Volatility Decomposition

The Australian economic minister would definitely want to watch the country's economic data for signs of overheating or cooling off. But the economies of other countries that depend on Australia for its natural resources—like China—must also be watched closely. How might you describe the economies of Australia and China?

No. Both countries have growing economies, so they're not volatile.

That's it! The two economies are correlated, as Australia depends on China to purchase its natural resources, and China relies on Australia to provide them. And when it comes to foreign investing, these types of relationships are important in understanding the volatility in currency risk.

That's not it. China depends on Australia for some of its natural resources.

To start understanding the risks associated with foreign investing, you need to break down the domestic-currency return into its two parts. Which part would contain risk?

Not quite. Although that's the return in the foreign currency, it's not going to capture the change versus the domestic currency.

No, actually. The spot change will capture the domestic currency impact, but there's also the foreign currency return to consider.

Yes! Both components contain some risk because they are exposed to that foreign country's economic variables and its currency. So to understand volatility, you need to start with both the foreign currency return and the percent change in the spot quote. That gives you the following approximate equation, which assumes that _R__{_FC_} and _R__{_FX_} are small. $$\displaystyle R_{DC} \approx R_{FC} + R_{FX}$$ _R__{_DC_} is the domestic currency return, _R__{_FC_} is the foreign currency return, and _R__{_FX_} is the percent change of the foreign currency against the domestic currency.

You can build the approximate equation out using the common variance equation $$\displaystyle \sigma^2(\omega_{x}X + \omega_yY) = \omega^2\sigma^2(X) +$$ $$\omega^2\sigma^2(Y) + 2\omega_x\omega_y\sigma(X)\sigma(Y)\rho(X,Y)$$ where _X_ and _Y_ are random variables, ω are weights attached to _X_ and _Y_, σ² is the variance of the random variable, σ is the standard deviation of the random variable, and ρ represents the correlation between two random variables.

Then, you can apply the specific domestic-currency return equation to the common variance equation to calculate the variance of the domestic-currency return. $$\displaystyle \sigma^2(R_{DC}) \approx \sigma^2(R_{FC}) + \sigma^2(R_{FX}) + 2\sigma(R_{FC})\sigma(R_{FX})\rho(R_{FC}R_{FX})$$ The equation shows the impact of the exchange rate exposure on the volatility of the domestic-currency return. If there were no exchange rate exposure, what would the variance of the domestic-currency return be?

No. Foreign-currency return makes part of the domestic-currency return.

Not quite. The standard deviation is the square root of the variance.

That's right! If there isn't foreign exchange risk, then that leaves the foreign-currency return variance as the only component of volatility. That means that by including the exchange rate exposure, it usually increases the domestic-currency return variance. And that's also the case for a portfolio of multiple foreign holdings. The variance of the domestic-currency returns for the overall foreign asset portfolio is the following. $$\displaystyle \sigma^2(\omega_1R_1 + \omega_2R_2) \approx \omega^{2}_{1}\sigma^2 (R_1) +$$ $$\omega^{2}_{2}\sigma^2 (R_2) + 2\omega_1\omega_2\sigma (R_1)\sigma(R_2) \rho(R_1,R_2)$$

You may be thinking that's a lot of variables, and actually, you're right. The variance of each foreign-currency return is made up of the impact of multiple currency exposures that lie within the portfolio. So, in reality, this equation doesn't even begin to capture the necessary inputs for a full picture of the domestic-currency return variance of the overall portfolio. But that doesn't mean that it's not useful for analysis. Since the foreign portfolio can have multiple exposures with various impacts, you need to know more than just the variance of each foreign-currency return and the exchange rate movement. What else will you need to know to calculate the domestic-currency return variance of the overall portfolio?

No. That's not going to capture all the necessary correlations.

Not so. GDP is only a small part of the large number of variables that impact currency holdings.

Exactly! You'll need to know all the interactions between the foreign-currency return and the exchange rates to accurately be able to calculate the domestic-currency return variance of the overall portfolio. That's because all the various interactions between the currencies and holdings will change and modify over time. The overall variance will also depend on the weights within the portfolio, and, if short selling is allowed, some of the portfolio weights can be negative. This can offer valuable diversification benefits to the investor.

The overall portfolio equation can also be used to predict expected domestic-currency portfolios, but that means that the asset manager will need to replace historical variances and covariance with expected future values. Besides the fact that it will involve lots of expected variables, many asset managers rely on historical price patterns and volatility/correlation measures. That can be very dangerous. Why do you think that is?

Clearly yes! Data points and sources change over time, so using historical data can be very dangerous if the historical information used is inconsistent with the current time period. Asset managers must use judgment to develop expectations. Typically, asset managers will turn to the market opinion to guide their thoughts on the variance of each foreign-currency asset return, the variance of future exchange rate movements, and the interaction of the variables over time. In most cases, computers and simulations can help guide the process.

No. Variances/correlations don't carry over through time. They can change.

No. Opinions of financial data can change over time, so that's not a consistent data source.

To sum it up: [[summary]]

Volatile

Correlated

Independent

Foreign-currency return

Percent change in the spot quote

Both the foreign-currency return and the percent change in the spot quote

The variance of the foreign-currency return

The standard deviation of the foreign-currency return

The standard deviation of the domestic-currency return

The economic data behind every currency holding

The GDP correlations between each currency holding

The interactions between the foreign-currency return and exchange rates

Data measures can drift over time

Variance/correlations carry over through time

Samples and surveys remain consistent over long periods

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