CFA CFA Level 3 Confusion on the Standard Deviation in Domestic Currency

# Confusion on the Standard Deviation in Domestic Currency

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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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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.

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@vincentt which chapter is this from? I’m not using Schweser…

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I don’t have all my books with me – is this a newer edited reading? I certainly don’t recall that equation from last time round and I can’t seem to find it in last year’s schweser books?

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Man, that is a good question. I can’t figure it out as well @vincentt‌ .

Other L3 candidates: @marc, @RaviVooda‌ , @Alta12‌ , @Jwa‌ , @AjFinance‌ – any idea on this?

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I’m kind of stumped too…

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@Jwa‌ no worries, i didn’t follow the order anyway, i started with asset classes then to the new portfolio stuff.

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Hi @vincentt‌, it basically follows directly from this: http://en.wikipedia.org/wiki/Standard_deviation#Identities_and_mathematical_properties – use the third equation SD(cX) = |c|SD(X). Since the foreign currency return is risk-free, you are guaranteed to have (1+Rfc) (multiplied by your initial amount of course) in foreign currency, a constant. Whatever this amount is in domestic currency depends on the exchange rate (so it depends on the foreign currency return).

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 🙂

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@reena in case you’re wondering it’s from schweser book 4 page 98 (SS14 R28).

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@AjFinance @jimmyg‌ there’s no numerical example in swcheser which is why it’s hard to understand if it’s actually a mistake by schweser.

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.

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@alta12 SS14 R28.

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@vincentt‌ – wait for this. I’m nearly there with an explanation hopefully as I’m scrambling to get the latest books. Leave this aside, will get back to you

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Hi @vincentt, I haven’t arrived at that particular reading yet, but had a quick look now and I found equation (1) on page 222 of CFA book 5 (CFA Institute Books). It reads Rdc = (1+Rfc)(1+Rfx)-1. This equals (1+Rfc) + Rfx * (1+Rfc) – 1. If we now take standard deviations we get SD(Rdc) = SD( (1+Rfc) + Rfx * (1+Rfc) -1). We use the rules for standard deviation (from the Wikipedia page I linked above) to arrive at the final equation: SD(Rdc) = SD( (1+Rfc) + Rfx * (1+Rfc) -1) = Sqrt ( Var (1+Rfc) + Var (Rfx * (1+Rfc)) + 2 SD(1+Rfc) * SD(Rfx * (1+Rfc)) * Corr (1+Rfc , Rfx * (1+Rfc))). Now if 1+Rfc is a constant, the Correlation term and first variance term are zero, so we are left with Sqrt (Var( Rfx * (1+Rfc)) which equals |1+Rfc|*SD(Rfx).

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! 🙂

Apologies for the format of the equations. I hope you can still understand them!

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the formula is using the concept of
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

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@vincentt @Reena Its hard to decipher this without a numerical example. But, I’ll give it a try.

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.

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@vincentt would you mind sharing the earlier part of this example? I had a long work through it and I don’t fully get it either.

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@vincentt – good, thanks, nice to know my memory isn’t totally failing. Sadly I haven’t got this far yet this year so have little to offer!

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@MM12‌ Thank you for your time on the explanation!

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?