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International arbitrage pricing theory: Relating risk premia
International
Arbitrage Pricing
Theory: Relating Risk Premia
DANA R. CLYMAN
It is elementary that in a world of certain exchange rates, the factor risk premia of different economies must be identical. Using standard arbitrage arguments, this paper demonstrates that when exchange rates become uncertain, the factor risk premia not only must be different, but more importantly, risk premia (of different economies) associated with the same underlying factor must differ by a particular amount. There are three immediate implications of this result: (1) the slopes of the CAPM (capital- asset-pricing-model) lines in different economies must be different, (2) no factor risk, including exchange-rate risk, can have a zero price in more than one economy, and (3) when making cost-of- capital comparisons, simply adjusting for differences in riskfree rates and comparing risk premia is not sufficient: one must also take into account these arbitragedetermined differences between the factor risk premia.
INTRODUCTION
Ever since the Arbitrage Pricing Theory (API) was first formulated in a single closed-economy model by Ross (1976), much has been written about how it is affected by the introduction of exchange-rate risk. Indeed two recent articles, Dumas and Solnik (1995) and Stultz (1995), include excellent and comprehensive reviews of this literature. This brief paper focuses on one small part of this literature, exploring how factor risk premia from different economies are related. It is elementary that, if the exchange rate between two economies is deterministic, then the factor risk premia (from the two economies) associated with the same underlying uncertain factor must be the same. This paper demonstrates that when exchange-rate fluctuations become uncertain, factor risk premia must still be related in a particular way. And, while it is no longer true that the factor risk premia from the two economies can be the same (except under highly serendipitous conditions), it is true that arbitrage continues to demand a particular relation between them. There are three immediate and important implications of this result. First, because the factor risk premia of different economies must, in general, be different, the security market lines of different countries must also, in general, be different. Furthermore, because one-factor models represent a special case, CAPM (capital-asset-pricing-model) lines in different economies must have different slopes. Second, in international-arbitrage-pricing-theory models that price exchange-rate risk, the price of that risk cannot, in general, be the same in more than one economy. In particular, Dana R. Clyman
l
The Darden School of The University
of Virginia.
International Review of Financial Analysis, Vol. 6, No. 1,197, pp. 13-20 Copyright 0 1997 by JAI PRESS Inc., All Rights of reproduction in any form reserved.
ISSN: 10574219
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/ Vol. 6(l)
therefore, it cannot be zero in more than one economy. This in turn implies that forward rates cannot be unbiased predictors of expected future spot rates. Lastly, when making cost-of-capital comparisons, it is not sufficient to adjust for differences in riskfree rates and then simply compare risk premia. Even in the absence of real cost-of-capital differences, there will be a difference between the risk premia of different economies, and any cost-of-capital comparisons must take this arbitrage-determined difference into account. The remainder of this paper is brief and direct. The next section builds the model and derives the main result. For each uncertain factor that affects asset returns measured in their own local currency numeraire, there is a (pairwise) arbitrage-determined relation between the factor risk premia from different economies that are associated with that factor. Furthermore, this result is equally true in simple international extensions of classic APT models, such as Solnik (1983), and more general international AFT models, like Ikeda (1991) where exchange-rate fluctuations do not need to have the same factor structure as asset returns. In the penultimate section we extend the argument to the risk premia associated with any exchange-rate risk not subject to the factor structure, as in models like Ikeda’s, by deriving a similar pairwise relation in any equilibria that prices the additional risk. The final section contains concluding remarks. AN ARBITRAGE
RELATION FOR ASSET-FACTOR
RISK PREMIA
In the standard framework commonly adopted for international-arbitrage-pricing-theory models, one postulates a world with a large number of countries, each of which has two assets: a locally riskless asset and a risky asset. In addition, one generally assumes that all risky-asset returns, when measured in their local-currency numeraire, follow a k-factor (k far smaller than the number of countries) return-generating process, and that the assets span the factors-that is, the market is complete. Ikeda (1991) pointed out that one could have as easily specified the theoretically equivalent framework of a world with few countries but many assets per country. We adopt this latter framework because we are concerned with pairwise relations, and this equivalent structure facilitates that view. For ease of exposition, we also assume, without loss of generality, that each local market is complete (i.e., the k common factors are spanned by assets naturally denominated in each local currency). This additional assumption of locally complete markets is free in the sense that, as long as the global market is compete, the local market can be made complete by dynamically maintaining currency-hedged versions of any “missing” foreign assets.1 Moreover, it allows us to simplify our discussion by focusing on securities from just two markets: one domestic and one foreign market. Nevertheless, all of the results are easily replicable in the equivalent model with two assets per country; our focus on two countries and locally complete markets merely simplifies the discussion and permits more parsimonious notation. Thus, we begin by postulating a world with several countries and many assets, all of which are freely traded in a complete international capital market. Each country has its own local currency in which asset returns are measured. Each country has many risky assets naturally denominated in the local currency, as well as one locally riskless asset. In addition, we assume that there are k arbitrarily correlated, common global factors (where k is small relative to the number of assets in each country) driving the local asset-return generating processes of the assets in each country. Finally, to simplify the notation and exposition, and without loss of generality, we also assume that each country has sufficiently many and diverse local assets to span the factor structure and make the local market complete.
