Currency management - domestic risk currency

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pollfre

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Hi guys,
example 7 reading 19 from curriculum states that the expected risk of the domestic-currency return is the volatility of the exchange rate times the foreign currency asset return. I do not understand why the foreign currency asset return should be accounted for…
Can somebody help?
 
Normally, to determine the standard deviation of returns in the domestic currency, you would have to analyze:
σrDC = σ[(1 + rFC)(1 + rFX)]
= σ[1 + rFC + rFX + (rFC)(rFX)]
This would require knowing the standard deviation of rFC and rFX, as well as the correlation of rFC with rFX. And it’s messy.
Here, however, rFC is the risk-free rate: it’s a constant, so its standard deviation is zero. This makes 1 + rFC a constant. One of the properties of standard deviation is that when k is a constant, σ(kX) = kσ(X). Another is that when c is a constant, σ(c + X) = σ(X). Putting all of this together:
σrDC = σ[(1 + rFC)(1 + rFX)]
= (1 + rFC)σ(1 + rFX)
= (1 + rFC)σ(rFX)
That’s the formula that they used: the standard deviation of returns in domestic currency is (1 + rFC) times the standard deviation of the currency exchange rate.
 
S2000magician wrote:
Click to expand...
Normally, to determine the standard deviation of returns in the domestic currency, you would have to analyze:
σrDC = σ[(1 + rFC)(1 + rFX)]
= σ[1 + rFC + rFX + (rFC)(rFX)]
This would require knowing the standard deviation of rFC and rFX, as well as the correlation of rFC with rFX. And it’s messy.
Here, however, rFC is the risk-free rate: it’s a constant, so its standard deviation is zero. This makes 1 + rFC a constant. One of the properties of standard deviation is that when k is a constant, σ(kX) = kσ(X). Another is that when c is a constant, σ(c + X) = σ(X). Putting all of this together:
σrDC = σ[(1 + rFC)(1 + rFX)]
= (1 + rFC)σ(1 + rFX)
= (1 + rFC)σ(rFX)
That’s the formula that they used: the standard deviation of returns in domestic currency is (1 + rFC) times the standard deviation of the currency exchange rate.
Not sure i get you Magician. In the CFAI material (example above), to calculate the EXPECTED RISK for the domestic-currency, the currency RISK is multiplied by the treasury bill RATE of RETURN. This is the first time i see RISK and RETURN are multiplied to get RISK. Im sure that was the original question by Pollfre
 
broadex wrote:
S2000magician wrote:Normally, to determine the standard deviation of returns in the domestic currency, you would have to analyze:
σrDC = σ[(1 + rFC)(1 + rFX)]
= σ[1 + rFC + rFX + (rFC)(rFX)]
This would require knowing the standard deviation of rFC and rFX, as well as the correlation of rFC with rFX. And it’s messy.
Here, however, rFC is the risk-free rate: it’s a constant, so its standard deviation is zero. This makes 1 + rFC a constant. One of the properties of standard deviation is that when k is a constant, σ(kX) = kσ(X). Another is that when c is a constant, σ(c + X) = σ(X). Putting all of this together:
σrDC = σ[(1 + rFC)(1 + rFX)]
= (1 + rFC)σ(1 + rFX)
= (1 + rFC)σ(rFX)
That’s the formula that they used: the standard deviation of returns in domestic currency is (1 + rFC) times the standard deviation of the currency exchange rate.
Click to expand...
Not sure i get you Magician. In the CFAI material (example above), to calculate the EXPECTED RISK for the domestic-currency, the currency RISK is multiplied by the treasury bill RATE of RETURN. This is the first time i see RISK and RETURN are multiplied to get RISK. Im sure that was the original question by Pollfre
Click to expand...
Not quite: the currency risk is multiplied by (1 + risk-free rate of return).
That’s exactly what I showed.
Suppose that you’re a USD investor holding a GBP-denominated risk-free bond: the GBP return is 3%, and the standard deviation of the USD/GBP exchange rate is 8%. Then the standard deviation of the USD return is:
(1 + rFC)σ(rFX) = 1.03(8%) = 8.24%
 
Before you dive into the statistics….what is is the intuition of this formula: RDC=(1+RFC)(1+RFX)−1.
My simple mind would tell me that if i am USD investor holding a GBP-denominated risk-free bond and the GBP return is 3%, and the standard deviation of the USD/GBP exchange rate is 8%, then the standard deviation of the USD return is just 8% since the bond is risk free. Ofcourse this is incorrect….as the you have pointed above, but i dont understand.
 
You started with a portfolio containing a bond worth, say, GBP1,000. A year later you have a portfolio worth GBP1,030: a GBP1,000 bond and GBP30 in cash. The exchange rate volatility applies to the entire GBP1,030, so the domestic currency return volatility is larger at the end of the year than it was at the beginning of the year, by exactly 3%.
 
As always… Magician is electric and swift. However, I take the liberty of all in just adding that the period is a single period and not multiple.
Incase of multiple period the RfD is no longer constant and the messy algebra will need to work through.
BTW, the property of sigma (variable+constant)= signma (variable)- this property is called translation invariance. sigma(constant x variable)=constant x sigma(variable) is called positive homogeniety.
Both statistical properties are essentail for risk management.
 
Good hint @ABAL.
Btw: refering to the same Example. The CFAI also shows a calculation with a leveraged position (one currency long, one currency short). The formula to calculate sigma squared R(DC), i.e., the variance of the domestic currency return, changes slightly: we do not add anymore 2 times standard deviation times standard deviation times correlation coeffiecient rather we substract it.
Could it be stated as a rule of thumb that if we have one position short and one long, then we have to substract the last part of our variance equation (thinking of diversification effect?!)?
Thanks!
 
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