Notes

Chapter 1

1. These results are in Brotherson et al. (2013), Da, Guo, and Jagannathan (2012), and Welch (2009).

2. See Brotherson et al. (2013).

3. See Luehrman and Heilprin (2009); Welch (2009); Donaldson, Kamstra, and Kramer (2010); and Fernandez, Aguirreamalloa, and Avendaño (2012).

4. The distinction between the global market index and the world market index is noted in Stulz (1995b).

5. See Krapl and O’Brien (2016).

6. See Bryan (2007) and Maldonado and Saunders (1983).

7. See Baruch, Karolyi, and Lemmon (2007). For trends in cross-listing, see Halling et al. (2008).

8. For dual-listed shares, see Froot and Dabora (1999) and de Jong, Rosenthal, and Van Dijk (2009). For Chinese A-shares and B-shares, see Mei, Scheinkman, and Xiong (2009). Also, see Arquette, Brown, and Burdekin (2008).

Chapter 2

1. FX exposure is explained in detail in O’Brien (2017).

2. The currency index could also be constructed to contain currency H, so that the index’s composition is the same for every currency, in which case there is no w_H term when finding . However, it is somewhat confusing to think about an index’s change against currency H when the index also contains currency H, so we restrict the currency index to consist of only foreign currencies from the perspective of currency H and include the adjustment for w_H.

3.

4. Stulz (1995a, 1995b) also “anchored” international GRP estimates to the local U.S. market risk premium estimate.

5. The two-factor ICAPM version represented by equations (2.1) and (2.2) is based on a simplification by Ross and Walsh (1983) of the “Solnik (1974) – Sercu (1980) special case” of the general ICAPM (Adler and Dumas 1983). Ignoring currency superscripts, the fundamental risk-pricing relation of the Solnik-Sercu model, adapted from equation (2.9) in Dumas (1994), is: , where RP_i is asset i’s required risk premium, equal to asset’s required rate of return, k_i, minus the nominal currency-H risk-free rate; R_i is asset i’s return, consisting of the asset’s local currency return and the change in the value of the asset’s currency versus currency H; R_G is the return in currency H on the unhedged global market index; x^H/C is the percentage change in currency C versus currency H; W_C is the wealth of the economy using currency C; q_C is the average degree of risk aversion of investors in the economy using currency C; W is total global wealth; and q is the global (harmonic mean) degree of risk aversion (over all economies, including H): 1/q = [Σ_C(W_C/q_C)]/W.

The currency risk factors may be aggregated into a portfolio, but with generally unobservable weights (Solnik 1997). However, assuming the same average degree of risk aversion across economies, q_C = Θ for all C (including H) implies that q in the risk-pricing equation also takes the value Θ, the global market price of risk. In this case, the weights in the currency portfolio are W_C/W (Ross and Walsh 1983).

To make the currency weights into foreign currency portfolio weights that sum to 1, let w_C = W_C/(W − W_H), the percentage of economy C’s wealth of the world wealth outside of economy H. Then W_C/W = (1 − w_H) W_C. The resulting simplified risk-pricing model in only two factors is: E(R_i) = Θcov(R_i,R_G) + (1 − Θ)(1 - w_H)cov(R_i,R_X),  where R_X is the return in currency H on a wealth-weighted portfolio of all other currencies, .

Equation (2.2) follows by noting that and and using , the definitions of the components of the global risk premium and foreign currency index risk premium. Equation (2.1) requires aggregating the simplified risk-pricing model twice, first over all assets to get the global market risk premium, and then over all currencies to get the foreign currency index risk premium. Equation (2.1) follows by solving the two aggregate equations simultaneously for Θ and (1 − Θ).

The equal risk aversion assumption may seem restrictive, but why would the average degree of risk aversion in the Eurozone be much different than that in the United States, or the United Kingdom, and so forth? It seems likely that China’s degree of risk aversion is relatively low, but this is inconsequential because the yuan does not have much volatility versus the US dollar. Moreover, the assumption is not nearly as unrealistic as the standard “homogeneous expectations” assumption of all CAPMs, that all investors have the same estimates of the expected rate of return and risk parameters for all assets. In his GCAPM analysis, Stulz (1995b, p. 12) seems to have no problem with the assumption that “investors are the same across countries in their preferences.”

