International Stock Market Integration: A Dynamic ... - FSA ULaval

International Stock Market Integration: A Dynamic General Equilibrium Approach∗ Harjoat S. Bhamra† December 2002

∗ I would like to thank my advisor Raman Uppal and Viral Acharya, Suleyman Basak, Francisco Gomes, Denis Gromb, Anna Pavlova, Paulo Volpin, and seminar participants at the LBS, Imperial College and CentER, Tilburg for valuable comments. † Ph.D. student at London Business School, 6 Sussex Place, Regent’s Park, London, United Kingdom NW1 4SA; Tel: 44-20-7262-5050 ext. 3487; Email: [email protected]. I am grateful for financial support from the ESRC and the Centre for Hedge Fund Research and Education at LBS.

Abstract In this article I study the effects of stock market liberalization on equity risk premia, stock return volatilities and the cross-country correlation of stock returns. In a two-country, continuoustime, dynamic economy, I solve for equilibrium in closed-form under different types and degrees of stock market integration. I find that opening the stock market of only one country to external investors results in a decrease in its equity risk premium and stock-return volatility and an increase in the cross-country stock-return correlation. The volatility of stock returns is increased in the country that does not liberalize its stock market. When both countries open their stock markets to external investors, risk premia and volatilities decrease in both markets and the correlation increases. Furthermore, when the country that opens its stock market is relatively small, the resulting decrease in risk premium is magnified. In contrast, the volatility effect is diminished. Correlation increases are magnified when just one country, which is small opens its stock market to external investors, but diminished when two countries of unequal size both open their stock markets. I also show how the dynamic CAPM changes with the level of stock market integration. Keywords: Asset pricing, capital asset pricing model, correlation, incomplete markets, stock market integration, volatility. JEL classification: D52, F36, G11, G12, G15.

Contents 1 Introduction

1

2 Model 2.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 7

3 Non-integrated stock markets 3.1 Agents’ endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Individual country consumption and portfolio choice . . . . . . . . . . . . . . . 3.3 Representative “world” investor, stochastic weight and cross-sectional wealth distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Equilibrium prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Capital asset pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 8 8 10 11 15

4 One-way stock market integration 4.1 Agents’ endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Individual country consumption and portfolio choice . . . . . . . . . . . . . . . 4.3 Representative “world” investor, stochastic weight and cross-sectional wealth distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Equilibrium prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Interest rate effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Risk premia and stock price effects . . . . . . . . . . . . . . . . . . . . . 4.4.3 Volatility and correlation effects . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Analysis of the comparison results for different parameter values . . . . 4.5 Capital asset pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 16

5 Integrated stock markets 5.1 Agents’ preferences and endowments . . . . . . . . . 5.2 Individual country consumption and portfolio choice 5.3 Equilibrium prices . . . . . . . . . . . . . . . . . . . 5.3.1 Interest rate effects . . . . . . . . . . . . . . . 5.3.2 Risk premia and stock price effects . . . . . . 5.3.3 Volatility and correlation effects . . . . . . . 5.4 Capital asset pricing model . . . . . . . . . . . . . .

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6 One-way partially integrated stock markets 6.1 Individual country consumption and portfolio choice 6.2 Equilibrium Prices . . . . . . . . . . . . . . . . . . . 6.2.1 Interest rate effects . . . . . . . . . . . . . . . 6.2.2 Risk premia and return effects . . . . . . . . 6.2.3 Volatility and correlation effects . . . . . . . 6.3 Capital asset pricing model . . . . . . . . . . . . . .

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7 Dynamic capital asset pricing models 7.1 Partial two way stock market integration . . . . . . . . . . . . . . . . . . . . . .

34 35

17 17 18 18 20 23 24

8 Conclusions

36

References

38

A Perfect Stock Market and Bond Market Segmentation A.1 Agents’ preferences and endowments . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Individual country consumption and portfolio choice . . . . . . . . . . . . . . . . . . A.3 Equilibrium prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 43 43

B Proofs

45

C Technical Appendix C.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 66

D Figures

69

International Stock Market Integration: A Dynamic General Equilibrium Approach

1

Introduction

In the 1980’s, many emerging markets’ countries opened their stock markets to foreign investors. In the 1990’s severe financial crises affected many of these countries. These events have helped fire a strong debate on the benefits of stock market liberalization and more generally on globalization (see Stiglitz (2002) and Soros (2002)). Economists are now seeking further guidance on policy-making, that looks beyond the standard argument that stock market liberalization leads to better risk-sharing and hence improves welfare.1 There is more interest in the effects of stock market liberalization on stock market volatility, correlations between markets, and equity risk premia (Kaminsky and Schmukler (2001), Bekaert and Harvey (2000), Henry (2000) and Kim and Singal (2000) conduct empirical studies). Opponents of stock market liberalization have even argued that it increases stock market volatility. My theoretical results show that when a country liberalizes its stock market by opening it to foreign investors, its risk premium and volatility decrease, whilst comovement with other stock markets rises. I undertake the analysis in a two country, continuous-time, dynamic general equilibrium economy with one good. Each country has its own exogenous output process for the perishable consumption good, a stock that is a claim on its output and a representative agent (countries, stocks and representative agents are labelled one and two), who derives utility from consuming this good. Prices are endogenous, so I can study the effects of various types of stock market integration on prices, returns, volatilities and correlation. I also derive a dynamic CAPM. Previous theoretical papers on stock market integration could not obtain results on volatilities or correlation, because they either worked in a static framework2 , assumed correlations 1 There is a large body of work examining the effects of international risk-sharing on welfare. For example, Backus, Kehoe and Kydland (1992), Obstfeld (1992), Cole and Obstfeld (1991) and Tesar (1995) show that global risk sharing would lead to a small welfare increase amounting to a negligible fraction of lifetime consumption. In contrast, Lewis (1996) , Obstfeld (1994) and van Wincoop (1994) show welfare gains close to 15% of lifetime consumption. 2 Black (1974), Subrahmanyam (1975), Stapleton and Subrahmanyam (1989), Stulz (1981), Errunza and Losq (1985), Errunza and Losq (1989), and Eun and Jarakiramanan (1986).

International Stock Market Integration

2

and volatilities were exogenous3 , or that stock prices were exogenous4 . Solving for volatilities and correlation in equilibrium presents technical challenges, which I discuss later. Assuming that individual country representative agents can always borrow/lend freely at the (endogenous) riskless rate, I consider the following cases of stock market integration: (i) Non-integrated stock markets (base case) — each agent holds only its own stock; (ii) One-way stock market integration — agent one can hold both stocks whilst agent two can only hold its own stock; (iii) Integrated stock markets — each agent may hold both stocks, and (iv) Partial one-way stock market integration — agent one can hold up to a certain proportion of its wealth in the other country’s stock market and agent two can only hold its own stock. Having outlined some of my results in the opening paragraph, I explain the underlying intuition. Opening up stock market two to external investors allows agent one to invest in stock two, whilst agent two is still restricted to holding only stock two. This leads to an increase in demand for stock two and a decrease in demand for stock one, relative to the base case. These “demand effects” lead to an increase in the price of stock two and a decrease in the price of stock one. Expected returns move in the opposite directions to prices. The riskless rate is raised, because better risk-sharing reduces the precautionary savings motive of investors. Both these effects act to reduce the equity risk premium in the liberalized market. This reduction becomes more pronounced when country two has a relatively smaller output size, and is consistent with the decrease seen in the data (see Henry (2000)). Considering only “risk-sharing effects” associated with the rise in the riskless rate as in Sellin and Werner (1993) will not accomplish this, because “risk-sharing effects” are skewed less by differences in country size. Furthermore, allowing for a difference in country output size produces theoretical results on volatility and correlation, which are consistent with the empirical findings of Bekaert and Harvey (2000), and Kim and Singal (2000). Opponents of stock market liberalization in emerging markets often state that it increases stock return volatility in the liberalized market. The model I use does not support this claim — there is a small increase in stock return volatility in the closed market and a decrease in the liberalized market. To understand this result, it is essential to note that markets are incomplete. Therefore, each country has its own market price of risk, which is also the volatility of its own country-specific state-price density. Opening up stock market two to agent one, allows 3 4

Basak (1996) Sellin and Werner (1993), Obstfeld (1994), Dumas and Uppal (2001).


3

agent one to absorb more volatility into her consumption stream. This raises the volatility of the country one state-price density. Using the Malliavin calculus (a stochastic calculus of variations)5 , I derive a relation between stock return volatility and state-price density volatility. This shows that the rise in state-price density volatility in country one results in an increase in stock return volatility in country one. The opposite effects occur in country two. Whilst the sizes of the state-price density volatilities have diverged upon stock market liberalization, the correlation between them has increased, raising the stock-return correlation. When country two is relatively smaller in output terms, the volatility effects in both countries are diminished and the increase in correlation is magnified. The intuition is that the size of country one’s stock market cushions against the increase in stock return volatility. This implies a smaller decrease in stock return volatility in country two, because the volatility of the aggregate “world” endowment is fixed. With correlation, it is as if country one drags country two into its orbit more powerfully as the relative size of country one increases — country two’s state price density is affected by agent one’s choices, but not vice versa. When both stock markets are opened up to external investors, “risk-sharing effects” associated with the decrease in precautionary savings motive dominate and risk premia in both countries are decreased. State-price density volatilities also decrease, driving down stock return volatilities. State-price densities become more highly correlated, driving up the stock return correlation. The effect of differences in country size is very intuitive — the larger country is affected less by the opening of stock markets and the rise in correlation is diminished. If stock market two is partially opened, so that agent one invests I1 of her wealth in stock two, the level of comovement between stock markets is increasing in I1 . When I1 is time-varying this provides a theoretical result to link time-variation in the level of stock market integration to time-variation in correlation, both of which are features of the data as shown by Bekaert and Harvey (1995) and Harvey (1991), respectively. I also provide theoretical support for the positive relation between foreign investment inflows and stock returns in emerging markets found by Froot, O’Connell and Seasholes (2001). 5 Many recent papers have exploited the Malliavian calculus for this purpose. In general equilibrium models where agents have heterogeneous beliefs, Gallmeyer (2000) and Gallmeyer and Hollifield (2002) use the Malliavin calculus to characterize the stochastic process for volatility. Serrat (2001) does the same in a two-country exchange economy, where markets are complete. Detemple and Zapatero (1991) exploit a Malliavin calculus based characterization of stock return volatility to obtain a 2-factor CAPM in an economy, where the representative agent has preferences exhibiting habit formation. The Malliavin calculus has also been used to compute numerically the hedging demands in optimal portfolios by Detemple, Garcia and Rindisbacher (2002). A thorough treatment of the Malliavin calculus can be found in the monograph by Nualart (1995).


4

I now discuss the technical contributions of the paper. The model I use has two stocks and even when markets are dynamically complete there is no exact closed-form representation for the stock prices. Using integral approximation methods (see Hinch (1991) and Judd (1998)), I derive highly accurate approximate closed-form expressions for stock prices. I must also solve for equilibria when markets are incomplete. This means solving a forward-backward stochastic differential equation system, which is essentially a fixed point problem involving stock prices and the endogenous wealth distribution (see Ma and Yong (1999)). I derive an approximate closed-form solution to this forward-backward stochastic differential equation system based on continuation methods and tools from non-linear functional analysis (see Zeidler (1986)). I show that this approach is as accurate as any purely numerical method. The technical difficulties outlined above do not arise in the single stock case studied by Basak and Cuoco (1998), where a simple expression was found for the stock price. The modelling framework in Shapiro (2002) is similar to the one used in this article. However, in Shapiro (2002) the equilibrium is not explicitly derived. Relative to Shapiro (2002), I make the additional assumption that the growth rates of output processes have constant means and volatilities. This assumption allows me to solve for equilibrium prices, returns and volatilities explicitly in closed-form and hence study the effects of stock market integration on these variables. Previous papers on incomplete markets such as Telmer (1993) and Heaton and Lucas (1996), do compute some equilibrium quantities numerically. However, neither compute stock return volatilities or correlation, which I do in closed-form. The paper proceeds as follows. Section 2 outlines the model. Section 3 solves for the equilibrium when stock markets are not integrated. Section 4 considers the case where one stock market is opened to external investment, but the other is not, whilst Section 5 considers the case where stock markets are perfectly integrated. Section 6 solves for the equilibrium where one stock market is partially opened to external investment, but the other stock market remains closed to outsiders. Section 7 generalizes the CAPM of Section 6 to the case where both stock markets are partially opened. Appendix A studies the case of autarky, where both stock and bond markets are not integrated. Proofs and additional results are given in Appendix B. Appendix C describes how I solve for equilibrium approximately in closed-form and shows how accurate my results are.


2

5

Model

I consider a continuous time economy on the finite time span [0, T ] modeled as follows. Uncertainty is represented by a filtered probability space (Ω, F, F, P) on which a 2-dimensional

Brownian motion W (t) = (W1 (t) , W2 (t))| is defined. A state of the world is described by ω ∈ Ω. The filtration F = {Ft }t∈[0,T ] is the augmentation (by the P-null sets in F W (T )) of

the filtration F W (t) = σ (W (s) ; 0 ≤ s ≤ t) for all t ∈ [0, T ], which is the filtration generated by W . I assume that F = F T , i.e. the true state of nature is completely determined by the sample paths of W on [0, T ] . The sigma-field, Ft , represents the information available at time t

and the probability measure, P, the agents’ common beliefs. All random processes are assumed progressively-measurable with respect to F. All inequalities and equalities involving random variables are understood to hold P-almost surely. There is a single perishable good (the numeraire). The agents’ consumption set C is defined as the set of nonnegative progressively measurable consumption rate processes C with RT 0 |C (t)| dt < ∞.

There are two countries i ∈ {1, 2} each with a representative agent having lifetime expected

utility functional E0

·Z

T

−βt

e

ui (Ci (t)) dt

0

¸

(1)

where ui (Ci (t)) = log Ci (t) and β is the rate of time preference for each agent. For each i, ui has a continuous and strictly decreasing inverse Ii that maps (0, ∞) onto itself. For each country i ∈ {1, 2} there is a strictly positive exogenous endowment (output) process ei such that ei (t) = ei (0) + µe

Z

t

ei (s) ds +

0

Z

0

t

ei (s) σ |e,i dW (s)

(2)

where σ e,1 = [σ e,11 , σ e,12 ]| , σ e,2 = [σ e,21 , σ e,22 ]| , kσ e,1 k = kσ e,2 k and µe is strictly positive. I

assume that [σ e,1 | , σ e,2 | ]| is non-singular. Let ρ denote the correlation between the growth rates of the two endowment processes. I also define the aggregate endowment process y (t) =

2 X i=1

ei (t) ∀t ∈ [0, T ] ,

(3)

and assume that the initial endowments of each country are such that e1 (0) = N e2 (0) for some N > 0. I define the relative output process N (t) = e1 (t) /e2 (t).


6

Investment opportunities are represented by three securities. The first is a locally riskless bond in zero net supply earning an instantaneous interest rate r over the finite time interval [0, T ]. The bond price process satisfies B (t) = exp

·Z

0

t

¸ r (s) ds ,

(4)

where the initial price of the bond has been normalized to unity. The are also two stocks each in a constant supply of one unit. Each stock is a claim on the endowment of a country. Let Sj denote the stock corresponding to the endowment of country j, ej . I assume that in equilibrium S = (S1 , S2 )| follows an Ito process Z t Z t [µ (s) IS (s) − e (s)] ds + S (s) σ (s) dW (s) S (t) = S (0) + 0

(5)

0

where µ (t) = (µ1 (t) , µ2 (t))| , IS (t) = diag (S1 (t) , S2 (t)), σ (t) = (σ |1 (t) , σ |2 (t)),

σ 1 (t) = (σ 11 (t) , σ 12 (t))| and σ 2 (t) = (σ 21 (t) , σ 22 (t))| . The interest rate process r, the drift coefficients µ and the volatility (diffusion) matrix, σ, are to be determined endogenously in equilibrium. Trading takes place continuously. I define trading strategies as vectors of the amounts invested in the bond, αi and in the risky assets, ϕi . A trading strategy (αi , ϕi ) finances a consumption plan Ci ∈ C if the corresponding wealth process Xi ≡ αi + ϕ|i 1 satisfies the dynamic budget constraint £ ¢ ¤ ¡ dXi (t) = αi (t) r (t) + ϕi (t)| µ (t) − r (t) 1 − Ci (t) dt + ϕi (t)| σ (t) dW (t)

(6)

where 1 ≡ (1, 1)| . A trading strategy is admissible if Xi (t) ≥ 0. The set of admissible trading strategies is denoted by Θ. I now define portfolio proportion processes. A portfolio proportion process π = (π 1 , π 2 )| is a vector of the proportions of wealth held by an investor in each risky financial asset. Given an admissible trading strategy (αi , ϕi ), which finances a consumption plan Ci ∈ C with corresponding wealth process Xi , the proportional portfolio process π is defined as follows ϕi (t) = Xi (t) π (t) , 0 ≤ t ≤ T ( ϕi (t) Xi (t) , 0 ≤ t ≤ τ p π (t) = 0, τ p ≤ t ≤ T where τ p = inf {t ∈ [0, T ] s.t. Xi (t) = 0} .


7

If (αi , ϕi ) ∈ Θ and finances Ci ∈ C, then the consumption portfolio process pair (Ci , π) is said to be admissible and lies in the set denoted by A.

2.1

Equilibrium

¡ ¢ Denote the initial endowment of country i by αi,0 , ϕi,0 . Firstly I define equilibrium for a ¡ ¢ country i, which is in autarky. I shall denote by Ei = ((Ω, F, F, P) , ei , Ui , αi,0 , ϕi,0 ) the

primitives for the economy of country i. An equilibrium for the economy Ei is a price process

(B, S1 ) or equivalently an interest rate-stock price process (r, S1 ) and a set {Ci∗ , (α∗i , ϕ∗i )} of consumption and admissible trading strategies such that: the trading strategy (α∗i , ϕ∗i )

finances Ci∗ ; consumption, Ci∗ , maximizes Ui over the set of consumption plans C ∈ C, which are financed by an admissible trading strategy (α, ϕ) ∈ Θ, with α (0) = αi,0 and ϕ (0) = ϕi,0 , and all markets clear. ¢ ¡ ¢ ¡ I shall denote by E = ((Ω, F, F, P) , e1 , e2 , U1 , U2 , α1,0 , ϕ1,0 , α2,0 , ϕ2,0 ) the primitives for

the “world” economy. An equilibrium for the “world” economy E is a price process (B, S1 , S2 ) or equivalently an interest rate-stock price process (r, S1 , S2 ) and a set {Ci∗ , (α∗i , ϕ∗i )}2i=1 of

consumption and admissible trading strategies such that: (α∗i , ϕ∗i ) finances Ci∗ for i ∈ {1, 2}; Ci∗ maximizes Ui over the set of consumption plans C ∈ C, which are financed by an admissible

trading strategy (α, ϕ) ∈ Θ, with α (0) = αi,0 and ϕ (0) = ϕi,0 for i ∈ {1, 2}, and all markets clear.

