Investment Risk Management and Correlations

Transcription

1 DISCUSSION PAPER SERIES IN ECONOMICS AND MANAGEMENT International Portfolio Choice when Correlations are Stochastic Nicole Branger, Matthias Muck, Stefan Weisheit Discussion Paper No GERMAN ECONOMIC ASSOCIATION OF BUSINESS ADMINISTRATION GEABA

2 International Portfolio Choice when Correlations are Stochastic Nicole Branger Matthias Muck Stefan Weisheit This version: April 30, 2015 Abstract We analyze optimal portfolios in an international multi-asset model with stochastic volatilities and stochastic correlations between stock returns and the exchange rate. We simultaneously derive optimal portfolios for a domestic and a foreign investor with constant relative risk aversion. First, we show that stochastic correlations between returns induce a significant hedging demand. Second, we find that the optimal allocation and the investor s utility from trading as well as from additionally trading derivatives significantly depends on his home country. Keywords: International portfolio choice, stochastic correlation, Wishart process, inflation, intertemporal hedging demand, derivatives JEL Classification: G11, G13 Finance Center Muenster, University of Muenster, Universitätsstr , Münster, Germany, [email protected] Chair of Banking and Financial Control, University of Bamberg, Kärntenstr. 7, Bamberg, Germany, [email protected], [email protected] A previous draft of this paper was presented at the 18th Conference of the Swiss Society for Financial Market Research 2015). We would like to thank the discussants and participants for valuable comments and suggestions.

3 1 Introduction The optimal portfolio of an investor depends on the characteristics of the assets he has access to. One important aspect of the investment opportunity set is the variance-covariance matrix of asset returns. It is by now widely accepted that return volatilities are stochastic, and that the investor earns a negative risk premium when he takes a long position in variance risk. There is by now also consentaneous evidence that correlations between asset returns are stochastic. Bollerslev et al. 1988), Longin and Solnik 1995), Ball and Torous 2000) and Goetzmann et al. 2005) document that correlations vary dynamically over time. This suggests that correlations must be treated as separate risk factors which earn specific risk premia, too. 1 We solve for the optimal portfolio in an international multi-asset model with stochastic volatilities and stochastic correlations between stock returns and the exchange rate. We find that the optimal portfolio and the overall utility gain of the investor from trading at all and from additionally trading derivatives significantly depends on his home country. Exchange rate risk has a significant impact on investors, and the different exposures of home and foreign investors to exchange rate risk has far-reaching implications for their optimal portfolios and overall well-being. Optimal portfolio planning problems belong to the most extensively studied research areas in finance. The continuous time fundament was developed by the seminal papers of Samuelson 1969) and Merton 1969, 1971, 1973). An important part of the portfolio choice literature has explored the impact of stochastic volatility. 2 The implications of stochastic correlations in multi-asset frameworks have recently gained attraction. Buraschi et al. 2010) model the joint dynamics of asset prices whose covariance matrix of returns is driven by a Wishart process. 3 Following Adler and Dumas 1983), there is also a huge strand of the literature that studies international asset allocation and the benefits from international investing. 4 Moreover, international portfolio choice problems are closely related to portfolio choice problems under 1 Driessen et al. 2009) find that correlation risk is priced based on index and individual option prices. The results provided by Krishnan et al. 2009) and Driessen et al. 2012) imply a sufficiently large, negative risk premium. 2 Among others, Liu et al. 2003), Liu and Pan 2003), Chacko and Viceira 2005), Liu 2007), Branger et al. 2008), and Muck 2010). 3 The Wishart process was first studied by Bru 1991) and applied by Gourieroux 2006) and Da Fonseca et al. 2007) for pricing derivative claims. 4 International portfolio choice models are analyzed by, e.g., Lioui and Poncet 2003), Smedts 2004) and Larsen 2010). Kaplanis and Schaefer 1991), Bekaert and Urias 1996) and Li et al. 2003) find empirical evidence on gains from international equity diversification. Driessen and Laeven 2007) find that international diversification depends on the investor s perspective and thus differs across countries. 1

4 inflation risk. 5 Dynamic asset allocation with inflation risk is analyzed by, e.g. Brennan and Xia 2002) and Munk et al. 2004). However, so far little attention has been paid to the role of correlation risk in international portfolio selection problems. An exception are Ang and Bekaert 2002) who study the impact of correlations in a dynamic asset allocation model with regime switches. The contributions of our paper are twofold. First, we propose a flexible dynamic portfolio model with stochastic second moments and inflation risk. We extend the model of Adler and Dumas 1983) to stochastic return volatilities and correlations and provide tractable solutions for the portfolio planning problem. Second, we document the impact of stochastic correlations in an international portfolio choice application with two investors. With the inclusion of stochastic state variables the optimal portfolio holdings decompose to myopic demands, driven by the risk premiums, and hedging demands. The inflation, which is also stochastically correlated to asset returns, affects both the optimal myopic and the hedging demand. Our framework allows to analyze the hedging demand for variance and covariance risk separately. We apply our framework to an international portfolio choice setting, i.e., we assume that the investment opportunity set comprises both the domestic and the foreign stock market) as well as the domestic and foreign money market account. 6 Our model allows us to analyze the optimal investment strategy of the domestic and the foreign investor simultaneously. Although both investors have the same preferences, their optimal portfolios and their utility gains from trading differ remarkably. First, the optimal portfolio holdings of the foreign money market account, which serves as a hedging vehicle for exchange rate risk, are of opposite signs. While the domestic investor holds a long position, the foreign investor optimally chooses a short position. Second, we find a large difference in the hedging components of the demand. The share of the hedging demand in the optimal positions in the stocks are of comparable order for the domestic and foreign investor. For the foreign bond, however, there are huge differences. The relative difference between the domestic and the foreign covariance hedging demands can be up to 40 percent. The volatility hedging demand becomes the dominant element for one investor, but plays a minor role for the other. These differences in the hedging demand also show up in its dependence on structural parameters that control the risk of the correlation process. Both the persistence of shocks and the uncertainty about the covariance matrix induce the investors to adjust their hedging portfolios in opposite directions. 5 Inflation problems usually give real prices that are obtained from dividing nominal prices by a price index. This is identical to dividing prices given in a measurement currency by the exchange rate. 6 A model with inflation risk can be reinterpreted as a model with exchange rate risk, so that our general framework can indeed be used in an international context, too. 2