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International Arbitrage Pricing Theory
Because our assumptions allow us to focus on just two markets, we simplify the discussion further by referring to one particular market as the domestic market, where, without loss of generality, we say that returns are measured in pounds, and to another market as the foreign market where we say that returns are measured in yen. Our construction begins with an arbitrary foreign asset. Write its local (yen) retum-generating process as
where 7%is the uncertain
return of this particular
foreign asset measured
Y in yen, ? IS the
expected component of that uncertain return, f;(i = 1, . . . . k) are the k arbitrarily-correlated global factors each with mean equal to zero and variance equal to one, bi are the factor loads of this particular asset, and E is the residual idiosyncratic risk of this particular asset. In addition, we assume that the E’S associated with each asset have zero mean and bounded variance and are unrelated to the common factors (e.g., E(Elfi) = 0, for all i and all E). Following the usual derivation of the classic APT, these conditions are sufficient to build a portfolio that zeros out the factor risks and diversifies away the idiosyncratic risk.2 Thus, by the fundamental theorem of the APT, there exists in this economy a set of risk prices, hy(i = l,...,k)such
that
where r: is the risk-free return of the local riskfree asset (shown by the subscriptm
when
measured in its own currency (shown by the superscript V). Equation (2) is the asset-pricing relation (also referred to as the pricing equation or market hyperplane) of the foreign economy. Ikeda (1991) in his extension of Solnik (1983) demonstrated that unless exchange-rate fluctuations also have the same factor structure as asset returns, the domestic return (in pounds) of this arbitrary asset cannot be priced by the domestic pricing equation, whereas a currency-hedged version of this asset can. To review, let the exchange rate measured in pounds per yen (f/Y) be denoted by
where the expected component
is Xiand i, is exchange-rate
risk not subject to the factor
structure. (Note, the primary difference between Solnik’s and Ikeda’s models is that in Solnik’s model this risk is presumed to be idiosyncratic and hence diversifiable, while in Ikeda’s more general model it is not. Indeed, Ikeda assumes only that it is conditionally independent of the other factors and argues that the inclusion of this extra risk should make fitting empirical models simpler. We include it in our derivations without requiring that it be diversifiable because our results are equally true regardless.) Using Ito’s lemma, the return on this asset in pounds can now be calculated: _f r
_Y
-
= r +X
E +cov
Y
1I _Y
-f
r ,X,.
(4)
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But as_Ikeda showed when this term is expanded, the expression contains the “nondiversifiable” fx; therefore, the return-generating process of this asset in pounds is different from that of other local assets whose return-generating processes depend only on the k common factors and diversifiable risk. Because of the inclusion of this extra term, therefore, this asset ccwww be priced by the local domestic pricing equation. As Ikeda also demonstrated, however, a currency-hedged version of this arbitrary foreign asset can be priced by the local domestic pricing equation (and it is for this reason that we are indifferent to the inclusion of the extra term in the exchange-rate process). To see this, form the portfolio (whose return will be subscripted by p for portfolio) consisting of a long position in the arbitrary foreign risky security, a short position of equal value in the risk-free foreign security, and a long position, also of equal value at the current spot-conversion rate in the riskfree domestic security. The instantaneous uncertain return of this portfolio in pounds is given by -f
rp = 7f-
_f
f
rj.%+ rf f
(5b) -
y
f
‘ry +rfE+blf,+...+b&+E.
