Cross hedges example 8 - Reading 10 Currency Management

Foreign-currency AUD asset return RFC = 4.0%
Foreign-currency NZD asset return RFC = 6.0%
Foreign-currency RFX= 5% for both
Asset risk σ(RFC) = 0% for both
Currency risk σ(RFX) AUD = 8%
Currency risk σ(RFX) NZD = 10%
Correlation (USD/AUD; USD/NZD) +0.85

Then is says: 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:
A The expected risk (i.e., standard deviation) of the domestic-currency
return for the Australian asset is equal to (1.04) × 8% = 8.3%.
B 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%.

Can someone explain why are we multiplying the return by std. dev. to get the currency risk (std. dev.) if it is already provided to us at the first place? I do recognize that under A and B they label it as expected risk of the domestic-currency r; however, it is not part of the formula for risk of the domestic currency:
σ2(RDC) ≈ σ2(RFC) + σ2(RFX) + 2σ(RFC)σ(RFX)ρ(RFC,RFX)

In example 1 in the beginning of this reading it is a simple plug and chug into the formula based on the data provided.
What follows is: σ2(RDC) = (0.5)2(8.3%)2 + (0.5)2(10.6%)2 + [(2)0.5(8.3%)0.5(10.6%)0.85]
= 0.8%
My take was that we would simply be inputting 8% and 10% instead of 8.3 and 10.6 computed in A and B. Thoughts?

At the end of one period you’ll have your original investment plus your return. That total amount is then subject to the exchange rate volatility.

This is the formula for a 2-asset portfolio, when the returns of those assets are added.

Here, you’re compounding the returns, not adding them. That formula doesn’t apply.

Now that you pointed this out I went back to the text and noticed that in example 1 they already provided the expected risk, while in example 6 they list it as “currency risk”, which appears to be the realized one. Thank you as always!

My pleasure.

Is this formula optional? As in can we asset weigh this?

Worked similar problem but was asset weighted. What is the case we ignore asset weight, other than lack of data about asset values?