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World Scientific Publishing Company
MARKET SEGMENTATION AND NOISE TRADER RISK
VIHANG ERRUNZA McGill University, Montreal, Canada
KED HOGAN
Barclays Global Investors, San Francisco, USA MAO-WEI HUNG
Department of International Business, College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan
E-mail : [email protected] Received 16 April 1999
A simple asset pricing model is developed to take into account two important charac-teristics in global investments: market segmentation and noise trader risk. Our results show the removal of international investment barriers and cross-border listings have not led to a fully integrated international capital market. We also show that different degree of investor rationality across borders induces an additional component of risk premium which is related to the “noise spill-over effect”.
Keywords: Asset pricing, market segmentation, noise trader.
1. Introduction
Global capital market segmentation has been the focus of a significant body of international asset pricing literature. For example, Stulz [9], Errunza and Losq [4], Eun and Janakiramanan [6], Padmanabhan [8] and Basak [1] have shown that obstacles to free flow of capital will in general lead to an additional risk premium on securities that are not freely accessible. Over the past decade, many types of investment barriers that limiting foreign capital flows to emerging markets have been removed. Bekaert and Harvey [2] present evidence which suggests that the transition from segmentation to integration has been anything but smooth. Despite the removal of investment barriers and significant cross-border listings, substantial market segmentation remains.
It is also widely believed that investors may differ in terms of investment horizon and their abilities to price securities rationally and thus induce an additional com-ponent of risk premium. DeLong, Shleifer, Summers, and Waldmann (DSSW) [3] argue that under irrational price behavior and finite investment horizon, irrational noise traders will induce a risk premium in security prices. In the context of U.S.
markets, the introduction of noise, may explain well-known anomalies, for example, the closed-end fund discount puzzle.
In this paper we incorporate noise into an international asset pricing model with one-way investment barrier. The added dimension of noise allows us to con-sider possible causes for the persistence of market segmentation despite significant steps toward globalization. Furthermore, the necessary condition of finite invest-ment horizon appears to be a more widely held belief for cross-border investors than perhaps for domestic investors.
Indeed, our model suggests that the removal of investment barriers and cross-listings are not sufficient for achieving full integration in the presence of noise traders. Although the one-way barrier leads to some integration in contrast to two-way barriers, it does not shield emerging markets from the effect of noise spill-over from developed markets. Thus, policies that reduce noise and encourage long horizon investing are necessary to obtain full benefits of liberalization aimed at global risk sharing.
Our paper is organized as follows. Section 2 describes the model setup. Steady state equilibrium asset prices are presented in Sec. 3. Section 4 discusses the noise trader risk and Sec. 5 concludes the paper.
2. Model
We adopt the two-period overlapping generations framework of DSSW in which investors’ first period endowment is exogenously determined, no labor decisions are made, and at the end of second period, all assets are liquidated to the “next generation” and the proceeds used for consumption by the old generation. The salient feature of the model which introduces noise to be priced is that the invest-ment horizon is sufficiently short so as to not outlive irrational pricing or noise effects.
There are two countries, domestic (D) and foreign (F ), in which two types of risky securities are traded. We assume each country has only two securities: one for each type.a Asset type “a” is priced absent of noise. Asset type “b” is priced with noise. There is a risk-free security across both countries with a constant rate of return, r. Asset supplies are fixed and normalized to be one. Security prices are assumed to have a joint normal distribution and foreign exchange risk is assumed not to exist.
Investors are either rational or noise traders. Investors type i(∈ I) are rational domestic investors, type n(∈ N) are domestic noise traders. Similarly, investors type j(∈ J) are rational foreign investors, type k(∈ K) are foreign noise traders. All investors have absolute risk aversion utility with possibly different Arrow–Pratt risk aversion coefficients: U (W ) = − exp(−2αW ), where α is the Arrow–Pratt aThe assumption of one security in each class of securities is not restricted. Extension to multiple
absolute risk aversion coefficient, W is the end of period wealth and exp is the exponential function.
In order to facilitate our discussion, we adopt the following notations: • Pda
t ≡ price for the domestic security without noise at time t.
• Pdb
t ≡ price for the domestic security with noise at time t.
• Pf a
t ≡ price for the foreign security without noise at time t.