(5c)
From equation (5b), we see how all exchange-rate uncertainty cancels out of the equation, and from equation (5~) we see that the currency-hedged foreign security has a retum-generating process of exactly the same form as that of any domestic security.3 In other words, the process consists of an expected component, a systematic component that is subject only to the k common global risk factors, and an idiosyncratic component-there is no extra exchange-rate factor.4 Therefore, be priced by the domestic-market
by the fundamental theorem of the APT, this portfolio can pricing equation, and there exists in this economy a set of
risk prices, hr(i = 1,..,k), measured in pounds, such that
= b&f
+
. + b,h:.
Using these general forms (equations 2 and 6), we now derive the main result wise differences of risk premia by repeating this derivation using a particular, arbitrary, foreign asset. Thus, rather than start with any foreign asset, we use the eign security-that is, the foreign security for which the factor loads are equal to # i. The return-generating process for this security may be written,
r; =?y +CiJj +
E;,
(6) for the pairrather than ith pure forzero for all j
(7)
where ci is used for the coefficient of the ith factor to make it simpler to identify as the factor load of the ith pure security. By the fundamental theorem of the APT (equation 2), v
‘pry
Y
Y
= Cjhi.
(8)
Similarly, after forming the currency-hedged version of this pure security and calculating its return in pounds (equation 5c), it follows that it can be priced by the domestic pricing equation (equation 6), as follows
17
International Arbitrage Pricing Theory
Y
r. +cov I
up
( 1 r,, I
Y
-r
Y
= cjh
P
f
i
(9)
The derivation is now almost complete. Into the left-hand side of equation (9), make two substitutions. First, write out and substitute for the covariance, which is given by
(10) where pij represents f, were diversifiable,
the (arbitrary)
correlation
between the ith and]Th factors. (Note, even if
the final term in equation
10 need not be zero, and in fact, cannot be Y Y zero under all possible sets of asset factors.) Second, use equation (8) to substitute for ri- r . P As a result of these substitutions, equation (9) becomes k
cixy +C j=l
from which it immediately
CiSjpij + E[&i
jx] = C;f,j,
follows that
1;- ky= i sjpij+ &E&l. j=l
I
(lib)
From the form of this relation, it is easily seen that both risk premia cannot, in general, be the same, unless (1) exchange-rate fluctuations are deterministic, (2) there is a highly serendipitous collection of parameters, or (3) exchange-rate risk is both strictly idiosyncratic ($ = 0, Vj) and uncorrelated with pure-asset residual risks.
EXTENSIONS
IN EQUILIBRIUM
The pairwise relation derived above encompasses only those pairs of risk premia associated with asset-factor risks because they are the only factors priced by the local arbitrage-pricing equations. When the factor structure is rich enough to incorporate the effects of all exchangerate risk as well as asset risk, as it is in Solnik (1983), then the preceding argument is sufficient to price exchange-rate risk. In models like Ikeda’s, however, where the exchange-rate fluctuations are not subject to the same factor structure as asset returns, a little more work is necessary to determine the difference between the risk premia associated with the extra exchange-rate risk. Furthermore, because the arbitrage-determined pricing equations do not price the extra risk, the relation cannot be determined by arbitrage. However, should equivalent pricing equations exist in any equilibrium that prices exchange-rate risk (e.g., that prices arbitrary foreign assets), then a similar relation, which must be true in the equilibrium, may be derived for the pair of risk premia associated with that additional risk. The argument is straightforward. If the domestic pricing equation can price any arbitrary foreign asset, then it can price the foreign, locally risk-free asset. Again using Ito’s lemma, the pound return on the foreign riskfree asset is given by
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(12) Now, by the domestic pricing equation that is presumed to exist in equilibrium,
(13) Similarly, consider the domestic riskfree security in the foreign economy. If it too is priced (this time by the foreign pricing equation that is also presumed to exist in equilibrium), then, after using Ito’s lemma to calculate
ii’f, which equals,
l/i:,
we obtain the corresponding
equation -Y
p=-(61h:+...+Skh:+6~a~)
r’ + X, - r fj
B
(14)
The sum of the left-hand sides of equations (13) and (14) is equal to the sum of the expected returns of the exchange-rate processes in both directions, which by Siegel’s (1972) paradox is equal to the variance of the exchange-rate process. The sum of the right-hand sides of the two equations is an exchange- rate-factor-weighted sum of the differences of the risk premia. Thus,
Using
the result
from the preceding
section,
however,
we can subtract
the value
of
from both sides of equation (15) for each i, and divide through by 6,, to obtain
iii ai-a:=a,- xk ,-&[&if,]. i=l
lx
(16)
Thus, even in models where the factor-structure is not robust enough to incorporate all exchange-rate risk, whenever there exists an equilibrium that prices that risk, the risk premia associated with the additional exchange-rate risk must still differ by a particular amount. This proves that it is also true that exchange-rate risks cannot in general have the same price in more than one economy. An immediate and important consequence of this result is that the risk premia associated with any additional exchange-rate risk cannot, in general, have a zero price in more than one economy. Furthermore, because the price cannot, in general, be zero, it follows that the forward rate cannot be an unbiased predictor of the expected spot rate.