5 Third, we compare the implications of the average return correlations and average exchange rate volatility on the certainty equivalent wealth CEW) of the domestic and foreign investor. We find that the average levels of the state variables have major impact on the certainty equivalents. However, investors are affected differently. For low correlations the domestic CEW dominates the foreign in our base case. The situation can become the entire opposite if the magnitude of the correlation is changed. The exchange rate volatility causes an analogous effect. Fourth, we determine the investors utility gains from trading additional non-redundant securities which dynamically complete the market. For both investors, a change in the correlations can cause utility improvements of more than 15 percent. While in most cases the utility increases of the foreign investor are larger, we show that for high absolute correlations the domestic improvements can potentially become dominant. The remainder of the paper is organized as follows: section 2 introduces the model setup. In section 3 we solve the investment problem and discuss the resulting implications. Our model is applied to an international portfolio choice problem in section 4. Finally, section 5 concludes. 2 The model We assume a very general model for portfolio choice in continuous time. We start with the dynamics of the stock market, discuss the risk premia and discuss the implications of the inflation rate. 2.1 Stock market The investor can trade n stocks and the money market account. The risk-free rate is denoted by R t. Under the empirical measure P the dynamics of the stock prices S R n are given by ds t = I S,t µ t dt + ) X t dzt X 1) with I S,t = diag S t ). Z X R n is a vector Brownian motion. The vector µ t denotes the expected stock returns, and X R n n is the variance-covariance matrix of returns. The variance-covariance matrix X is assumed to be stochastic. It follows a Wishartprocess: dx t = Ω X Ω X + M X X t + X t M X ) dt + X t db X t Q X + Q X db X t Xt, 2) 3

6 where B X R n n is a matrix-valued Brownian motion. Ω X, M X, Q X R n n are square matrices, and we assume that Ω X is invertible. The Wishart process describes the dynamics of positive semi-definite matrices. It was first analyzed by Bru 1991), and represents the matrix-analogue to the Cox et al. 1985) process. The matrix M X can be interpreted as the negative of the mean-reversion speed, while Q X of volatility. Throughout the paper we set Ω X Ω X denotes the multivariate volatility = k X Q X Q X for k X > n 1, which guarantees that X has full rank. Thus, X is indeed a viable variance-covariance matrix, and variances as well as covariances are stochastic. More precisely, the instantaneous volatilities of stock returns are given by X t,ii dt. The correlations between stock returns are determined by [ dst,i ρ t,ij Corr, ds ] t,j = S t,i S t,j X t,ij Xt,ii X t,jj, 3) where X ij denotes the element {i, j} of the matrix X, for i, j = {1,..., n}. Due to the dependence on the elements of X, the return correlation is stochastic. Our model can also capture the empirically well-documented negative correlation between returns and their volatilities. The Brownian motions Z X and B X which drive stock returns and their variance-covariance matrix, respectively, are related via dzt X = dbt X ρ X + dwt X 1 ρ X ρ X 4) with ρ X R n such that ρ X ρ X 1. W X is a vector Brownian motion independent of B X. The correlation between X ii and the stock return ds i /S i is then given by [ ] dst,i Corr, dx t,ii S t,i = Q X ρ X) i Q X Q X ) ii. 5) In contrast to the stochastic correlation between stock returns, the correlation between stock returns and their local variances is constant Risk premia We now turn to the market prices of risk for the Wiener processes and thus to the premia paid for stock price risk, co-)variance and correlation risk. For the Wiener process B X, which drives the variance-covariance matrix, it holds that db X,Q t = db X t + X t η X dt 6) 7 This is also true for the WASC model proposed by Da Fonseca et al. 2007). In contrast, Branger and Muck 2012) study a model where all correlations are stochastic. 4

7 where the superscript Q denotes a quantity given under the risk-neutral pricing) measure and where η X R n. The dynamics of X under the risk-neutral measure Q are dx t = Ω X Ω X + M X X t + X t M X ) dt + X t db X,Q t Q X + Q X db X,Q Xt 7) t where M X = M X Q X η X. From the dynamics of X under P see Equation 2)) and Q, we deduce the compensation for variance-covariance risk. More precisely, the risk premia for taking on an exposure to X are given by CovRP t 1 dt Et [dx t ] E Q t [dx t ] ) = Q X η X X t + X t η X Q X. 8) There is convincing evidence that the variance risk premium is negative. 8 However, less is known about the premium for covariance risk. Instead, we focus on the compensation for correlation risk. The correlation dynamics follow follow by applying Itô s Lemma to the correlation given by Equation 3). The correlation risk premium is given by 1 Et [d ρ t,ij ] E Q t [ ρ t,ij ] ) = ρ ij CovRP t,ii + ρ ij CovRP t,jj + ρ ij CovRP t,ij. 9) dt X ii X jj X ij The correlation risk premium is a linear combination of the premia for variance and covariance risk. Empirical evidence suggests that a long exposure to correlation risk earns negative premium. 9 For the Brownian motion W X the dynamics under the risk-neutral measure are dw X,Q t = dw X t + X t ξ X dt. 10) The resulting risk-neutral dynamics of Z X, which is defined in Equation 4) and drives stock prices, are then given by where dz X,Q t = dz X t + X t Λ X dt 11) Λ X = η X ρ X + ξ X 1 ρ X ρ X. 12) 8 See, among others, Bakshi and Kapadia 2003a), Bakshi and Kapadia 2003b), Carr and Wu 2009) and Todorov 2010). These studies find that, on average, option-implied variances are higher than realized variances. A long variance position therefore earns a negative average risk premium. 9 See Krishnan et al. 2009), Driessen et al. 2009) and Driessen et al. 2012). A long position provides a hedge against upwards movements in the correlation of returns. As investors usually fear increases in the correlation, they accept a negative expected excess return on a long correlation position. 5

