To consider the validity of the Perold and Schulman argument on theoretical grounds,note that it implies that foreign investors short the domestic currency for an amount equal to their holdings of domestic currency risky assets.Domestic investors must be the counterparties of these short sales,so that if foreign investors hedge all their domestic exchange rate risk,domestic investors must hold an amount of the domestic risk-free asset equal to the short position in domestic risky assets of foreign investors.In this case,foreign investors take no foreign exchange risk.This makes sense if foreign investors receive no reward for being long the domestic currency,which requires that the expected excess return of the domestic currency risk- free asset over the foreign risk-free asset is zero. The excess return of the domestic currency risk-free asset over the foreign risk-free asset is rdt +o.dt-de/e-r*dt.Foreign investors will not take a long position in the domestic risk- free asset if the expected excess return of the domestic currency risk-free asset over the foreign risk-free asset for that investor is zero.That is,the following condition must hold:rdt +o.dt- uedt-r*dt=0.For the Perold and Schulman argument to be theoretically valid,it must be that when this condition is satisfied,domestic investors do not take exchange rate risk either. However,the excess return on the foreign risk-free asset over the domestic risk-free asset for a domestic investor is r*dt de/e-rdt,so that the expected excess return for a domestic investor on a long position in the foreign risk-free asset is r*dt uedt-rdt.If the expected excess return of holding the domestic risk-free asset is zero for foreign investors,then the expected excess return of holding the foreign risk-free asset is positive and equal to o2 dt for a domestic investor. Consequently,it is not possible for both domestic and foreign investors to contemplate an 14
14 To consider the validity of the Perold and Schulman argument on theoretical grounds, note that it implies that foreign investors short the domestic currency for an amount equal to their holdings of domestic currency risky assets. Domestic investors must be the counterparties of these short sales, so that if foreign investors hedge all their domestic exchange rate risk, domestic investors must hold an amount of the domestic risk-free asset equal to the short position in domestic risky assets of foreign investors. In this case, foreign investors take no foreign exchange risk. This makes sense if foreign investors receive no reward for being long the domestic currency, which requires that the expected excess return of the domestic currency riskfree asset over the foreign risk-free asset is zero. The excess return of the domestic currency risk-free asset over the foreign risk-free asset is rdt + 2 σ e dt – de/e – r*dt. Foreign investors will not take a long position in the domestic riskfree asset if the expected excess return of the domestic currency risk-free asset over the foreign risk-free asset for that investor is zero. That is, the following condition must hold: rdt + 2 σ e dt – µedt – r*dt = 0. For the Perold and Schulman argument to be theoretically valid, it must be that when this condition is satisfied, domestic investors do not take exchange rate risk either. However, the excess return on the foreign risk-free asset over the domestic risk-free asset for a domestic investor is r*dt + de/e – rdt, so that the expected excess return for a domestic investor on a long position in the foreign risk-free asset is r*dt + µedt – rdt. If the expected excess return of holding the domestic risk-free asset is zero for foreign investors, then the expected excess return of holding the foreign risk-free asset is positive and equal to 2 σ e dt for a domestic investor. Consequently, it is not possible for both domestic and foreign investors to contemplate an
expected excess return of zero when investing in the other country's risk-free asset.In a symmetric world,one would think neither domestic nor foreign investors would be rewarded for bearing exchange rate risk since rewarding one group of investors would seem to penalize the other group of investors.Jensen's inequality makes this reasoning incorrect.If foreign investors receive no reward for being long the foreign currency risk-free asset,then domestic investors are rewarded for being long the foreign currency risk-free asset.In equilibrium,in a symmetric world where consumption opportunity sets differ across countries,there will therefore exist some reward for taking currency risk,so that exposures to foreign exchange rates will be priced in assets. To present a key insight of Solnik's model,let's use his assumption that the domestic risky assets have domestic currency returns uncorrelated with the exchange rate and that the foreign risky assets have foreign currency returns uncorrelated with the exchange rate.In this case,the domestic investor can hedge her holdings of the foreign risky assets against exchange rate risk by borrowing an amount in foreign currency equal to her holdings