CONCLUSIONS
This paper uses an arbitrage argument to relate the factor-risk premia of multiple countries. In particular, it demonstrates that asset-factor risk premia cannot, in general, be the same across
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economies. Because this result applies to one-factor models as a special case, it follows immediately that the slopes of the CAPM lines in different countries must be different. Next, even though the preceding result is only for risk premia associated with factors that affect asset pricing in their own local numeraire, a similar result holds for risk premia associated with any exchange-rate risk that might not be subject to the factor structure, as long as there exists an equilibrium that prices arbitrary foreign assets. Furthermore, in such equilibria, because the risk premia associated with any such exchange-rate risk cannot be the same, it also follows that the price of exchange-rate risk cannot be zero in more than one economy. This in turn implies that the forward rate cannot be an unbiased predictor of the expected spot rate. Finally, these results have important implications for cost-of-capital comparisons. Because arbitrage requires that, even in a perfectly integrated capital market, like the one modeled here, where there cannot exist cost-of-capital differences, there must exist differences between the factor risk premia; therefore, it is not sufficient to note that the riskfree rates of different countries are different and then simply compare the risk premia. Cost-of-capital comparisons must also take this arbitrage-determined difference into account.
NOTES
1. This is demonstrated in Ikeda (1991) and it will be readily apparent from the derivation in this section as well. 2. Note that while it is often customary to require that the E’S be independent of one another, they only need to be sufficiently perpendicular as a set to enable this diversification. 3. Note that the factor loads are identical in the domestic and foreign return-generating processes. This occurs because we were able to use a foreign, locally risk-free asset in the currency hedge. Were no such asset available, it would still be possible through currency hedging to eliminate the uncertain component of exchange-rate risk, but it might not be possible to obtain identical factor loads. 4. It is because of this derivation that the assumption of locally complete markets is free. Once one market is presumed to be complete, all others can be completed using currencyhedged versions of any missing assets.
REFERENCES
Adler, M. & Dumas. B. (1983). International portfolio choice and corporation finance: A synthesis. Journal of Finance, 38, (June), 925-984. Clyman, D., Edleson, M., and Miller, R. (1993). Arbitrage, Exchange Rates and the Cost of Capital. Darden School Working Paper #93-14. Dumas, B. & Solnik, B. (1995). The world price of foreign exchange risk. Journal of Finance, SO, (June), 445479. Ikeda, S. (1991). Arbitrage asset pricing under exchange risk. Journal of Finance, 46, (March) 447-455. Kleidon, A. & Pfleiderer, P. (1983). Discussion of Solnik’s International arbitrage pricing theory. Journal of Finance, 38, (May), 470-472. Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13, (December), 341-360.
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Siegel, J. (1972). Risk, interest rates and the forward exchange. Quurterly Journal of&onomics, 86, (May), 303-309. Solnik, B. (1983). International arbitrage pricing theory. Journal of Finance, 38, (May), 449457. Stulz, R.M. (1995). International portfolio choice and asset pricing: An integrative study. In R.A. Jarrow, V. Maksimovic & W.T. Ziemba, (Eds.), Handbooks in OR & MS: Finance. Vo19, B.V.: Elsevier Science, 201-223. Stulz, R.M. (1984). Pricing capital assets in an international setting: An introduction. Journal of Inter nutionul Business Studies, (Winter), 55-72. Stulz, R.M. (1981). A model of international asset pricing. Journal of Financial Economics, 9, (December), 383-406.