6. See Krapl and O’Brien (2016).

7. Sercu (1980) and Ross and Walsh (1983) show that if the GCAPM holds from one currency perspective, the consistent risk-return model from any other currency perspective must be a two-factor model, with the FX rate between the currencies being the second factor.

Chapter 3

1. See “Bank of America Roundtable on Evaluating and Financing Foreign Direct Investment” (1996).

2. This point is made in Pettit, Ferguson, and Gluck (1999) and Block (2003). Also, see Krüger, Landier, and Thesmar (2015).

3. Weston (1973) pioneered the idea using the traditional CAPM for finding an operation-specific cost of capital.

4. Desai (2006) uses this formula in his Harvard case on the cost of capital for AES Corporation.

5. Other country beta estimates are in Perold (2004).

6. See Lessard (1996). The only difference is that the betas in the original Lessard analysis are relative to the local market index, whereas the betas in equation (3.2) are relative to the global market index.

7. For further details, see Bekaert et al. (2016).

8. Abuaf (2015) advocates Φ_i = 0.35, 0.70, and 1 for the low, medium, and high political risk exposure categories, but he also uses the full sovereign CDS yield for the political risk premium. Basically, the Abuaf (2015) model makes the simplifying assumption that all countries have the same ratio of political risk premium to sovereign risk premium of about 0.70.

9. Damodaran (2003) pioneered the idea of this type of exposure, but called the factor “country risk” instead of political risk, and used the exposure notation λ_i instead of Φ_i. Technically, country risk includes the equity market’s volatility, and so political risk is an even smaller portion of country risk than of sovereign risk, for example, Click and Wiener (2010).

10. Equation (3.4) adapts ideas from diverse places in the literature, such as Zenner and Akaydin (2002), Godfrey and Espinosa (1996), Harvey (2000), and Damodaran, op. cit. Also, see Bekaert, et al., op. cit. for insights and a review of this literature.

Chapter 4

1. To derive equation (4.1), start with equation (1.1a): . Multiply out the right-hand side to get . Take expectations to get . The term is equal to , so we substitute and get , which simplifies to . Because we get that . Expressing the expectations as equilibrium concepts, we get the cost of capital conversion formula in equation (4.1), where $ = H and € = C.

2. To derive equation (4.3), start with the definition: . Thus, . Rewrite the covariance term: where the new covariance term is the covariance between the percentage deviation of from , and the percentage deviation of from . Using the approximation, , the covariance term becomes , which is also equal to . The last expression may be restated in terms of two familiar variables: (a) the FX operating exposure to the US dollar, and (b) the volatility of the euro versus the US dollar, . By definition, . Substituting, we get that , which simplifies to , which is approximated by equation (4.3).

Chapter 5

1. Some of the material is based on Butler, O’Brien, and Utete (2013).

2. Siegel’s paradox is based on the notion E(X/Y) cannot be equal 1/E(Y/X), even though X/Y = 1/(Y/x). Equation (5.2) is in Solnik (1993).

3. Estimates of time-0 FX misvaluation are often based on purchasing power parity (PPP) violations, as in the Big Mac Index and adjusted Big Mac Index. See O’Brien (2016), who also shows how to bring in the uncovered interest rate parity (UIRP) condition. This approach could be adjusted to use a version of the UIRP condition with a currency risk premium adjustment like that covered in Chapter 4.

4. Luehrman and Quinn (2010) address this issue in a Harvard case, basing intrinsic FX forecasts on purchasing power parity.

5. See Harris and Ravenscraft (1991); Dewenter (1995); and Erel, Liao, and Weisbach (2012). See also Baker, Foley, and Wurgler (2009), who report that FDI flows increase sharply with source-country stock market valuations.

Chapter 6

1. Start with the linear return-generating model, , where is asset i’s FX exposure to the euro. Take the covariance of both sides of this equation with the return on the global market (in US dollars), to get . Divide both sides by the variance of , and note that the definition of a beta is covariance with , divided by the variance of , to get . Repeat for asset j to get Subtract the second beta equation from the first and note that if assets i and j have different betas only because of FX exposure, then The result is equation (6.1), where H is $ and C is €.