the special cases we discuss in the first two parts of this section.We conclude the section with a discussion of the empirical evidence on the predictions of the perfect markets model for security returns. 2.A.Identical Consumption Opportunity Sets Across Countries Consider a world where goods and financial markets are perfect,so that we have no transportation costs,no tariffs,no taxes,no transaction costs,and no restrictions to short sales. Grauer,Litzenberger,and Stehle (1976)modeled such a world using a state-preference framework.We assume further that there is only one consumption good.?In such a world,every investor has the same consumption and investment opportunity sets regardless of where she resides.Further,the law of one price holds for the consumption good,so that if e(t)is the price of foreign currency at date t,P(t)is the price of the good in the domestic country,and p*(t)is the price in the foreign currency,it must be that P(t)=e(t)P*(t).In such a world,an investor can use the consumption good as the numeraire,so that all prices and returns are expressed in units of the consumption good. We now consider a one-period economy in which real returns are multivariate normal and there is one asset that has a risk-free return in real terms,earning r.Investors care only about the distribution of their real terminal wealth.The properties of the multivariate normal distribution (see Fama,1976,Chapters 4 and 8)imply that: E(g)-r=&+B[E(a)-r], (1) 2 The consumption good can be a basket of goods where the spending proportions on each good are constant. 4
4 the special cases we discuss in the first two parts of this section. We conclude the section with a discussion of the empirical evidence on the predictions of the perfect markets model for security returns. 2. A. Identical Consumption Opportunity Sets Across Countries Consider a world where goods and financial markets are perfect, so that we have no transportation costs, no tariffs, no taxes, no transaction costs, and no restrictions to short sales. Grauer, Litzenberger, and Stehle (1976) modeled such a world using a state-preference framework. We assume further that there is only one consumption good.2 In such a world, every investor has the same consumption and investment opportunity sets regardless of where she resides. Further, the law of one price holds for the consumption good, so that if e(t) is the price of foreign currency at date t, P(t) is the price of the good in the domestic country, and P*(t) is the price in the foreign currency, it must be that P(t) = e(t)P*(t). In such a world, an investor can use the consumption good as the numéraire, so that all prices and returns are expressed in units of the consumption good. We now consider a one-period economy in which real returns are multivariate normal and there is one asset that has a risk-free return in real terms, earning r. Investors care only about the distribution of their real terminal wealth. The properties of the multivariate normal distribution (see Fama, 1976, Chapters 4 and 8) imply that: [ ] d E(r ) r E(r ) r i i id −= + − α β , (1) 2 The consumption good can be a basket of goods where the spending proportions on each good are constant
where E(.)denotes an expectation,ri is the real return on asset i,ra is the real return on the domestic market portfolio,Ba is the domestic beta of asset i defined as Cov(r a)where Var(ra) Cov(.,)denotes a covariance and Var(.)denotes a variance,and o is a constant.If domestic investors can only hold domestic assets and if foreign investors cannot hold domestic assets,then we know that o must be equal to zero for all assets because then the capital asset pricing model (CAPM)must hold in the domestic country.However,in a world where domestic investors have access to foreign assets,the CAPM must hold in real terms using the world market portfolio. Consequently: E(c)-r=B"[E(w)-r], (2) where rw is the real return on the world market portfolio,B is the world beta of asset i defined as CovWe call qution (2)the"world CAPMto conrast it with the traditional Var(r) implementation of the CAPM that uses the domestic market portfolio,which we call the domestic CAPM. We now build on Stulz(1995)to analyze the mistake in using the domestic CAPM when the world CAPM is appropriate.Equations(1)and(2)imply that oi must satisfy: &=[B-Ba][E(w)-r] (3) Equation(3)shows that systematic mistakes are possible when one uses the domestic CAPM and when domestic investors have access to world markets.At the same time,though,equation (3)puts a bound on the economic importance of these mistakes.Since the domestic market 5
