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Confusion on the Standard Deviation in Domestic Currency
Equation 3 in the image below is actually the formula to calculate standard deviation for a portfolio with 2 assets.
Rdc = Asset return in domestic currency
Rfc = Asset return in foreign currency
Rfx = Foreign Exchange return
p = correlation
I understand that with a risk free foreign asset you would have 0 standard deviation (SD) hence the first (SD Rfc) and last item in the addition (2 * SD Rfc * SD Rfx * correlation) will be 0 which leave with just SD of Rfx.
What I don't understand is why do we have to multiply the total SD with (1 + Rfc) in Equation 4?
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Other L3 candidates: @marc, @RaviVooda , @Alta12 , @Jwa , @AjFinance - any idea on this?
In this case we are aware of the return on the asset (The constant Risk free rate). So the volatility of the foreign currency would be multiplied with the risk-free rate of the asset, in order to account for the magnitude of std deviation in Domestic currency. This being a simplified equation, would provide a direct answer.
Usually, we are not aware of the return that the foreign asset would generate. So it would be unwise to account for the return in the equation. Over here, it seems possible.
Not sure if that clears your doubt, but like I said, in the absence of a numeric illustration, this is what I can think of.
The formula before equation 3 is the basic formula for variance of a two asset portfolio W1 * SD1 ^2 + W2 * SD2^2 + 2 * W1 * W2 * SD1 * SD2 * (correlation of 1 & 2).
Following are the remaining text from the book which I think doesn't explain much about the doubts I have above.
"The exposures (weights) to Rfc and Rfx are each 100% with the weights in the formula expressed as 1.0. The formula becomes equation 3."
"The standard deviation of Rdc is the square root of this variance. Examining the equation indicates risk to our domestic investor:
- Depends on the standard deviation of Rfc and Rfx.
- May be higher for our domestic investor because standard deviation of Rfx is an additive term in the equation.
- However, correlation also matters. If the correlation between Rfc and Rfx is negative, the third component of the calculation becomes negate. The correlation measures the interaction of Rfc and Rfx.
- If the correlation is positive, then Rfc returns are amplified by Rfx returns, increasing the volatility of return to our domestic investor.
- If the correlation is negative, then Rfc returns are dampened by Rfx returns, decreasing the volatility of return to our domestic investor.
"
http://www.finquiz.com/blog/2013/08/19/changes-to-the-2014-level-3-cfa-curriculum/
"Reading 35 in SS14, “Currency Risk Management” has been replaced with Reading 28, “Currency Management: An Introduction.” It looks like the material is a little more basic though many of the LOS look the same. You’ll probably recognize most of the material if you took last year’s exam but don’t neglect the reading because there is quite a bit of new stuff here."
The intuitive way of thinking about this is that when you buy 1 EUR using USD as your domestic currency, and you keep that 1 EUR in your wallet, whatever it is worth at the end of the period depends solely on the return of the EUR vs the USD. The standard deviation of your return on this 'investment' is exactly the standard deviation of the EUR vs USD return. On the other hand, instead of putting the 1 EUR in our wallet, we now invest it at the risk-free rate, so we know that it will become 1+Rfc EUR after one period. This is exactly the same situation as before, but now all the numbers are scaled by 1+Rfc, so our standard deviation is multiplied by |1+Rfc| (which was already positive, so the absolute value changes nothing).
I hope this clears things up for you! Good luck in your studies :-)
I understand that the asset in foreign currency return (Rfc) is risk-free, hence there shouldn't be any correlation with the FX return, so why do we have to include the scale up of (1 + Rfc) into the SD of the FX return?
Note that the original equation 3 that you mention in your first post has the approximation sign - there is no precise equality. In the event that Rfc is risk-free (i.e. 100% certain), then this changes because of the reasons I mentioned before.
Hope this helps! :-)
[edit]Apologies for the format of the equations. I hope you can still understand them!
if a random variable v has a SD of x
another random variable that is k * v has a SD of k * x.
its variance will be k^2 * x*2