3

Non-integrated stock markets

In this section, I examine the base case where stock markets are not integrated, but bond markets are perfectly integrated. This corresponds to the situation where two countries or regions of the world can borrow/lend freely at the riskless rate, but are prohibited from investing in each others’ stock markets. I suppose that agent one and agent two can borrow and lend to each other at the riskless rate, but are only allowed to invest in their own stock markets. I label this economy EN I . In this case markets are incomplete and the competitive equilibrium is not Pareto optimal. This model differs from those with market incompleteness in a domestic setting such as Telmer (1993), because each agent faces a country-specific stochastic investment opportunity set. To solve for the equilibrium I construct a representative agent with progressively measurable


8

stochastic weights as in Cuoco and He (1994). It is important to note that the weights are stochastic, not deterministic as would be the case if the competitive equilibrium were Pareto optimal. This formulation will reduce the search for the equilibrium to finding the stock prices and stochastic weights. For the two-investor problem under consideration here, there is only one stochastic weight to find. For logarithmic investors this weight is the ratio of individual country representative agents’ wealth processes.6 This ratio is determined by a forward stochastic differential equation in terms of the exogenous parameters, the price of stock one and its volatility. The price of stock one is the conditional expectation of an integral in terms of exogenous variables and the stochastic weight. This system of equations can be written as a forward-backward stochastic differential equation, which cannot be solved exactly in closed-form. Such an equation system can be regarded as a fixed point problem. I solve this forward-backward stochastic differential equation approximately using techniques from nonlinear functional analysis (see Zeidler (1986)).

3.1

Agents’ endowments

Each individual country agent is endowed with all the stocks in its country and amount zero of the bond.

3.2

Individual country consumption and portfolio choice

Both agents face incomplete markets. Hence, to characterize optimal consumption for each individual agent I use the duality approach of Cvitanic and Karatzas (1992). A textbook treatment can be found in Karatzas and Shreve (1998). This approach relies on embedding the constrained optimization problem for each agent in a larger family of unconstrained optimization problems. For each agent the solution of one and only one of the unconstrained problems in the family gives the unique state-price density for the agent in the constrained problem. Therefore each agent i faces her own unique country-specific state-price density, ξ i (·). These two state-price densities, one for each agent, formalize the intuition that the representative agents of each country have access to different investment opportunities. I shall use these two state-price densities to derive other quantities, in an analogous way to the complete 6

The stochastic weight λ (t) is given by λ (t) = X2 (t) /X1 (t) and the cross-sectional wealth distribution is given by x (t) = X1 (t) / (X1 (t) + X2 (t)). There is therefore a bijective mapping between cross-sectional wealth distrubutions taking values in the open interval (0, 1) and stochastic weights taking values in the open interval (0, ∞).


9

markets case. Proposition 1 The optimal consumption-portfolio process pair (Ci , π i ) of the representative agent of country i with (Ci , π i ) ∈ A satisfies Ci (t) =

e−βt ψ i ξ i (t)

(7)

and µi (t) − r (t) 1i , (8) σ |i (t) σ i (t) ¡ ¢ where 11 = (1, 0)| , 12 = (0, 1)| . The constant ψ i = 1 − e−βT /βXi (0) is the Lagrange RT multiplier associated with the static budget constraint of agent i, E 0 ξ i (t) Ci (t) dt = Xi (0). π i (t) =

The state-price density process faced by agent i is ½ Z t ¾ Z Z t 1 t | 2 ξ i (t) = ξ i (0) exp − r (u) du − θi (u) dW (u) − kθi (u)k du 2 0 0 0

(9)

with the market price of risk process for country i given by θi (t) =

σ i (t) (µ (t) − r (t)) 1i . σ i (t)| σ i (t) i

(10)

Because each agent is logarithmic (myopic) and holds only her own stock, the portfolio proportions take the simple mean-variance form shown in (8). When the individual country representative agent i follows the optimal policy, ξ i (t, ω) can be interpreted as the agent’s Arrow-Debreu price (per unit probability P) of one unit of consumption good at state ω and time t. At the optimum, the marginal benefit from an additional unit of consumption at state ω and time t is proportional to the cost of that unit. Investors from different countries face different investment opportunity sets in incomplete markets, so for each individual country representative agent the cost of a unit of consumption is different. Hence there is a different price of risk in each country as seen in equation (10).7 These different country-specific prices of risk define different state-price densities for each country, shown in equation (9). 7

King, Sentana and Wadhwani (1990) provide empirical evidence that the price of risk is not common across countries.


3.3

10

Representative “world” investor, stochastic weight and cross-sectional wealth distribution

I now outline the construction of a state-dependent “world” representative investor, which is used to obtain the optimal consumption policies of each country in terms of the stochastic weight or cross-sectional wealth distribution (see footnote 6 on page 8). The use of a statedependent utility to characterize non-Pareto efficient equilibria in incomplete markets has been used recently in a variety of models. For example, Basak and Cuoco (1998), Basak and Gallmeyer (1999), Basak (2000), Basak and Croitoru (2000), Gallmeyer and Hollifield (2002) and Detemple and Serrat (2002). Following Cuoco and He (1994), I construct the state-dependent “world” representative investor U (x; λ (t)) = E

Z

T

e−β(s−t) u (C (s) , λ (s)) ds ,

(11)

λ (t) C (t) C (t) + λ (t) ln 1 + λ (t) 1 + λ (t)

(12)

t

where u (C (t) , λ (t)) = ln

and C(t) is the consumption policy of the “world” representative agent. Of course at the optimum, the “world” representative agent consumes the entire "world" endowment, so C(t) = y(t). Lemma 1 The optimal consumption allocations for the representative agent for each country can be characterized in terms of the stochastic weight λ(t) C1 (t) =

y (t) λ (t) y (t) and C2 (t) = 1 + λ (t) 1 + λ (t)

(13)

or in terms of the cross-sectional wealth distribution x(t) C1 (t) = x (t) y (t) and C2 (t) = (1 − x (t)) y (t)

(14)

This lemma characterizes consumption-sharing across countries as variable proportions of the aggregate “world” endowment process y. These variable proportions must be determined endogenously in equilibrium. This entails finding the stochastic weight or equivalently the cross-sectional wealth distribution. Once these processes have been determined, I shall be able to derive the mean and variance of consumption growth in each country as shown in Lemma B1 in Appendix B.


3.4

11

Equilibrium prices

Proposition 2 The equilibrium interest rate is given by |

|

r (t) = β + µy (t) + µx (t) − (σ y (t) − σ x (t)) (σ y (t) − σ x (t)) + σ x (x) σ y (x)

(15)

where 1 | | [c (t) p1 (t) σ 1 (t) − σy (t) ] [x (t) σ y (t) + 2 1 − x (t) σx (t) = c (t) p1 (t) σ 1 (t) − σ y (t) S1 (t) p1 (t) = y (t) x (t) β c (t) = 1 − e−β(T −t) µx (t) =

µ

¶ 1 − x (t) c (t) p1 (t) σ 1 (t)] (16) 2 (17) (18) (19)

The equilibrium price of stock one is determined by the following system of equations8 Z

T

e1 (s) ds y (s) x (s)

(20)

dx (t) | = µx (t) dt + σ x (t) dW (t) , x (0) = x0 . x (t)

(21)

S1 (t) = y (t) x (t) Et

e−β(s−t)

t

The price of stock two is given by S2 (t) = y (t) (1 − x (t)) Et

Z

T

e−β(s−t) t

e2 (s) ds. y (s) (1 − x (s))

(22)

This proposition gives the equilibrium riskless interest rate in terms of the price of stock one, S1 , it’s volatility, σ 1 , the cross-sectional wealth distribution, x, (or stochastic weight) and exogenous variables. The price of stock one and the cross-sectional wealth distribution are determined by a system of equations which can be written as forward-backward stochastic differential equation (FBSDE). In this case, unlike Basak and Cuoco (1998), the FBSDE cannot be decoupled into a separate backward stochastic differential equation for the stock price and a forward stochastic differential equation for the cross-sectional wealth distribution (or equivalently for the stochastic weight). The FBSDE must be solved as a system. Such FBSDE’s have been used to characterize equilibria in Basak and Gallmeyer (1999) who address issues concerning differential taxation. They solve for the FBSDE by assuming a Markovian framework to obtain a partial differential equation for the stock price, which can then be solved numerically. Following Basak and Gallmeyer (1999), I characterize the price of stock one as the solution to a partial differential equation. 8 To obtain the integral representation of the stock price, I have assumed that the discounted gains price (a local martingale) is a martingale.


12

Corollary 1 Defining the state variable G (t) = e1 (t) /y (t) the price of stock one is given by S1 (t) = y (t) x (t) eβt φ (t)

(23)

where φ (t) solves the quasilinear9 parabolic partial differential equation ¶ µ G (t) ∂ + L φ (t, G, x) + e−βt , φ (T, G (T ) , x (T )) = 0 0= ∂t x (t)

(24)

and L is the differential generator of (G (t) , x (t)) . Corollary 1 exploits the Markovian structure of the economy to write the backward stochastic differential equation for the stock price as a Kolmogorov backward equation. The Kolmogorov backward equation is a quasilinear parabolic partial differential equation. Unlike Basak and Gallmeyer (1999) who use a numerical algorithm so solve for the stock price, I solve for the stock price in closed-form using the method of continuation with respect to a parameter (see Zeidler (1986)). This is one of the contributions of this paper. The solution approach hinges on the observation that the partial differential equation for the stock price can be solved exactly when the cross-country correlation in endowment returns, ρ, is unity. Expanding around the known solution for the stock price when, ρ = 1, enables the construction of a solution for the stock price when ρ ∈ [−1, 1] . Using this solution I can obtain approximate closed-form expressions for stock returns, volatilities, individual country market prices of risk and the riskless rate. These expressions can be computed to arbitrarily high order in the di√ mensionless perturbation parameter, = 1 − ρ/2 (using computer algebra packages).10 This approach offers several advantages over numerical methods, such as finite-difference schemes or the method of lines. For example, the solution can be computed for all possible combinations of model parameters at once. Numerical stability issues do not arise and no accuracy is lost in computing derivatives. Whilst perturbation methods are ubiquitous in mathematics and the 9

The coefficients of the highest derivatives depend on φ and its first partial derivatives, making the partial differential equation quasilinear. p 10 The expression for some variable, η, can be written as a power series in (1 − ρ) /2 r (1 − ρ) (1 − ρ) η1 + η 2 + ... η = η0 + 2 2 Ãµ r ¶3/2 ! (1 − ρ) (1 − ρ) 1−ρ η1 + η2 + O = η0 + 2 2 2 where the big-Oh notation, O

³¡

¢ 1−ρ 3/2 2

´

· q , simply means that η − η 0 + (1−ρ) η1 + 2

(1−ρ) η2 2

¸

/

¡ 1−ρ ¢3/2 2

< ∞.


13

physical sciences (see Hinch (1991) and Kevorkian and Cole (1996)), they have only recently begun to gain ground in finance and economics (see Chan and Kogan (2002), Kogan (2001), Kogan and Uppal (2001) for applications and Judd (1996) for a survey). A principal concern is accuracy, an issue, which I address in Appendix C. It can be shown that the solution I obtain for φ (t) is as accurate as any purely numerical method. I also take care to show that the solution I seek exists (see Appendix C). Proposition 3 This proposition describes the equilibrium in economy ENI . 1. In equilibrium, the vector of optimal proportions of wealth invested by each agent i in stocks is given by πi (t), where

³ ´ π i (t) = 1i + O (1 − ρ)2

(25)

2. The equilibrium interest rate is given by ³ ´ 3 r (t) = β + µe − σ2e + O (1 − ρ)

(26)

3. The market price of risk in country one is given by θ1 (t) = σ e

Ã

1 11 − 2

r

1 − ρ N (t) 1 1 1 − ρ N (t) 12 − 11 2 N (t) + 1 x (t) 4 N (t) + 1 x (t)

!

³ ´ + O (1 − ρ)2

(27)

and the market price of risk in country two is given by Ã ! r ³ ´ 1−ρ 1 1 1 1 1−ρ θ2 (t) = σe 11 − 12 − 11 + O (1 − ρ)2 (28) 2 N (t) + 1 1 − x (t) 4 N (t) + 1 x (t)

4. The risk premia on stocks one and two are given by ³ ´ µi (t) − r (t) = σ 2e + O (1 − ρ)3 , i ∈ {1, 2}

(29)

5. The prices of stocks one and two are given by Si (t) =

³ ´ 1 − e−β(T −t) 4 , i ∈ {1, 2} ei (t) + O (1 − ρ) β

(30)

and cumulative stock returns are given by ³ ´ 4 µi (t) = β + µe + O (1 − ρ) , i ∈ {1, 2}

(31)


14

Conditional stock return volatilities are given by Ã ! r ³ ´ 1−ρ 1−ρ (1 − ρ)2 3 12 − 11 − 11 + O (1 − ρ) σ 1 (t) = σ e 11 + 2 4 32 Ã ! r 2 ³ ´ 1−ρ (1 − ρ) 1−ρ σ 2 (t) = σ e 11 − 12 − 11 + 11 + O (1 − ρ)3 2 4 32

(32) (33)

6. The conditional variances of stock returns for stocks one and two are given by ³ ´ Vi (t) = σ 2e + O (1 − ρ)3 , i ∈ {1, 2}

(34)

7. The correlation in cross-country stock returns is given by ³ ´ 3 ρ12 (t) = ρ + O (1 − ρ)

8. The cross-sectional wealth distribution is given by dx (t) = x (t) (µx (t) dt + σ x (t)| dW (t)) , x (0) = x0

(35)

where ´ 1 − 2x (t) 2 ³ 3 + (1 − ρ) σ e (N (t) + 1)2 x (t) ³ ´ ´ ³ r  N (t) 1−x(t) +1 +1 N (t) x(t)−1 x(t) x(t) 1 − ρ σ x (t) = σ e  12 + (1 − ρ) 11  2 N (t) + 1 N (t) + 1 ³ ´ 2 +O (1 − ρ) N (t)

µx (t) = (1 − ρ)

(36)

(37)

The relative sizes of the initial endowment processes do not affect the equilibrium prices, returns or volatilities. This is a direct consequence of stock market segmentation. Assuming that x (t) ≈ N (t) /(1 + N (t)), it follows that µx (t) ≈ (1 − ρ) and σ x (t) ≈ σ e

1 − N (t) 2 σe (N (t) + 1)2

r

1−ρ 12 . 2

From these expressions, it can be seen that country size acts as a cushion against wealth redistribution. However, the way the dispersion of wealth across the “world” economy changes is little affected by country-size. In Sections 4, 5 and 6, I shall compare and contrast the equilibrium under non-integrated stock markets with the equilibria when stock markets are integrated.


3.5

15

Capital asset pricing model

In this section, I derive a dynamic capital asset pricing model when stock markets are not integrated, but bond markets are integrated. This result is more general than previous results: it is valid for all progressively measurable endowment processes. Proposition 4 The equity premia are given by Et (µ1 (t) − r (t)) = Et (µ2 (t) − r (t)) =

¶ µ dS1 (t) s (t) V art x (t) S1 (t) ¶ µ dS2 (t) 1 − s (t) V art 1 − x (t) S2 (t)

(38) (39)

where s (t) /x (t) and 1 − s (t) /1 − x (t) are the ratios of stock market wealth to financial wealth for countries one and two, respectively. Note that the bond market is integrated, whereas stock markets are not. As as result, covariance risk between stock returns across countries is unpriced in both stock markets. Only local variance risk is priced. Increasing the proportion of wealth held in stocks at the individual country level increases the loading on this local variance risk. These loading factors are present, because markets are incomplete, so investors domiciled in a particular country adjust their portfolio holdings subject to exogenous shocks to their economy.

4

One-way stock market integration

In this section I examine the case where stock market one is closed to external investors, stock market two is open to external investors and bond markets are perfectly integrated. This mirrors the case where an emerging markets country has opened up its stock market to external investors, but residents of this emerging markets country cannot invest abroad. That is, I suppose that agent one can invest freely in both stock markets, agent two can only invest in its own stock market, while both agents can borrow and lend to each other at the riskless rate. This corresponds to the mild integration/segmentation case considered by Errunza and Losq (1985). I therefore label this economy EMI . In this case markets are incomplete and the competitive equilibrium is not Pareto optimal. I use the same approach to derive the equilibrium as in Section 3. The aim of deriving the equilibrium for economy EMI is to determine how opening the stock market of country two affects equilibrium prices, returns, volatilities and correlations


16

relative to the base case in economy ENI , where neither stock market was open to external investors. In my comparison, I consider both the symmetric case, where both countries have endowment processes with the same initial size, identical growth rates and volatilities and the non-symmetric case, where initial endowment sizes differ. This analysis is motivated by the empirical observation that emerging markets’ countries have much smaller economies that developed countries (see Table I below). This table shows US average GDP output over the period 1991-2000, relative to the countries shown . Source: Economic Research Service. Country Canada Japan Australia New Zealand EU Czech Republic Hungary Poland

N (t) 10.0 2.0 17.9 127.8 0.7 232.0 207.0 93.0

Country China Korea Indonesia Malaysia Philippines Thailand Vietnam India

N (t) 9.0 17.5 40 98.8 125.8 53.2 635.8 15.5

Country Mexico Argentina Brazil Iran Iraq Saudia Arabia Turkey Algeria

N (t) 20.5 32.6 11.8 9.2 168.2 52.6 34.6 97.9

Table I

US GDP output relative to other countries

4.1

Agents’ endowments

Each individual country agent is endowed with all the stocks in the country and amount zero of the bond.

4.2


Agent 1 faces complete markets, whereas agent 2 faces incomplete markets. To characterize optimal consumption for each individual agent I use the duality approach of Cvitanic and Karatzas (1992) as in Section 3. Proposition 5 The optimal consumption-portfolio process pair (Ci , π i ) of the representative


17

agent of country 1 with (Ci , π i ) ∈ A satisfies Ci (t) =

e−βt ψi ξ i (t)

(40)

¢ ¡ π 1 (t) = (σ (t)| σ (t))−1 µ (t) − r (t) 1 (41) µ2 (t) − r (t) (42) π 2 (t) = σ |2 (t) σ 2 (t) ¡ ¢ where 1 = (1, 1)| , 12 = (0, 1)| . The variable ψ i = 1 − e−βT /βXi (0) is the Lagrange multiRT plier associated with the static budget constraint of agent i, E 0 ξ i (t) Ci (t) dt = Xi (0). The state-price density process faced by agent i is ½ Z t ¾ Z Z t 1 t | 2 ξ i (t) = ξ i (0) exp − r (u) du − θi (u) dW (u) − kθi (u)k du 2 0 0 0

with the market price of risk process for country i given by ( ¡ ¢ σ (t)−1 µ (t) − r (t) 1 , i = 1 θi (t) = σ2 (t) (µ2 (t) − r (t)) 12 , i = 2 σ2 (t)| σ2 (t)

(43)

(44)

Each agent is logarithmic (myopic). Equation (41) shows that agent 1 can invest in both stocks and holds the combined mean-variance portfolio, whereas equation (42) shows that agent 2 can invest in stock two only and holds the mean-variance portfolio for that stock.

4.3

Representative “world” investor, stochastic weight and cross-sectional wealth distribution

As in Section 3, I follow Cuoco and He (1994) in constructing a state-dependent “world” representative investor. Optimal consumption policies take the same form as in Lemma 1. However, the distribution of the stochastic weight will be different.

