Correlation risk and international portfolio choice

• Author: Nicole Branger,Matthias Muck,Stefan Weisheit
• Date: 01 January 2019
• Published date: 01 January 2019
• DOI: http://doi.org/10.1002/fut.21941

Received: 30 June 2017

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Revised: 28 May 2018

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Accepted: 1 June 2018

DOI: 10.1002/fut.21941

RESEARCH ARTICLE

Correlation risk and international portfolio choice

Nicole Branger

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Matthias Muck

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Stefan Weisheit

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1

Finance Center Muenster, University

of Muenster, Universitätsstr.

14‐16, Muenster, Germany

2

Chair of Banking and Financial Control,

University of Bamberg, Bamberg,

Germany

Correspondence

Matthias Muck, Chair of Banking and

Financial Control, University of Bamberg,

Kärntenstr. 7, 96045 Bamberg, Germany.

Email: matthias.muck@uni-bamberg.de

Variance‐covariance risk of the exchange rate is highly relevant for interna-

tional investors. This paper addresses optimal asset allocation with stochastic

variances and covariances in a Wishart Affine Stochastic Correlation (WASC)

model in incomplete and complete markets. We show that the (hedging)

demand for exchange rate variance‐covariance risk can differ significantly

between international investors. Local correlations with the exchange rate can

affect the utilities of international investors differently while the impact of

correlations between stocks can be symmetric. Depending on the current local

exchange rate correlations domestic investors can benefit more or less than

foreign investors from international trading.

KEYWORDS

International asset allocation, stochastic correlation, Wishart processes, dynamic trading strategies,

derivatives

JEL CLASSIFICATION

G11, G13

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INTRODUCTION

Empirical evidence suggests that return variances and return correlations are stochastic, and that stochastic second

moments are priced. This has been documented within the stock market, across asset classes, and also on international

markets.

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Stochastic second moments have far‐reaching implications for the asset allocation of investors because

expected returns and risk exposures are no longer deterministic, and investment opportunity sets become stochastic.

While stochastic variances have attracted a lot of attention in the literature, approaches to the modeling of stochastic

correlations are relatively new. Within the class of affine models, so‐called Wishart processes allow to specify the joint

dynamics of variances and covariances directly.

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Examples in the literature on portfolio choice are Buraschi, Porchia

and Trojani (2010) and Da Fonseca, Grasselli and Ielpo (2011) who consider the analytically tractable Wishart Affine

Stochastic Correlation (WASC) model.

In this paper we study the asset allocation of international investors. We are interested in how stochastic covariances

and stochastic correlations affect the portfolio choice of domestic and foreign investors. We use a Wishart process to

model the variances and covariances of domestic and foreign stock markets and the foreign exchange rate, and compare

the optimal asset allocation decisions of domestic and foreign investors. Furthermore, we consider benefits from

international trading and compare utility improvements of domestic and foreign investors in particular for different

levels of correlation.

J Futures Markets. 2019;39:128–146.wileyonlinelibrary.com/journal/fut128

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© 2018 Wiley Periodicals, Inc.

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Bollerslev, Engle and Wooldridge (1988), Longin & Solnik (1995), Ball & Torous (2000) and Goetzmann, Li and Rouwenhorst (2005) document that international market correlations vary dynamically

over time. Moreover, Driessen, Maenhout and Vilkov (2009) find that correlation risk is priced based on index and individual option prices. The results provided by Krishnan, Petkova and Ritchken

(2009) and Driessen, Maenhout and Vilkov (2012) imply a sufficiently large, negative risk premium for correlation risk. Further studies that deal with the pricing of variance and correlation risk are

Hollstein & Simen (2017), Buss, Schoenleber and Vilkov (2018).

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Wishart processes were first studied by Bru (1991). Gourieroux (2006) and Da Fonseca, Grasselli and Tebaldi (2007) study the pricing of derivative claims in Wishart frameworks.

