Currency risk - what if R fc is risk free

[thespacebar]

In Reading 28, page 98 of Schweser, how do they derive that equation 4??
std(R_DC) = std(R_fx) (1 + R_fc)

 

[S2000magician]

Because R_fc is risk-free, it is a constant. A general rule about standard deviation is:
σ(kX) = kσ(X)

 

[thespacebar]

Still not clear
we have Var(R_dc) = Var(R_FX) + Var(R_FC) + 0 . (0 is due to zero correlation)
then what happens?

 

[cpk123]

var(r_fc) = 0 too…
r_dc = (1+rfc) ( 1+rfx) = 1 + rfc + rfx (1+rfc)
var(r_dc) = 0 + 0 + (1+rfc)^2 VAR(rfx) <– because (1+rfc) is a constant now.
= (1+rfc)^2 VAR(rfx)
OR
σ(rdc) = (1+rfc) σ(rfx)

 

[S2000magician]

Consider this set of returns: 5%, 7%, 11%, 1%, 10%.
The standard deviation of these returns is 3.6%. (You can check this if you like.)
Now consider this set of returns: 10%, 14%, 22%, 2%, 20%; each return is twice the corresponding return in the first list.
These standard deviation of these returns is 7.2%: twice that of the first set. (Again, you can check this if you like.)
Thus, when you multiply each number in the list by a constant, the standard deviation of the new list is the standard deviation of the old list multiplied by that same constant.
In the case you gave, we start with a series of FX returns: rfx1, rfx2, …, rfxn. They have a standard deviation std(rfx). Each of the domestic currency returns is the FX return times (1 + rfc); (1 + rfc) is a constant. Thus,
rdc1 = rfx1(1 + rfc)
rdc2 = rfx2(1 + rfc)
.
.
.
rdcn = rfxn(1 + rfc)
The standard deviation of the domestic currency returns, std(rdc) is thus that same constant (1 + rfc) times the standard deviation of the FX returns:
std(rdc) = std(rfx) × (1 + rfc).

 

[stunnerrunner]

I’m still not grasping the concept - I guess I’m confused as to why Stdev (R_DC) = Stdex (R_FX)*(1+R_FC) and not Stdev (R_FX)*(1+R_FX).
Or maybe what’s the difference between R_FC and R_FX? I’m looking at example #8 on pg. 270 in the curriculum.

 

[S2000magician]

R_FC is the (fixed, constant) return on the foreign risk-free bond (in foreign currency)
R_FX is the (variable) return on the foreign currency exchange rate (in domestic currency).
The (variable) exchange rate return is magnified by the (fixed) risk-free foreign bond return; thus, the standard deviation of the (variable) exchange rate return is magnified by the same (constant) factor: (1 + R_FC).
Try this: in Excel, in one column put in a bunch (say, 50) of random FX returns in the range of -5% to +5%; you can do this with “= rand()*10% – 5%”. In the next column, multiply those returns by 1.04, simulating the return on a 4% risk-free bond. Finally, compute the standard deviation (STDEVP) of those two columns, then divide the second column’s standard deviation by the first column’s standard deviation; you’ll get 1.04.