• Pf b
t ≡ price for the foreign security with noise at time t.
• Ct(l, m)≡ Covt(pjt+1, p k
t+1), l, m∈ (da, db, fa, fb)
• Ω2t≡ domestic security variance — covariance matrix
≡ "
Ct(da, da) Ct(da, db)
Ct(db, da) Ct(db, db)
#
• Ω4t≡ global security variance — covariance matrix
≡
Ct(da, da) Ct(da, db) Ct(da, f a) Ct(da, f b)
Ct(db, da) Ct(db, db) Ct(db, f a) Ct(db, f b) Ct(f a, da) Ct(f a, db) Ct(f a, f a) Ct(f a, f b) Ct(f b, da) Ct(f b, db) Ct(f b, f a) Ct(f b, f b) .
There are mainly three types of international capital market structure analyzed in the literature: full integration, segmentation and mild segmentation. In the paper, we focus on a mildly segmented market which prohibits domestic investors from investing in foreign assets while foreign investors have access to both foreign and domestic assets. Note that polar cases of integration and segmentation are just special cases of our model. In such a setting, domestic investors face the following budget constraint when choosing their portfolios:
Wt+1= λs,dat (p da t+1) + λ s,db t (p db t+1) + (1 + r)(Wt− λs,dat p da t − λ s,db t p db t ) , (2.1)
where λs,L are the demand function of investor type s∈ (I & N) for security L ∈
(da, db) and Wtis the investor’s wealth at time t.
Foreign investors without restricted investment opportunities, face the following budget constraint: Wt+1= λs,dat (pdat+1) + λ s,db t (pdbt+1) + λ s,f a t (p f a t+1) + λ s,f b t (p f b t+1) + (1 + r)(Wt− λs,dat p da t − λ s,db t p db t − λ s,f a t p f a t − λ s,f b t p f b t ) , (2.2)
where λs,L are the demand function of investor type s ∈ (J & K) for security
Given exponential utility functions and assuming risky securities are jointly normal, expected utility can be equivalently expressed as
Et(Ut) = Et(Wt+1)− α · Vart(Wt+1) , (2.3)
where Et and Vart are conditional expectation and conditional variance,
respectively.
This implies that rational investors i∈ I determine asset demands by maximiz-ing the followmaximiz-ing expression:
Et(Uti) = " λi,dat λi,dbt #0" Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) # + Wt(1 + r)− αi " λi,dat λi,dbt #0 Ω2t " λi,dat λi,dbt # (2.4)
and foreign rational investors j∈ J maximize
Et(Utj) = λj,dat λj,dbt λj,f at λj,f bt 0 Et(pdat+1)− p da t (1 + r) Et(pdbt+1)− p db t (1 + r) Et(p f a t+1)− p f a t (1 + r) Et(pf bt+1)− p f b t (1 + r) + Wt(1 + r)− αj λi,dat λi,dbt λi,f at λi,f bt 0 Ω4t λi,dat λi,dbt λi,f at λi,f bt . (2.5)
We further assume that noise traders misperceive the expected price of pdb t+1and
pf bt+1, by random noise factors ρ D
t and ρFt, respectively.