8 The compensation for stock diffusion risk equity risk premium) is thus 1 Et [I 1 S,t dt ds t] E Q t [I 1 S,t ds t] ) = µ t R t = X t Λ X. 13) Due to its dependence on the X, the equity risk premium is stochastic. 2.3 Inflation The prices S are nominal prices. The dynamics of the price index I are di t I t = π t dt + δ X X t dz X t + δ Y Y t dz Y t 14) where π t denotes the expected inflation rate. Y R n n is a second state variable, and Z Y another n-dimensional vector Brownian motion which is independent of B X, W X, and Z X. For the parameters it holds that δ X, δ Y realized inflation rate over the next instant of time. R n. The term di/i can be interpreted as the The second state variable Y is assumed to follow a Wishart process, too. Analogous to the dynamics for X, it holds that dy t = Ω Y Ω Y + M Y Y t + Y t M Y ) dt + Y t db Y t Q Y + Q Y db Y t is Y t, 15) where Ω Y, Q Y, M Y R n n, and where we assume that Ω Y Ω Y = k Y Q Y Q Y with k Y > n 1. B Y R n n is a matrix-valued Brownian motion. The Brownian motion Z Y is defined by dz Y t = db Y t ρ Y + dw Y t 1 ρ Y ρ Y 16) with ρ Y R n such that ρ Y ρ Y 1. W Y is a vector Brownian motion independent of B Y. The market prices of risk for B Y, W Y, and Z Y depend on η Y, ξ Y, and Λ Y, respectively. Inflation thus not only depends on the same risk factors stock returns are exposed to, but it is also driven by Z Y. For δ Y 0 inflation risk is not spanned by the traded assets. The state variable Y drives the inflation variance and the correlation between inflation and stock returns. The inflation variance is δ X Xδ X + δ Y Y δ Y ) dt. The correlation is given in the following lemma. Lemma 1. The correlation between the inflation rate and the stock returns is given by [ dsi Corr, di ] = S i I Xδ X ) i Xii δ X Xδ X + δ Y Y δ Y. 17) 6

9 Proof. The covariance follows directly from Equations 1) and 14). The correlation between stock returns and the inflation rate depends on all elements of the matrices X and Y. Note, however, that the correlation becomes zero if δ X = 0. For a correlation between returns and inflation we thus require spanned inflation risk. Unspanned inflation risk whose magnitude is scaled by the vector δ Y > 0 reduces the correlation. that The state variables X and Y also drive the risk-free rate and expected inflation. It holds RX, Y ) = α R 0 + tr [ β R XX ] + tr [ β R Y Y ] 18) πx, Y ) = α π 0 + tr [β π XX] + tr [β π Y Y ], 19) where the constants βx R, βr Y, βπ X, βπ Y Rn n. Furthermore, we assume that α0 R 0 and that βx R and βr Y are positive definite matrices. This guarantees that the interest rate is positive. The dependence of the short rate and expected inflation on Y implies that changes in these variables are not perfectly correlated with stock prices. Interest rate risk and inflation risk are thus not completely spanned by the stock market. The distinctive feature of our model is that return correlations as well as the correlations between returns and the inflation rate are stochastic. We thereby extend the model studied by Adler and Dumas 1983) to additional state variables, i.e., stochastic return volatilities and correlations. Moreover, for zero inflation and n = 2 assets, our setting nests the model recently analyzed Buraschi et al. 2010) and Da Fonseca et al. 2011) in a portfolio choice context. 3 Portfolio planning problem Before we provide the solution and a discussion to the investment problem, we determine the wealth dynamics in real terms. We build upon the model dynamics described in the previous section. 3.1 Wealth dynamics The investor can trade n stocks with nominal prices S and the money market account, which earns the short rate RX, Y ). The trading strategy ω t R n gives the portfolio weights of the risky assets. The portfolio weight of the money market account is given by 1 1 ω t. The 7

10 resulting dynamics of the nominal wealth Π t are dπ t Π t = R t + ω tx t Λ X) dt + ω t Xt dz X t 20) The deflated, real wealth is defined by Π = Π/I. We provide its dynamics in the following lemma. Lemma 2. The dynamics of real wealth Π are given by dπ t Π t = { α0 R α0 π + tr [β RY β πy + δ Y δ Y ) ] Y t [ +tr βx R βx π + Λ X δ X) ω t + δ X δ X ) ] } X t dt 21) + ω t δ X) X t dz X t δ Y Y t dz Y t Proof. See Appendix A.1. The real wealth of the investor is exposed to the 2 n risk factors Z X and Z Y. With n risky stocks and one money market account, which is risk-free in nominal terms and risky in real terms, the investor cannot achieve any specific exposure to these 2 n risk factors. Thus, the market is incomplete. From Lemma 2 we also get that the real return on wealth is µ Π,t = R t π t + ω tx Λ X δ X) + δ X Xδ X + δ Y Y δ Y. 22) The real return depends on the expected nominal portfolio return R t + ω tx t Λ X ) and the inflation. However, inflation can have different impact: i) the expected inflation rate π t ) reduces the real return, ii) the variance of the inflation rate δ X Xδ X + δ Y Y δ Y ) increases the real return. Finally, iii) the covariance of the nominal portfolio return and the inflation dynamics ω tx t δ X ) may increase or decrease the real return on wealth, depending on the magnitudes of the spanned inflation parameter δ X and the portfolio strategy ω t. 3.2 Optimal portfolios We consider an investor with constant relative risk aversion CRRA). The investor s objective is to maximize the expected utility of terminal real wealth at time T > 0, i.e., max E ω t,0 t T [ ] Π T )1 γ, 23) 1 γ 8