The analysis is done both in an incomplete market where the investors can only trade stocks and bonds, and in a

complete market. In a complete market arbitrary exposures to variance‐covariance risk can be attained by trading

derivatives. As it is, for example, discussed by Liu & Pan (2003) this can lead to significant utility improvements for

investors. Mathematically, the particular types of contracts that complete the market do not matter as long as the

derivatives provide an exposure to variance‐covariance risk. In practice, though, it might be advantageous to trade

derivatives whose prices are directly connected to elements of the variance‐covariance matrix. Natural candidates to

trade stock and exchange rate variance risk are stock options and exchange rate options. Quanto contracts and options

on several assets provide an exposure to covariance risk. Besides options, one might also trade futures and swaps on

variances, covariances, and correlations. An advantage of the WASC model is that its affine structure results in quasi

closed form solutions for standard derivatives which allow for an efficient calculation of prices and Greeks. In practice,

this facilitates the application of the model discussed in this paper.

Our paper makes several contributions. First, we consider the joint dynamics of foreign and domestic stock prices

in their local currencies and the (direct) exchange rate when variances, covariances, and correlations are stochastic.

Given this starting point, we derive the investment opportunity sets of domestic and foreign investors who can trade in

international stocks and bonds. The stochastic variance‐covariance matrix is driven by a Wishart process.

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Second, we determine the optimal portfolios of international investors and their utility gains from trading

international stocks and bonds. We assume that investors have constant relative risk aversion (CRRA). While the

demand for stocks is similar for domestic and foreign investors, the demand for foreign bonds –which allow to trade

exchange rate risk –differs significantly between the investors. We also find that the benefits from international trading

are different for domestic and foreign investors. Utility gains depend on the current levels of variances and correlations.

In our example, both certainty equivalent returns increase in the correlation between the stocks, but decrease in the

correlations between the stocks and the exchange rate. Furthermore, the difference between the certainty equivalent

returns can change sign depending on the latter correlations.

Third, we assess the impact of market completion, that is, we consider a market in which international investors have

access to derivatives. Investors can disentangle stock and bond market risk from variance‐covariance risk and exploit risk

premia for stochastic variances and covariances. In our numerical example we find that, in contrast to, for example, the

exposure to the covariance between the stocks, the exposures to the covariance between stocks and the exchange rate are

significantly different and even change sign. The benefits from trading depend on the stochastic correlations. As in the

incomplete market, changes in stock correlations have a symmetric impact on domestic and foreign investors, while changes

in the correlation between stocks and exchange rates might have different consequences.

Our paper is related to several strands of the literature. Following Merton (1969, 1971, 1973) numerous papers deal

with asset allocation problems that take additional risk factors like stochastic volatility or jumps into account. Examples

include Liu, Longstaff and Pan (2003), Chacko & Viceira (2005) and Liu (2007). The implications of market

completeness are explored by, for example, Liu & Pan (2003), Branger, Schlag and Schneider (2008), Muck (2010),

Egloff, Leippold and Wu (2010) and Da Fonseca et al. (2011). Furthermore, Branger, Muck, Seifried and Weisheit (2017)

consider the impact of jumps in variances and covariances in a WASC model for stocks.

Alternatives to the WASC model exist in the literature as well. For example, Engle (2002) proposes the Dynamic

Conditional Correlation (DCC) model. This approach facilitates the econometric estimation of the dynamics of the

correlation matrix. However, in this framework we cannot distinguish between complete and incomplete markets.

Moreover, the pricing of derivatives is more involved. Another possibility is to consider latent state variables (different

from a Wishart process) to describe the variances and covariances. This is, for example, the case in the multi‐Heston

model of De Col, Gnoatto and Grasselli (2013). The multi‐Heston models also results in quasi closed form solutions for

derivative prices. However, covariances are usually not represented by distinct state variables in this approach, but are

driven by the same state variables that determine the variances.

Our paper is also related to the literature on international asset allocation and the benefits from international

investing. Empirical evidence on gains from international equity diversification is provided by, among others, Kaplanis

& Schaefer (1991), Bekaert & Urias (1996) and Li, Sarkar and Wang (2003). In the spirit of Adler & Dumas (1983), a

number of papers study the implications of international asset allocation in dynamic models.

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Lioui & Poncet (2003)

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Our setup nests the special case in which stock prices and exchange rates are driven by a WASC model.

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Adler & Dumas (1983) show that international portfolio selection is closely related to portfolio choice problems under inflation risk. The latter usually consider real prices obtained from dividing

nominal prices by a price index, which is identical to dividing prices given in a measurement currency by the exchange rate. Examples for models with inflation risk are Brennan & Xia (2002) and

Munk, Sørensen and Nygaard Vinther (2004).

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