5 where E(.) denotes an expectation, ri is the real return on asset i, rd is the real return on the domestic market portfolio, d βi is the domestic beta of asset i defined as i d d Cov(r ,r ) Var(r ) where Cov(.,.) denotes a covariance and Var(.) denotes a variance, and αi is a constant. If domestic investors can only hold domestic assets and if foreign investors cannot hold domestic assets, then we know that αi must be equal to zero for all assets because then the capital asset pricing model (CAPM) must hold in the domestic country. However, in a world where domestic investors have access to foreign assets, the CAPM must hold in real terms using the world market portfolio. Consequently: [ ] w E(r ) r E(r ) r i iw −= − β , (2) where rw is the real return on the world market portfolio, w βi is the world beta of asset i defined as i w w Cov(r , r ) Var(r ) . We call equation (2) the “world CAPM” to contrast it with the traditional implementation of the CAPM that uses the domestic market portfolio, which we call the domestic CAPM. We now build on Stulz (1995) to analyze the mistake in using the domestic CAPM when the world CAPM is appropriate. Equations (1) and (2) imply that αi must satisfy: [ ] w wd α β ββ i i di w =− − E(r ) r . (3) Equation (3) shows that systematic mistakes are possible when one uses the domestic CAPM and when domestic investors have access to world markets. At the same time, though, equation (3) puts a bound on the economic importance of these mistakes. Since the domestic market
portfolio has a beta of one,the weighted average of the alphas in equation(3)must be equal to zero.To understand the nature of the mistakes,we have to understand how the world beta of asset i,B,differs from the product of the world beta of the domestic market,B,and of the domestic beta of asset i,Bd.Using the multivariate normal distribution,we can write the return of asset i as: 5-r=&+B[-r]+e. (4) Similarly,the return of the domestic market portfolio can be written as: ra -r=B rw -r e. (5) Substituting equation(5)into equation(4),we have: 5-r=&+[B[w-r]+ea+e. (6) The world market beta of asset i is therefore: B=阳+ Cov(ed, (7) Var(r) Substituting(7)into(3),we get the result of Stehle(1977)that the pricing mistake from using the domestic CAPM when the global CAPM is appropriate is: 6
6 portfolio has a beta of one, the weighted average of the alphas in equation (3) must be equal to zero. To understand the nature of the mistakes, we have to understand how the world beta of asset i, w βi , differs from the product of the world beta of the domestic market, w βd , and of the domestic beta of asset i, d βi . Using the multivariate normal distribution, we can write the return of asset i as: [ ] d d i i id i r r rr −= + − + α β ε . (4) Similarly, the return of the domestic market portfolio can be written as: [ ] w w d dw d r r rr −= − + β ε . (5) Substituting equation (5) into equation (4), we have: [ ] dw w d i i i dw d i r r rr −= + − + + α β β ε ε . (6) The world market beta of asset i is therefore: d w dw i w i id w Cov( , r ) Var(r ) ε β ββ = + . (7) Substituting (7) into (3), we get the result of Stehle (1977) that the pricing mistake from using the domestic CAPM when the global CAPM is appropriate is:
=Cov(e)[E()-r]. (8) Var(r) Equation(8)shows that the domestic CAPM understates the return of assets whose market model residual is positively correlated with the world market portfolio.However,the domestic CAPM correctly prices those assets whose market model residual is uncorrelated with the world market portfolio.If high domestic beta stocks have risk diversifiable internationally but not domestically,using the domestic CAPM inappropriately would lead one to conclude that the security market line is too flat.One would expect multinational corporations to have returns correlated with foreign markets in such a way that the domestic market portfolio return does not capture all their systematic risk. Our analysis so far shows that,even when the assumptions required for the CAPM to hold are made,the domestic CAPM does not hold in global markets.One inevitably makes pricing mistakes using the domestic CAPM as long as Cov()is not zero for every security.There exists an upper bound on the absolute value of the pricing mistakes.Let Rbe the R-square of a regression of the return of asset i on asset j.With this notation,the bound can be written as: 风≤-R-心E)-1 (9) Consider an asset that has a variance that is four times the variance of the world market portfolio.If the variance of security i is four times the variance of the world market,the R-square of a regression of the security on the domestic market portfolio is 0.3,the R-square of a regression of the world market on the domestic market is 0.2,and the world market risk premium is 6%,the bound is 9%.Consider now an asset with the same volatility but in the U.S.,where a 7