11 where his relative risk aversion is 0 < γ To solve the investment problem, we follow Merton 1971): we define the indirect utility function J and then derive the corresponding Hamilton-Jacobi-Bellman equation. The solution to the investment problem and the optimal portfolio strategy are summarized in the next proposition. Proposition 1. The value function for the portfolio planning problem is given by J t, Π, X, Y ) = Π ) 1 γ and the optimal portfolio weights are ω t = 1 γ 1 γ exp { tr A X τ) X t ) + tr A Y τ) Y t ) + C τ) }, 24) Λ X + δ X γ 1) + 2A X τ) Q X ρ X ), 25) where τ = T t. The functions A X, A Y equations A X τ) τ A Y τ) τ C τ) τ and C solve the system of ordinary differential = A X Γ X + Γ X A X + 2A X Q X I + 1 γ ) γ ρx ρ X Q X A X + ζ X 26) = A Y Γ Y + Γ Y A Y + 2A Y Q Y Q Y A Y + ζ Y 27) = 1 γ) ) ) ) α0 R α0 π + tr Ω X Ω X A X + tr Ω Y Ω Y A Y 28) subject to the terminal conditions A X 0) = 0, A Y 0) = 0, C 0) = 0 and Γ X = M X + 1 γ γ ζ X = 1 γ 2γ QX ρ X Λ X δ X) [ 2γ β R X β π X) + Λ X δ X) Λ X δ X) + γ Γ Y = M Y + 1 γ) Q Y ρ Y δ Y [ ζ Y = 1 γ) βy R βy π + δ Y δ Y 1 γ )] 2 Proof. See Appendix A.2. Λ X δ X + δ X Λ X )] The system of matrix Riccati equations 26) and 27) can be solved analytically by the linearization method as, e.g., proposed by Da Fonseca et al. 2007) or Grasselli and Tebaldi 2008). Once we have determined a solution to the matrix-valued functions A X τ) and A Y τ), the function C τ) follows by direct integration of Equation 28). 10 We do not consider the special case γ = 1 representing the log-investor. 9

12 3.3 Structure of optimal demand Proposition 1 provides the optimal portfolio weights ω t = 1 γ Λ X + δ X γ 1) + 2A X τ) Q X ρ X ). 29) The optimal portfolio first comprises a myopic speculative) demand. It is determined by the vector Λ X and reflects the investor s desire to earn the risk premium. The second component is controlled by the vector δ X and refers to the investor s demand for inflation hedging. 11 The optimal speculative demand is thus corrected by an inflation-induced part, which is in line with the result of Adler and Dumas 1983). Note that only spanned inflation risk influences the optimal portfolio decision. 12 The third term denotes the intertemporal hedging demand. It is driven by the desire of the investor to hedge unanticipated movements of the state variables over the investment horizon. The hedging demand reflects the impact of the variances X ii and covariances X ij on the indirect utility function J in Equation 24). To analyze the role of the parameters that determine the size of the hedging demand, consider a market with two risky assets. The hedging component of the portfolio weights can then be rewritten as ) ω h,t = 2 γ A Q 11 ρ 1 + Q 21 ρ γ A 22 ) + 2 γ A Q 12 ρ 1 + Q 22 ρ Q 11 ρ 1 + Q 21 ρ 2 ) 0 30) Q 12 ρ 1 + Q 22 ρ 2 The size of the hedging demand is driven by two factors. First, A X ii and A X ij are the sensitivities of the value function J with respect to shocks in the variances and covariances of stock returns. The hedging demand can thus be decomposed into a variance hedging demand and a covariance hedging demand. The second component, which depends on Q X and ρ X, is proportional to the correlations between the stock returns and their local variances. 4 International portfolio selection The results of section 3 show that real prices are obtained from dividing nominal prices by the price index I. Adler and Dumas 1983) show that this is identical to dividing prices 11 Adler and Dumas 1983) call this component the inflation hedge portfolio. Since it does not depend on the length of the investment horizon, we categorize this component as myopic, too. 12 The unspanned inflation risk, i.e., the components associated to Z Y and the state variable Y, enter the indirect utility function see Proposition 1) through the function A Y and C. 10

13 given in a measurement currency by the exchange rate. In this section we therefore analyze an international portfolio choice problem. We focus on the optimal portfolio holdings, the hedging demands, the certainty equivalent wealth, and utility gains from market completion of investors from different countries. Our analysis extents the model proposed by Adler and Dumas 1983) in that we allow for stochastic second moments of returns. Moreover, the correlation between returns and the exchange rate is stochastic, too. Our model therefore enables to analyze whether investor s benefits from international trading depend on their home country, a stylized fact recently revealed by Driessen and Laeven 2007). 4.1 Overview We consider two countries. The US represents the home country, while Europe is defined to be the foreign country. The exchange rate F is given in US-$ per unit of the foreign currency, i.e., Euro. In the following, we consider both the domestic and the foreign investor. Both investors have CRRA preferences and maximize the expected utility of terminal wealth without intermediate consumption. The investment opportunity set for the domestic investor comprises the domestic) money market account and three risky assets: the domestic stock with price S US, the foreign stock with price S EU, and the foreign money market account with price P EU. All prices are denoted in the domestic currency, i.e., US-$. Under the common empirical measure the joint dynamics of asset prices are given by dst US /St US ds EU t dp EU t /S EU t /P EU t = r US 1 + X t Λ X) dt + X t dz X t. 31) We assume that the domestic risk-free rate is constant, i.e., we set α0 R = r US and βx R = 0. The dynamics of X are given in Equation 2). The portfolio problem thus describes a standard n = 3 asset case. From the perspective of a US investors the prices are already given in real terms or that the investor maximizes the expected utility of nominal consumption). Formally, we set the price index equal to one and all coefficients δ X, δ Y, α0, π βx π, βπ Y which govern its dynamics equal to zero. We are also interested in the optimal portfolio holdings of the foreign investor. considers all prices in Euro, which follow from dividing the prices in US-$ by the exchange rate F. Formally, this can be interpreted as dividing nominal US-$ prices by the price index F. The resulting real returns are the returns in Euro. The dynamics of the foreign exchange He 11

14 rate or price index) are given by df t F t = [ r US r EU + tr β π XX t ) ] dt + δ X X t dz X t 32) where we have set α π 0 = r US r EU, δ Y = 0 and β π Y = 0.13 We assume that the Euro interest rate is constant, too. The remaining parameters δ X and βx π of the exchange rate dynamics follow from the dynamics of the European money market account in US-$, i.e., from the dynamics of the third asset in Equation 31). This gives δ X = ) βx π = 0.5 δ X Λ X + Λ X δ X ). With this specification of F it indeed holds true that the dynamics of the European money market account measured in Euro are given by dp EU t = r EU Pt EU dt. The wealth dynamics from the foreign perspective follow from Lemma 2. In the last step, we have to account for the relation between the market prices of risk as seen by the domestic investor and the foreign investor and the dynamics of the exchange rate. It holds that M EU t F t M US t 33) where Mt EU and Mt US denote the stochastic discount factors in Europe and the US, respectively. We denote the market prices of risk from the perspective of the foreign European) investor for B X and W X by η X X and ξ X X, respectively. Equation 33) implies which results in [ tr η X ] X t dbt X [ = tr η X ] X t dbt X ξ X X t dw X t ξ X X t dw X t + δ X X t dz X t η X η X + δ X ρ X = 0 ξ X ξ X + 1 ρ X ρ X δ X = To reduce the number of parameters and for the ease of simplification, the additional state variable Y does not have an impact on the exchange rate. 12

