I have hard time understanding the following:
Nguyen now turns her attention to calculating the portfolio’s investment risk [σ(RDC)]. To calculate the expected risk for the domestic-currency return, the currency risk of RFX needs to be multiplied by the known return on the treasury bills. The portfolio’s investment risk, σ(RDC), is found by calculating the standard deviation of the right-hand-side of:
RDC = (1 + RFC)(1 + RFX) – 1
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:
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%
The standard deviation of this amount—that is, σ(RDC)—is 9.1%. Note that in the expression, all of the units are in percent, so for example, 8.3% is equivalent to 0.083 for calculation purposes. The careful reader may also note that Nguyen is able to use an exact expression for calculating the variance of the portfolio returns, rather than the approximate expressions shown in Equations 3 and 5. This is because, with risk-free foreign-currency assets, the variance of these foreign-currency returns σ2(RFC) is equal to zero.
Nguyen now considers an alternative scenario in which, instead of an equally weighted portfolio (where the ωi = 0.5), the fund has a long exposure to the New Zealand asset and a short exposure to the Australian asset (i.e., the ωi are +1 and −1, respectively; this is similar to a highly leveraged carry trade position). Putting these weights into Equations 2 and 4 leads to
RDC = –1.0(9.2%) + 1.0(11.3%) = 2.1%
σ2(RDC) = (1.0)2(8.3%)2 + (1.0)2(10.6%)2 + [–2.0(8.3%)(10.6%)0.85]
= 0.3%
The standard deviation—that is, σ(RDC)—is now 5.6%, less than either of the expected risks for foreign-currency asset returns (results A and B). Nguyen concludes that having long and short positions in positively correlated currencies can lead to much lower portfolio risk, through the benefits of cross hedging. (Nguyen goes on to calculate that if the expected correlation between USD/AUD and USD/NZD increases to 0.95, with all else equal, the expected domestic-currency return risk on the long–short portfolio drops to 3.8%.)
(Institute 361-362)
Institute, CFA. 2016 CFA Level III Volume 3 Economic Analysis and Asset Allocation. CFA Institute, 07/2015. VitalBook file.
Nguyen now turns her attention to calculating the portfolio’s investment risk [σ(RDC)]. To calculate the expected risk for the domestic-currency return, the currency risk of RFX needs to be multiplied by the known return on the treasury bills. The portfolio’s investment risk, σ(RDC), is found by calculating the standard deviation of the right-hand-side of:
RDC = (1 + RFC)(1 + RFX) – 1
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:
- The expected risk (i.e., standard deviation) of the domestic-currency return for the Australian asset is equal to (1.04) × 8% = 8.3%.
- 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%.
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%
The standard deviation of this amount—that is, σ(RDC)—is 9.1%. Note that in the expression, all of the units are in percent, so for example, 8.3% is equivalent to 0.083 for calculation purposes. The careful reader may also note that Nguyen is able to use an exact expression for calculating the variance of the portfolio returns, rather than the approximate expressions shown in Equations 3 and 5. This is because, with risk-free foreign-currency assets, the variance of these foreign-currency returns σ2(RFC) is equal to zero.
Nguyen now considers an alternative scenario in which, instead of an equally weighted portfolio (where the ωi = 0.5), the fund has a long exposure to the New Zealand asset and a short exposure to the Australian asset (i.e., the ωi are +1 and −1, respectively; this is similar to a highly leveraged carry trade position). Putting these weights into Equations 2 and 4 leads to
RDC = –1.0(9.2%) + 1.0(11.3%) = 2.1%
σ2(RDC) = (1.0)2(8.3%)2 + (1.0)2(10.6%)2 + [–2.0(8.3%)(10.6%)0.85]
= 0.3%
The standard deviation—that is, σ(RDC)—is now 5.6%, less than either of the expected risks for foreign-currency asset returns (results A and B). Nguyen concludes that having long and short positions in positively correlated currencies can lead to much lower portfolio risk, through the benefits of cross hedging. (Nguyen goes on to calculate that if the expected correlation between USD/AUD and USD/NZD increases to 0.95, with all else equal, the expected domestic-currency return risk on the long–short portfolio drops to 3.8%.)
(Institute 361-362)
Institute, CFA. 2016 CFA Level III Volume 3 Economic Analysis and Asset Allocation. CFA Institute, 07/2015. VitalBook file.