Hence, domestic noise traders n∈ N determine asset demands by maximizing
Et(Utn) = " λn,dat λn,dbt #0" Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− ρDt − pdbt (1 + r) # +Wt(1 + r)− αn " λn,dat λn,dbt #0 Ω2t " λn,dat λn,dbt # (2.6)
and foreign noise traders k∈ K determine asset demands by maximizingb Et(Utk) = λk,dat λk,dbt λjkf at λk,f bt 0 Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− ρDt − pdbt (1 + r) Et(pf at+1)− p f a t (1 + r) Et(pf bt+1)− ρ F t − p f b t (1 + r) + Wt(1 + r)− αk λk,dat λk,dbt λk,f at λk,f bt 0 Ω4t λk,dat λk,dbt λk,f at λk,f bt . (2.7)
3. Equilibrium Asset Prices
Given expected utilities in the previous section, the asset prices can be derived. The equilibrium asset prices for each class of securities under mild segmentation are stated in the following theorem:
Theorem 3.1. Under mild segmentation the equilibrium asset prices for each class of securities can be expressed as follows:
Et(pdat+1)− pdat (1 + r) = αwCt(pdat+1, Vt+1M ) (3.1) Et(pdbt+1)− pdbt (1 + r) = αwCt(pdbt+1, Vt+1M )− αw αnk ρDt (3.2) Et(p f a t+1)− p f a t (1 + r) = αjkCt(pf at+1, V M t+1)− αwαjk αin "C t(da, f a) Ct(da, f a) #0 Ω−12t " Ct(pdat+1, Vt+1M ) Ct(pdbt+1, Vt+1M ) # − αjk αn − αwαjk αinαnk "C t(da, f a) Ct(db, f a) #0 Ω−12t " 0 ρD t # (3.3) Et(pf bt+1)− p f b t (1 + r) = αjkCt(p f b t+1, V M t+1)− αwαjk αin "C t(da, f b) Ct(db, f b) #0 Ω−12t " Ct(pdat+1, Vt+1M ) Ct(pdbt+1, Vt+1M ) # − αjk αn − αwαjk αinαnk "C t(da, f b) Ct(db, f b) #0 Ω−12t " 0 ρDt # − αjk αk ρFt , (3.4)
bThe relaxation of the implicit assumption of similar price misperceptions by both domestic and
where VD
t ≡ pdat + pdbt is domestic market value, VtF ≡ p f a t + p
f b
t is foreign
mar-ket value, VM
t ≡ VtD + VtF is global market value, αi ≡ coefficient of absolute
risk aversion for investor i, and aggregate measures of risk aversion are defined as follows: αw ≡ X i∈I 1 αi + X n∈N 1 αn + X j∈J 1 αj + X k∈K 1 αk −1 , αk ≡ X k∈K 1 αk !−1 , αin≡ X i∈I 1 αi + X n∈N 1 αn !−1 , αnk≡ X n∈N 1 αn + X k∈K 1 αk !−1 , αn ≡ X n∈N 1 αn !−1 αij≡ X i∈I 1 αi + X j∈J 1 αj −1 .
Proof. See Appendix A.
Equation (3.1) in Theorem 3.1 states that the presence of noise in other asset markets will not alter the familiar past results. That is, with a one-way barrier, the domestic security without noise is priced as if the market was fully integrated, i.e. only a function of world covariance risk. Equation (3.2) is essentially identical to DSSW which shows the price for the domestic security with noise is a function of period t’s misperception by noise traders. Equation (3.3) is the pricing equation for the foreign security without noise. In addition to the world covariance risk, there are two additional terms in Eq. (3.3). Both terms are related to the ability of domestic securities to replicate the foreign security without noise. These two terms will be further studied in Theorem 3.2. Similarly, Eq. (3.4) states that price of the foreign security with noise will depend on the world covariance risk, the risk related to the replication by domestic securities, and its own noise term.
The interpretation of Eq. (3.3) can be made more intuitive by expressing it in a more familiar form as stated in the following theorem. This will help us interpret the results along the lines of previous papers in the literature.
Theorem 3.2. Pricing kernel for foreign security traded without noise under mild segmentation can be expressed in the following terms:
Et(p f a t+1)− p f a t (1 + r) = αwCt(p f a t+1, V M t+1)− (αjk− αw)[Ct(p f a t+1, V F t+1) − Ct(pf at+1, V A t+1)]− αjk αn − αwαjk αinαnk β3ρDt , (3.5)
where β3 is the covariance between pf a and pdb conditional on pda, VA is the
“ad-justment” portfolio which is the portfolio of domestic securities most highly corre-lated with the portfolio of foreign securities.
Proof. See Appendix B.
Theorem 3.2 states that price for the foreign security without noise is a function of (i) its covariance with the world market portfolio; (ii) the “pure” foreign market risk premium which is defined in [6] as the difference between covariance of pf a
and VF, and covariance of pf awith VA; and (iii) a “noise spill-over” risk premium
resulting from domestic noise. Notice that unlike the domestic asset without noise, the foreign asset without noise is influenced by the noise originating in the pric-ing of the domestic asset pdb. This “noise spill-over” risk premium has not been
documented in the previous literature. The immediate implication of which is that governments cannot shield domestic financial markets from international sources of noise by creating a one-way barrier.
