350 RISK BUDGETING Combining the local factor model with equation (20.
26), we can write the linear factor model in terms of total returns: R(t) = B'it-lfi'it) + u\t) + E^t) + xc(t) (20.27) where xc{t) = Ke{t) X E (t) is a cross term between local returns and exchange rate returns. Equation (20.27) allows us to explain the cross section of international asset returns. So, for example, we can identify a set of factors that explain the cross-sectional dispersion of U.S., European, and Japanese stock returns. Finally, note that in equation (20.27), Fe(t) is not restricted exclusively to so-called local factor returns. As we show later, Fi(t) may include returns to global factors such as the Global Industry Classification Standard (GICS) classifications. Having explained the basic framework for the local and global models, we will now describe how the asset exposures in these models are constructed. Asset Exposures In the linear cross-sectional factor model, exposures are defined at the asset level and then aggregated to generate portfolio exposures. Each asset is related to (i.e., has exposure to) some factor. For example, an asset can have exposure to: 11 Itself. 11 A particular industry or sector. II A country (local market). II A currency. II Investment styles and/or risk factors. Examples Of Asset Exposures An asset's exposure to a particular factor depends on the type of exposure we are dealing with. For example, typically an asset's exposure to an industry is either one (the asset belongs to an industry) or zero (the asset does not belong to an industry). On the other hand, consider the calculation of an asset's exposure to volatility. When computing this exposure, three steps are usually involved: 1. We compute some measure of historical volatility for each asset. This is known as the raw exposure. 2. We define an estimation universe and compute the average volatility exposure across all assets, as well as the standard deviation of volatility exposure (again, across all assets). 3. We standardize the value of each raw volatility exposure by subtracting the mean and then dividing by the standard deviation. The next section discusses various types of exposures covering industries, investment styles, countries, and currencies. Industry Exposures Probably the easiest set of exposures to understand is industry exposures. An asset's exposure to an industry is usually one if it is in that indus-
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