Arbitrage Pricing
Theory: Relating Risk Premia
DANA R. CLYMAN
It is elementary that in a world of certain exchange rates, the factor risk premia of different economies must be identical. Using standard arbitrage arguments, this paper demonstrates that when exchange rates become uncertain, the factor risk premia not only must be different, but more importantly, risk premia (of different economies) associated with the same underlying factor must differ by a particular amount. There are three immediate implications of this result: (1) the slopes of the CAPM (capital- asset-pricing-model) lines in different economies must be different, (2) no factor risk, including exchange-rate risk, can have a zero price in more than one economy, and (3) when making cost-of- capital comparisons, simply adjusting for differences in riskfree rates and comparing risk premia is not sufficient: one must also take into account these arbitragedetermined differences between the factor risk premia.
INTRODUCTION
Ever since the Arbitrage Pricing Theory (API) was first formulated in a single closed-economy model by Ross (1976), much has been written about how it is affected by the introduction of exchange-rate risk. Indeed two recent articles, Dumas and Solnik (1995) and Stultz (1995), include excellent and comprehensive reviews of this literature. This brief paper focuses on one small part of this literature, exploring how factor risk premia from different economies are related. It is elementary that, if the exchange rate between two economies is deterministic, then the factor risk premia (from the two economies) associated with the same underlying uncertain factor must be the same. This paper demonstrates that when exchange-rate fluctuations become uncertain, factor risk premia must still be related in a particular way. And, while it is no longer true that the factor risk premia from the two economies can be the same (except under highly serendipitous conditions), it is true that arbitrage continues to demand a particular relation between them. There are three immediate and important implications of this result. First, because the factor risk premia of different economies must, in general, be different, the security market lines of different countries must also, in general, be different. Furthermore, because one-factor models represent a special case, CAPM (capital-asset-pricing-model) lines in different economies must have different slopes. Second, in international-arbitrage-pricing-theory models that price exchange-rate risk, the price of that risk cannot, in general, be the same in more than one economy. In particular, Dana R. Clyman
l
The Darden School of The University
of Virginia.
International Review of Financial Analysis, Vol. 6, No. 1,197, pp. 13-20 Copyright 0 1997 by JAI PRESS Inc., All Rights of reproduction in any form reserved.
ISSN: 10574219
14
INTERNATIONAL
REVIEW
OF FINANCIAL
ANALYSIS
/ Vol. 6(l)
therefore, it cannot be zero in more than one economy. This in turn implies that forward rates cannot be unbiased predictors of expected future spot rates. Lastly, when making cost-of-capital comparisons, it is not sufficient to adjust for differences in riskfree rates and then simply compare risk premia. Even in the absence of real cost-of-capital differences, there will be a difference between the risk premia of different economies, and any cost-of-capital comparisons must take this arbitrage-determined difference into account. The remainder of this paper is brief and direct. The next section builds the model and derives the main result. For each uncertain factor that affects asset returns measured in their own local currency numeraire, there is a (pairwise) arbitrage-determined relation between the factor risk premia from different economies that are associated with that factor. Furthermore, this result is equally true in simple international extensions of classic APT models, such as Solnik (1983), and more general international AFT models, like Ikeda (1991) where exchange-rate fluctuations do not need to have the same factor structure as asset returns. In the penultimate section we extend the argument to the risk premia associated with any exchange-rate risk not subject to the factor structure, as in models like Ikeda’s, by deriving a similar pairwise relation in any equilibria that prices the additional risk. The final section contains concluding remarks. AN ARBITRAGE
RELATION FOR ASSET-FACTOR
RISK PREMIA
In the standard framework commonly adopted for international-arbitrage-pricing-theory models, one postulates a world with a large number of countries, each of which has two assets: a locally riskless asset and a risky asset. In addition, one generally assumes that all risky-asset returns, when measured in their local-currency numeraire, follow a k-factor (k far smaller than the number of countries) return-generating process, and that the assets span the factors-that is, the market is complete. Ikeda (1991) pointed out that one could have as easily specified the theoretically equivalent framework of a world with few countries but many assets per country. We adopt this latter framework because we are concerned with pairwise relations, and this equivalent structure facilitates that view. For ease of exposition, we also assume, without loss of generality, that each local market is complete (i.e., the k common factors are spanned by assets naturally denominated in each local currency). This additional assumption of locally complete markets is free in the sense that, as long as the global market is compete, the local market can be made complete by dynamically maintaining currency-hedged versions of any “missing” foreign assets.1 Moreover, it allows us to simplify our discussion by focusing on securities from just two markets: one domestic and one foreign market. Nevertheless, all of the results are easily replicable in the equivalent model with two assets per country; our focus on two countries and locally complete markets merely simplifies the discussion and permits more parsimonious notation. Thus, we begin by postulating a world with several countries and many assets, all of which are freely traded in a complete international capital market. Each country has its own local currency in which asset returns are measured. Each country has many risky assets naturally denominated in the local currency, as well as one locally riskless asset. In addition, we assume that there are k arbitrarily correlated, common global factors (where k is small relative to the number of assets in each country) driving the local asset-return generating processes of the assets in each country. Finally, to simplify the notation and exposition, and without loss of generality, we also assume that each country has sufficiently many and diverse local assets to span the factor structure and make the local market complete.