A more precise analysis of the impact of noise on assets prices requires steady state solutions for prices. We assume that ρD
t ∼ N(ρmD, σρD2 ), ρFt ∼ N(ρmF, σρF2 )
and the unconditional distribution of assets prices are equal to their conditional dis-tributions. Given these assumptions, steady state solutions implied by Theorem 3.1 for equilibrium prices can be derived by recursion.
Theorem 3.3. Under mild segmentation, the steady state equilibrium asset prices for each class of assets can be expressed as follows:
pdat = dda r − αw r Ct(p da t+1, V M t+1) (3.6) pdbt = ddb r − αw r Ct(p db t+1, V M t+1) + αw αnk · (ρD t − ρmD) 1 + r + αw αnk · ρmD r (3.7) pf at = df a r − αw r Ct(p f a t+1, V M t+1)− (αjk− αw) r Ct(p f a t+1, V F t+1) +(αjk− αw) r Ct(p f a t+1, V A t+1) + Aβ3(ρDt − ρmD) 1 + r + Aβ3 r ρ mD (3.8) pf bt = df b r − αw r Ct(p f b t+1, V M t+1)− (αjk− αw) r Ct(p f b t , V F t+1) +(αjk− αw) r Ct(p f b t+1, V A t+1) + Aβ4(ρDt − ρmD) 1 + r + Aβ4 r ρ mD +αw(ρ F t − ρ mF ) αk(1 + r) + αw αkr ρmF, (3.9) where A = (αjk αn − αwαjk αinαnk), β3 is the covariance of p db and pf aconditional on pda
and β4 is the covariance of pdb and pf b conditional on pda.
From Theorem 3.3, we can immediately assess the impact of changes in noise trader sentiment on asset prices. Equation (3.6) states that an increase in domestic noise trader bullishness, i.e. increase in ρD
without noise. Equation (3.7) states that the impact of an increase in domestic noise bullishness on the domestic asset with noise is proportional to the risk aversion of all traders relative to that of noise traders, and the discount rate r.
From Eq. (3.8), we can assess the conditions under which the price of the foreign asset without noise may increase, decrease, or remain unchanged, given an increase in domestic noise trader bullishness. We state these conditions in the following theorem:
Theorem 3.4. Under mild segmentation and assuming international asset prices are positively correlated and all investors are equally risk adverse, an increase in domestic noise bullishness will decrease (increase) the price for the foreign asset without noise if the ratio of noise traders to total traders is higher (lower) in the foreign market than that in the domestic market.
Proof. If all assets are positively related, then β3 in Eq. (3.8) is strictly positive.
Hence, the sign of A determines the relationship between domestic noise ρDt and
foreign asset price pf at . Note that A reflects two characteristics: (1) the average risk
aversion coefficient of each investor type, and (2) the number of investors of each investor type. Assuming investor risk preferences are identical across investor types, the sign of A depends only on the measure of noise traders to rational traders. To see this, recall that
A = αjk αn − αwαjk αinαnk = αwαjk 1 αnαw − 1 αinαnk = αwαjk X n∈N 1 αn ! X i∈I 1 αi + X n∈N 1 αn + X j∈J 1 αj + X k∈K 1 αk − X i∈I 1 αi + X n∈N 1 αn ! X n∈N 1 αn + X k∈K 1 αk !) = αwαjk X n∈N 1 αn X j∈J 1 αj − X i∈I 1 αi X k∈K 1 αk .
Let α denote the common constant risk aversion coefficient across all investors. Then A can be written as follows:
A = αwαjk N α · J α· I α· K α .
Let P = N +II , i.e. the percent of domestic investors which are rational; Q =
J
J+K, i.e. the percent of foreign investors which are rational; D = N +I, i.e. the total
Using these definitions, A can be expressed as A = αwαjk (1− P )D α · QF α − P D α · (1− Q)F α
which can be rewritten as
A = αwαjkDF
α2 (Q− P ) .
Hence, A < 0 is equivalent to Q < P . That is A < 0 is equivalent to the ratio of noise traders to total traders is higher in the foreign market than that in the domestic market.