15 This finally implies that η X = η X ρ X ξ X = ξ X ρ X ρ X The optimal portfolio weights for both the domestic and the foreign investor follow from Proposition 1. For the foreign investor, ω has to be reinterpreted. It gives the weights in the US stock, the European stock, and the risk-free) European money market account. The weight of the risky US money market account is given by ω 4 = 1 3 ω i, 34) i=1 where ω i denotes the ith entry of vector ω. The weights of the risky assets are then given by ω 1, ω 2, ω 4 ). A similar argument applies to the hedging demand. Again, the foreign investor considers the third asset as riskless. Therefore, we calculate the hedging demand for the risky) US money market account as: 3 ω h,4 = ω h,i 35) where ω h,i denotes the hedging demand for the ith asset and where we assume that all hedging demands add to zero. For a myopic investor or a zero investment horizon) the hedging demand in the risk-free asset is zero, i.e., the total demand equals the myopic demand. However, if hedging demands arise, then it will be financed on behalf of the myopic risk-free investment. 14 For this reason, we give the optimal hedging demand relative to the myopic demand in subsection 4.3. i=1 4.2 Parametrization The parameters for our base case are given in Table 1. The return volatilities and correlations follow from the matrix X 0. We set X 0 equal to its long-run average, which is controlled by the mean-reversion matrix M, the vol-of-vol matrix Q, and the Gindikin coefficient k. The correlation between the returns of the domestic and foreign risky asset is 0.55, which 14 More precisely, we define the hedging demand to be the difference between the total demand and myopic demand. 13

16 reflects the high integration of the US and the European financial markets. 15 The correlation between returns on the domestic foreign) stock and the exchange rate is approximately The matrix η captures the risk premia for variance and covariance risks. Here, we rely on findings from the relevant empirical literature. First, we consider negative variance risk premia, in line with, e.g. Pan 2002), Bakshi and Kapadia 2003a,b), or Carr and Wu 2009). Second, we compute the covariance risk premia such that the premia for correlation risks become negative and large. 16 The equity risk premia are controlled by the coefficient Λ, which is given in Equation 12) and driven by η and ξ. We choose the vector ξ such that the risk premia for the first two assets 5% and 5.3%) are distinctly higher than the premium for the third asset 1%). 17 This reflects almost identical risk-return characteristics of the domestic and the foreign asset, which allows us to focus on the impact of the characteristics of the foreign bond exchange rate) in our analysis. Both the foreign and the domestic investor have constant relative risk aversion. In the base case we set γ = 5. The length of the investment horizon is τ = Optimal demands Figure 1 gives the optimal portfolio weights for the US investor left column) and the EU investor right column). There is no significant difference between the domestic and the foreign stock holdings. For a five-year investment horizon and a relative risk-aversion of γ = 5, both investors devote approximately 10.2% and 12.5% of their wealth to the US stock and the EU stock, respectively. The small difference between both stock holdings is due to the higher equity risk premium of the European stock, which makes the US stock slightly less attractive. The optimal holdings of the investors differ significantly when it comes to the foreign bond. 18 While the US investor puts 6.8% of his wealth in the European bond, the EU investor optimally chooses a short position in the US bond and puts 8.7% of his wealth into this bond. These positions are surprisingly large and cannot be explained by the risk premia only. Recall that the US investor earns a premium of 1% on the European bond only. The main reason for holding foreign bonds is to exploit the gains from international diversification. 15 A very basic 40-day rolling window computation gives a mean correlation of historical S&P500 and DAX30 returns of approximately We considered data from 01/2002 to 10/2013, retrieved from Thomson Reuters Datastream. 16 Among others, Krishnan et al. 2009) and Driessen et al. 2012) report significantly negative prices for correlation risk. 17 The risk premia are given in the domestic currency. 18 Note that we refer to the foreign bond as the foreign money market account. From the perspective of the domestic investor the foreign money market account is a risky asset due to its exposure to the exchange rate. 14

17 Both investors hold significant shares in the foreign money market account in order to hedge the risk accompanied by the exchange rate. Next we analyze the optimal hedging demands in more detail. Figure 2 gives the variance and the covariance hedging demand 19 for the US and the EU investor. In addition, Table 2 shows the optimal covariance and variance hedging demands for different combinations of investment horizon and relative risk aversion. Both in the figure and in the table, the hedging demand is given as percentage of the respective myopic demand.consider, for example, an investment horizon of τ = 5 and a relative risk aversion of 5. From the US EU) perspective the total average covariance hedging demand in the stocks is 4.7% 5.4%), while the total average variance hedging demand is 5.1% 4.8%). For both investors, the covariance hedging demand for US stocks is larger than for European stocks, while it is the other way round for the variance hedging demand. Again, the major difference between both investors stems from the foreign bond holdings. For the EU investor, the total hedging demand in the foreign money market can be up to four times higher than for the US investor. The covariance hedging demand of the EU investor exceeds the one of the US investor by up to 40% for investment horizons of more than 1.5 years, while it is smaller for shorter investment horizons. The most pronounced differences are seen for the variance hedging demand. While it is below 1% of the myopic demand for the US investor, it amounts to around 13% of the myopic demand for the EU investor see lower panels of Figure 2). To gain more insights into the driving forces of the hedging demand, we look at the comparative statistics with respect to selected model parameter. We focus on the parameters that influence the level of the matrix A X τ), i.e., the sensitivity of the indirect utility function with respect to changes in the state variables. The matrix M determines the persistence of shocks in the variance-covariance matrix of returns, while Q determines the variance of shocks in the variance-covariance matrix and also influences the leverage. In the following we focus on the parameters M 31 and Q 23, as they primarily determine the co-evolution of the first second) asset with the exchange rate. 20 Figure 3 shows the comparative statics for the total hedging demand with respect to the parameters M 31 and Q 23. Again, there are significant differences between the US and the EU investor. First, the hedging demands of the US investor are increasing functions of these two parameters, while they are decreasing functions for the EU investor. The ability of the investor to hedge against adverse changes in the state variables thus increases for the US 19 See section 3.3 for a detailed discussion and for the decomposition of the hedging demand. 20 Due to the structure of the Wishart process these parameters have impact on several elements of the sensitivity matrix A X. 15