15
International Arbitrage Pricing Theory
Because our assumptions allow us to focus on just two markets, we simplify the discussion further by referring to one particular market as the domestic market, where, without loss of generality, we say that returns are measured in pounds, and to another market as the foreign market where we say that returns are measured in yen. Our construction begins with an arbitrary foreign asset. Write its local (yen) retum-generating process as
where 7%is the uncertain
return of this particular
foreign asset measured
Y in yen, ? IS the
expected component of that uncertain return, f;(i = 1, . . . . k) are the k arbitrarily-correlated global factors each with mean equal to zero and variance equal to one, bi are the factor loads of this particular asset, and E is the residual idiosyncratic risk of this particular asset. In addition, we assume that the E’S associated with each asset have zero mean and bounded variance and are unrelated to the common factors (e.g., E(Elfi) = 0, for all i and all E). Following the usual derivation of the classic APT, these conditions are sufficient to build a portfolio that zeros out the factor risks and diversifies away the idiosyncratic risk.2 Thus, by the fundamental theorem of the APT, there exists in this economy a set of risk prices, hy(i = l,...,k)such
that
where r: is the risk-free return of the local riskfree asset (shown by the subscriptm
when
measured in its own currency (shown by the superscript V). Equation (2) is the asset-pricing relation (also referred to as the pricing equation or market hyperplane) of the foreign economy. Ikeda (1991) in his extension of Solnik (1983) demonstrated that unless exchange-rate fluctuations also have the same factor structure as asset returns, the domestic return (in pounds) of this arbitrary asset cannot be priced by the domestic pricing equation, whereas a currency-hedged version of this asset can. To review, let the exchange rate measured in pounds per yen (f/Y) be denoted by
where the expected component
is Xiand i, is exchange-rate
risk not subject to the factor
structure. (Note, the primary difference between Solnik’s and Ikeda’s models is that in Solnik’s model this risk is presumed to be idiosyncratic and hence diversifiable, while in Ikeda’s more general model it is not. Indeed, Ikeda assumes only that it is conditionally independent of the other factors and argues that the inclusion of this extra risk should make fitting empirical models simpler. We include it in our derivations without requiring that it be diversifiable because our results are equally true regardless.) Using Ito’s lemma, the return on this asset in pounds can now be calculated: _f r
_Y
-
= r +X
E +cov
Y
1I _Y
-f
r ,X,.
(4)
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But as_Ikeda showed when this term is expanded, the expression contains the “nondiversifiable” fx; therefore, the return-generating process of this asset in pounds is different from that of other local assets whose return-generating processes depend only on the k common factors and diversifiable risk. Because of the inclusion of this extra term, therefore, this asset ccwww be priced by the local domestic pricing equation. As Ikeda also demonstrated, however, a currency-hedged version of this arbitrary foreign asset can be priced by the local domestic pricing equation (and it is for this reason that we are indifferent to the inclusion of the extra term in the exchange-rate process). To see this, form the portfolio (whose return will be subscripted by p for portfolio) consisting of a long position in the arbitrary foreign risky security, a short position of equal value in the risk-free foreign security, and a long position, also of equal value at the current spot-conversion rate in the riskfree domestic security. The instantaneous uncertain return of this portfolio in pounds is given by -f
rp = 7f-
_f
f
rj.%+ rf f
(5b) -
y
f
‘ry +rfE+blf,+...+b&+E.
