R21 Currency Blue box example 7

CFA Level III Economics
#1

This text below is from the book. Can someone explain how they calculated the standard deviation of the domestic currency return for each foreign asset?

Why do you multiply return and standard deviation?

Although RFX is a random variable—it is not known in advance—the RFC term is in fact known in advance because the asset return is risk-free. Because of this Nguyen can make use of the statistical rules that, first, σ(kX) = kσ(X), where X is a random variable and k is a constant; and second, that the correlation between a random variable and a constant is zero. These results greatly simplify the calculations because, in this case, she does not need to consider the correlation between exchange rate movements and foreign-currency asset returns. Instead, Nguyen needs to calculate the risk only on the currency side. Applying these statistical rules to the above formula leads to the following results:

  1. The expected risk (i.e., standard deviation) of the domestic-currency return for the Australian asset is equal to (1.04) × 8% = 8.3%.
  2. The expected risk (i.e., standard deviation) of the domestic-currency return for the New Zealand asset is equal to (1.06) × 10% = 10.6%.
#2

If the FC is a risk free asset, the formula says to do (1 + Rfc) x σ(fx)

Look at the list of formulas, its all there.

#3

I have the same question as foshizzle. Why do we multiply returns on LC with currency risk? I am trying to understand the formula. It does not sense to me.
Can anyone explain?

#4

The standard deviation of the returns in domestic currency is computed as:

σ\left[r_{DC}\right] = σ\left[\left(1 + r_{FC}\right)\left(1 + r_{FX}\right) - 1\right] = σ\left[1 + r_{FX} + r_{FC} + r_{FC}r_{FX} - 1\right]
= σ\left[r_{FX} + r_{FC} + r_{FC}r_{FX}\right]

Recalling that r_{FC} is a constant (so 1 + r_{FC} also is a constant), and noting that σ\left[X + k\right] = σ\left[X\right], we have:

σ\left[r_{DC}\right] = σ\left[r_{FX} + r_{FC} + r_{FC}r_{FX}\right] = σ\left[r_{FX} + r_{FC}r_{FX}\right]
= σ\left[\left(1 + r_{FC}\right)r_{FX}\right] = \left(1 + r_{FC}\right)σ\left[r_{FX}\right]
#5

Thanks magician. So it’s just an algebraic derivation?

#6

Yup.

Note that it works only because r_{FC} is a constant. When it isn’t, then you have to take into account the correlation between r_{FC} and r_{FX} and things become quite messy.

#7

Yes. I understand. This formula is applicable only when risk of foreign investment is zero? Thanks

#8

You got it!