18 investor, but decreases for the EU investor. Second, the investors again differ when it comes to the hedging demand in the foreign bond. For the US investor, the hedging demand in the foreign bond hardly depends on the two parameters M 31 and Q 23. For the EU investor, on the other hand, there is a pronounced dependence of the hedging demand for the foreign bond on M 31 and Q Certainty equivalent wealth Next we look at the certainty equivalent wealth CEW) levels and thus the well-being of the US and the EU investor. The certainty equivalent wealth is defined as the amount of wealth for which the investor is indifferent between receiving this deterministic) amount of wealth and following the optimal risky) investment strategy. Lemma 3. The investor s certainty equivalent wealth is given by { 1 [ ) ] } Π CEW = Π 0 exp tr A X τ) X 0 + C τ). 36) 1 γ The functions A X τ) and C τ) are provided in Proposition 1. We are interested in the dependence of the CEW on the average variances and covariances. Figure 4 shows the CEW as a function of the average exchange rate volatility and of the average correlation between the exchange rate and stock returns. 21 The upper left panel of Figure 4 presents the dependence of Π CEW on the average exchange rate volatility X 33. When varying X 33, we assume that the correlations between the exchange rate and the stock returns remain constant and thus change X 13 = X 31 and X 23 = X 32 accordingly. For low volatility levels the certainty equivalent of the US investor is above that of the EU investor. As X 33 increases, the US certainty equivalent remains rather constant, while the European CEW increases remarkably. The implications of a higher exchange rate volatility are thus rather different depending on the home country of the investors. The dependence of Π CEW on the correlation between the stock returns is shown in the upper right panel of Figure 4. Here, we vary X 12 = X 21 such that the correlation varies between 1 and 1. The CEW for both investors is nearly identical. It decreases in the correlation of returns, since a higher covariance of returns reduces the gains from international diversification. 21 Formally, we change the long run average of X and keep the excess returns constant. This implies that the structural parameter ξ can differ compared to the base case. See Liu and Pan 2003) for an equivalent approach in a single-asset economy. 16

19 The lower panels of Figure 4 show Π CEW as a function of the average correlations between the foreign exchange rate and the US stock returns left panel) and the EU stock return right panel), respectively. Again, we vary the corresponding correlations in the interval [ 1, 1] by adapting the covariances X 13 = X 31 and X 23 = X 32. The effect on the certainty equivalent is similar for both correlations. The CEW is rather constant for moderate levels of the correlations, but increases significantly when the absolute correlation exceeds 0.5. Furthermore, the CEW of the European investor is significantly higher than that of the US investor for low correlations. In contrast, for correlations above 0.25, the situation is reversed. For the correlation between the US stock returns and the exchange rate exceeding 0.7, the situation changes again. This supports our previous findings: the effect of stochastic correlations and the volatility of the exchange rate on international portfolios can be large and is by no means the same for investors in different countries. 4.5 Economic value of derivatives Up to now, we have focused on an incomplete market economy where the investor s opportunity set contains the risky assets with return dynamics given in Equation 31) and the domestic money market account. While the investor can trade stock market diffusion risk, he cannot fully) trade variance-covariance risk. Now we turn to the complete market. The investment opportunity set additionally comprises enough non-redundant derivatives such that the market is dynamically completed, i.e., the investor can attain arbitrary exposures θ B R n n and θ W R n to the risk factors B X and W X. The solution to the corresponding investment problem is summarized in the next Proposition. Proposition 2. The value function is given by J t, Π, X) = Πc ) 1 γ 1 γ exp {tr Ac τ) X t ) + C c τ)} and the optimal exposures are θt B = 1 η X + 2A c τ) Q X ) γ θ W t = 1 γ ξx 17

20 where τ = T t. The functions A c and C c are given by A c τ) = F 22 τ) 1 F 21 τ) 37) [ ] C c τ) = 1 γ) r US τ γkx 2 tr lnf 22 τ) + Γ τ 38) subject to the terminal conditions A c 0) = 0, C c 0) = 0 and F 11 τ) F 21 τ) Proof. See Appendix A.3. F 12 τ) F 22 τ) ) := exp [ Γ = M X + 1 γ γ ζ = 1 γ 2γ τ Γ 2 γ QX Q X ζ X Γ QX η X η X η X + ξ X ξ X ). In contrast, an incomplete market investor always holds a package of risk factors through the choice of the portfolio weights. His utility will therefore always be lower than that of the complete-market counterpart. To assess the investor s gain from having access to derivatives that complete the market), we follow Liu and Pan 2003) and rely on the annualized, continuously compounded increase in certainty equivalent wealth. Lemma 3 gives the investor s CEW on an incomplete market. Equivalently, Π CEW c It is given by denotes the certainty equivalent wealth for a complete market investor. Π CEW c )] { } 1 = Π 0 exp 1 γ [tr Ac τ) X 0 ) + C c τ)], 39) where the functions A c τ) and C c τ) are disclosed in Proposition 2. Hence, the annualized gain R from trading derivatives is R = 1 τ [ ) )] ln Π CEW c ln Π CEW. 40) Figure 5 gives R for the US and the European investor as a function of the average volatility of the exchange rate, the average correlations as well as of risk aversion and the investment horizon. The utility gain is around 1%-3% for the base case, but can reach up to 20% if correlations change. Furthermore, for most values of X, the utility gain is larger for the EU investor than for the US investor. The upper and the middle panels of Figure 5 show the dependence of R on the average 18

