Currency risk - what if R fc is risk free

In Reading 28, page 98 of Schweser, how do they derive that equation 4??

std(R_DC) = std(R_fx) (1 + R_fc)

Because R_fc is risk-free, it is a constant. A general rule about standard deviation is:

σ(kX) = kσ(X)

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Still not clear

we have Var(R_dc) = Var(R_FX) + Var(R_FC) + 0 . (0 is due to zero correlation)

then what happens?

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)

= (1+rfc)^2 VAR(rfx)

OR

σ(rdc) = (1+rfc) σ(rfx)

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, . . ., rfx_n_. 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)

.

.

.

rdc_n_ = rfx_n_(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).

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.

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.

Sorry, I’m trying to understand how to get from

to

If var(r_dc) = var(r_fc)+var(r_fx)+2σ(r_dc)σ(r_fx)rho(rfc,rfx) and var(r_fc) =0 b/c it’s a rf asset, then wouldn’t var(r_dc)=var(r_fx)?

I understand the intuition as explained by sir S2000, but Im lost on the algebra… please help.