of foreign currency risky assets. Similarly,foreign investors hedge their currency risk by borrowing in domestic currency an amount equal to their holdings of domestic currency risky assets.If there is no reward for taking foreign currency risk,domestic investors will short the foreign currency risk-free asset for an amount exactly equal to their investment in foreign currency risky assets and will bear no foreign exchange rate risk.As the reward for taking foreign exchange rate risk increases,domestic 6 This result holds more generally and is known as Siegel's Paradox(Siegel(1972)).This Paradox states that it is not possible for the forward exchange rate to be an unbiased predictor of the spot exchange rate from the perspective of investors in the domestic as well as in the foreign country.If the forward exchange rate is F and the spot exchange rate at maturity of the contract is e,the forward exchange rate is an unbiased predictor from the perspective of the domestic country if F =E(e).However,from the perspective of investors in the foreign country,the forward exchange rate is an unbiased predictor if(1/F)=E(1/e). Because of Jensen's inequality,E(e)is not equal to 1/E(1/e)as long as e is random.Black(1990)argues that investors bear exchange rate risk in equilibrium because of Siegel's paradox.Solnik (1993)clarifies the implication of Siegel's paradox for the pricing of exchange rate risk. 15
15 expected excess return of zero when investing in the other country’s risk-free asset.6 In a symmetric world, one would think neither domestic nor foreign investors would be rewarded for bearing exchange rate risk since rewarding one group of investors would seem to penalize the other group of investors. Jensen’s inequality makes this reasoning incorrect. If foreign investors receive no reward for being long the foreign currency risk-free asset, then domestic investors are rewarded for being long the foreign currency risk-free asset. In equilibrium, in a symmetric world where consumption opportunity sets differ across countries, there will therefore exist some reward for taking currency risk, so that exposures to foreign exchange rates will be priced in assets. To present a key insight of Solnik’s model, let’s use his assumption that the domestic risky assets have domestic currency returns uncorrelated with the exchange rate and that the foreign risky assets have foreign currency returns uncorrelated with the exchange rate. In this case, the domestic investor can hedge her holdings of the foreign risky assets against exchange rate risk by borrowing an amount in foreign currency equal to her holdings of foreign currency risky assets. Similarly, foreign investors hedge their currency risk by borrowing in domestic currency an amount equal to their holdings of domestic currency risky assets. If there is no reward for taking foreign currency risk, domestic investors will short the foreign currency risk-free asset for an amount exactly equal to their investment in foreign currency risky assets and will bear no foreign exchange rate risk. As the reward for taking foreign exchange rate risk increases, domestic 6 This result holds more generally and is known as Siegel’s Paradox (Siegel (1972)). This Paradox states that it is not possible for the forward exchange rate to be an unbiased predictor of the spot exchange rate from the perspective of investors in the domestic as well as in the foreign country. If the forward exchange rate is F and the spot exchange rate at maturity of the contract is e, the forward exchange rate is an unbiased predictor from the perspective of the domestic country if F = E(e). However, from the perspective of investors in the foreign country, the forward exchange rate is an unbiased predictor if (1/F) = E(1/e). Because of Jensen’s inequality, E(e) is not equal to 1/E(1/e) as long as e is random. Black (1990) argues that investors bear exchange rate risk in equilibrium because of Siegel’s paradox. Solnik (1993) clarifies the implication of Siegel’s paradox for the pricing of exchange rate risk
investors decrease the extent to which they hedge the foreign exchange risk associated with their holdings of assets risky in foreign currency. We assume that there are N risky assets for each investor.Let the N-th risky asset be the foreign currency risk-free asset for a domestic investor and the domestic currency risk-free asset for a foreign investor.Define w as the fraction of her wealth the i-th domestic investor puts in the N-th domestic risky asset,which is the foreign currency risk-free asset for her,and w be the fraction of her wealth the j-th foreign investor puts in the N-th foreign risky asset,which is the domestic currency risk-free asset for her.Solving for these asset demands,we have: (15) =rr (16) where T and T'are,respectively,the relative risk tolerances of the domestic investor and of the foreign investor,and w and wa are,respectively,the fraction of the wealth of the i-th domestic investor invested in foreign risky assets and of the wealth of the i-th foreign investor invested in domestic risky assets.Let Te be the wealth-weighted average of the risk-tolerances of the domestic investors and T'the corresponding quantity for foreign investors.Further,let Wd be the aggregate wealth of domestic investors,W the aggregate wealth of foreign investors,and Ww equals the sum of wd and w. We can use equations(15)and (16)to obtain the equilibrium risk premium on an investment in a foreign risk-free bond financed through risk-free borrowing in the domestic currency. Assuming that there is no net supply of the risk-free asset,the domestic holdings of the domestic risk-free asset must correspond to the short position of foreign investors in the same asset. 16