21 volatility of the foreign exchange rate and the average correlations. 22 R is a U-shaped function of the exchange rate volatility. While the utility gain increases sharply for low volatilities, it also increases slightly when the volatility reaches very high levels. For the correlation between returns it holds that the utility gain is mainly decreasing in ρ 12 and only starts to increase for very high correlations close to one. The smaller the potential for international diversification, the smaller not only the CEW as seen in Figure 4), but the smaller also the utility gains from having access to derivatives. Finally, the dependence on the correlation between the exchange rate and the stock return is similar for US stocks and EU stocks. Utility gains are a U-shaped function of this correlation. They sharply increase for both very low and very high correlations. The utility gain from derivatives is largest for extreme correlations. When stock returns have a correlation below 0.75 or when the correlation between the foreign exchange rate and stock returns exceeds 0.75 in absolute terms, both the US and the EU investor gain more than 10% annually from having access to derivatives. 5 Conclusion In this paper, we study a multi-asset model for optimal portfolio selection. The variancecovariance matrix of returns is driven by a Wishart process, such that all volatilities and correlations between asset returns are stochastic. We furthermore include inflation risk, which can be re-interpreted as exchange rate risk. We apply our general setup to an international portfolio selection problem. First, we find that the inclusion of additional state variables determining return volatilities and correlations has a significant impact on optimal portfolios. It induces a noticable hedging demand, and changes in average volatilities and correlations can have a large impact on the certainty equivalent wealth and utility gains from having access to derivatives. Second, the implications of variance and correlation risk can differ remarkably between investors from different countries. This holds true, for example, for the sign of the optimal position in the foreign bond, or for the sign of the impact of stochastic correlations on the hedging demand. Furthermore, the utility gain from trading in stocks and bonds at all, and the utility gains from having access to derivatives depend on the home country of the investor. An issue for future research is the estimation of our model in an international context. To this end, it would also be interesting to apply the model to the pricing of stock and foreign exchange options in order to shed light to the premia for correlation risk. Finally, 22 The computation of R as a function of the volatility of the exchange rate and the correlations is done analogously to the computation of the CEW in Section

22 the inclusion of jump risk, as proposed by Leippold and Trojani 2010), might reveal further interesting results for the investor s optimal decision on asset allocation. 20

23 A Appendix A.1 Proof of Lemma 2 The real wealth is defined by Π = Π/I. For simplicity, the subscripts denoting the dependence to time t are omitted. The dynamics of dπ follow by an application of Itô s lemma: dπ Π = ) R X, Y ) + ω XΛ X dt + ω XdZ X π X, Y ) dt + δ X XdZ X + δ Y Y dz Y + [ tr δ X δ X X ) + tr δ Y δ Y Y ) tr ) ωδ X X )] dt After plugging in the definitions of R X, Y ) and π X, Y ) we obtain the stated result. A.2 Proof of Proposition 1 We solve the investment problem by applying the stochastic control approach. The Hamilton- Jacobi-Bellman for the indirect utility function of wealth is: { J [ 0 = max ω t + J Π Π α0 R α0 π + tr βy R βy π + δ Y δ Y ) ]) Y [ +J Π Π tr βx R βx π + Λ X δ X) ω + Λ X δ X ) ] X J Π Π Π ) 2 tr [X ω δ X) ω δ X) + Y δ Y δ Y ] [ +tr Ω X Ω X + M X X + XM X ) )] D X + 2 XD X Q X Q X D X J [ +tr Ω Y Ω Y + M Y Y + Y M Y ) )] D Y + 2 XD Y Q Y Q Y D Y J +J Π Π tr [XD X Q X ρ X ω δ X) + X ω δ X) ] ρ X Q X D X ] +J Π Π tr [Y } D Y Q Y ρ Y δ Y + Y δ Y ρ Y Q Y D Y where the arguments of the function J are suppressed for the ease of notation. The terms J Π and J Π Π denote short-hand notation for the partial derivatives with respect to Π. The elements of the matrix D X refer to the partial derivatives with respect to X, i.e., Dij X = X ij for i, j {1,..., n}. This applies analogously to the matrix D Y. Inserting the first and second 21

24 order partial derivatives into the previous equation and dividing by J gives { ) ) 0 = max tr ω t AX X + tr t AY Y + t C + 1 γ) ) α0 R α0 π 41) [ + 1 γ) tr βy R βy π + δ Y δ Y ) ] Y 1 [X 2 γ 1 γ) tr ω δ X) ω δ X) + Y δ Y δ Y ] [ + 1 γ) tr βx R βx π + Λ X δ X) ω δ X) + Λ X δ X ) ] X [ +tr Ω X Ω X + M X X + XM X ) )] A X + 2 XA X Q X Q X A X [ +tr Ω Y Ω Y + M Y Y + Y M Y ) )] A Y + 2 XA Y Q Y Q Y A Y + 1 γ) tr [XA X Q X ρ X ω δ X) + X ω δ X) ] ρ X Q X A X ] + 1 γ) tr [Y } A Y Q Y ρ Y δ Y + Y δ Y ρ Y Q Y A Y Applying the first-order condition with respect to ω yields the optimal portfolio weights in Proposition 1. Finally, the system of matrix) ordinary differential equations, which is solved by the functions A X, A Y and C, follows if we insert the optimal weights into Equation 41). Separating variables then completes the proof. A.3 Proof of Proposition 2 On a complete market the investor can disentangle the structure of the risk factors. He can pick arbitrary exposures to the risk factors driving the economy. The stochastic discount factor SDF) provides information on the pricing of these risk factors. The SDF for domestic investor is given by dm US t M US t = r US dt tr η X X t dbt X ) ξ X X t dw X t with initial value M US 0 = 1. The coefficients η X and ξ X control the market prices of risk for B X and W X, respectively. Let Π c be the wealth of the investor on a complete market. We thus write the dynamics of wealth as dπ c t Π c t [ = r US dt + tr θt B X t η X dt + X t dbt X )] + θ W t X t ξ X dt + X t dw X t where θ B t and θ W t denote the exposures to the risk factors B X and W X, respectively. The CRRA investor maximizes utility of terminal wealth. We proceed according to Merton 1971). The Hamilton-Jacobi-Bellman equation reads ) 22