4.4

Equilibrium prices

I solve for the equilibrium in the same way as in Section 3. The forward-backward stochastic differential equation (FBSDE) characterizing stock prices and the cross-sectional wealth distribution is given in Proposition 2 in Appendix B. Corollary B1 gives the partial differential equation for the price of stock two. For the sake of brevity, the resulting closed-form characterization of the equilibrium is given in Proposition B4 in Appendix B. I now discuss the results of the comparison between the equilibrium in the case where stock market two has been


18

opened to external investors (economy EMI ) and the base case, where both stock markets are closed to external investors (economy ENI ). Recall that the relative output process N (t) is defined by N (t) = e2 (t) /e1 (t). I shall use the notation ∆MI η (t) to denote the difference, η MI (t) − η NI (t), where η MI (t) is the value of η at time t in economy EMI and η NI (t) is the value of η at time t in the base-case economy, ENI . 4.4.1

Interest rate effects

Proposition 6 The riskless rate is changed by ¶¸ · µ ´ ³ 1 − x (t) 2N (t) 2 2 σ . 1 − N (t) + O (1 − ρ) ∆MI r (t) = (1 − ρ) e x (t) (1 + N (t))2 ³ ´ If x (t) = N (t) /(1 + N (t)) + O (1 − ρ)2 , then the riskless rate is increased by Ã ! ³ ´ −β(T −t) σ 2e N (t) 4e (1 − ρ)2 3 2 2 (1 − ρ) + O . − βT σ β + 4σ e e 4 β 1 − e−β(T −t) (1 + N (t))2

(45)

(46)

As one would expect, making N (t) large in equation (46) shows that opening up the stock market of a smaller country to investors from a larger country will only lead to a small improvement in risk-sharing in the “world” economy and hence a small increase in the riskless rate. This increase compensates for the reduction in the precautionary savings motive of investors, which arises from the improvement in risk-sharing. I label this effect as a “risksharing” effect. This result has been found in a two-period setting by Basak (1996) for the special case N (t) = 1, where both countries are the same size. In a one-country, domestic setting, Basak and Cuoco (1998) show that full stock market participation increases the equilibrium riskless rate in a continuous-time equilibrium model. A similar result has also been found by Telmer (1993), who computes the equilibrium numerically. Both Basak and Cuoco (1998) and Telmer (1993) highlight this result as a possible way of resolving the “riskfree rate puzzle” of Weil (1989). Basak (1996) and Telmer (1993) do not consider the effect of varying the current output size of country one relative to country two, N (t), on their results. Basak and Cuoco (1998) work in a single country framework, where the effect of varying N (t) cannot be considered. 4.4.2

Risk premia and stock price effects

Proposition 7


19

1. The risk premium on stock one is increased by ∆MI [µ1 (t) − r (t)] = (1 − ρ)

³ ´ 1 2N (t) − (2N (t) + 1) x (t) 2 3/2 , σe + O (1 − ρ) x (t) 1 + N (t)

(47)

whilst the risk premium on stock two is decreased by ∆MI [µ2 (t) − r (t)] = − (1 − ρ)

³ ´ N (t) σ 2e + O (1 − ρ)3/2 . 1 + N (t)

(48)

2. The price of stock one is decreased by ∆MI S1 (t)

(49)

1 − (1 + β (T − t)) e−β(T −t) = − (1 − ρ) y (t) σ 2e 3 [2N (t) − (1 + N (t)) x (t)] β2 (1 + N (t)) ³ ´ +O (1 − ρ)2 . N

The price of stock two is increased by ∆MI S2 (t) = (1 − ρ)

N

(1 + N (t)) ³ ´ 2 +O (1 − ρ) .

3

[2N (t) − (1 + N (t)) x (t)]

1 − (1 + β (T − t)) e−β(T −t) y (t) σ2e β2 (50)

3. Cumulative returns on stock one are increased by ∆MI µ1 (t) = (1 − ρ) σ 2e

´ ³ 2N (t) − (1 + N (t)) x (t) + O (1 − ρ)3/2 , (1 + N (t)) x (t)

(51)

whilst cumulative returns on stock two are decreased by ∆M I µ2 (t) = − (1 − ρ) σ 2e N (t)

´ ³ 2N (t) − (1 + N (t)) x (t) + O (1 − ρ)3/2 . (1 + N (t)) x (t)

(52)

Assuming that the output of country two is less than the output of country one, i.e. N (t) > 1, one can see from equation (48) that opening the stock market of a smaller country (country two) results in a decrease in the equity risk premium in that country. The greater the disparity in country output sizes, the greater this effect will be. The theoretical results I find concur well with empirical findings: Henry (2000) reports cumulative abnormal returns of the order of 40 per cent in a 12 month window prior to liberalization for the emerging markets’ countries he considers. Table II and Figure 2 show the results of simple calibrations based on my theoretical results. The results of Proposition 7 fit together in a highly intuitive way. Opening stock market two to external investors leads to a drop in agent two’s demand for stock one and a rise in her


20

demand for stock two. The price of stock two is driven up by this. Agent one invests less in stock one, so there is less demand for it, causing its price to fall. Conditional expected stock returns move in the opposite directions to prices. I label these effects as “demand effects”. The effect of size, i.e. N (t) on these results has a very intuitive interpretation: prices and conditional expected returns in a large country will be changed by a small amount when a small country opens up its stock market to investors in the large country. However, the effects on prices and conditional expected returns in the smaller country (country two) will be magnified as the relative disparity in country output sizes, N (t), increases. This can be seen from equations (49), (50), (51) and (52). It is interesting to note that risk premia move in the same direction as expected stock returns, despite the rise in the riskless rate. This is because “demand effects” outweigh “risksharing effects”. 4.4.3

Volatility and correlation effects

Proposition 8 1. The size of the market price of risk in country one is changed by 2N (t)

2

∆MI kθ1 (t)k = (1 − ρ)

2

(1 + N (t))

"

N (t)

1 − x (t) x (t)

2

2

#

³ ´ 2 − 1 σ 2e + O (1 − ρ) .

(53)

The market price of risk in country one is increased iff x (t)2 < N (t) / (1 + N (t)). The size of the market price of risk in country two is decreased by ∆MI kθ2 (t)k2 = − (1 − ρ)

³ ´ 2N (t) 2 σ e + O (1 − ρ)2 . 1 + N (t)

(54)

2. The conditional variance of returns on stock one is increased by ∆MI V1 (t) = (1 − ρ)

2

4N (t)2 3

(1 + N (t)) ³ ´ 3 +O (1 − ρ) ,

1 − x (t) x (t)

2

µ ¶ β (T − t) e−β(T −t) σ4e 1− β 1 − e−β(T −t) (55)

whilst the conditional variance of returns on stock two is decreased by ∆MI V2 (t) 2N (t) 2 = − (1 − ρ) (1 + N (t)) ´ ³ 3 +O (1 − ρ)

µ 1−

N (t) 2 1 + N (t) x (t)

¶ µ ¶¶ µ N (t) β (T − t) e−β(T −t) σ 4e 1+ 1− x (t) β 1 − e−β(T −t) (56)


21

Proposition 8 shows that conditional variance of returns on stock two has been decreased by allowing agent one to invest in stock two. The conditional variance of returns on stock one has increased. I also show that the size of the market price of risk (i.e. the size of the volatility of the state-price density) in country two is decreased, whilst in country one it increases for most realizations of the cross-sectional wealth distribution (see Figure 3). These results concerning the volatilties of stock returns and state-price densities can be linked together once a suitable representation for return volatility is obtained. This is possible using the Malliavin calculus. Proposition 9 links stock return volatility in country i to properties of the state-price density in country i, the cross-sectional wealth distribution and macroeconomic factors, namely aggregate output and individual country output. Proposition 9 If the price of stock i, Si , is given by discounting future dividends, ei , using some state-price density, ξ i , which depends on the aggregate endowment, y, and the cross-sectional wealth distribution, x,.i.e. Z T ξ i (s) ei (s) ds (57) ξ i (t) S1 (t) = Et t

where

ξ i (s) = ξ i (s, y (s) , x (s))

(58)

then the volatility of returns on stock i is given by RT RT ξ (s) ξ (s) Et t ξ i (s) ei (s) ξi,y(s) Dt y (s)| ds Et t ξ i (s) ei (s) ξi,x(s) Dt x (s)| ds i i + (59) σi (t) = θi (t) + RT RT Et t ξ i (s) ei (s) ds Et t ξ i (s) ei (s) ds RT (s)| Et t ξ i (s) ei (s) Dteeii(s) ds + RT Et t ξ i (s) ei (s) ds

I briefly explain the notation used. The Malliavin derivative of the process y (s) is denoted

by Dt y (s) = (D1t y (s) , D2t y (s)). The process Dit y (s) measures the effect of an innovation in the Brownian motion Wi at time t on the process y at time s ≥ t. In a discrete-time setting, one would use impulse response functions for this purpose (see Ljungqvist and Sargent (2000)). In this economy, the only exogenous shocks are those to the endowment processes. The first line on the right-hand side of equation (59) shows the effects of exogenous shocks on the stateprice density, ξ i . The first term is the current conditional expected volatility of the state-price density. The second term shows how shocks to current endowments affect the future variability of the state-price density, holding the future path of x (t) fixed. The third term shows how exogenous endowment shocks affect the future variability of the state-price density through the cross-sectional wealth distribution. Note that shocks affecting the future variability of the state-price density can be interpreted as affecting the future variability of returns, which the


22

empirical literature (see Campbell (1991) and Cochrane (1991)) documents as the main source of return volatility. The last term shows the effect of future variability in dividend growth on return volatility. Equation (53) shows that opening up stock market two to external investors increases the volatility of the state-price density for country one, ξ 1 , if x (t)2 < N (t) /(1 + N (t)), whilst equation (54) shows that the volatility of the state-price density for country two, ξ 2 , is decreased. Proposition 9 shows how the volatility of the state-price density for country i affects the stock return volatility in country i. In country one, the increase in state-price density volatility drives the increase in stock return volatility. In country two, both the decrease in state-price density volatility and cross-sectional wealth distribution volatility drive the decrease in stock return volatility. When the country that opens its stock market is small in output terms, as emerging markets’ countries are (i.e. N (t) and x (t) are large), the change in stock return variance becomes much smaller (see Figure 6). This brings my theoretical result into line with some recent empirical findings (see Kim and Singal (2000) and Bekaert and Harvey (2000)), which document that stock market liberalization in emerging markets has a negligible effect on stock return volatilities. In contrast, Kaminsky and Schmukler (2001) find an increase in volatility in a three-year window after liberalization, which is not supported by my theoretical result. In the long-term, however, this increase fades away. Proposition 10 The cross-country correlation in stock returns is increased by ∆MI ρ12 (t) 2N (t) [2N (t) − x (t) (1 + N (t))] x (t) (1 + N (t))2 ´ ³ +O (1 − ρ)3 .

= (1 − ρ)2

Ã

1 − [1 + β (T − t)] e−β(T −t) 1 − e−β(T −t)

!

(60) σ 2e β (61)

The correlation between stock returns is increased by opening stock market two to external investors. This is driven by the increased correlation between individual country state-price densities — they become more aligned as risk-sharing in the “world” economy improves. As is clear from equation (60) and Figure 7, this increase is magnified by increasing N (t) — the increase in correlation which occurs when a small country liberalizes its stock market varies between 0.05 to 0.1( under the highly plausible assumption that the larger country holds most


23

of the aggregate wealth). This tallies well with the empirical findings of Bekaert and Harvey (2000), who document an increase in correlation of 0.045. Table II

Changes in equilibrium variables in country two This table shows how equilibrium variables in country two change when stock market two opens, for various N (t), assuming that x(t) = N (t)/(1 + N (t)). The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞. N=1 N=10 N=100 N=1000

4.4.4

% change in risk premium -31.7% -52.7% -56.3% -56.7%

% change in kθ2 k -26.6% -41.7% -43.4% -43.5%

% change in V2 -6.5% -7.0% -6.5% -6.5%

change in ρ12 0.03 0.06 0.06 0.06

Analysis of the comparison results for different parameter values

For N (t) ≥ 1, I now study the effects of varying the time-horizon remaining, τ = T − t on the comparison of the equilibria in economies EMI and EN I . Before describing these effects it is important to make clear that the proportional drop in the risk premium in country two, resulting from opening stock market two to external investors is large for all N (t) ≥ 1 and all τ ≥ 0. The most important observation is that increasing the time remaining, τ , magnifies the impact on the increase in the cross-country return correlation arising from opening stock market two to external investors. It is important to bear in mind that for large N (t) this increase in cross-country correlation is already large for small τ ; for larger τ , this effect becomes even larger. Increasing τ has a negligible effect on the changes in risk premia and cumulative returns resulting from opening stock market two to external investors. However the slight increase in the riskless rate, which occurs when stock market two is opened to external investors is magnified when τ increases. The changes in stock prices that occur when stock market two is opened to external investors are also magnified when τ increases. For large values of τ , the proportional changes in stock prices can be close to 3%. Increasing τ also magnifies the changes in stock return variances. Increasing the rate of time preference, β, has the opposite directional effects to increasing τ.


24

The proportional changes in equilibrium variables are of course decreased as the crosscountry correlation in output returns, ρ, is increased. This is because a higher ρ implies that there is less idiosyncratic risk that can be hedged against by increased risk-sharing opportunities. Hence the proportional effects of these increased risk-sharing opportunities on equilibrium variables are smaller. Increasing the level of individual country growth rate volatilities will increase the proportional changes in all equilibrium variables with the exception of risk premia. In fact, raising the country growth rate volatility (assumed to be identical for both countries), diminishes the proportional changes in risk premia, moving from economy ENI to economy EMI .This is because the risk premia in the base case (economy ENI ) are now higher, whereas the absolute changs in risk premia are not affected much. Other equilibrium variables change more proportionally, moving from economy ENI to economy EMI . This is because a higher growth rate volatility implies a higher level of idiosyncratic risk, that can be hedged against when there is increased stock market integration.

4.5


In this section, I derive a dynamic capital asset pricing model when stock markets are integrated on one direction only. This result does not rely on the assumption that individual country endowment processes follow geometric Brownian motions-it is valid for all progressively measurable endowment processes. Proposition 11 The equity premia are given by " ¶ ¶−1 # µ µ (t) dS2 (t) dS s (t) 2 Et (µ1 (t) − r (t)) = − det σ (t)2 V art x (t) V art x (t) S2 (t) S2 (t) ¶ µ dS1 (t) dS2 (t) + (1 − s (t)) Covt , S1 (t) S2 (t) ¶ ¶ µ µ dS2 (t) dS1 (t) dS2 (t) + s (t) Covt , Et (µ2 (t) − r (t)) = (1 − s (t)) V art S2 (t) S1 (t) S2 (t)

(62) (63)

Allowing agent one to invest in stock two not only leads to the pricing of covariance risk in the returns on asset one. It leads to covariance risk being priced in the returns of asset two.


5

25

Integrated stock markets

In this section I consider the case, where both stock and bond markets are perfectly integrated. This mirrors the case where two countries undergo full economic union, essentially becoming one country from an economic perspective. Hence, I assume that the representative agents for each country can borrow and lend to each other freely at the riskfree rate. Furthermore, I assume that each representative agent is free to trade in the stock market of the other country. Hence financial markets are dynamically complete in the sense of Duffie and Huang (1985). I label this economy EI .

5.1

Agents’ preferences and endowments

Assuming that each agent has logarithmic preferences and equal initial endowments, I solve for the equilibrium in closed-form. Following Huang (1987) or Karatzas, Lehoczky and Shreve (1990), the two agents can be replaced with a single “world” representative agent endowed with the aggregate supply of securities and with the following utility functional Z T U (C; λ) = E e−βt u (C (t) , λ) dt ,

(64)

0

where C (t) is the consumption policy of the “world” representative agent. Aggregation in the sense of Rubinstein (1974) implies that u (C (t) , λ) = ln C (t). Consuming the aggregate world endowment is optimal for this “world” representative agent, i.e. C (t) = y (t), so the marginal rate of substitution process is given by ξ (t) = e−βt

5.2

uc (C (t) , λ) C (0) = e−βt . uc (C (0) , λ) C (t)

(65)


The assumptions that both investors are logarithmic and markets are complete make it simple to derive the optimal consumption and portfolio policies. Proposition 12 The optimal consumption policy of each agent is identical and is given by β 1 C (t) = y (t) = X (t) −β(T −t) 2 1−e

(66)

where X (t) =

1 − e−β(T −t) y (t) 2β

(67)


26

is the wealth held by each agent. The optimal proportions of individual wealth invested by each agent in stocks is given by π (t) = [π 1 (t) , π 2 (t)]| ¢ ¡ = (σ (t) σ (t)| )−1 µ (t) − r (t) 1

(68)

where π i (t) is the proportion of wealth invested in stock i.

5.3

Equilibrium prices

Using the approach of Cox and Huang (1989), one can characterize the equilibrium riskfree rate, market price of risk, equity premia, stock return volatilities and stock prices. There are two stocks in the economy. Hence, to derive the equilibrium variables explicitly in terms of exogenous variables it is necessary to assume that individual country endowment processes follow explicit processes. In fact, I assume that they follow geometric Brownian motions. The stock prices can be represented in the standard way as the conditional expectations of integrals and computed approximately in closed-form using Laplace’s method (see Hinch (1991) or Judd (1998)). The resulting closed-form approximations to the stock prices are shown to be highly accurate (see Table V). For the sake of brevity, the equilibrium is described in Proposition B5 in Appendix B. I now summarize the results of comparing the equilibrium in economy EI , where stock markets are not integrated with the equilibrium in economy ENI , where stock markets are integrated. Recall that the relative output process N (t) is defined by N (t) = e2 (t) /e1 (t). I shall use the notation ∆I η (t) to denote ηI (t) − η NI (t), where ηI (t) is the value of η at time t in economy EI and η NI (t) is the value of η at time t in the base-case economy, ENI . 5.3.1


Proposition 13 The riskless is rate is increased by ∆I r (t) = (1 − ρ) σ 2e

´ ³ 2N (t) 2 . + O (1 − ρ) (1 + N (t))2

(69)

Recall that N (t) is the current size of country one output relative to country two output. In the symmetric case, N (t) = 1, both countries have economies of the same size, with the same growth rate mean and variance. The only heterogeneity is in the output shocks — across


27

countries, these shocks are not perfectly correlated. From equation (69), one can see that integrating stock markets raises the riskless rate. As for the case where just one stock market was opened up to external investors (see Section 4), greater risk-sharing opportunities increase the attractiveness of investing in risky stocks and the precautionary savings motive of investors declines. Hence, to induce investors to invest in the bond, it’s rate of return, r, must be raised. Increasing N (t) diminishes the effect that integrating stock markets has on the riskless rate: it increases less for higher N (t). The state-price density gives us the world pricing kernel, which for large N (t) is determined almost entirely by agent one. Agent one lives in a country one, which has much larger output and a much larger stock market than country two. Hence when agent one can invest in stock market two, the increase in portfolio diversification opportunities is small, resulting in only a small increase in the riskless rate 5.3.2

Risk premia and stock price effects

Proposition 14 1. The risk premium on stock one is decreased by ∆I [µ1 (t) − r (t)] = − (1 − ρ) σ 2e

´ ³ 1 + O (1 − ρ)3/2 , 1 + N (t)

(70)

whilst the risk premium on stock two is decreased by ∆I [µ2 (t) − r (t)] = − (1 − ρ) σ 2e

´ ³ N (t) 3/2 . + O (1 − ρ) 1 + N (t)

(71)

2. The price of stock one is decreased by i N (t) (N (t) − 1) σ 2e h −β(T −t) (1 + β (T − t)) y (t) 1 − e 3 β2 (1 + N (t)) ³ ´ 2 +O (1 − ρ) .

∆I S1 (t) = − (1 − ρ)

(72)

The price of stock two is increased by

i N (t) (N (t) − 1) σ 2e h −β(T −t) (1 + β (T − t)) y (t) 1 − e 3 β2 (1 + N (t)) ³ ´ +O (1 − ρ)2 .