4. Noise Trader Risk
The goal of this section is to analyze the impact of noise trader risk on asset prices. Since our focus is on the noise trader risk, we will suppress fundamental risk and concentrate on noise trader risk. The impact of noise trader risk on asset prices is summarized in the following theorem.
Theorem 4.1. Under mild segmentation, steady state equilibrium asset prices aris-ing from noise are as follows:
pdat (noise) = 0 , (4.1) pdbt (noise) = αw αnk · (ρD t − ρmD) (1 + r) + αw αnk ρmD r − αw αnk · 1 r(1 + r)2 α2 w αnk + Aβ2 σρD2 , (4.2) pf at (noise) = Aβ3 1 + r(ρ D t − ρmD) + Aβ3 r ρ mD−αjk r A2β23 (1 + r)2 + A2β3β4 (1 + r)2 σ2ρD, +β2αjk− (1 + β2)αw r Aβ3 (1 + r)2 αw αnk σρD2 , (4.3) pf bt (noise) = Aβ4(ρDt − ρmD) (1 + r) + Aβ4 r ρ mD +αjk(ρ F t − ρmF) αk(1 + r) + αjk αk· r ρmF −αjk r A2β 3β4 (1 + r)2 + A2β2 4 (1 + r)2 σρD2 +β2αjk− (1 + β2)αw r Aβ4 (1 + r)2 αw αnk σρD2 −αjk r 1 (1 + r)2 α2 jk α2 k α2ρF. (4.4)
In order to see the impact of one-way barrier on asset pricing, let’s compare Eq. (4.2) to the pricing function in DSSW (Eq. (12) in p. 710). The major difference is the influence of σρD2 to the price. One-way barrier creates the additional term,
Aβ2. Parameter A represents the risk preference. However, β2 measures the ability
of eligible security with noise to replicate the foreign portfolio.
Similarly, the impact of noise trader risk on the volatilities of asset prices are summarized in the following theorem.
Theorem 4.2. Under mild segmentation, the incremental volatilities of asset prices arising from the existence of noise trader are as follows:
∆Var(pdat ) = 0 , (4.5) ∆Var(pdb t ) = αw αnk 2 1 (1 + r)2σ 2 ρD, (4.6) ∆Var(pf at ) = Aβ3 1 + r 2 σρD2 , (4.7) ∆Var(pf bt ) = Aβ4 1 + r 2 σ2 ρD+ αjk αk 2 1 (1 + r)2σ 2 ρF. (4.8)
Equations (4.5) and (4.6) are the well-known results. Equations (4.7) and (4.8) again demonstrate the spill-over effect from domestic noise term to the volatility of foreign securities. The magnitudes of influence depends on two factors: risk pref-erence parameter, A and the ability of replication of foreign security by domestic security with noise, β3and β4.
5. Conclusion
In this paper we incorporate noise trader’s mispricing in an international setting with investment barriers. The subcases of our general model include international CAPM, international asset pricing model with investment barrier as well as do-mestic noise trader’s pricing model. The model allows us to analyze how irrational pricing behavior affects international asset pricing across alternative market struc-tures. In general, we find that noise traders induce an additional risk premium which can further segment international capital markets over and beyond capital barriers. This result suggests that the removal of capital barriers is not a sufficient condition to achieve integrated world markets. This result is consistent with the recent empirical evidence on the evolution of market integration.
Appendix A. Proof of Theorem 3.1
Representative investors i, n, j and k maximize Eqs. (2.4)–(2.7) when choosing asset demands, respectively. The corresponding first order conditions for each class
of investors are as follows: " Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) # = αiΩ 2t " λi,dat λi,dbt # , i∈ I (A.1) " 0 ρD t # + " Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) # = αnΩ 2t " λn,dat λn,dbt # , n∈ N (A.2) Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) Et(pf at+1)− p f a t (1 + r) Et(p f b t+1)− p f b t (1 + r) = αjΩ 4t λj,dat λj,dbt λj,f at λj,f bt , j ∈ J (A.3) 0 ρD t 0 ρF t + Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) Et(pf at+1)− p f a t (1 + r) Et(pf bt+1)− p f b t (1 + r) = αkΩ 4t λk,dat λk,dbt λk,f at λk,f bt , k∈ K . (A.4)
Equilibrium prices are determined by summing over individual demands and setting aggregate asset demand equal to aggregate supply (normalize to be 1) for each security s ∈ da, db, fa, fb. The solution can most easily be obtained by expressing aggregate investor demand function in a common 4× 4 matrix.