25 { J [ 0 = max θ ) t + J Π cπc r US + tr θ B Xη X + Xξ X θ W )] J Π c Π c Πc ) 2 tr [Xθ B θ B + Xθ W θ W ] [ +tr Ω X Ω X + M X X + XM X ) )] D X + 2 XD X Q X Q X D X J [ ] } +J Π cπ c tr XD X Q X θ B + Xθ B Q X D X. The remaining procedure is as follows: first, we insert the partial derivatives from the value function. Second, applying the first-order condition with respect to θ ) yields optimal factor exposures. Finally, re-inserting these exposures and separating variables with respect to A c and C c completes the proof. 23

26 References Adler, M., Dumas, B., International Portfolio Choice and Corporation Finance: A Synthesis. Journal of Finance 38, Ang, A., Bekaert, G., International Asset Allocation With Regime Shifts. Review of Financial Studies 15, Bakshi, G., Kapadia, N., 2003a. Delta Hedged Gains and the Negative Market Volatility Risk Premium. Review of Financial Studies 16, Bakshi, G., Kapadia, N., 2003b. Volatility Risk Premiums Embedded in Individual Equity Options: Some New Insights. Journal of Derivatives 11, Ball, C.A., Torous, W.N., Stochastic correlation across international stock markets. Journal of Empirical Finance 7, Bekaert, G., Urias, M.S., Diversification, Integration and Emerging Market Closed-End Funds. Journal of Finance 51, 835. Bollerslev, T., Engle, R.F., Wooldridge, J.M., A Capital Asset Pricing Model with Time-Varying Covariances. Journal of Political Economy 96, Branger, N., Muck, M., Keep on smiling? The pricing of Quanto options when all covariances are stochastic. Journal of Banking & Finance 36, Branger, N., Schlag, C., Schneider, E., Optimal portfolios when volatility can jump. Journal of Banking & Finance 32, Brennan, M.J., Xia, Y., Dynamic Asset Allocation under Inflation. Journal of Finance 57, Bru, M.F., Wishart Processes. Journal of Theoretical Probability 4, Buraschi, A., Porchia, P., Trojani, F., Correlation Risk and Optimal Portfolio Choice. Journal of Finance 65, Carr, P., Wu, L., Variance Risk Premiums. Review of Financial Studies 22, Chacko, G., Viceira, L.M., Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets. Review of Financial Studies 18,

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29 Table 1 Base parameters Interest rates r US 0.04 r EU 0.04 State variables X Q M k Risk premia Η Ξ Ρ This table shows the parameters for the base case in the numerical analysis. The matrix X 0 is set to its long-run mean. 27

30 Table 2 Optimal hedging demands Γ 3m 6m 1y 2y 5y Myopic US EU US EU US EU US EU US EU US EU Covariance hedging myopic Variance hedging myopic The table shows the optimal hedging demand in the US stock, the EU stock, and the foreign bond for variance and covariance risk for different investment horizons τ and risk aversion coefficients γ. All hedging demands are calculated as a percentage of the myopic demand provided in the last column. 28

31 Figure 1 Optimal portfolio holdings 0.15 US: Portfolio weights 0.15 EU: Portfolio weights 0 0 US stock EU stock foreign bond Τ: Investment horizon Τ: Investment horizon Γ: Risk aversion Γ: Risk aversion This figure shows optimal portfolio weights as functions of the investment horizon τ upper panels) and the coefficient of relative risk aversion γ lower panels), both for the domestic investor left panels) and the foreign investor right panels). The parameters are given in Table 1. For the upper lower) panels, we set γ = 5 τ = 5 years). 29

32 Figure 2 Optimal hedging demands 9 US: Covariance hedging demand 9 EU: Covariance hedging demand hedging demand 6 3 hedging demand 6 3 US stock EU stock foreign bond Τ: Investment Horizon Τ: Investment Horizon 14 US: Variance hedging demand 14 EU: Variance hedging demand hedging demand 9 4 hedging demand Τ: Investment Horizon Τ: Investment Horizon This figure shows the intertemporal hedging demand as functions of the investment horizon τ for the domestic investor left panels) and the foreign investor right panels). The variance lower panels) and covariance hedging demand upper panels) is given as percentage of the myopic demand. The parameters are given in Table 1. The coefficient of relative risk aversion is to set γ = 5. 30

33 Figure 3 Intertemporal hedging demand: comparative statistics 15 US: Hedging demand 35 EU: Hedging demand US stock hedging demand 10 5 hedging demand EU stock foreign bond M 31 M US: Hedging demand 25 EU: Hedging demand hedging demand 9 6 hedging demand Q 23 Q 23 This figure shows the hedging demands as a function of the structural parameters M 31 and Q 23, where Z ij denotes the element {i, j} of the matrix Z. Hedging demands are calculated as percentage of the myopic demand, both for the domestic investor left panels) and the foreign investor right panels). The parameters are given in Table 1. The investment horizon is τ = 5 years, the coefficient of relative risk aversion is γ = 5. 31

34 Figure 4 Certainty equivalent wealth CEW 1.05 CEW 1.08 US Investor EU Investor X 3,3 : Exchange rate volatility Ρ 1,2 : Correlation of returns asset 1 & CEW 1.06 CEW Ρ 1,3 : Correlation of returns asset 1 & Ρ 2,3 : Correlation of returns asset 2 & 3 The figure shows the impact of the exchange rate volatility and the average return correlations on the certainty equivalent wealth for an initial wealth level of Π 0 = 1. The parameters are given in Table 1. The investment horizon is τ = 1 year, the relative risk aversion is γ = 5. 32

35 Figure 5 Portfolio improvement 5 20 Improvement CEW 3 Improvement CEW 10 US Investor EU Investor X 3,3 : Exchange rate volatility Ρ 1,2 : Correlation of returns asset 1 & Improvement CEW 10 5 Improvement CEW Ρ 1,3 : Correlation of returns asset 1 & 3 Ρ 2,3 : Correlation of returns asset 2 & Improvement CEW 5 Improvement CEW Γ: RRA Coefficient Τ: Investment horizon The figure shows the impact of the average return correlations, the average exchange rate variance, the risk aversion γ, and the investment horizon τ on the improvement in CEW. The parameters are given in Table 1. The investment horizon is τ = 5 years and the coefficient of relative risk aversion is to set γ = 5. 33