7 [ ] d i w i w w Cov( , r ) E(r ) r Var(r ) ε α = − . (8) Equation (8) shows that the domestic CAPM understates the return of assets whose market model residual is positively correlated with the world market portfolio. However, the domestic CAPM correctly prices those assets whose market model residual is uncorrelated with the world market portfolio. If high domestic beta stocks have risk diversifiable internationally but not domestically, using the domestic CAPM inappropriately would lead one to conclude that the security market line is too flat. One would expect multinational corporations to have returns correlated with foreign markets in such a way that the domestic market portfolio return does not capture all their systematic risk. Our analysis so far shows that, even when the assumptions required for the CAPM to hold are made, the domestic CAPM does not hold in global markets. One inevitably makes pricing mistakes using the domestic CAPM as long as d Cov( ,r ) i w ε is not zero for every security. There exists an upper bound on the absolute value of the pricing mistakes. Let 2 Rij be the R-square of a regression of the return of asset i on asset j. With this notation, the bound can be written as: [ ] 2 2 ( ) (1 )(1 ) ( ) ( ) i i id wd w w Var r R R Er r Var r α ≤− − − . (9) Consider an asset that has a variance that is four times the variance of the world market portfolio. If the variance of security i is four times the variance of the world market, the R-square of a regression of the security on the domestic market portfolio is 0.3, the R-square of a regression of the world market on the domestic market is 0.2, and the world market risk premium is 6%, the bound is 9%. Consider now an asset with the same volatility but in the U.S., where a
regression of the world market portfolio on the domestic market portfolio would have an R- square of at least 0.8.In this case,the bound would be 4.49%.If we are willing to make assumptions about the correlation between the domestic market model residual of an asset and the return of the world market portfolio that is not explained by the domestic market portfolio,we can compute the limits of the pricing mistakes.Consider,for example,a correlation of 0.2.In this case,the asset in a country whose market portfolio is poorly explained by the world market portfolio has a pricing mistake no greater than 1.8%,while the U.S.asset has a pricing mistake of 0.88% It follows from this analysis that using a domestic CAPM is more of a problem in countries whose market has a lower beta relative to the world market portfolio.Since the U.S.is a country where the R-square of a regression of the world market portfolio on the domestic market portfolio is high,the mistakes made by using the domestic market portfolio for U.S.risky assets are smaller than for risky assets of other countries where the R-square of similar regressions is much smaller.Further,the mistakes are likely to be less important for larger firms simply because the domestic market portfolio explains more of the return of these firms. We now turn to the issue of the determinants of the market risk premium.We keep the same assumptions,but now we have an investor who chooses her portfolio to maximize her expected utility of terminal real wealth,E[U(W)],with U(W)strictly increasing and concave in W. Consider first an investor who can only hold U.S.assets,so that the investor holds the U.S. market portfolio.Using a first-order Taylor-series expansion,the first-order conditions of the investor's portfolio choice problem imply that: E(ra)-r=TRVar(ra), (10) 8
8 regression of the world market portfolio on the domestic market portfolio would have an Rsquare of at least 0.8. In this case, the bound would be 4.49%. If we are willing to make assumptions about the correlation between the domestic market model residual of an asset and the return of the world market portfolio that is not explained by the domestic market portfolio, we can compute the limits of the pricing mistakes. Consider, for example, a correlation of 0.2. In this case, the asset in a country whose market portfolio is poorly explained by the world market portfolio has a pricing mistake no greater than 1.8%, while the U.S. asset has a pricing mistake of 0.88%. It follows from this analysis that using a domestic CAPM is more of a problem in countries whose market has a lower beta relative to the world market portfolio. Since the U.S. is a country where the R-square of a regression of the world market portfolio on the domestic market portfolio is high, the mistakes made by using the domestic market portfolio for U.S. risky assets are smaller than for risky assets of other countries where the R-square of similar regressions is much smaller. Further, the mistakes are likely to be less important for larger firms simply because the domestic market portfolio explains more of the return of these firms. We now turn to the issue of the determinants of the market risk premium. We keep the same assumptions, but now we have an investor who chooses her portfolio to maximize her expected utility of terminal real wealth, E[U(W)], with U(W) strictly increasing and concave in W. Consider first an investor who can only hold U.S. assets, so that the investor holds the U.S. market portfolio. Using a first-order Taylor-series expansion, the first-order conditions of the investor’s portfolio choice problem imply that: R E(r ) r T Var(r ) d d − = , (10)