.</p>

<p>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.</p>

<p>$$\sigma=\sqrt{(16^2)+(5^2)}$$</p>

CC BY-SA 3.0
2016-04-30T18:22:28.533Z
I'm voting to close this question as off-topic because this is clearly a question about basic statistics. (Perhaps a migration to stats.SE is in order?)
Giskard
CC BY-SA 3.0
2016-05-04T11:09:40.253Z
A lot of clarifications needed: What I understand from what you write is that you have made an investment $W$ denominated in foreign currency (USD) , and you want to know what will be its volatility (vague term - since you use percentage I guess you mean the standard deviation over mean value ratio "coefficient of variation"), _in your local currency_. But then you are looking at the entity $W\cdot S$ where $S$ is the "local currency per unit of USD" exchange rate. Independent they may (assumed to) be as random variables, but you have a product of ind. rv's, not a sum.
Alecos Papadopoulos
CC BY-SA 3.0
2016-05-11T03:21:04.453Z
It does not make sense to multiply them: suppose one had volatility 0, and the other 1. What would be the total volatility? 0? You can also see it in logs, log(ab)= log(a)+log(b). So that var(log(ab))=var(log(a))+var(log(b)) if hey are independent
Fix.B.
<finance><statistics>
Volatility of investment: Joint-Volatility of mutually independent variables
egikm
# Volatility of investment: Joint-Volatility of mutually independent variables

(Asked by: username_0 on 2016-04-30)

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.

$$\sigma=\sqrt{(16^2)+(5^2)}$$

**Tags:** finance, statistics

> **Comments:**
> **username_1**: I'm voting to close this question as off-topic because this is clearly a question about basic statistics. (Perhaps a migration to stats.SE is in order?)  
> **username_2**: A lot of clarifications needed: What I understand from what you write is that you have made an investment $W$ denominated in foreign currency (USD) , and you want to know what will be its volatility (vague term - since you use percentage I guess you mean the standard deviation over mean value ratio "coefficient of variation"), _in your local currency_. But then you are looking at the entity $W\cdot S$ where $S$ is the "local currency per unit of USD" exchange rate. Independent they may (assumed to) be as random variables, but you have a product of ind. rv's, not a sum.  
> **username_3**: It does not make sense to multiply them: suppose one had volatility 0, and the other 1. What would be the total volatility? 0? You can also see it in logs, log(ab)= log(a)+log(b). So that var(log(ab))=var(log(a))+var(log(b)) if hey are independent  

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## 1 answers

### Answer by username_4 on 2016-05-02. Score: 1

[The variances of the sum of independent random variables are additive](https://en.wikipedia.org/wiki/Variance#Sum_of_uncorrelated_variables_.28Bienaym.C3.A9_formula.29). Therefore, if X and Y are independent, Var(X+Y)= Var(X) + Var(Y). The standard deviation is defined as $\sigma\_z = \sqrt{Var(z)}$. Therefore, if X and Y are independent, $\sigma\_{X+Y} = \sqrt{Var(X+Y)} = \sqrt{Var(X) + Var(Y)}$. This gives the result above.


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