∆I S2 (t) = (1 − ρ)

(73)

3. Cumulative returns on stock one are increased by ∆I µ1 (t) = (1 − ρ) σ2e

N (t) − 1

2

(1 + N (t))

³ ´ 2 + O (1 − ρ) ,

(74)


28

whilst cumulative returns on stock two are decreased by ∆I µ1 (t) = − (1 − ρ) σ 2e

N (t) (N (t) − 1) (1 + N (t))

2

³ ´ 2 + O (1 − ρ) .

(75)

When the output of each country is the same, integrating stock markets has no effect on stock price levels. Even though investors now hold diversified portfolios, providing perfect insurance against swings in individual country output, the overall demand for each stock has not changed. Essentially, the decrease in demand for stock i from one agent resulting from portfolio diversification has been balanced by the increase in demand for stock i from the other agent, which is also driven by portfolio diversification. Expected stock returns move in the opposite directions to prices. The equity risk premium in each country is decreased by stock market integration. When N (t) = 1, “demand effects” play only a small role in this. This is unlike the situation in Section 4, where only one stock market was opened to external investors. The decrease in equity risk premia is mainly a “risk-sharing effect”, driven by the increase in the riskless rate. Whilst in absolute terms, the decreases in equity premia are small (this is a manifestation of the well-known equity premium puzzle of Mehra and Prescott (1985)), in proportional terms they are large—30 per cent (see figures 9 and 10). The intuition underlying the effect of increasing N (t) on the risk premium effects is the same as for the interest rate. Increasing N (t) decreases the size of the drops in equity risk premia that occur in each country as a result of stock market integration. From figures 9 and 10, it can be seen that as N (t) approaches 100 the risk premium decline for stock one becomes negligible, whilst the risk premium decline for stock two becomes much larger-it is more than halved. For N (t) > 1, stock price one is decreased by stock market integration and stock price two is increased. Essentially, when N (t) > 1, “demand effects” outweigh risk-sharing effects. The increased demand for stock two from agent one more than compensates for the drop in demand for stock two from agent two. Hence the price of stock two is driven up. Similarly, the drop in demand for stock one from agent one is not fully compensated for by the increase in demand for stock one from agent two. Hence, the price of stock one falls. 5.3.3


Proposition 15


29

1. The size of the market price of risk in both countries is decreased ∆I kθ1 (t)k2

= ∆I kθ2 (t)k2 = − (1 − ρ) σ2e

2N (t) (1 + N (t))2

³

+ O (1 − ρ)

2

´

(76)

2. If N(t)=1, then conditional variance of returns on stocks one and two is decreased by σ4 ∆I V1 (t) = ∆I V2 (t) = − (1 − ρ) e 4β 2

¶ µ ³ ´ β (T − t) e−β(T −t) 3 + O (1 − ρ) . 1− 1 − e−β(T −t)

(77)

In both countries, the conditional variance of stock returns is decreased by stock market integration. This can be explained with reference to state-price densities. In economy, EN I , where stock markets were not integrated, financial markets were incomplete and each country i ∈ {1, 2} had it’s own state-price density ξ i , the volatility of which defined a country-specific market price of risk θi . In economy, EI , stock markets are integrated and financial markets are complete. Hence there is now only one state-price density, ξ, for each country, which defines the market price of risk, θ, which is the same in each country. Using the Malliavin calculus representation for volatilities in Proposition 9, I can explain why stock return variances decrease in each country when stock markets are integrated. Firstly, there is better risk sharing, so the state-price density does not depend on the wealth distribution. Hence, the market price of risk in each country is lowered (the size of term 1 in equation (59) is decreased). Furthermore, current shocks to output, which affect the future wealth distribution do not affect stock return volatilities (term 3 in equation (59) vanishes). This is because markets are complete, so stateprice densities are independent of the wealth distribution. Hence stock return variances are decreased. For the case where N (t) > 1, it is possible to obtain approximate closed-form expressions for the changes in conditional stock return variances, ∆I Vi (t), i ∈ {1, 2}. However, for large N (t), the magnitudes of these expressions are very small — in fact they are of the same order of magnitude as the errors in Table IV. Hence, in this case, the closed-form expressions will not give reliable information on the directions of changes in stock return variances. All that can be said is that increasing N (t) diminishes the volatility effects that result from opening both stock markets.

International Stock Market Integration Proposition 16 The cross-country correlation in stock returns is increased by Ã ! ³ ´ 2 4N (t) β (T − t) e−β(T −t) 2 σe 3 2 ∆I ρ12 (t) = (1 − ρ) + O (1 − ρ) . 1 − σ e β (1 + N (t))2 1 − e−β(T −t)

30

(78)

The increase in the cross-country correlation in stock returns is driven by two factors. The first is the decrease in the conditional variance of cross-country stock returns. The second is the increase in the covariance between stock returns across countries. This second effect is driven by the replacement of country-specific state-price densities with a single state-price density. The rise in correlation becomes smaller as N (t) increases — this is not what occurs when just stock market two is opened to external investment. In this case, “risk-sharing effects” are dominant and these are less pronounced when there is a large disparity in country output sizes. This decreases the correlation effect. The effects of varying parameter values, such as the effective time horizon, τ , the rate of time preference, β, the volatility of output growth, σ e on the comparison of the equilibria between economies ENI and EI are the same as in Section 4.4.4.

5.4


The capital asset pricing model for the case where markets are perfectly integrated and dynamically complete is standard, so I omit it.

6

One-way partially integrated stock markets

In this Section, I consider the case where stock market one is closed to external investors, stock market two is partially open to external investors and bond markets are integrated. Agent one can invest up to a certain proportion, I1 , of her wealth in stock market two and agent two can only invest in her own stock market. Both agents can borrow/lend freely at the riskless rate. Therefore, stock markets are partially integrated in one direction. I label this economy EP I . I now briefly discuss the motivation behind imposing a constraint on agent one’s investment in stock two. Imposing this constraint can represent several conceptually distinct scenarios. The first is a limitation on the proportion of their wealth that external investors can hold in stock market two, imposed in some way by the government of country two. The second is the same form of limitation imposed on the residents of country one, but this time by their own government. The third is home-bias in equity portfolios, which in the modelling framework


31

described here, is clearly exogenously imposed.11 Hence, because of home-bias, agent one only invests the proportion, I1 , of her wealth in stock market two, even though she may theoretically invest more and diversify her portfolio more fully. There is empirical evidence that investment constraints at the international level do matter and that as a result markets are not fully integrated (see Errunza and Losq (1985), Errunza, Losq and Padmanabhan (1992) and Bekaert and Harvey (1995)). It is not the aim of this paper to address why such investment constraints exist, but rather to study their effects.

6.1


As in Section 3.2, I use the duality approach of Cvitanic and Karatzas (1992) to characterize optimal consumption and portfolio choice under constraints. Proposition 17 The optimal consumption-portfolio process pair (Ci , π i )of the representative agent of country i with (Ci , π i ) ∈ A satisfies Ci (t) =

e−βt ψ i ξ i (t)

(79)

and π i (t) = [π i1 (t) , π i2 (t)]|

(80)

where (81) π 11 (t) = (σ (t) σ (t)| )−1 [µ1 (t) − r (t)] ¸ · | σ 1 (t) σ 2 (t) (82) π 12 (t) = (σ (t) σ (t)| )−1 I1 − (µ1 (t) − r (t)) σ 1 (t)| σ 2 (t) π 21 (t) = 0 (83) µ2 (t) − r (t) π 22 (t) = (84) σ 2 (t)| σ 2 (t) ¢ ¡ and 11 = (1, 0)| , 12 = (0, 1)| . The constant ψ i = 1 − e−βT /βXi (0) is the Lagrange multiRT plier associated with the static budget of agent i E 0 ξ i (t) Ci (t) dt = Xi (0). The state-price density process faced by agent i is ½ Z t ¾ Z Z t 1 t | 2 ξ i (t) = ξ i (0) exp − r (u) du − θi (u) dW (u) − kθi (u)k du 2 0 0 0

(85)

11 French and Poterba (1991) provide empirical evidence that investors underweight foreign stocks in their portfolios


32

with the market price of risk process for country i given by θi (t), where θ1 (t) =

σ 1 (t) I1 (µ1 (t) − r (t)) + σ (t)−1 12 | | σ 1 (t) σ 1 (t) σ 1 (t) σ 1 (t)

(86)

σ 2 (t) (µ (t) − r (t)) σ 2 (t)| σ 2 (t) 2

(87)

and θ2 (t) =

I have assumed both agents are logarithmic (myopic). From equations (81) and (82), it is clear that agent one can freely invest in stock market one and is constrained in the long position she can hold in stock market two. In contrast, equations (83) and (84) show that agent two can invest freely in stock market two but cannot take a in position in stock market two.

6.2

Equilibrium Prices

The equilibrium can be characterized a FBSDE, which cannot be decoupled, as in Sections 4 and 5. The FBSDE characterizing stock prices and the cross-sectional wealth distribution is given in Proposition B6 in Appendix B. I solve this system in the same way as in Sections 3 and 4. The closed-form characterization of the equilibrium is rather complicated and is hence omitted for the sake of brevity.12 Relative to economy EMI in Section 4, where agent one could invest freely in stock two, the only difference in economy EP I is that agent one can now only invest up to I1 of her wealth in stock two. Hence, the following discussion is restricted to describing the effects of varying the parameter I1 on the comparison of equilibria between economies EP I and ENI . In general, as I1 is increased, the changes in equilibrium variables in economy EP I relative to the base case in economy ENI are increased. The rate of increase is locally linear, until the constraint comes close to being binding. Before the constraint starts to bind the effect of increasing I1 begins to level off. 6.2.1


Allowing agent two to invest up to I1 of her wealth in stock two, increases the riskless rate. This rise in the riskless less rate is increasing and concave in I1 . Relaxing the constraint decreases the precautionary savings motive, so the riskless rate must be increased to compensate for this. 12 Mathematica code giving the approximate closed-form characterization of the solution is available from the author upon request


33

As the constraint is relaxed, the marginal benefit of the resulting increased risk-sharing in the economy decreases, so the rise in the riskless rate is concave in I1 . The closed-form expression for the interest rate change, ∆rP I (t), is omitted because of its complexity. 6.2.2

Risk premia and return effects

¡ ¢ Proposition 18 For small I1 , if N(t) = 1 and x(t) = 1/2 + O I12 , then the risk premia and

conditional expected returns on stocks one and two are changed by

σ 2e (T −t) −σ 2e (T −t) e

∆I [µ1 (t) − r (t)] = ∆I µ1 (t) = −∆I [µ2 (t) − r (t)] = −∆I µ2 (t) = I1 e

¡ ¢ − e−β(T −t) +O I12 . −β(T −t) 1−e (88)

Increasing I1 magnifies the changes in risk premia resulting from the partial opening of stock market two to external investors: the risk premium in country one increases more, and the decrease in the risk premium in country two is larger. This proposition shows how flows drive up returns and the risk premium in the liberalized market and provides a possible theoretical explanation for the positive relation between portfolio inflows and returns in emerging markets found by Froot, O’Connell and Seasholes (2001). 6.2.3


The changes in volatility are in the same direction as when stock market two if fully opened to external investors. Closed-form expressions are rather complicated and hence omitted. The following proposition show how the cross-country correlation in stock returns behaves for small I1 . Proposition 19 The cross-country correlation in stock returns is given by ´ i i x (t) (N (t) + 1)2 + N (t) (N (t) − 1) h 2 h ³ −σ 2e (T −t) 2 −β(T −t) σ + σ e − β 1 − e ρ12 (t) = ρ + (1 − ρ) I1 e e N (t) (β + σ 2e ) ´ ³ (89) +O (1 − ρ)2 As I1 increases, the correlation increases. This reflects the fact that as agent one can invest more in stock two, risk-sharing in the economy improves, and the state-price densities in each country become more closely correlated. This drives up the return correlation. For larger N (t), the effect of increasing I1 is more pronounced. It is as if by investment, the larger


34

country drags the smaller country into its orbit, increasing the level of comovement across the two markets. Note that this proposition would hold if I1 were stochastic, but uncorrelated with the endowment shocks. Hence, this result shows theoretically how time-variation in the level of integration (see Bekaert and Harvey (1995)) can lead to time-varying correlation (see Harvey (1991)).

6.3


The following proposition shows a dynamic international CAPM for one-way partially integrated stock markets. Proposition 20 In equilibrium the risk premia are given by ¸ ¶ ¶ · µ µ s (t) dS1 (t) dS1 (t) dS1 (t) dS2 (t) Et − r (t) = V art + I1 Covt , S1 (t) x (t) S1 (t) S1 (t) S2 (t) ¸ µ ¶ ¶ · µ 1 − s (t) x (t) I1 dS2 (t) dS2 (t) Et − r (t) = − V art S2 (t) 1 − x (t) 1 − x (t) S2 (t)

(90) (91)

This result is generalized in the following section.

7

Dynamic capital asset pricing models

In this section I derive a two-beta, dynamic capital asset pricing model for the case where both countries are constrained in their investment in external stock markets. This CAPM is valid when individual country endowment processes follow any Ito process with progessively measurable coefficients and constraints are stochastic, but uncorrelated with endowment shocks. This CAPM provides a way to calculate the cost of capital for firms, without assuming perfect integration or segmentation of financial markets, as in Stulz (1995) and Stulz (1999). In the framework I employ, markets are incomplete, unlike the framework used by Duffie and Zame (1989) to derive a multiagent consumption-based CAPM.13 Cuoco (1997) derives a consumption-based CAPM, where markets are incomplete, but agents all face the same investment constraint. I consider the case where agents face different investment opportunity sets. In this sense, the CAPM I derive is similar to Shapiro (2002), but with one difference—all agents face incomplete markets. 13 Duffie (1992) outlines how this can be extended to incomplete markets, with all agents facing the same investment opportunity set.


7.1

35

Partial two way stock market integration

The following proposition shows a two-beta, dynamic international CAPM for two-way partially integrated stock markets, where agent i can invest up to Ii of her wealth in stock market j. Proposition 21 In equilibrium the risk premia are given by µ µ ¶ ¶ dSi (t) dSi (t) dSj (t) , µi (t) − r (t) = β i (t) V art + β j (t) Covt , i ∈ {1, 2} , j 6= i (92) Si (t) Si (t) Sj (t) where si (t) − y (t) xj (t) Ij xi (t) β j (t) = y (t) xi (t) Ii β i (t) =

x1 (t) = x (t) , x2 (t) = 1 − x (t) Si (t) . si (t) = S1 (t) + S2 (t) Stock returns in country i ∈ {1, 2} are priced by their own variance and their variance with non-local stock market returns. The local variance component (term one in equation (92)) influences pricing more, if a greater proportion of financial wealth in country i is held in its stock market. Increasing the proportion of aggregate “world” financial wealth held in country i will decrease the importance of local factors and increase the importance of non-local factors in pricing returns. A novel feature of this CAPM is the prescence of “cross-effects”- constraints on investors in one country affect investors in other countries too. I now compare the discrete-time version of (92), with the CAPM of Bekaert and Harvey (1995). Using intuitive arguments, Bekaert and Harvey (1995) show that Et−1 [Ri,t − rt ] = (1 − Ii,t−1 ) λi,t−1 V art−1 (Ri,t ) + Ii,t−1 λt−1 Covt−1 (Ri,t , Rw,t )

(93)

where Ri,t denotes returns in country i, Rw,t denotes returns on the world market portfolio, rt is the riskless rate, Ii is the level of integration of country i with the world, λi is the country i market price of risk, λ is the world market price of risk. I show formally that µ ¶ si,t−1 − Ai,t−1 V art−1 (Ri,t ) + Aj,t−1 Covt−1 (Ri,t , Rj,t ) Et−1 [Ri,t − rt ] = xi,t−1 where Ak,t = yt xk,t Ik

(94)


36

Here, si,t is the proportion of aggregate financial wealth held in country i, yt is the aggregate endowment or output size and Ik is the proportion of wealth in country k that can be invested externally in stocks. The econometric specification of CAPM tested in Bekaert and Harvey (1995) is close to the CAPM, which I formally derive. The variance-covariance structure is the same, but I am able to account for “cross-effects” from constraints. I also relate the prices of covariance risk to the proportions of aggregate wealth held in each country and the proportions of aggregate wealth in the stock markets of each country. Hence, for countries where stock market capitalization is very different from net financial wealth, the CAPM I derive will be significantly different from the CAPM tested in Bekaert and Harvey (1995).

8

Conclusions

In this article I have derived equilibria in closed-form for a world economy with two countries under various market structures, including both complete and incomplete financial markets in a continuous-time setting. I have shown that when just one country opens its stock market to external investors, the equity risk premium and volatility of stock returns decrease in that country. In contrast, the risk premium and stock return volatility increase in the market that remains closed. The correlation in stock returns is increased. When both stock markets are opened to external investors different effects arise: equity risk premia and stock-return volatilities fall in both markets. The correlation in stock returns is still increased. The results on volatility and correlation have not appeared before in the theoretical literature on stock market integration. In a domestic framework, Allen and Gale (1994) show that increased stock market participation can increase stock return volatility. My results show that the opposite occurs in a country that liberalizes its stock market. In addition to showing that stock market integration umambiguously increases cross-country stock return correlations, I show that this effect is much larger in magnitude than the effect on volatilities. This is consistent with the empirical results of Bekaert and Harvey (1997) on emerging markets. I also show that as the level of integration (measured by external investment) increases, the cross-country stock return correlation rises, which provides theoretical underpinning for the findings of Goetzmann, Li and Rouwenhorst (2002), who document the long-term behavior of correlation. Another recent empirical result for which I provide theoretical support is the positive relation between foreign investment inflows and stock returns in


37

emerging markets documented by Froot, O’Connell and Seasholes (2001). By allowing for differences in country output size, simple calibrations show that my theoretical results on the effects of stock market liberalization on risk premia, volatility and correlation are consistent with the empirical findings of Bekaert and Harvey (2000), Henry (2000) and Kim and Singal (2000). I have also derived a dynamic international capital asset pricing model, which allows for partial integration of stock markets at the international level. This CAPM provides theoretical justification for and extends the CAPM tested in Bekaert and Harvey (1995). To derive these results, I have made use of continuation methods (see Zeidler (1986)) and have shown that they can be highly accurate in computing equilibria approximately in closedform in economies with two risky assets for both complete and incomplete markets.