This requires expressing Eqs. (A.1) and (A.2), which are 2× 2 matrices, into an equivalent “augmented” 4× 4 matrix form. Following this procedure, Eq. (A.1) can be expressed as follows:
Ω4t X i∈I λi,dat X i∈I λi,dbt 0 0 =X i∈I 1 αi Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) 0 0 + 0 0 Ct(f a, da) X i∈I λi,dat + Ct(f a, db) X i∈I λi,dbt Ct(f b, da) X i∈I λi,dat + Ct(f b, db) X i∈I λi,dbt (A.5)
and Eq. (A.2) can be expressed as follows: Ω4t X n∈N λn,dat X n∈N λn,dbt 0 0 = X n∈N 1 αn Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) 0 0 + 0 0 Ct(f a, da) X n∈N λn,dat + Ct(f a, db) X n∈N λn,dbt Ct(f b, da) X n∈N λn,dat + Ct(f b, db) X n∈N λn,dbt +X n∈N 1 αN 0 ρD t 0 0 . (A.6)
Equation (A.3) after aggregating individual demands over investor j∈ J becomes
Ω4t X j∈J λj,dat X j∈J λj,dbt X j∈J λj,f at X j∈J λj,f bt =X j∈J 1 αj Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) Et(pf at+1)− p f a t (1 + r) Et(pf bt+1)− p f b t (1 + r) . (A.7)
Similarly, Eq. (A.4) after aggregating individual demand over investor k ∈ K becomes Ω4t X k∈K λk,dat X k∈K λk,dbt X k∈K λk,f at X k∈K λk,f bt = X k∈K 1 αk λk,dat λk,dbt λk,f at λk,f bt +X k∈K 1 αk 0 ρDt 0 ρF t . (A.8)
Summing up Eqs. (A.5), (A.6), (A.7) and (A.8), as well as recognizing that market clearing condition requires that total demand for each security must be
equal to total supply, we have the following expression: Ω4t 1 1 1 1 = α−1w Et(pdat+1)− α−1w pdat (1 + r) α−1w Et(pdbt+1)− α−1w pdbt (1 + r) α−1jkEt(pf at+1)− α−1jkp f a t (1 + r) α−1jkEt(pf bt+1)− α−1jkpf bt (1 + r) + 0 α−1nkρD t 0 α−1k ρF t + 0 0 Ct(f a, da) X i∈I λi,dat + X n∈N λn,dat + Ct(f a, db) X i∈I λi,dbt +X n∈N λn,dbt Ct(f b, da) X i∈I λi,dat + X n∈N λn,dat + Ct(f b, db) X i∈I λi,dbt + X n∈N λn,dbt . (A.9) The pricing kernel for the domestic security without noise can be easily obtained as the solution to the first row of Eq. (A.9). Hence, we have
Et(pdat+1)− p da
t (1 + r) = αwCt(pdat+1, V M
t+1) . (A.10)
This is Eq. (3.1) in Theorem 3.1.
Similarly, solving for the second row of Eq. (A.9), we have the pricing kernel for the domestic security with noise:
Et(pdbt+1)− p db t (1 + r) = αwCt(pdbt+1, V M t+1)− αw αnk ρDt . (A.11) This is Eq. (3.2).
To solve the pricing kernels for da and db, we derive the aggregate demand function for investors s∈ I, N for securities da and db from Eqs. (A.1) and (A.2) as follows: X s∈I & N λi,dat X i∈I & N λi,dbt = α−1inΩ−12 " Et(pdat+1)− pdat (1 + r) Et(pdbt+1)− pdbt (1 + r) # + α−1n Ω−12 " 0 ρD t # . (A.12)
Substituting Eq. (A.12) into Eq. (A.9) and solving for the third row, we have the following pricing kernel for the foreign asset without noise:
Et(pf at+1)− p f a t (1 + r) = αjkCt(p f a t+1, V M t+1)− αwαjk αin " Ct(da, f a) Ct(db, f a) #0 Ω−12t " Ct(pdat+1, Vt+1M ) Ct(pdbt+1, Vt+1M ) # − αjk αn − αwαjk αinαnk "C t(da, f a) Ct(db, f a) #0 Ω−12t 0 ρD t . (A.13)
Similarly, solving the fourth row of Eq. (A.9) after substituting in Eq. (A.12), we have the following pricing kernel for the foreign asset with noise:
Et(p f b t+1)− p f b t (1 + r) = αjkCt(pf bt+1, V M t+1)− αwαjk αin "C t(da, f b) Ct(da, f b) #0 Ω−12t " Ct(pdat+1, Vt+1M ) Ct(pdbt+1, Vt+1M ) # − αjk αn − αwαjk αinαnk "C t(da, f a) Ct(db, f a) #0 Ω−12t 0 ρD t − αjk αk ρFt . (A.14)
This is Eq. (3.4) in text.