16 investors decrease the extent to which they hedge the foreign exchange risk associated with their holdings of assets risky in foreign currency. We assume that there are N risky assets for each investor. Let the N-th risky asset be the foreign currency risk-free asset for a domestic investor and the domestic currency risk-free asset for a foreign investor. Define di wN as the fraction of her wealth the i-th domestic investor puts in the N-th domestic risky asset, which is the foreign currency risk-free asset for her, and fj wN be the fraction of her wealth the j-th foreign investor puts in the N-th foreign risky asset, which is the domestic currency risk-free asset for her. Solving for these asset demands, we have: * di d di e Ni f 2 e r r wT w µ σ + − = − , (15) 2 fj f fj e e Nj d 2 e r r* wT w µ σ σ − +− = − , (16) where d Ti and f Tj are, respectively, the relative risk tolerances of the domestic investor and of the foreign investor, and di wf and fi wd are, respectively, the fraction of the wealth of the i-th domestic investor invested in foreign risky assets and of the wealth of the i-th foreign investor invested in domestic risky assets. Let Td be the wealth-weighted average of the risk-tolerances of the domestic investors and Tf the corresponding quantity for foreign investors. Further, let Wd be the aggregate wealth of domestic investors, Wf the aggregate wealth of foreign investors, and Ww equals the sum of Wd and Wf . We can use equations (15) and (16) to obtain the equilibrium risk premium on an investment in a foreign risk-free bond financed through risk-free borrowing in the domestic currency. Assuming that there is no net supply of the risk-free asset, the domestic holdings of the domestic risk-free asset must correspond to the short position of foreign investors in the same asset
Another way to put this is that a dollar invested in the domestic risk-free asset by domestic investors is a dollar that foreign investors borrow from domestic investors.Equilibrium in the market for the domestic risk-free asset therefore requires that: Wd-waW"+T'w r*+ue-r Twd +Tw o (17) where w is the weight of domestic currency risky assets in the world market portfolio.The numerator on the right-hand side of this equation is the domestic currency expected excess return on an investment in the foreign risk-free bond.Alternatively,it is the expected gain from a long forward position to buy the foreign currency.This is the forward risk premium for a domestic investor. An important implication of equation(17)is that the forward risk premium can be different from zero when(1)the foreign exchange rate is uncorrelated with equities in their own currency and(2)there is no net supply of riskless bonds in either domestic or foreign currency.?In such a world,there would be no risk premium for foreign exchange in the world CAPM in real terms. The domestic currency return on foreign stocks would depend on the exchange rate,but the real return of foreign stocks would not.Consequently,the beta of foreign exchange with respect to the world market portfolio would be zero.In contrast,foreign exchange rate risk would be priced in Solnik's model so that the markets for the foreign currency and domestic currency risk-free assets are in equilibrium.Using the domestic CAPM leads to pricing mistakes in this model not only 7One might argue that assuming no riskless bonds outstanding is unrealistic since there are government bonds outstanding.However,government bonds are obligations of domestic residents,so that in a world where the Ricardian equivalence result holds,one can view domestic residents as supplying these bonds (see Barro,1974). 17
17 Another way to put this is that a dollar invested in the domestic risk-free asset by domestic investors is a dollar that foreign investors borrow from domestic investors. Equilibrium in the market for the domestic risk-free asset therefore requires that: d ww ff d e dd ff 2 e W w W T W r* r TW TW µ σ − + +− = + , (17) where w wd is the weight of domestic currency risky assets in the world market portfolio. The numerator on the right-hand side of this equation is the domestic currency expected excess return on an investment in the foreign risk-free bond. Alternatively, it is the expected gain from a long forward position to buy the foreign currency. This is the forward risk premium for a domestic investor. An important implication of equation (17) is that the forward risk premium