38

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39

Cox, J. C., and C.-f. Huang, 1989, “Optimal consumption and portfolio policies when assets prices follow a diffusion process,” Journal of Economic Theory, 49, 33—83. Cuoco, D., 1997, “Optimal consumption and equilibrium prices with portfolio constraints,” Journal of Economic Theory, 72, 33—73. Cuoco, D. and H. He, 1994, “Dynamic equilibrium in infinite-dimensional economies with incomplete financial markets,” Wharton School, University of Pennsylvania, Working Paper. Cvitanic, J. and I. Karatzas, 1992, “Convex duality in constrained portfolio optimization,” Annals of Applied Probability, 2, 767—818. Detemple, J., R. Garcia and M. Rindisbacher, 2002, “A Monte-Carlo method for optimal portfolios,” Journal of Finance, forthcoming. Detemple, J., and S. Murthy, 1997, “Equilibrium asset prices and no arbitrage with portfolio constraints,” Review of Financial Studies, 10, 1133—1174. Detemple, J., and S. Murthy, 2002, “Dynamic equilibrium with liquidity constraints,” Review of Financial Studies, forthcoming. Detemple, J., and F. Zapatero, 1991, “Asset prices in an exchange economy with habit formation,” Econometrica, 6, 1633 —1657. Duffie, D., 1992, Dynamic asset pricing theory, Princeton University Press, Princeton, N.J. Duffie, D., and C.-F. Huang, 1985, “Implementing Arrow-Debreu equilibria by continuous trading of few long-lived securities,” Econometrica, 53, 1337 —1356. Duffie, D., and W. Zame, 1989, “The consumption-based capital asset pricing model,” Econometrica, 57, 1279 —1297. Dumas, B. and R. Uppal, 2001, “Global diversification, growth, and welfare with imperfectly integrated markets for goods,” Review of Financial Studies, 14, 277—305. Errunza, V., and E. Losq, 1985, “International asset pricing under mild segmentation: theory and test,” Journal of Finance, 40, 105—124. Errunza, V., and E. Losq, 1989, “Capital flow controls, international asset pricing, and investors’ welfare: a multi-country framework,” Journal of Finance, 44, 1025—1037. Errunza, V., E. Losq and P. Padmanabhan, 1992, “Tests of integration, mild segmentation and segmentation hypotheses,” Journal of Banking and Finance, 16, 949—972. Eun, C., and S. Jarakiramanan, 1986, “A model of international asset pricing with a constraint on the foreign equity ownership,” Journal of Finance, 41, 1025—1037. French, K. and J.M. Poterba, 1991, “Investor diversification and international equity markets,” American Economic Review, 81, 222—226. Friedman, A., 1964, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, NJ.


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Froot, K., O’Connell, P., and M. Seasholes, 2001, “The portfolio flows of international investors,” Journal of Financial Economics, 59, 151—193. Gallmeyer, M., 2000, “Beliefs and volatility,” Working paper, GSIA, Carnegie-Mellon University. Gallmeyer, M., and B. Hollifield, 2002, “An examination of heterogeneous beliefs with a shortsale constraint,” Journal of Business, 61, 275—298. Goetzmann, W., L. Li and K. Rouwenhorst, 2002, “Long-term global market correlation,” Working Paper, Yale University. Harvey, C., 1991, “The world price of covariance risk,” Journal of Finance, 46, 111—157. Heaton, J., and D. Lucas, 1996, “Evaluating the effects of incomplete markets on risk sharing and asset pricing,” Journal of Political Economy, 104(3), 443—487. Henry, P., 2000, “Stock market liberalization, economic reform and emerging market equity prices,” Journal of Finance, 55, 529—564. Hinch, E., 1991, Perturbation methods, Cambridge University Press, Cambridge. Judd, K. L., 1996, “Approximation, perturbation, and projection methods for economic growth models,” in H. Amman, D. Kendrick, and J. Rust (ed.), Handbook of Computational Economics, Elsevier, Amsterdam. Judd, K. L., 1998, Numerical methods in economics, , The MIT Press. Huang, C.-F., 1987, “An intertemporal general equilibrium asset pricing model: the case of diffusion information,” Econometrica, 55, 117—142. Kaminsky, G. and S. Schmukler, 2001, “On booms and crashes: financial liberalization and stock market cycles,” Working paper. Karatzas, I., J.P. Lehoczky and S. Shreve, 1996, “Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model,” Mathematics of Operations Research, 15, 80—128. Karatzas, I., S. Shreve, 1998, Methods of Mathematical Finance, Springer-Verlag, New York. Kevorkian, J., and J. Cole, 1996, Multiple scale and singular perturbation methods, SpringerVerlag, New York. Kim, E. H., and V. Singal, 2000, “Stock market openings: experience of emerging economies,” Journal of Business, 73, 25—66. King, M., E. Sentana and S. Wadhwani, 1990, “Volatility and links between national stock markets,” Econometrica, 62, 901—933. Kogan, L., 2001, “An equilibrium model of irreversible investment,” Journal of Financial Economics, 62, 201—245. .


41

Kogan, L., and R. Uppal, 2001, “Risk aversion and optimal portfolio policies in partial and general equilibrium economies,” Working paper, MIT. Lewis, K., 1996, “What can explain the apparent lack of international consumption risk sharing?,” Journal of Poltical Economy, 104, 267—297. Ljungqvist, L., and T. Sargent, 2000, Recursive macroeconomic theory, The MIT Press. Ma, J., and J. Yong, 1999, Forward-backward stochastic differential equations and their applications, Springer-Verlag, Heidelberg, Berlin. Mehra, E., and E. Prescott, 1985, “The equity premium puzzle,” Journal of Monetary Economics, 15, 145—161. Nualart, D., 1995, The Malliavin Calculus and Related Topics, Springer-Verlag. Obstfeld, M., 1992, “International risk-sharing and capital mobility,” Journal of International Money and Finance, 11, 115—121. Obstfeld, M., 1994, “Risk-taking, global diversification, and growth,” The American Economic Review, 84, 1310—1329. Rubinstein, M., 1974, “An aggregation theorem for securities markets, ” Journal of Financial Economics, 1, 225—244. Sellin, P., and I. Werner, 1993, “International barriers in general equilibrium, ” Journal of International Economics, 34, 2107—2138. Serrat, A., 2001, “A dynamic equilibrium model of international portfolio holdings,” Econometrica, 69, 1467—1489. Shapiro, A., 2002, “The investor recognition hypothesis in a dynamic general equilibrium: theory and evidence,” Review of Financial Studies, 15, 97—141. Soros, G., 2002, George Soros on Globalization, Public Affairs Ltd., Oxford. Stapleton, R. C., and M. Subrahmanyam, 1977, “Market imperfections, capital market equilibrium and corporation finance, ” Journal of Finance, 32, 307—319. Stiglitz, J., 2002, Globalization and its discontents, The Penguin Press, London. Stulz, R., 1981, “On the effects of barriers to international investment,” Journal of Finance, 36, 923—934. Stulz, R., 1995, “Globalization and the cost of capital: the case of Néstle,” European Financial Management, 8, 30—38. Stulz, R., 1999, “Globalization of equity markets and the cost of capital,” Working paper, Dice Center, Ohio-State University. Subrahmanyam, M., 1975, “On the optimality of international capital market integration,” Journal of Financial Economics, 2, 3—28.


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Telmer, C., 1993, “Asset-pricing puzzles and incomplete markets,” Journal of Finance, 48.5, 1803—1832. Tesar, L., 1995, “Evaluating the gains from international risksharing,” Carnegie-Rochester Conference Series on Public Policy, 42, 95—143. van Wincoop, E., 1994, “Welfare gains from international risksharing,” Journal of Monetary Economics, 34, 175—200. Weil, P., 1989, “The equity premium puzzle and the riskfree rate puzzle,” Journal of Monetary Economics, 24, 401—421. Zeidler, E., 1986, “Nonlinear functional analysis and its applications, volume I,” SpringerVerlag, New York.


A

43

Perfect Stock Market and Bond Market Segmentation

In this section I consider the case, where both stock and bond markets are perfectly segmented. Hence, within each country investors can only trade in their own stock and borrow and lend at the riskless rate at the national level. Thus, at the international level, each country is in autarky. I label this economy EA . A.1

Agents’ preferences and endowments

I assume each individual country agent has logarithmic preferences given by ui (c) = log c, i ∈ {1, 2} .

(A1)

The initial endowment gives the representative agent of each country all the stocks for that country. A.2


In this case, given that there is autarky, each agent consumes his endowment and holds shares in only his own stock market with no international lending and borrowing.

Proposition A1 The optimal consumption policy of the representative agent of country i is Ci (t), where Ci (t) = ei (t) ,

(A2)

and the vector of optimal proportions of country i’s wealth invested in stocks is given by π i (t), where · ¸| µ1 (t) − r (t) , 0 = [1, 0]| , (A3) π1 (t) = [π 11 (t) , π 12 (t)]| = | σ 1 (t) σ 1 (t) · ¸ and µ (t) − r (t) (A4) = [0, 1]| . π2 (t) = [π 21 (t) , π 22 (t)]| = 0, 2 σ 2 (t) σ 2 (t)| A.3

Equilibrium prices

Using the approach of Cox and Huang (1989) one can derive the equilibrium riskfree rate, market price of risk, equity premia, stock return volatilities and stock prices. These expressions are well-known for the case of autarky.

Proposition A2 For all times t ∈ [0, T ] 1. The riskless rate in country i is given by ri (t), where ri (t) = β + µe,i (t) − σ e,i (t)| σe,i (t)

(A5)

2. The market price of risk in country i is given by θi , where θi (t) = σe,i (t)

(A6)

3. The risk premium in country i is given by |

µi (t) − ri (t) = σ e,i (t) σe,i (t)

(A7)


44

4. The price of stock i is given by Si (t), where 1 − e−β(T −t) ei (t) β Cumulative stock returns for stock i ∈ {1, 2} are given by µi (t), where Si (t) =

µi (t) = β + µe,i (t)

(A8)

(A9)

Stock return volatilities for stock i ∈ {1, 2} are given by σ i (t), where σ i (t) = σ e,i (t)

(A10)

5. The cross-country correlation in stock returns is given by ρ12 (t) = ρ.

(A11)


B

45

Proofs

Proof of Proposition 1. Define the non-empty closed convex set © ª | K1 = α ∈ R2 : α = (α1 , 0) s.t. α1 ∈ R

Agent one faces constraints on it’s portfolio proportion process π1 such that she cannot invest in stock two, i.e. π 1 (t) ∈ K for Lebesgue-a.e. t ∈ [0, T ]

This is in addition to the standard conditions that a consumption and portfolio process pair (C1 , π 1 ) are admissible. I can now define the set A (K1 ), which constrains the set of admissible consumption-portfolio proportion pairs ( ) Z T −βt A (K1 ) = (C1 , π 1 ) ∈ A : π1 (t) ∈ K for Lebesgue-a.e. t ∈ [0, T ] ; E0 e u1 (C1 (t)) dt > −∞ 0

Agent one faces the following optimization problem # "Z T −βt sup E0 e u1 (C1 (t)) dt (C,π1 )∈A(K1 )

0

Note that the static budget constraint arising from the initial wealth held by the agent is implicit in A (K1 ), because portfolio proportion processes have been defined in terms of admissible trading strategies which finance a feasible consumption plan. Cvitanic & Karatzas (1992) show that this problem can be solved via an appropriately defined dual minimization program associated with the constraint set K1 . Let ν = (ν 1 , ν 2 )| . For v ∈ R2 the support function of -K1 is defined by δ (ν) = sup − (α1 ν 1 + α1 ν 2 ) α∈K1

The effective domain of this support function is given by

Hence

e 1 = {ν : δ (ν) < ∞} K

e 1 = {ν : ν 1 = 0 and ν 2 ∈ R} K

e1. Note that δ (ν) = 0 on K Let H be the Hilbert space of {Ft } −progressively measurable processes ν : [0, T ] × Ω −→ R2 with Z T the inner product hν 1 , ν 2 i = E ν 1 (t)| ν 2 (t) dt 0

which induces the norm k·kH given by

kνk2H = E

Z

0

T

kν (t)k2 dt < ∞

e 1 with Denote by D the subset of H consisting of processes ν : [0, T ] × Ω −→ K Z T E δ (ν (t)) dt < ∞ 0

For each ν ∈ D, there is a fictitious unconstrained economy under an EMM Qv in which the agent faces the state-price density ½ Z t ¾ Z t Z 1 t v ζ v (t) = ζ v (0) exp − (r (u) + δ (v (u))) du − θv| (u) dW (u) − kθ (u)k2 du 2 0 0 0 where θν = σ −1 (µ + v − r)


46

Following Cvitanic and Karatzas (1992), I solve the static variational problem in the fictitious markets by considering the following dual minimization problem min

where Jν (ψ 1 ) = E

Z

0

and

{ψ 1 ∈R+ ,ν∈D} T

e1 (t, ψ 1 ζ v (t)) dt + ψ 1 E U

Jν (ψ 1 ) ,

"Z

#

T

ζ v (t) δ (ν (t)) dt + ψ 1 ξ ν (0) X1 (0)

0

e1 (t, z) = sup (U1 (t, x) − xz) U x

e1 , of U1 . For U1 (t, x) = e−βt log x defines the convex dual, U

e1 (t, z) = −e−βt (1 + log z + βt) U

and when the agent cannot invest in the second risky asset θv = σ−1 (µ + v − r) where v=

·

¸

0 v2

Then I can write the state-price density ζ v (t) as ½ Z t ¾ Z t Z 1 t v 2 v| ζ v (t) = ζ v (0) exp − r (u) du − θ (u) dW (u) − kθ (u)k du 2 0 0 0 e 1 . The FOC wrt ψ 1 gives Recall that δ (v (·)) = 0 on K −βT 1 c = 1−e ψ 1 β ξ ν (0) X1 (0) I now seek to minimize Jν (ψ 1 ) wrt v. Hence I seek to pointwise minimize 1 v kθ (u)k2 2 The FOC gives Σ12 (µ − r) µ2 + vb2 − r = Σ11 1 where the matrix Σ is defined by Σ = σσ |

Therefore θ1

:

v b

=θ =

"

−1 = Σ−1 Σ 11 σ

σ 11 σ 211 +σ 212 σ 12 ·σ211 +σ212

µ1 − r 0

#

(µ1 − r) ¸

Note that the optimal portfolio proportion process in the fictitious unconstrained economy is ¸ · 1 µ1 − r Σ−1 σθ1 = | σ1 σ1 0 which coincides with the optimal portfolio proportion process in the actual constrained economy. Similarly for agent 2 who cannot trade in the first risky asset " # σ 21

θ2

= θwb =

σ 221 +σ 222 σ 22 σ 221 ·+σ222

−1 = Σ−1 Σ 22 σ

(µ2 − r) ¸

0 µ2 − r


47

As for agent one the optimal portfolio proportion process · in the¸fictitious unconstrained economy 1 0 Σ−1 σθ2 = | σ 2 σ 2 µ2 − r coincides with the optimal portfolio proportion process in the actual constrained economy. I have thus identified the unique state-price density for the representative agent of each country in terms of the riskless rate, conditional expected stock returns and the conditional volatilities of stock returns, which are to be determined endogenously in equilibrium. I have also obtained the optimal portfolio proportion processes for each agent. For agent one I can obtain the optimal consumption plan as a simple function of the state-price density ζ vb −βt b1 (t) = e . C b ζ (t) ψ 1 v b Similarly for agent two −βt b2 (t) = e C . b ζ (t) ψ 2 w b

Proof of Lemma 1. Consuming the aggregate endowment process is optimal for the representative agent. Hence the marginal rate of substitution process is given by uc (y (t) , λ (t)) 1 + λ (t) y (0) = e−βt ξ (t) = e−βt uc (y (0) , λ (t)) 1 + λ (0) y (t) Optimal consumption allocations are given by y (t) λ (t) y (t) , C ∗ (t) = I2 (uc (y (t) , λ (t)) /λ (t)) = C1∗ (t) = I1 (uc (y (t) , λ (t))) = 1 + λ (t) 2 1 + λ (t) Following Cuoco and He (1994), the stochastic weight is given by u0 (C ∗ (t)) ψ ξ (t) C ∗ (t) X2 (t) λ (t) = 10 1∗ = 2 2 = 2∗ = u2 (C2 (t)) ψ 1 ξ 1 (t) C1 (t) X1 (t) Therefore I can write the stochastic weight in terms of the cross-sectional wealth distribution x (t) = X1 (t) / (X1 (t) + X2 (t)), i.e. λ (t) = (1 − x (t)) /x (t) and x (t) = 1/ (1 + λ (t)). Hence C1∗ (t) = x (t) y (t) and C2∗ (t) = (1 − x (t)) y (t). Lemma B1 Individual country wealth processes Xi are given by dXi (t) = Xi (t) µXi (t) dt + Xi (t) σ Xi (t)| dW (t) , i ∈ {1, 2} where

µ

Si (t) µXi (t) = r (t) − c (t) + c (t) y (t) xi (t) β Si (t) σ Xi (t) = σ i (t) 1 − e−β(T −t) y (t) xi (t) β c (t) = 1 − e−β(T −t) for x1 = x, x2 = 1 − x.

¶2

|

σ 1 (t) σ 1 (t)

(B12)

(B13) (B14) (B15)

This lemma characterizes the drifts and volatilities of each individual country’s wealth in terms of the equilibrium riskless rate, the price and volatility of its own stock, the cross-sectional wealth distribution and exogenous variables. Both investors are logarithmic, so the volatility of the wealth growth rate, σXi (t), is the same as the volatility of the consumption growth rate. Suppose that the endowment processes in each country have the same growth rates, growth rate variances, initial wealth levels are equal, i.e. x1 (0) = x2 (0) and that each country faces the same kind of investment restrictions. It follows that S1 (0) = S2 (0) and σ 1 (0) = σ 2 (0). Therefore the consumption growth rate volatility is directly tied to stock market volatility in each country. Hence an increase in consumption growth rate volatility at time, t = 0, will drive up stock market volatility at time, t = 0.