Appendix B. Proof of Theorem 3.2 We first make use of the following fact:
αwαjk
αin
= αjk− αw> 0 . (B.1)
Substituting Eq. (B.1) into Eq. (3.3) and noting that Vt+1M = V D t+1+V F t+1, we obtain: Et(p f a t+1)− p f a t (1 + r) = [(αjk− αw) + αw]Ct(p f a t+1, V M t+1) − (αjk− αw) " Ct(da, f a) Ct(db, f a) #0 Ω−12t " Ct(p f a t+1, Vt+1D ) Ct(pf bt+1, Vt+1D ) # − (αjk− αw) " Ct(da, f a) Ct(db, f a) #0 Ω−12t " Ct(p f a t+1, Vt+1F ) Ct(pf bt+1, Vt+1F ) # − αjk αn − αwαjk αinαnk "C t(da, f a) Ct(db, f a) #0 Ω−12t 0 ρD t . (B.2) Again noting that VM
t+1= Vt+1D + Vt+1F , we obtain: Et(pf at+1)−p f a t (1 + r) = αwCt(pf at+1, V M t+1)+(αjk− αw)Ct(pf at+1, V F t+1)+(αjk− αw) × ( Ct(pf at+1, V D t+1)− " Ct(da, f a) Ct(db, f a) #0 Ω−12t " Ct(p f a t+1, Vt+1D ) Ct(pf bt+1, Vt+1D ) #) − (αjk− αw) " Ct(da, f a) Ct(db, f a) #0 Ω−12t " Ct(p f a t+1, Vt+1F ) Ct(pf bt+1, Vt+1F ) # − αjk αn − αwαjk αinαnk "C t(da, f a) Ct(db, f a) #0 Ω−12t 0 ρD t . (B.3)
To interpret the third and fourth terms of Eq. (B.3), consider the following projection of VF on pdaand pdb:
VF = β1pda+ β2pdb+ e . (B.4)
Define the adjustment portfolio VAas the fitted values of Eq. (B.4):
VF = β1pda+ β2pdb. (B.5)
Covariance between VA and pf acan be expressed as:
Covt(p f a t+1, V A T +1) = Covt(p f a t+1, β1(pdat+1) + β2(pdbt+1)) (B.6) = β1Covt(p f a t+1, p da t+1) + β2Covt(p f a t+1, p db t+1) (B.7) = " Covt(pf at+1, pdat+1) Covt(p f a t+1, p db t+1) #0" β1 β2 # (B.8) = " Ct(f a, da) Ct(f a, db) #0 Ω−12t " Ct(pdat+1), V F t+1 Ct(pdbt+1), Vt+1F # . (B.9) Similarly, Covt(pf at+1, V D T +1) = " Ct(f a, da) Ct(f a, db) #0 Ω−12t " Ct(pdat+1), Vt+1D Ct(pdbt+1), Vt+1D # . (B.10)
Here, we made use of the fact that the linear projection of VDon its components
pdaand pdbis just itself. Finally, we consider the following multiple repression:
pf a= βpda+ β3pdb+ ε (B.11)
and note that
β3ρDt = " Ct(da, f a) Ct(da, f b) #0 Ω−12t " 0 ρD t # . (B.12)
Substituting Eqs. (B.9), (B.10) and (B.11) into Eq. (B.3), we obtain
Eq. (3.5).
Acknowledgment
M.-W. Hung thanks NSC for financial support.
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