(5c)
From equation (5b), we see how all exchange-rate uncertainty cancels out of the equation, and from equation (5~) we see that the currency-hedged foreign security has a retum-generating process of exactly the same form as that of any domestic security.3 In other words, the process consists of an expected component, a systematic component that is subject only to the k common global risk factors, and an idiosyncratic component-there is no extra exchange-rate factor.4 Therefore, be priced by the domestic-market
by the fundamental theorem of the APT, this portfolio can pricing equation, and there exists in this economy a set of
risk prices, hr(i = 1,..,k), measured in pounds, such that
= b&f
+
. + b,h:.
Using these general forms (equations 2 and 6), we now derive the main result wise differences of risk premia by repeating this derivation using a particular, arbitrary, foreign asset. Thus, rather than start with any foreign asset, we use the eign security-that is, the foreign security for which the factor loads are equal to # i. The return-generating process for this security may be written,
r; =?y +CiJj +
E;,
(6) for the pairrather than ith pure forzero for all j
(7)
where ci is used for the coefficient of the ith factor to make it simpler to identify as the factor load of the ith pure security. By the fundamental theorem of the APT (equation 2), v
‘pry
Y
Y
= Cjhi.
(8)
Similarly, after forming the currency-hedged version of this pure security and calculating its return in pounds (equation 5c), it follows that it can be priced by the domestic pricing equation (equation 6), as follows
17
International Arbitrage Pricing Theory
Y
r. +cov I
up
( 1 r,, I
Y
-r
Y
= cjh
P
f
i
(9)
The derivation is now almost complete. Into the left-hand side of equation (9), make two substitutions. First, write out and substitute for the covariance, which is given by
(10) where pij represents f, were diversifiable,
the (arbitrary)
correlation
between the ith and]Th factors. (Note, even if
the final term in equation
10 need not be zero, and in fact, cannot be Y Y zero under all possible sets of asset factors.) Second, use equation (8) to substitute for ri- r . P As a result of these substitutions, equation (9) becomes k
cixy +C j=l
from which it immediately
CiSjpij + E[&i
jx] = C;f,j,
follows that
1;- ky= i sjpij+ &E&l. j=l
I
(lib)
From the form of this relation, it is easily seen that both risk premia cannot, in general, be the same, unless (1) exchange-rate fluctuations are deterministic, (2) there is a highly serendipitous collection of parameters, or (3) exchange-rate risk is both strictly idiosyncratic ($ = 0, Vj) and uncorrelated with pure-asset residual risks.
EXTENSIONS
IN EQUILIBRIUM
The pairwise relation derived above encompasses only those pairs of risk premia associated with asset-factor risks because they are the only factors priced by the local arbitrage-pricing equations. When the factor structure is rich enough to incorporate the effects of all exchangerate risk as well as asset risk, as it is in Solnik (1983), then the preceding argument is sufficient to price exchange-rate risk. In models like Ikeda’s, however, where the exchange-rate fluctuations are not subject to the same factor structure as asset returns, a little more work is necessary to determine the difference between the risk premia associated with the extra exchange-rate risk. Furthermore, because the arbitrage-determined pricing equations do not price the extra risk, the relation cannot be determined by arbitrage. However, should equivalent pricing equations exist in any equilibrium that prices exchange-rate risk (e.g., that prices arbitrary foreign assets), then a similar relation, which must be true in the equilibrium, may be derived for the pair of risk premia associated with that additional risk. The argument is straightforward. If the domestic pricing equation can price any arbitrary foreign asset, then it can price the foreign, locally risk-free asset. Again using Ito’s lemma, the pound return on the foreign riskfree asset is given by
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(12) Now, by the domestic pricing equation that is presumed to exist in equilibrium,
(13) Similarly, consider the domestic riskfree security in the foreign economy. If it too is priced (this time by the foreign pricing equation that is also presumed to exist in equilibrium), then, after using Ito’s lemma to calculate
ii’f, which equals,
l/i:,
we obtain the corresponding
equation -Y
p=-(61h:+...+Skh:+6~a~)
r’ + X, - r fj
B
(14)
The sum of the left-hand sides of equations (13) and (14) is equal to the sum of the expected returns of the exchange-rate processes in both directions, which by Siegel’s (1972) paradox is equal to the variance of the exchange-rate process. The sum of the right-hand sides of the two equations is an exchange- rate-factor-weighted sum of the differences of the risk premia. Thus,
Using
the result
from the preceding
section,
however,
we can subtract
the value
of
from both sides of equation (15) for each i, and divide through by 6,, to obtain
iii ai-a:=a,- xk ,-&[&if,]. i=l
lx
(16)
Thus, even in models where the factor-structure is not robust enough to incorporate all exchange-rate risk, whenever there exists an equilibrium that prices that risk, the risk premia associated with the additional exchange-rate risk must still differ by a particular amount. This proves that it is also true that exchange-rate risks cannot in general have the same price in more than one economy. An immediate and important consequence of this result is that the risk premia associated with any additional exchange-rate risk cannot, in general, have a zero price in more than one economy. Furthermore, because the price cannot, in general, be zero, it follows that the forward rate cannot be an unbiased predictor of the expected spot rate.