48

Remark B1 Endogenizing prices makes it possible for consumption volatility and stock return volatility to move in the same direction. With exogenous stock prices, as in Obstfeld (1994), consumption volatility is inversely proportional to stock volatility. To see this suppose that stock price processes are exogenous. From the expressions for consumption volatility σ Ci = π i | σ i

(B16)

The optimal portfolio weights are given by πi =

µi − r 1i σi | σi

(B17)

It follows that

µi − r σ ii , i ∈ {1, 2} (B18) σi | σi Clearly, increasing stock market volatility decreases consumption volatility, if prices are exogenous. This implication of Obstfeld (1994) runs counter to the data (see Bekaert, Harvey and Lundblad (2002) and Bekaert and Harvey (1997)), where countries with higher consumption volatility tend to have higher stock market volatility. By endogenizing prices it is possible to break the inverse relationship between consumption volatility and stock volatility found in the model of Obstfeld (1994) σ Ci =

Proof of Lemma B1 and Proposition 2. First I derive an expression for the equilibrium riskless rate as in Cuoco and He (1994). I know the optimal consumption plans satisfy ¢ ¡ Ci (t) = Ii ψ i eβt π i (t) and

Therefore

C1∗ (t) = I1 (uc (y (t) , λ (t))) , C2∗ (t) = I2 (uc (y (t) , λ (t)) /λ (t)) e−βt uc (y (t) , λ (t)) = ψ 1 π1 (t)

Consider the utility function u with the cross-sectional wealth distribution as the second argument rather that the stochastic weight, I can write e−βt uc (y (t) , x (t)) = ψ 1 π1 (t) Applying Ito’s Lemma, and equating drift terms, I can show that the equilibrium riskfree rate is given ¤ £ by D e−βt uc (y (t) , x (t)) , r (t) = − −βt e uc (y (t) , x (t)) where D (·) denotes the drift of the process, which is it’s argument. For the case where the representative agents of both countries have logarithmic preferences, this reduces to |

|

r (t) = β + µy (t) + µx (t) − (σ y (t) − σ x (t)) (σ y (t) − σ x (t)) + σ x (x) σ y (x)

I now start to derive the dynamics of the cross-sectional wealth distribution. The cross-sectional wealth distribution is defined by the ratio of agent one’s wealth to the aggregate wealth in the world economy. i.e. x (t) = X1 (t) / (X1 (t) + X2 (t)). Market clearing in the consumption good market implies that C1 (t) + C2 (t) = y (t), where Ci is the amount consumed by agent i. Both agents are logarithmic, so β Ci (t) = 1−e−β(T −t) Xi (t) . Hence X1 (t) + X2 (t) =

1 − e−β(T −t) y (t) β

£¡ ¢ ¤ I choose to write the cross-sectional wealth variable x (t) as x (t) = βX1 (t) / 1 − e−β(T −t) y (t) . By Ito’s lemma I can show that the following stochastic differential describes the cross-sectional wealth distribution dx (t) | = µx (t) dt + σ x (t) dW (t) , x (t)


49

where βe−β(T −t) µx (t) = + µX1 (t) − µy (t) + σ y (t)| (σ y (t) − σ X1 (t)) 1 − e−β(T −t) σ x (t) = σ X1 (t) − σ y (t) . Recalling that µX1 (t) = r (t) + and

µ

β

S1 (t) −β(T −t) y (t) x (t) 1−e

¶2

σ1 (t)| σ 1 (t) −

β 1−

e−β(T −t)

β S1 (t) σ 1 (t) 1 − e−β(T −t) y (t) x (t) using the expression for the equilibrium riskless rate, I can show that µ ¶2 β S1 (t) µx (t) = σ 1 (t)| σ1 (t) − (σ y (t) − σx (t))| (σ y (t) − σ x (t)) − σ y (t)| σ x (t) 1 − e−β(T −t) y (t) x (t) and β S1 (t) σx (t) = σ 1 (t) − σ y (t) 1 − e−β(T −t) y (t) x (t) Hence I have determined the cross-sectional wealth distribution in terms of the price of stock one, the conditional volatility of returns on stock one and exogenous parameters. To determine the drift I note that from Cuoco and He (1994) it is known that the stochastic weight satisfies the forward stochastic differential equation σX1 (t) =

dλ (t) = −θ2 (t)| (θ1 (t) − θ2 (t)) λ (t) dt − (θ1 (t) − θ2 (t))| λ (t) dW (t) , λ (t) = λ (0) Note also that the stochastic weight λ (t) is given by λ (t) = X2 (t) /X1 (t). Hence λ (t) = x (t) Applying Ito’s Lemma gives

−1

−1.

|

dλ (t) = (1 + λ (t)) [(σ |x (t) σ x (t) − µx (t)) dt − σ x (t) dW (t)] . Therefore comparing coefficients from the two stochastic differential equations for λ (t) σ |x (t) σ x (t) − µx (t) = θ2 (x)| σ x (t) (θ1 (t) − θ2 (t)) λ (t) = (1 + λ (t)) σ x (t) Hence I can show that

|

|

µx (t) = σx (t) σ x (t) + θ2 (t) σ x (t) where

σ 2 (t) (µ (t) − r (t)) σ 2 (t)| σ 2 (t) 2 I know that the portfolio weight for the representative investor for country two is given by µ (t) − r (t) β S2 (t) π2 (t) = 2 | = −β(T −t) (1 − x (t)) y (t) σ 2 (t) σ 2 (t) 1−e Therefore S2 (t) σ 2 (t) β θ2 (t) = 1 − e−β(T −t) (1 − x (t)) y (t) All the wealth in the world is held in equities, so 1 − e−β(T −t) y (t) µy S1 (t) σ1 (t) + S2 (t) σ 2 (t) = β Hence I can show that the cross-sectional wealth distribution and the stochastic weight satisfy the forward stochastic differential equations ¸ · dx (t) 1 β S1 (t) | | (t) − σ (t) [x (t) σ y (t) = σ 1 y x (t) 1 −µx (t) 1 −¶e−β(T −t) y (t) x (t) 1 β S1 (t) +2 − x (t) σ1 (t)]dt −β(T −t) x (t) y (t) 1−e · 2 ¸ S1 (t) β | | + (t) − σ (t) dW (t) , x (0) = x0 σ 1 y 1 − e−β(T −t) y (t) x (t) θ2 (t) =


50

and

¸ · 1 + λ (t) 1 + λ (t) β β σ (t) S (t) − y (t) σ (t) [ (1 [ 1 1 y y (t) λ (t) y (t) 1 − e−β(T −t) 1 − e−β(T −t) +λµ(t))σ 1 (t) S1 (t) − y (t) σ y (t)]dt ¶ β | | + y (t) σ y (t) − (1 + λ (t)) σ 1 (t) S1 (t) dW (t)], λ (0) = λ0 1 − e−β(T −t) respectively. I define the gains process for stock one Z t ξ 1 (s) e1 (s) ds G1 (t) = ξ 1 (t) S1 (t) + dλ (t) =

0

Applying Ito’s Lemma

|

|

dG1 (t) = ξ 1 (t) S1 (t) [(µ1 (t) − σ 1 (t) θ1 (t) − r (t)) dt + (σ1 (t) − θ1 (t)) dW (t)] I now note that

|

Therefore G1 (t) is a local martingale

σ 1 (t) θ1 (t) = µ1 (t) − r (t) |

dG1 (t) = ξ 1 (t) S1 (t) (σ 1 (t) − θ1 (t)) dW (t) If this local martingale is a martingale, then from applying the terminal boundary condition S1 (T ) = 0, Z T it follows that ξ 1 (s) e1 (s) ds. S1 (t) = Et ξ 1 (t) t Therefore Z T Z T e1 (s) (1 + λ (s)) e1 (s) y (t) S1 (t) = y (t) x (t) Et e−β(s−t) e−β(s−t) ds = Et ds. y (s) x (s) 1 + λ (t) y (s) t t Then I can find the price of stock two via the condition 1 − e−β(T −t) y (t) S1 (t) + S2 (t) = β or by defining a gains process for stock two and thereby obtaining an integral representation. Proof of Corollary 1. Exploiting the Markovian structure of the economy, with state variables y (t), x (t) and G (t) = e1 (t) /y (t) and defining φ (t) via S1 (t) = y (t) x (t) eβt φ (t) it follows from Ito’s Lemma that σ 1 (t) = σy (t) + σ x (t) + x (t) σx (t)

φx (t) φ (t) + G (t) σG (t) G . φ (t) φ (t)

I know that

βeβt φ (t) σ 1 (t) − σ y (t) 1 − e−β(T −t) 1 Therefore I can solve for both σ x (t) and σ 1 (t). From the representation Z T e1 (s) e−βs φ (t) = Et ds y (s) x (s) t σ x (t) =

it follows that φ (t) solves the partial differential equation (24). Proof of Proposition 3. The main step of the proof consists of expanding the operator for the partial differential equation (24) and p the function φ on which it operates as power series in the dimensionless parameter , where = (1 − ρ) /2. This decomposes the nonlinear pde for φ into a series of linear pde’s Li φi = 0, φi (T ) = 0, i = 0, 1, 2, ... It is possible to solve these linear partial differential equations exactly in closed-form for i = 0, 1, 2, 3, 4. This gives approximate equilibrium stock prices as a polynomials in . In fact 1 − e−β(T −t) e1 (t) φ0 (t) = eβt , φi (t) = 0, i ∈ {1, 2, 3, 4} . β y (t)


51

From equilibrium stock prices other financial variables such as returns, volatilities etc can be derived using simple but tedious algebra. Proof of Proposition 4. Market clearing in stock market i implies that c (t) π1i (t) x (t) + π2i (t) (1 − x (t)) = Si (t) , i ∈ {1, 2} , y (t) β where c (t) = . 1 − e−β(T −t) Substituting in the optimal portfolio policies µ (t) − r (t) µ (t) − r (t) , π 12 (t) = 0, π21 (t) = 0, π 22 (t) = 2 | π 11 (t) = 1 | σ 1 (t) σ1 (t) σ 2 (t) σ 2 (t) and solving for the risk premia gives the CAPM relationships.

Proof of Proposition 5. After defining the sets in which admissable consumption-portfolio processes are constrained to lie, the proof proceeds in the same way as the proof of Proposition 1. Proposition B3 The equilibrium interest rate is given by |

(B19)

µx (t) = σx (t)| σ x (t) + θ2 (t)| σ x (t)

(B20)

where and

|

r (t) = β + µy (t) + µx (t) − (σ y (t) − σ x (t)) (σ y (t) − σ x (t)) + σ x (t) σ y (t)

c (t) p1 (t) (1 − 2c (t) p1 (t)) σ 1 (t) + σ y (t) − σ y (t) x (t) 1 − c (t) p1 (t) µ ¶ σ 1 (t)| σ 2 (t) θ2 (t) = c (t) p1 (t) (t) − σ (t) + σ y (t) σ 2 1 | σ 1 (t) σ 2 (t) σ y (t) − c (t) p1 (t) σ 1 (t) σ 2 (t) = 1 − c (t) p1 (t) σ x (t) =

S1 (t) y (t) x (t) β c (t) = 1 − e−β(T −t) The equilibrium price of stock one is determined by the following system of equations Z T y (t) e2 (s) − y (t) (1 − x (t)) Et ds S1 (t) = c (t) y (s) (1 − x (s)) t dx (t) = µx (t) dt + σ x (t)| dW (t) , x (0) = x0 x (t) or using an equivalent formulation in terms of the stochastic weight Z T y (t) λ (t) y (t) (1 + λ (s)) e2 (s) S1 (t) = − Et ds c (t) 1 + λ (t) λ (t) y (s) t p1 (t) =

|

dλ (t) = (1 + λ (t)) [(σ |x (t) σ x (t) − µx (t)) dt − σ x (t) dW (t)] , λ (0) = λ0

(B21)

(B22) (B23)

(B24) (B25)

(B26)

(B27) (B28)

Market clearing in the consumption good market implies that the price of stock two is given by Z T e2 (s) y (t) S2 (t) = − S1 (t) = y (t) (1 − x (t)) Et ds c (t) t y (s) (1 − x (s))

Proof of Proposition B3. the proof of Proposition 2.

Details are omitted, because the proof follows the same approach as


52

Corollary B1 Using state variables x (t) and G (t) = e1 (t) /y (t) the price of stock one is given by y (t) − y (t) (1 − x (t)) eβt φ (t) c (t) where φ (t) solves the quasilinear ¶ parabolic partial differential equation µ ∂ 1 − G (t) + L φ (G, x, t) + e−βt , φ (G (T ) , x (T ) , T ) = 0 0= ∂t 1 − x (t) L is the differential generator for (G, x) and the volatility of stock one is given by ³ ´ σy φx φG x(t) βt c(t) − (1 − x (t)) e φ (G, x, t) σ y − 1−x(t) σ x (t) + σ G φ + σ x (t) φ σ1 (t) = 1 βt c(t) − (1 − x (t)) e φ (G, x, t) S1 (t) =

(B29)

(B30)

Proof of Corollary B1. The proof proceeds in the same way as the proof of Corollary 1. Proposition B4 The equilibrium variables at time t in economy EMI are given by 1. In equilibrium, the vector of optimal proportions of wealth invested by each agent i in stocks is given by πi (t), where π 1 (t) = [π 11 (t) , π 12 (t)]| (B31) N (t) π 11 (t) = (1 + N (t)) x (t) ¤ £ 2N (t) 2N (t) − x (t) (1 + N (t)) 1 − e−β(T −t) (1 + β (T − t)) σ 2e − (1 − ρ) 3 2 β 1 − e−β(T −t) (1 +Ń (t)) x (t) ³ 2 +O (1 − ρ) (B32) N (t) π 12 (t) = 1 − (1 + N (t)) x (t) ¤ £ 2N (t) 2N (t) − x (t) (1 + N (t)) 1 − e−β(T −t) (1 + β (T − t)) σ 2e + (1 − ρ) β 1 − e−β(T −t) (1 +Ń (t))3 x (t)2 ³ 2 +O (1 − ρ) (B33)

while, for the second investor

π2 (t) = [0, π 22 (t)]| π 22 (t) = 1 − (1 − ρ)

´ ³ N (t) + O (1 − ρ)2 1 + N (t)

(B34) (B35)

2. The equilibrium interest rate is given by

µ ¶ 1 − x (t) r (t) = β + µe − + (1 − ρ) 1 − N (t) σ 2e 2 x¶(t) (1 (t)) µ+ N 4σ2e 4e−βT N + (1 − ρ)2 σ 2e T σ2e + 1 − 2 β 1 − e−βT (1 ³ ´ + N) 3 +O (1 − ρ) σ 2e

2N (t)

(B36)

3. The market price of risk in country r one is given by

µ ¶ 2N (t) 1 1−ρ θ1 (t) = σe 11 + − 1 σ e 12 2 1+ µ N (t) x (t) ¶ ³ ´ x (t) − 1 N (t) 3/2 + (1 − ρ) + 1 σ e 11 + O (1 − ρ) 1 + N (t) x (t) and the market price of risk in country two is given by r ´ ³ 1 − ρ 1 + 5N (t) 1−ρ θ2 (t) = σ e 11 − σe 12 + σ e 11 + O (1 − ρ)3/2 2 4 1 + N (t)

(B37)

(B38)


53

4. The risk premia on stocks one and two are given by

· ¸ 2N (t) (1 − x (t)) 1 µ1 (t) − r (t) = σ 2e + (1 − ρ) σ 2e −1 1 + N (t) x (t) ´ ³ 3/2 +O (1 − ρ) ´ ³ N (t) + O (1 − ρ)3/2 µ2 (t) − r (t) = σ 2e − (1 − ρ) σ 2e 1 + N (t)

(B39) (B40)

5. The conditional volatilities of stocks returns are given by r

³ ´ 1−ρ 1−ρ 2 12 − 11 ] + O (1 − ρ) r 2 4 ³ ´ 1−ρ 1−ρ 3/2 σ2 (t) = σ e [11 − 12 − 11 + O (1 − ρ) 2 4

σ1 (t) = σ e [11 +

(B41) (B42)

6. The conditional variances of stock returns for stocks one and two are given by

µ ¶ 2 4N (t) 1 − x (t) 1 − [1 + β (T − t)] e−β(T −t) σ 4e V1 (t) = σ 2e + (1 − ρ)2 3 2 β 1 − e−β(T −t) ³ ´(1 + N (t)) x (t) 3 +O (1 − ρ) (B43) 2 V2 (t) = σ e ¶ µ µ ¶¶ µ N (t) 2N (t) 2 N (t) β (T − t) e−β(T −t) σ 4e 2 − (1 − ρ) 1− 1+ 1− (1 ´ + N (t)) 1 + N (t) x (t) x (t) β 1 − e−β(T −t) ³ 3 (B44) +O (1 − ρ)

7. The correlation in cross-country stock returns is given by (1 − ρ)2 ρ12 (t) = ρ + β

¶ µ ³ ´ β (T − t) e−β(T −t) 3 2 + O (1 − ρ) σ 1− e 1 − e−β(T −t)

(B45)

8. The cross-sectional wealth distribution is given by dx (t) = x (t) (µx (t) dt + σ x (t)| dW (t)) , x (0) = x0 where

³ ´ 2σ 2e 3/2 + O (1 − ρ) µx (t) = (1 − ρ) 2 r (1 + N (t)) ³ ´ 2σ e 1−ρ 1−ρ σ x (t) = [ 12 + 11 ] + O (1 − ρ)3/2 1 + N (t) 2 2

(B46) (B47) (B48)

Proof of Proposition B4 . The proof proceeds along the same lines as the proof of Proposition 3, so details are omitted.

Proof of Propositions 6, 7, 8 and 10. These propositions follows directly from propositions 3 and B5.

Proof of Proposition 9 . Note that the SPD’s, ξ i , satisfy the FSDE’s dξ i (t) = −r (t) dt − θi (t)T dW (t) , ξ i (0) = 1 ξ i (t) Assume that the price of stock i is given by Z T ξ i (s) Si (t) = Et ei (s) ds ξ i (t) t


54

Apply Ito’s Lemma

"

dξ (t) Si (t) + ξ (t) dSi (t) + dξ (t) dSi (t) = d Et Define the L2 (P) martingale M (t) M (t) = Et

Z

Z

T

#

ξ (s) ei (s) ds

t

(B49)

T

ξ (t) ei (t) dt

0

Let D1,2 denote the closure of the class of smooth random variables, S, with respect to the norm h ³ ´ ³ í1/2 , F ∈ S. kF k1,2 = E |F |2 + E kDF k2L2 (T ) R T −βt For 0 e ξ (t) ei (t) dt ∈ D1,2 , applying the Clark-Ocone formula (see page 42 of Nualart (1995)) gives # " Z Z t T Es Ds ξ (u) ei (u) du dW (s) M (t) = M (0) + 0 0 # " Z t Z T = M (0) + Es Ds [ξ (u) ei (u)] du dW (s) , s ≤ u ≤ T, 0 ≤ s ≤ t 0

s

Hence

dM (t) = Et Note that Et

Z

"Z

Dt [ξ (u) ei (u)] du , t ≤ u ≤ T

t

T

t

ξ (s) ei (s) ds = M (t) −

Equating diffusion terms in (B49) yields |

|

Z

−ξ (t) θ (t) Si (t) + ξ (t) σ i (t) Si (t) = Et Therefore

Et σ (t) =

Note that and

#

T

hR

T t

t

ξ (s) ei (s) ds

0

"Z

#

T

t

i Dt [ξ (u) ei (u)]| du ξ (t) Si (t)

Dt [ξ (u) ei (u)] du + θ (t)

Dt [ξ (u) ei (u)] = ei (u) Dt ξ (u) + ξ (u) Dt ei (u) , t ≤ u ≤ T Dt ξ (u) = ξ y (u) Dt y (u) + ξ x (u) Dt x (u) = ξ y (u)

Hence

2 X

k=1

Dt ek (u) + ξ x (u) Dt x (u)

Dt [ξ (u) ei (u)] = ξ y (u) ei (u) Dt y (u) + ξ x (u) ei (u) Dt x (u) + ξ (u) Dt ei (u)

One can then show that RT RT RT ξy (u) (u)| x (u) Et t ξ (u) ei (u) ξ(u) Dt y (u)| du Et t ξ (u) ei (u) ξξ(u) Dt x (u)| du Et t ξ (u) ei (u) Dteeii(u) du + + +θ (t) . σ i (t) = RT RT RT Et t ξ (u) ei (u) du Et t ξ (u) ei (u) du Et t ξ (u) ei (u) du

Proof of Proposition 11. The proof proceeds in the same way as the proof of proposition 4. Proof of Proposition 12. The optimal consumption and portfolio policies are derived in the standard way, using either the wealth distribution of the “world” representative agent or from the first order conditions of the Hamilton-Jacobi-Bellman equation.