can be different from zero when (1) the foreign exchange rate is uncorrelated with equities in their own currency and (2) there is no net supply of riskless bonds in either domestic or foreign currency.7 In such a world, there would be no risk premium for foreign exchange in the world CAPM in real terms. The domestic currency return on foreign stocks would depend on the exchange rate, but the real return of foreign stocks would not. Consequently, the beta of foreign exchange with respect to the world market portfolio would be zero. In contrast, foreign exchange rate risk would be priced in Solnik’s model so that the markets for the foreign currency and domestic currency risk-free assets are in equilibrium. Using the domestic CAPM leads to pricing mistakes in this model not only 7 One might argue that assuming no riskless bonds outstanding is unrealistic since there are government bonds outstanding. However, government bonds are obligations of domestic residents, so that in a world where the Ricardian equivalence result holds, one can view domestic residents as supplying these bonds (see Barro, 1974)
because the market portfolio should include foreign assets,but also because it omits the foreign currency risk factor when the assumptions that lead to Solnik's model hold. To understand the determinants of the risk premium on foreign exchange,consider an increase in domestic wealth,Wd.As a result of this increase,the demand for foreign stocks by domestic investors increases.Absent increased borrowing in foreign currency,the foreign currency exposure of domestic investors increases.To reduce this foreign currency exposure, domestic investors want to borrow more abroad to hedge holdings of foreign stocks.Keeping all expected returns the same,there is now excess demand for borrowing in foreign currency.To reduce that excess demand,the reward for bearing foreign currency risk has to increase for domestic investors,so that they borrow less to hedge their foreign stock holdings.An increase in the value of domestic stocks means that the weight of domestic stocks in the portfolios of foreign investors increases and hence these investors want to borrow more in domestic currency to reduce their foreign currency exposure.To reduce the excess demand for borrowing in domestic currency by foreign investors,the expected excess return of the domestic risk-free asset in foreign currency has to increase,which is equivalent to a decrease of the expected excess return of the foreign risk-free asset in domestic currency. This pricing of foreign exchange risk is the central feature of Solnik's model.It is important to note,however,that his model takes exchange rate dynamics as exogenous.In an equilibrium model,the exchange rate and its dynamics would be functions of more primitive quantities.In such a model,there would be no exchange rate risk premium-the risk premium paid for taking on foreign exchange rate risk would be the risk premium for taking on exposure to the primitive variables that affect the exchange rate.However,it would still be correct that taking on foreign 18
18 because the market portfolio should include foreign assets, but also because it omits the foreign currency risk factor when the assumptions that lead to Solnik’s model hold. To understand the determinants of the risk premium on foreign exchange, consider an increase in domestic wealth, Wd . As a result of this increase, the demand for foreign stocks by domestic investors increases. Absent increased borrowing in foreign currency, the foreign currency exposure of domestic investors increases. To reduce this foreign currency exposure, domestic investors want to borrow more abroad to hedge holdings of foreign stocks. Keeping all expected returns the same, there is now excess demand for borrowing in foreign currency. To reduce that excess demand, the reward for bearing foreign currency risk has to increase for domestic investors, so that they borrow less to hedge their foreign stock holdings. An increase in the value of domestic stocks means that the weight of domestic stocks in the portfolios of foreign investors increases and hence these investors want to borrow more in domestic currency to reduce their foreign currency exposure. To reduce the excess demand for borrowing in domestic currency by foreign investors, the expected excess return of the domestic risk-free asset in foreign currency has to increase, which is equivalent to a decrease of the expected excess return of the foreign risk-free asset in domestic currency. This pricing of foreign exchange risk is the central feature of Solnik’s model. It is important to note, however, that his model takes exchange rate dynamics as exogenous. In an equilibrium model, the exchange rate and its dynamics would be functions of more primitive quantities. In such a model, there would be no exchange rate risk premium – the risk premium paid for taking on foreign exchange rate risk would be the risk premium for taking on exposure to the primitive variables that affect the exchange rate. However, it would still be correct that taking on foreign