CONCLUSIONS
This paper uses an arbitrage argument to relate the factor-risk premia of multiple countries. In particular, it demonstrates that asset-factor risk premia cannot, in general, be the same across
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International Arbitrage Pricing Theory
economies. Because this result applies to one-factor models as a special case, it follows immediately that the slopes of the CAPM lines in different countries must be different. Next, even though the preceding result is only for risk premia associated with factors that affect asset pricing in their own local numeraire, a similar result holds for risk premia associated with any exchange-rate risk that might not be subject to the factor structure, as long as there exists an equilibrium that prices arbitrary foreign assets. Furthermore, in such equilibria, because the risk premia associated with any such exchange-rate risk cannot be the same, it also follows that the price of exchange-rate risk cannot be zero in more than one economy. This in turn implies that the forward rate cannot be an unbiased predictor of the expected spot rate. Finally, these results have important implications for cost-of-capital comparisons. Because arbitrage requires that, even in a perfectly integrated capital market, like the one modeled here, where there cannot exist cost-of-capital differences, there must exist differences between the factor risk premia; therefore, it is not sufficient to note that the riskfree rates of different countries are different and then simply compare the risk premia. Cost-of-capital comparisons must also take this arbitrage-determined difference into account.
NOTES
1. This is demonstrated in Ikeda (1991) and it will be readily apparent from the derivation in this section as well. 2. Note that while it is often customary to require that the E’S be independent of one another, they only need to be sufficiently perpendicular as a set to enable this diversification. 3. Note that the factor loads are identical in the domestic and foreign return-generating processes. This occurs because we were able to use a foreign, locally risk-free asset in the currency hedge. Were no such asset available, it would still be possible through currency hedging to eliminate the uncertain component of exchange-rate risk, but it might not be possible to obtain identical factor loads. 4. It is because of this derivation that the assumption of locally complete markets is free. Once one market is presumed to be complete, all others can be completed using currencyhedged versions of any missing assets.
REFERENCES
Adler, M. & Dumas. B. (1983). International portfolio choice and corporation finance: A synthesis. Journal of Finance, 38, (June), 925-984. Clyman, D., Edleson, M., and Miller, R. (1993). Arbitrage, Exchange Rates and the Cost of Capital. Darden School Working Paper #93-14. Dumas, B. & Solnik, B. (1995). The world price of foreign exchange risk. Journal of Finance, SO, (June), 445479. Ikeda, S. (1991). Arbitrage asset pricing under exchange risk. Journal of Finance, 46, (March) 447-455. Kleidon, A. & Pfleiderer, P. (1983). Discussion of Solnik’s International arbitrage pricing theory. Journal of Finance, 38, (May), 470-472. Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13, (December), 341-360.
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Siegel, J. (1972). Risk, interest rates and the forward exchange. Quurterly Journal of&onomics, 86, (May), 303-309. Solnik, B. (1983). International arbitrage pricing theory. Journal of Finance, 38, (May), 449457. Stulz, R.M. (1995). International portfolio choice and asset pricing: An integrative study. In R.A. Jarrow, V. Maksimovic & W.T. Ziemba, (Eds.), Handbooks in OR & MS: Finance. Vo19, B.V.: Elsevier Science, 201-223. Stulz, R.M. (1984). Pricing capital assets in an international setting: An introduction. Journal of Inter nutionul Business Studies, (Winter), 55-72. Stulz, R.M. (1981). A model of international asset pricing. Journal of Financial Economics, 9, (December), 383-406.