Proposition B5 For every time, t ∈ [0, T ] 1. The riskless rate is given by |

r (t) = β + µy (t) − σ y (t) σ y (t)

(B50)


55

2. The market price of risk is given by θ (t) = σ y (t)

3. The risk premium is given by

(B51)

|

µ (t) − r (t) 1 = σ (t) σ y

(B52)

4. The prices of stocks one and two are given by

where

for

´ ³ p 1−ρ = S10 (t, N (t)) + S1 t, N (t) , 1 − ρ S12 (t, N (t)) 2 µ ¶2 ³ ´ 1−ρ 3 + S14 (t, N (t)) + O (1 − ρ) ´ ´ ³ ³ 2 p p S2 t, N (t) , 1 − ρ = S1 t, N (t)−1 , − 1 − ρ

(B53) (B54)

1 − e−β(T −t) N (t) y (t) S10 (t, N (t)) = β 1 + N (t) i h N (t) (N (t) − 1) 2 2 σe S12 (t, N (t)) = 2 1 − e−β(T −t) (1 + β (T − t)) 3 β (1 + N (t)) h i´ 2 ³ 2 S14 (t, N (t)) = 3 2 − e−β(T −t) 1 + (1 + β (T − t)) F (t, N (t)) σ 4e β −5

F (t, N (t)) = N (t) [1 + N (t)] [−1 h + 11N (t) i 2 3 2 −β(T −t) 2 + 2β (T − t) + β 2 (T − t) ) −11N (t) + N (t) ](2 − e

(B55) (B56)

and their dynamics are given by

where

for

and

dS1 (t) + e1 (t) dt = S1 (t) [µ1 (t) dt + σ 1 (t)| dW (t)] , i ∈ {1, 2}

(B57)

´ ³ p 1−ρ = µ10 + µ1 t, N (t) , 1 − ρ µ (t, N (t)) 2¶2 12 µ ³ ´ 1−ρ 3 + µ14 (t, N (t)) + O (1 − ρ) ³ ³ 2 ´ ´ p p µ2 t, N (t) , 1 − ρ = µ1 t, N (t)−1 , − 1 − ρ

(B58)

µ10 = β + µe N (t) − 1 2 µ12 (t, e1 (t) , e2 (t)) = 2 2 σe (1 + N (t))

(B60) (B61)

µ14 (t, e1 (t) , e2 (t)) (N (t) − 1) (N (t) − 3) 1 − e−β(T −t) (1 + β (T − t)) σ 4e = 4N (t) β 1 − e−β(T −t) (1 + N (t))4 r µ ¶ ´ ³ p 1−ρ 1−ρ = σ 10 + σ 11 + σ 12 σ 1 t, N (t) , 1 − ρ 2 µ µ ¶23/2 ¶2 1−ρ 1−ρ + σ 13 (t, N (t)) + σ 14 2 ³2 ´ 5/2 +O³ (1 − ρ) ´ ´ ³ p p σ 2 t, N (t) , 1 − ρ = σ 1 t, N (t)−1 , − 1 − ρ

(B59)

(B62)

(B63) (B64) (B65)


56

where = [σ e , 0]| (B66) | = h[0, σ e ] i| (B67) σe = − , σe (B68) 2 1 − e−β(T −t) (1 + β (T − t)) N (t) (N (t) − 3) 3 | ¡ ¢ σ13 (t, e1 (t) , e2 (t)) = −4 σ e [0, 1] (B69) 3 −β(T −t) β 1 − e (N (t) + 1) h σ i| e σ 14 = − , 0 (B70) 8 The conditional volatilities of returns for stocks one and two are given by ¢µ ¡ ¶2 ´ ³ p (N (t) − 3) N (t) 1 − e−β(T −t) [1 + β (T − t)] 1−ρ 2 £ ¤ = σe − 8 σ 4e V1 t, N (t) , 1 − ρ 3 −β(T −t) 2 β 1 − e (N (t) + 1) ³ ´ +O (1 − ρ)3 (B71) σ 10 σ 11 σ 12

´ ´ ³ ³ p p −1 V2 t, N (t) , 1 − ρ = V1 t, N (t) , − 1 − ρ

and

respectively.

(B72)

5. The cross-country correlation in stock returns is given by

£ ¤ ¶2 1−ρ 16N (t) 1 − 1 + β (T − t) e−β(T −t) σ 2e ρ12 (t) = ρ + 2 β 1 − e−β(T −t) ³ 2 ´(1 + N (t)) 3 +O (1 − ρ) µ

Remark B2 The results for the equilibrium riskless rate, equity premia and the market price of risk are standard. The expressions for equilibrium prices, returns and volatilities are new and to my knowledge have not appeared before in the literature. These results make it is easy to see that prices, returns and volatilities for the stocks are different at a given instant if and only if the instantaneous values of the individual country endowment processes are different. If at a given instant, the endowment process in country i is higher than in country j, the price of the stock in country i is higher than in country j. As one would expect, at that instant, conditional stock returns in country i are lower than in country j. The behavior of size of stock return volatilities is quite different. Both stocks have the same size of conditional volatility of returns, regardless of endowment fluctuations. Thus, the sensitivity of stock returns and the size of return volatilities to endowment fluctuations exhibit different properties in the cross-section: a temporary imbalance in the size of individual country endowment processes affects the cross-section of country prices and returns but not the cross-section of sizes of individual return volatilities. Proof of Proposition B5. I first note that the individual country endowments are identical in all respects apart from their shocks. The endowment processes for each country satisfy the stochastic differential equations dei (t) = µe dt + σ e dZi (t) , ei (0) = ei0 , i ∈ {1, 2} ei (t) where e10 = e20 and Z1 (t), Z2 (t) are Brownian motions such that p p Z1 (t) = 1 − 2 W1 (t) + W2 (t) , Z2 (t) = 1 − 2 W1 (t) − W2 (t) where

∈ [0, 1]. Note that

dZ1 (t) .dZ2 (t) = ρdt, ρ = 1 − 2

2

where ρ is the correlation between the shocks to the two endowment processes. I can therefore write the aggregate endowment process y as ¸ ¶ ·µ p 1 2 y (t) = exp µe − σ e t + σ e 1 − 2 W1 (t) (e1 (0) exp [σ e W2 (t)] + e2 (0) exp [−σe W2 (t)]) 2


57

This world endowment process satisfies the following stochastic differential equation dy (t) = µy dt + σ y (t)| dW (t) , y (0) = e1 (0) + e2 (0) y (t) · √ · √ ¸ ¸ where 2 2 e1 (t) e2 (t) 1 − 1 − + σe µy = µe , σ y (t) = σ e − y (t) y (t) The dynamics of the marginal rate of substitution process are given by ¤ ¢ £¡ dξ (t) = −ξ (t) β + µy − σ |y σ y dt + σ |y dW (t) It is also known that

|

dξ (t) = −ξ (t) [r (t) dt + θ (t) dW (t)]

where r is the riskless rate and θ is the market price of risk. Hence

r (t) = β + µy − σ y (t)| σ y (t) and θ (t) = σ y (t) . The price of stock one, i.e.the claim on the endowment of country one is given by Z T ξ (u) e1 (u) du S1 (t) = ξ (t)−1 Et t

Hence,

S1 (t) = y (t) Et Note that

¶ ·µ p 1 2 y (u) = exp µe − σ e (u − t) + σe 1 − 2 +e2 (t) exp [−σ e (W2 (u) − W2 (t))])

Z 2

T

e−β(u−t) t

e1 (u) du. y (u) ¸

(W1 (u) − W1 (t)) (e1 (t) exp [σ e (W2 (u) − W2 (t))]

and

·µ ¶ p 1 2 e1 (u) = e1 (t) exp µe − σ e (u − t) + σ e 1 − 2

2

¸

(W1 (u) − W1 (t)) + σ e (W2 (u) − W2 (t)) ,

where is a dimensionless parameter that lies in the range [0, 1]. I can hence simplify the expression for the price of stock one to obtain Z T 1 S1 (t) = y (t) Et e−β(u−t) du. e2 (t) 1 + e1 (t) exp [−2σ e (W2 (u) − W2 (t))] t Similarly, the price of stock two is given by Z T e−β(u−t) S2 (t) = y (t) Et t

Define the integral I (t, k, σ e , ) = E

Z

t

1 1+

e1 (t) e2 (t)

T

e−β(u−t)

exp [2σ e (W2 (u) − W2 (t))]

1 h i du, p 1 + k exp 2σ e (u − t)Z

du.

where Z ∼ N [0, 1]. Note that I (t, k, σ e , ) = I (t, k, σ e , − ). Hence, I can write µ ¶ µ ¶ e2 (t) e1 (t) S1 (t) = y (t) I t, and S2 (t) = y (t) I t, , σe , , σe, e1 (t) e2 (t) iínterchange the expectation and integration in I (t, k, σ e , ). h I now show that I³can use Fubini’s Theorem to p The function 1/ 1 + k exp 2σ e (u − t)z is monotonic for z ∈ R. Therefore 1 h i ≤ 1. p 1 + k exp 2σ e (u − t)z

International Stock Market Integration Hence E

Z

T 0

58

¯ ¯ ¯ ¯ Z T ¯ ¯ −β(u−t) 1 − e−βT 1 ¯ ¯e h i du ≤ E e−β(u−t) du = < ∞. p ¯ ¯ T 0 ¯ 1 + k exp 2σ e (u − t)Z ¯

Therefore I can apply Fubini’s Theorem to obtain Z T e−β(u−t) E I (t, k, σ e , ) = t

Note that 1

1 h i=√ E p 2π 1 + k exp 2σe (u − t)Z

1 h i du. p 1 + k exp 2σ e (u − t)Z

· ¸ 1 2 1 h i dz. exp − z p 2 1 + k exp 2σ (u − t)z e −∞ Z∞

I use Laplace’s Method (see Hinch (1991) or Judd (1998)) to construct an analytic approximation to this expectation. Hence (using a Taylor’s series centered at zero) ∞ X 1 ∂ 2i 1 1 h i= h i |z=0 . + E p 2i 1 + k i=1 ∂z 1 + k exp 2σ p(u − t)z 1 + k exp 2σ e (u − t)Z e Truncating the infinite sum at the n’th term, I obtain N X 1 1 1 ∂ 2i 1 h i= h i |z=0 + ε2N , E + p p 2i 1 + k (2i)! ∂z 1 + k exp 2σe (u − t)Z 1 + k exp 2σ (u − t)z i=1 e where the upper bound on the size of the error term is given by ¯ ¯ ¯ ¯ 2N +2 ¯ ¯ 1 ∂ 1 ¯ h i ¯¯ . |ε2N | ≤ sup ¯ p 2N+2 (2N + 2)! ∂z z∈R ¯ 1 + k exp 2σe (u − t)z ¯

For the parameter specification σ e = 1% (per annum), = 1, u − t = 5 years, k = 1.5 it can be shown that |ε0 | ≤ .0001, |ε2 | ≤ 2 × 10−8 and |ε4 | ≤ 4 × 10−14 . Higher order errors are shown in Table III. Table III

Approximating the expectation: Absolute errors

N 0 2 4 6 8 10 12 14 16 18 20 22

Maximum Error, |ε2N | 0.0001 2 × 10−8 4 × 10−14 8 × 10−16 2 × 10−19 4 × 10−23 4 × 10−27 1.5 × 10−30 3 × 10−34 6 × 10−38 1.5 × 10−41 3 × 10−45

The errors in this table were computed using the parameter values σ e = 1% (per annum), = 1, T = 5 years, t = 0, e2 (t) /e1 (t) = 1.5, β = 0.006, y (t) = 1. After multiplying by the factor e−β(u−t) , I integrate the polynomial approximation to the expectation over the interval [0, T ] to obtain an analytic approximation to the integral I (t, k, σ e , ). From this


59

I can obtain analytic approximations to the stock prices. For the case where I truncate the series for the expectation at N = 4, I obtain the expressions ¡ ¢ S1 (t, e1 (t) , e2 (t) , ) = S10 (t, e1 (t)) + 2 S12 (t, e1 (t) , e2 (t)) + 4 S14 (t, e1 (t) , e2 (t)) + O 6 where

S10 (t, e1 (t)) = S12 (t, e1 (t) , e2 (t)) = S14 (t, e1 (t) , e2 (t)) = for

1 − e−β(T −t) e1 (t) β i e (t) e (t) (e (t) − e (t)) 2 h 1 2 2 1 −β(T −t) 1 − e (1 + β (T − t)) σ 2e 2 2 β ³ (e (t) + e (t)) 1 2 h i´ 2 −β(T −t) 1 + (1 + β (T − t))2 F (t, e1 (t) , e2 (t)) σ 4e 3 2−e β −4

3

2

F (t, e1 (t) , e2 (t)) = e1 (t) e2 (t) [e1 (t) +he2 (t)] [−e1 (t) + 11e1 (t) ie2 (t) 2 3 2 −11e1 (t) e2 (t) + e2 (t) ](2 − e−β(T −t) 2 + 2β (T − t) + β 2 (T − t) )

and using the fact the the aggregate wealth on the world is held in equities S1 (t) + S2 (t) = X1 (t) + X2 (t) Market clearing in the consumption good market implies that C1 (t) + C2 (t) = y (t) Noting that optimal consumption can be written as β Ci (t) = X (t) , i ∈ {1, 2} −β(T −t) i 1−e it follows that 1 − e−β(T −t) y (t) S1 (t) + S2 (t) = β Hence S2 (t, e1 (t) , e2 (t)) = S1 (t, e2 (t) , e1 (t))

as would be expected from the symmetry of the problem. It is straightforward to derive an upper bound on the absolute error for a general N -th order approximation to the stock price. The upper bound on the error for the analytic approximation to the integral I (t, k, σ e , ) is given by ¯ ¯ ¯ ¯ 2N+2 ¯ ¯ 1 ∂ 1 −β(u−t) ¯ h i ¯¯ e T sup p ¯ (2N + 2)! ∂z 2N +2 z∈R,u∈[t,T ] ¯ 1 + k exp 2σ e (u − t)z ¯

Therefore the upper bound on the absolute error in the N -th order analytic approximation to the stock price is given by ¯ ¯ ¯ ¯ 2N +2 ¯ ¯ 1 ∂ 1 −β(u−t) ¯ ¯ h i ε2N (S) = y (t) T sup e p ¯ ¯ (2N + 2)! ∂z 2N+2 z∈R,u∈[t,T ] ¯ 1 + k exp 2σe (u − t)z ¯

For the parameter values σ e = 1% (per annum), = 1, T = 5 years, t = 0, k = 1.5, β = 0.006, y (t) = 1, I obtain the errors for the stock price shown in Table IV. I define the variable g (t) = ee21 (t) (t) . Hence g (u) = and

e2 (t) exp [−2σ e (W2 (u) − W2 (t))] , t ≤ u ≤ T e1 (t) dg (t) = µg (t) dt + σg (t) dW (t) g (t)


60

Table IV

Stock prices: Absolute errors

N 0 2 4 6 8 10 12 14 16 18 20 22

Maximum Absolute Error, ε2N (S) 0.0004 2 × 10−8 2 × 10−12 2 × 10−16 2 × 10−20 1 × 10−24 4 × 10−28 1.5 × 10−32 3 × 10−36 6 × 10−40 1.5 × 10−44 3 × 10−48

The errors in this table were computed using the parameter values σ e = 1% (per annum), = 1, T = 5 years, t = 0, k = e2 (t) /e1 (t) = 1.5, β = 0.006, y (t) = 1. for some µg (t) and σ g (t). The following (backward) stochastic differential equation dS1 (t) + e1 (t) dt = S1 (t) [µ1 (t) dt + σ 1 (t)| dW (t)] , S1 (T ) = 0, gives the price of stock one, hence using the Markovian assumption, with state variables variables y (t) and g (t) = e2 (t) /e1 (t) and applying Ito’s Lemma to S1 (·) , it can be shown that the coefficient σ 1 is µ ¶ given by ∂S1 ∂S1 + gσ g σ 1 (t, y (t) , g (t, )) = S1−1 yσ y ∂y ∂g where the parameter dependence of the right hand sides of the above expression has been suppressed. By substituting the perturbation expansion for S1 (·) into the above expression it can be shown that ¡ ¢ σ 1 (t, e1 (t) , e2 (t) , ) = σ10 + σ 11 + 2 σ 12 + 3 σ 13 (t, e1 (t) , e2 (t)) + 4 σ 14 + O 5 where

|

σ 10 σ 11 σ 12

= [σ e , 0] | = h[0, σe ] i| σe = − , σe 2 1 − e−β(T −t) (1 + β (T − t)) (3e2 (t) − e1 (t)) e1 (t) e2 (t) 3 ¡ ¢ σ13 (t, e1 (t) , e2 (t)) = −4 σ e [0, 1]| 3 −β(T −t) β 1 − e (e (t) + e (t)) h σ i| 1 2 e σ 14 = − , 0 8 and from applying Ito’s Lemma to the condition S1 (t) + S2 (t) =

1 − e−β(T −t) y (t) β

and equating diffusion coefficients σ 2 (t, e1 (t) , e2 (t) , ) = σ 1 (t, e2 (t) , e1 (t) , − ) Markets are complete, so the usual relationship between risk premia and volatilities holds µ (t) − r (t) 1 = σ (t)| θ (t)


61

Hence µ1 (t, e1 (t) , e2 (t) , ) = µ10 +

2

µ12 (t, e1 (t) , e2 (t)) +

4

µ14 (t, e1 (t) , e2 (t)) + O

where

¡ 6¢

µ10 = β + µe e2 (t) − e1 (t) µ12 (t, e1 (t) , e2 (t)) = 2 e2 (t) σ2e e1 (t) + e2 (t) e1 (t) e2 (t) (e2 (t) − e1 (t)) (3e2 (t) − e1 (t)) 1 − e−β(T −t) (1 + β (T − t)) σ 4e 4 β 1 − e−β(T −t) (e1 (t) + e2 (t)) and applying Ito’s Lemma to the condition 1 − e−β(T −t) S1 (t) + S2 (t) = y (t) β and equating drift coefficients gives µ14 (t, e1 (t) , e2 (t)) = 4

µ2 (t, e1 (t) , e2 (t) , ) = µ1 (t, e2 (t) , e1 (t) , − ) I can also derive approximate analytic expressions for risk premia. Noting that the conditional variance of returns on stock i is given by Vi (t, e1 (t) , e2 (t) , ) = kσ i k2 , it follows that ¢ ¡ (3e2 (t) − e1 (t)) e1 (t) e2 (t) 1 − e−β(T −t) [1 + β (T − t)] 4 4 2 £ ¤ V1 (t, e1 (t) , e2 (t) , ) = σ e − 8 σe β 1 − e−β(T −t) (e1 (t) + e2 (t))3 ¡ 6¢ +O and

V2 (t, e1 (t) , e2 (t) , ) = V1 (t, e2 (t) , e1 (t) , − )

Optimal portfolio proportion processes are obtained by substituting expressions for returns, volatilities and the riskless rate into ¸ · ¡ ¢−1 σ 1 (t)| σ 2 (t) π 1 (t) = 1 − ρ212 (t) (t) − r (t)) µ1 (t) − r (t) − (µ | σ 2 (t) σ 2 (t) 2 ¸ · and ¡ ¢−1 σ 1 (t)| σ 2 (t) 2 π 2 (t) = 1 − ρ12 (t) µ2 (t) − r (t) − (µ (t) − r (t)) | σ 1 (t) σ 1 (t) 1 | where σ1 (t) σ 2 (t) p ρ12 (t) = p σ 1 (t)| σ 1 (t) σ 2 (t)| σ2 (t)

Proof of Propositions 13, 14, 15 and 16. This proposition follows directly from Propositions

2 and B5.

Proof of Proposition 17.

After defining the sets in which admissable consumption-portfolio processes are constrained to lie, the proof proceeds in the same way as the proof of Proposition 1.

Proposition B6 The equilibrium interest rate is given by r (t) = β + µy (t) + µx (t) − (σ y (t) − σ x (t))| (σ y (t) − σ x (t)) + σ x (x) σ y (x)| The equilibrium price of stock one is determined by the following system of equations Z T y (t) e2 (s) S1 (t) = − y (t) (1 − x (t)) Et ds c (t) t y (s) (1 − x (s)) dx (t) | = µx (t) dt + σ x (t) dW (t) , x (0) = x0 . x (t) or using an equivalent formulation in terms of the stochastic weight Z T y (t) λ (t) y (t) (1 + λ (s)) e2 (s) S1 (t) = − Et ds c (t) 1 + λ (t) λ (t) y (s) t

(B73)

(B74) (B75)

(B76)


62 |

where

dλ (t) = (1 + λ (t)) [(σ |x (t) σ x (t) − µx (t)) dt − σ x (t) dW (t)] , λ (0) = λ0

(B77)

µx (t) = σx (t)| σ x (t) + θ2 (t)| σ x (t)

(B78)

and 1 − c (t) p1 (t) x (t) − I1 x (t) 1 − c (t) p1 (t) x (t) − I1 − σ y (t) 1 − c (t) p1 (t) x (t) 1 − c (t) p1 (t) x (t) [1 − c (t) p1 (t) x (t) − I1 x (t)] [σy (t) − c (t) x (t) p1 (t) σ 1 (t)] θ2 (t) = (1 − x (t)) (1 − c (t) x (t) p1 (t))

σ x (t) = c (t) p1 (t) σ 1 (t)

p1 (t) = c (t) = The price of stock two is given by S2 (t) =

S1 (t) y (t) x (t) β 1 − e−β(T −t)

y (t) − S1 (t) c (t)

= y (t) (1 − x (t)) Et

Proof of Proposition B6.

Z

(B79) (B80)

(B81) (B82)

(B83) T t

e2 (s) ds y (s) (1 − x (s))

Details are omitted, because the proof follows the same approach as

the proof of Proposition 2.

Corollary B2 Assuming a Markovian structure with state variables x (t) and G (t) = e1 (t) /y (t) the price of stock one is given by y (t) (B84) − y (t) (1 − x (t)) eβt φ (t) S1 (t) = c (t) where φ (t) solves the quasilinear parabolic partial differential equation µ ¶ ∂ 1 − G (t) 0= + L φ (G, x, t) + e−βt , φ (G (T ) , x (T ) , T ) = 0 ∂t 1 − x (t) L is the differential generator for (G, x) and the volatility of stock one is given by σ1 (t) =

σy c(t)

³ − (1 − x (t)) eβt φ (G, x, t) σ y − 1 c(t)

Proof of Propositions 18 and 19.

x(t) 1−x(t) σ x

(t) + σ G φφG + σ x (t) φφx

− (1 − x (t)) eβt φ (G, x, t)

´

(B85)

The proof proceeds in the same way as the proof of

Proposition 2.

Proof of Propositions 20 and 21. Proposition 11.

The proof proceeds in the same way as the proof of


C

63

Technical Appendix

In this section I show that the quasilinear parabolic partial differential equations characterizing stock prices when markets are incomplete are indeed solvable. I also show that the method I use to obtain approximate closed-form solutions for the equations is valid. Furthermore, I demonstrate how accurate these solutions are. C.1

Existence

It should be mentioned that in order to write stock prices as the conditional expectations of integrals I have assumed that the local martingales describing discounted gains processes are martingales. To show that the partial differential equations I use to derive stock prices have solutions, I use a two-step approach. Firstly I show that for fixed output processes and wealth distribution, the relevant partial differential equations can be solved (Proposition C7). In the proof of Proposition C8, I then use a fixed point argument based on the Leray-Schauder Principle (Theorem C1) to extend the existence results to the case where the output processes and wealth distribution are stochastic and furthermore, the wealth distribution is dependent on the stock price and stock return volatilities. Firstly I define key technical concepts such as the Banach spaces of Hölder continuous functions. Let f : H → RN ⊂ R be a function defined on a bounded region H. Let |·| be the Euclidean norm. For 0 < α ≤ 1, define the Hölder constants f (p) − f (q) Hα (f ) = sup (C86) α |p − q| p, q∈H, p6=q

Then f is called Hölder continuous on H iff Hα (f ) < ∞. Set X X ¯ ¯ ¢ ¡ kf km,α = sup ¯Dj f (p)¯ + Hα Dj f . |j|≤m

p∈H

(C87)

|j|=m

The first sum is over all partial derivatives of orders 0, 1, . . . , m and the second sum is over all partial derivatives of order m. We set D0 f = f . Now set o ¡ ¢ n ¡ ¢ (C88) C m,α H = f ∈ C m H : kf km,α < ∞ .

¡ ¢ C m,α H is the Banach space of m-times Hölder continuous differentiable functions. Set P = (t, G, x) ∈ D = H × (0, T ) and define the norm ´ ³ 2 2 1/2 d (P, P 0 ) = |t − t0 | + |G − G0 | + |x − x0 |

(C89)

Define the Hölder constants by

Hα (f )∗ = Set kf k∗α = maxP ∈D |f (P )| + Hα (f )∗ and kuk∗2,α = kuk∗α +

sup P,P 0 ∈D, P 6=P 0

X i,j

f (P ) − f (P 0 ) α d (P, P 0 )

kDi uk∗α + kDi Dj uk∗α + kut k∗α .

(C90)

(C91)

The summation is over¡ all¢ the first and second derivatives with respect to the space variables x and y. ¡ ¢ Let C∗α D and C∗2,α D be equal to the set of all real functions with kf k∗α < ∞ and kuk∗2,α < ∞, respectively. Let X ∗ ∗ ∗ kuk1,α = kukα + kDi ukα . (C92) i

Here the summation is over all the first derivatives with respect to the space variables.

To show existence of a solution to the quasilinear parabolic partial differential equations, such as (24) I first start with the following existence result for a linear parabolic partial differential equation.


64

Proposition C7 The function φ (t, G, x) satisfies the linear parabolic partial differential equation. ∂φ + Lφ + f = 0 on H × (0, T ) ∂t φ (T, G, x) = 0 on H φ (t, G, x) = b (t, G, x) on ∂H × [0, T ]

(C93) (C94) (C95)

where L = a1 φx + a2 φG − (a11 φxx + 2a12 φxG + a22 φGG ).

Set D = H × (0, T ) and suppose further that:

1. H is a bounded convex region in R2 with ∂H ∈ C 2,α , 0 < α < 1 and ∂H consists of a single line, which has positive curvature; ¡ ¢ ¡ ¢ 2. f ∈ C∗α D and b ∈ C∗2,α D ;

3. b is compatible with the differential equation, i.e., b satisfies the differential equation for all (G, x) ∈ ∂H and t = 0. ¡ ¢ 4. aij , ai ∈ C α H ,i, j = 1, 2;

5. there is a fixed number d > 0 such that, for all (G, x) ∈ H and all (ς, η) ∈ R2 ¡ ¢ a11 (G, x) ς 2 + 2a12 (G, x) ςη + a22 (G, x) η 2 ≥ d ς 2 + η2 .

(C96)

¡ ¢ Then there is exactly one solution φ ∈ C∗2,α D .

Proof of Proposition C7. This is a simple adaptation of the proof in Friedman (1964). I now state the Leray-Schauder Principle (see page 245 of Zeidler (1986)). This principle is used to show that quasilinear parabolic partial differential equations, such as (24) have a solution. Zeidler (1986) uses the same method of proof to show that a quasilinear elliptic partial differential equation has a solution. After stating the Leray-Schauder Principle, I briefly describe the main ideas behind the existence proof. I then state the existence Proposition and give a proof.

Theorem C1 (Leray-Schauder Principle, page 245, Zeidler (1986)) Suppose that 1. the operator T : X → X is compact, with X a Banach space; 2. (a priori estimate) there exists an r > 0 such that if x = εT (x) with 0 < ε < 1 then kxk ≤ r.

(C97)

Then the equation x = T (x) is solvable. I now consider the quasilinear parabolic partial differential equation φt + A1 φx + A2 φG + A11 φxx + 2A12 φxG + A22 φGG + f (G, x) = 0

(C98)

where φ (T, G, x) = 0 on H and (G, x) ∈ H and φ (t, G, x) = b (t, G, x) on ∂H × [0, T ]. The coefficient functions are of the form Ai = Ai (G, x, φ, φx , φG ), Aij = Aij (G, x, φ, φx , φG ). The existence proof uses the idea of converting a quasilinear partial differential equation into a fixed point problem. The arguments of the functions Aij, Ai are fixed, converting the quasilinear partial differential equation into a linear partial differential equation. For example the equation F (φ, φ) = 0

(C99)


65

is replaced by the equation F (z, φ) = 0

(C100)

where z is fixed. If for fixed z this equation is linear, then φ = Tz

(C101)

for some operator T . By solving the fixed point problem (using Theorem C1) φ = Tφ

(C102)

a solution to the original problem is obtained.

Proposition C8 Suppose that 1. H is a bounded convex region in R2 with ∂H ∈ C 2,α , 0 < α < 1 and ∂H consists of a single curve, which has positive curvature;

2. D = H × (0, T ); 3. The coefficient functions Ai and Aij belong to C α (M ), where M = D × R4 , and for the given boundary values, g ∈ C 2,α (∂H) ;

4. The equation (C98) is uniformly elliptic, i.e., there is a constant c > 0 such that ¡ ¢ A11 (P ) ς 2 + 2A12 (P ) ςη + A22 (P ) η 2 ≥ c ς 2 + η2

(C103)

for all P ∈ M and (ς, η) ∈ R2 .

Then the quasilinear parabolic partial differential equation with the specified boundary conditions ¡ ¢ has a solution φ ∈ C∗2,α D .

Proof of Proposition C8. This is an adaptation of the proof of Proposition 6.12 on page 247 of Zeidler (1986) to the case of parabolic quasilinear pde’s. Consider the modified problem

φt + A1 φx + A2 φG + A11 φxx + 2A12 φxG + A22 φGG + f (G, x) = 0 (C104) φ (T, G, x) = 0 on H φ (t, G, x) = εb (t, G, x) on ∂H × [0, T ] where Ai Aij

= Ai (G, x, z, zG , zx ) = Aij (G, x, z, zG , zx )

¡ ¢ with fixed z ∈ C∗1,β D , 0 < β < 1, and fixed ε ∈ [0, 1]. ¡ ¢ Then Aij ∈ C∗γ D and b ∈ C∗2,γ (∂H), where γ = αβ. Hence by Proposition C7 the modified ¡ ¢ problem has a unique solution φ ∈ C∗2,α D . Now define an operator ¡ ¢ ¡ ¢ Tε : C∗1,β D → C∗2,γ D by φ = Tε z. ¡ ¢ φ is the solution to a linear problem, so Tt = εT , where T = T1 . Set X = C∗1,β D . ¡ ¢

1. T maps bounded sets X into bounded sets in C 1,β H . For if z is such that kzk∗1,β ≤ k, then kAij (z)k∗γ ≤ k1 , with a constant independent of z. From Proposition C7 kT zk∗2,γ ≤ k2 .


66

2. T maps bounded sets in X into relatively compact sets in X. This follows from the above and ¡ ¢ ¡ ¢ the compact embedding C∗2,γ D ⊆ C∗1,β D .

3. T is continuous. Suppose that zn → z in X as n → ∞. Then by the statement above, there is a subsequence (zn ) such that the sequence of elements φn = T zn converges in X to an element φ. Consider (C104) with zn replacing z and φn replacing φ, then equation (C104) holds as n → ∞, i.e. φ = T z.

Therefore T : X → X is a compact linear operator and (C98) is reduced to the solution of the operator equation φ = εT φ, φ ∈ X with ε = 1.

¡ ¢ I now deal with the issue of an a priori estimate for a solution φ ∈ C∗2,α D of (C98). Here it is possible to use a very different argument from Zeidler (1986). From the integral representation of φ ¡ ¢ ∗ it follows that if φ ∈ C∗2,α D is a solution of (C104), with z = φ, ε ∈ [0, 1], then kφk1,β ≤ c with a constant c independent of ε. ¡ ¢ Now I apply the Theorem C1, the Leray-Schauder Principle. With X = C∗1,β D and ε = 1. It is known that T φ ∈ C∗2,γ , so that φ ∈ C∗2,γ . The coefficients Aij , Ai containing φ hence belong to C∗α . Using Proposition C7, φ ∈ C∗2,α .

C.2

Accuracy

Having established existence of a solution to the quasilinear parabolic partial differential equations, which I use to derive stock prices, it is possible to discuss how I obtain approximate closed-form solutions. In addition, I show how accurate these solutions are. Consider the quasilinear14 parabolic partial differential equation ∂φ (G, x, t) 0= + Lφ (G, x, t) + Ψ (G, x, t) , φ (G (T ) , x (T ) , T ) = 0 (C105) ∂t where ∂ ∂ + G (t) µG (t) L = x (t) µx (t) ∂x ∂G 1 ∂2 ∂2 ∂2 1 + x (t)2 σ x (t)| σ x (t) 2 + x (t) G (t) σx (t)| σ G (t) + + G (t)2 σ G (t)| σ G (t) 2 ∂x ∂x∂G 2 ∂G2 The variables state variables G (t), x (t), the coefficients µG (t), µx (t), σG (t), σ x (t) and φ (G, x, t) p depend on the dimensionless parameter = (1 − ρ) /2. When = 0, (C105) can be solved exactly in closed-form. All the partial differential equations which I use to solve for stock prices are of this form. I solve (C105) by expanding both φ (G, x, t) and the operator L in power series in , centered on = 0 to obtain Ã∞ ! !Ã ∞ ∞ ∞ X X X ∂ X i i i i 0= φi (G, x, t) + Li φi (G, x, t) + Ψi (G, x, t) = 0 ∂t i=0 i=0 i=0 i=0

By comparing coefficients (see Proof of Proposition 3), I obtain a series of linear partial differential equations Li φi (G, x, t) = 0, φi (G (T ) , x (T ) , T ) = 0, i = 1, 2, 3, . . .

14 The coefficients of the highest derivatives depend on φ and its first partial derivatives, making the partial differential equation quasilinear.


67

The linearity of these partial differential equations makes them much easier to deal with then the quasilinear partial differential equation (C105). By solving the first N of these equations (for the equations I consider this is possible in closed-form for N = 4), yields the polynomial N X

i

φi (G, x, t) ,

i=0

which is an approximation to the solution of (C105). Two main questions arise concerning this approach:

1. How accurate is the polynomial

N X

i

φi

i=0

for given N, as a solution to the original pde (C105)?15

2. Is this approach of expanding the operator L and the function φ on which it operates in power series in and then comparing coefficients to obtain a series of linear pde’s a valid way of constructing a solution to (C105)? The first question can be answered easily in a brute-force way: just substitute the polynomial approximation for φ into the original pde (24) for N = 1, 2, 3, 4 etc.and check how close the right-hand side is to zero for ∈ [0, 1] and whether the error in the approximation is decreasing in N .

The second question is somewhat more subtle, but it can be shown (see Zeidler (1986), page 363) that it is a valid way of constructing a solution for in some open neighborhood of zero. To determine this neighborhood analytically is not possible in general, but to check in which neighborhood the approximate solution is good is straightforward as has been explained above. In fact, this is exactly how one would check the accuracy of a solution obtained by purely numerical methods. The following tables show how accurate the closed-form approximations I have derived in this paper are.

Table V

Non-integrated stock markets: Absolute errors ε 0.1 0.2 0.3 0.4 0.5 1

Log10 of Absolute Error −11 −10 −9 −8 −7 −4

The errors in this table were computed using the a fourth order polynomial approximation to φ, and parameter values σ e = 1% (per annum), T = 5 years, t = 0, x (t) = G (t) = 1/2, β = 0.02, y (t) = 1.

15

Polynomial approximants can be used to construct Pade approximants, which have improved convergence properties (see Judd (1998)). I do not do this, because the polynomial approximants I construct are already good approximations.


68

Table VI

One-way stock market integration: Absolute errors

ε 0.1 0.2 0.3 0.4 0.5 1


The errors in this table were computed using the a fourth order polynomial approximation to φ, and parameter values σ e = 1% (per annum), T = 5 years, t = 0, x (t) = G (t) = 1/2, β = 0.02, y (t) = 1.

Table VII

One-way partially integrated stock markets: Absolute errors

ε 0.1 0.2 0.3 0.4 0.5 1


The errors in this table were computed using the a fourth order polynomial approximation to φ, and parameter values I1 /ε2 = 0.1, σ e = 1% (per annum), T = 5 years, t = 0, x (t) = G (t) = 1/2, β = 0.02, y (t) = 1.


D

69

Figures

40 20 0 -20

1000 800 600 0.4

400 0.6 x

N

200 0.8 1

Figure 1. This figure shows the % change in risk premium in country one, relative to the base case, ENI , when stock market two is opened to external investment. The risk premium is increased when the cross-sectional wealth distribution parameter, x, is less than a certain critical value, xc (N ). This percentage increase rises with N . For x > xc (N ), the risk premium is decreased, but the size of this decrease declines with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.


70

-40 1000

-60

800

-80

600 0.4

400 0.6 x

N

200 0.8 1

Figure 2. This figure shows the % change in risk premium in country two, relative to the base case, ENI , when stock market two is opened to external investment. The risk premium is unambiguously decreased. The magnitude of this decrease rises with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.

60 40 20 0

1000 800 600 0.4

400 0.6 x

N

200 0.8 1

Figure 3. This figure shows the percentage change in the size of the market price of risk in country one, relative to the base case, ENI , when stock market two is opened to external investment. The market price of risk in country one is increased. The percentage increase declines with x and rises with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.


71

0

1000

-20

800

-40

600 N 400

0.2 0.4 0.6 x

200 0.8 1

Figure 4. This figure shows the percentage change in the size of the market price of risk in country two, relative to the base case, ENI , when stock market two is opened to external investment. The market price of risk in country two is decreased. The percentage decrease declines with x and rises with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.

0.4 1000 0.2

800

0

600 400

0.4 x

N

200

0.6 0.8

Figure 5. This figure shows the % change in the variance of returns for stock one, relative to the base case, ENI , when stock market two is opened to external investment. The % change is small for all x and N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.


72

0 -20 -40 -60 -80

1000 800 600

0.4

400

N

0.6 200 x

0.8 1

Figure 6. This figure shows the % change in the variance of returns for stock two, relative to the base case, ENI , when stock market two is opened to external investment. The variance of returns for stock two is unambiguously decreased. The percentage decrease declines with x and rises with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.

0.4

1000

0.2

800

0 0.2

600 400

0.4 0.6 x

N

200 0.8 1

Figure 7. This figure shows the change in the cross-country correlation in stock returns, ρ12 , relative to the base case, EN I , when stock market two is opened to external investment. The correlation, ρ12 , is increased. The percentage increase declines with x and rises with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.


73

2.5 2 1.5 1 0.5 N 20

40

60

80

100

Figure 8. This figure shows the % change in the riskless rate, relative to the base case, ENI , when both stock markets are opened up to external investment. The riskless rate is unambiguously decreased. The magnitude of this increase declines with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.

N 2

4

6

8

10

-10 -15 -20 -25 -30

Figure 9. This figure shows the % change in the risk premium for stock one, relative to the base case, ENI , when both stock markets are opened up to external investment. The risk premium for stock one is unambiguously decreased. The magnitude of this decrease declines with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.


74

-30 -35 -40 -45 -50 -55 N 20

40

60

80

100

Figure 10. This figure shows the % change in the risk premium on stock two, relative to the base case, ENI , when both stock markets are opened up to external investment. The risk premium on stock two is unambiguously decreased. The magnitude of this decrease rises with N . The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.

N 20

40

60

80

100

-2.5 -5 -7.5 -10 -12.5 -15

Figure 11. This figure shows the % change in the magnitude of the market price of risk in countries one and two, relative to the base case, ENI , when both stock markets are opened up to external investment. The size of the market price of risk in each country is unambiguously decreased by the same amount. The magnitude of this decrease declines with N. The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.


75

0.03 0.025 0.02 0.015 0.01 0.005 N 20

40

60

80

100

Figure 12. This figure shows the change in the cross-country correlation of stock returns, relative to the base case, ENI , when both stock markets are opened up to external investment. The increase in correlation diminishes as N , the output size of country one relative to country two increases. The following set of parameter values is used: β = .01, σ e = .03, ρ = 0.4, τ → ∞.