How past market movements affect correlation and volatility

Accepted Manuscript How past market movements affect correlation and volatility Christoph Becker, Wolfgang M. Schmidt

PII:

S0261-5606(14)00138-7

DOI:

10.1016/j.jimonfin.2014.09.003

Reference:

JIMF 1469

To appear in:

Journal of International Money and Finance

Received Date: 10 February 2014 Revised Date:

9 July 2014

Accepted Date: 4 September 2014

Please cite this article as: Becker, C., Schmidt, W.M., How past market movements affect correlation and volatility, Journal of International Money and Finance (2014), doi: 10.1016/j.jimonfin.2014.09.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights

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Highlights • We study the realized market trend or volatility as driver of covariances.

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• We find increase in correlations and volatilities in downturns.

• The model captures the dynamics of covariances particularly well in a crisis.

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• Our model offers an alternative to the concept of exceedance correlations.

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• The models provides superior portfolio allocations in crisis scenarios.

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How past market movements affect correlation and volatility

Abstract

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July 9, 2014

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The influence of past stock price movements on correlations and volatilities is essential for understanding diversification and contagion in financial markets. We develop a model that makes the influence of past returns, aggregated into driving factors for correlations and volatilities, explicit. Employing information about recent market movements leads to a more realistic model for the behavior of stock returns in a downturn than conventional models. Our approach offers a fresh perspective on the behavior of stock markets, and provides an alternative to the concept of exceedance correlation. For a US investor we find that international diversification in China or the UK remains beneficial in a crisis.

Keywords: correlation, volatility, financial contagion, diversification, exceedance correlation, GARCH models

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JEL: C13, C32, C58, G11, G12

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Introduction

This paper attempts a more realistic analysis of the determinants of correlations and volatilities of stock returns than had been available with conventional methods. As is well-known, volatilities and correlations are not static over time, but tend to increase in periods of market

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stress. We develop a new approach to investigate the dynamic dependence of correlations and volatilities on each of the following three factors: the realized market trend, the realized market volatility and the realized volatility of the CBOE Volatility Index (VIX). The dynamics of correlations and volatilities is known to be asymmetric in downturn versus normal markets,

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exhibiting strongly non-linear behavior in a crisis. Therefore, we propose a flexible dependence on these factors that is able to capture these non-linearities and makes the influence of

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realized market movements explicit. Special emphasis is put on the influence of these factors during market downturns. Realized market movements are taken from a past window whose length, the “market memory”, is flexible and estimated from data. The idea that correlations (albeit not volatilities) depend on the market trend goes back at least to Karolyi and Stulz [1996]. The dependence of volatilities and correlations on today’s return is a standard feature of conventional models, but these models lack flexibility to capture

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the influence of returns observed in the past. The explicit dependence of correlations and volatilities on the recently observed market trend or market volatility has not been investigated in a systematic way. Our analysis shows that realized market trend or volatility provide

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valuable information for explaining the dynamics of correlations and volatilities in a crisis. Understanding the determinants of correlations and volatilities during market stress is

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vital to analyze financial contagion and the efficiency of diversification, see, for example, Forbes and Rigobon [2002], Forbes [2012]. The methodology that we develop offers a fresh perspective on long-lasting, controversial debates on the benefits of diversification and the existence of contagion in financial markets. For the benefits of diversification see, for example, seminal studies like Grubel [1968] and Solnik [1995]. Both studies emphasize the benefits of diversification, while current research like Bartram and Bodnar [2009] find that diversification breaks down when markets fall, except for particular markets, see Berger et al. [2011]. For financial contagion, for example, Forbes and Rigobon [2002] argue in a seminal paper that there is no contagion in international financial markets, while Corsetti et al. [2005] question their methodology. 2

ACCEPTED MANUSCRIPT We investigate US large caps and find, that a shift of the market trend from its empirical median to its 95% quantile yields an increase of volatilities from the level taken at the median by 200% and of correlations by 100% on average. Volatilities respond most strongly to changes in the market trend, slightly less to changes in the market volatility or to changes in the

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volatility of the VIX. Correlations, on the other hand, respond stronger to changes in the volatility of the VIX. Across stock there are large differences in observed increases in both volatilities and correlations that can be explained by the stocks’ factor loading on the HMLfactor from Fama and French [1993], suggesting that this variation is due to the different

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behavior of value and growth stocks.

Our analysis also shows that the power of diversification to reduce portfolio risk is a highly nonlinear function of the realized market trend, market volatility or volatility of VIX. How

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diversification benefits on the one hand and correlations and volatilities on the other hand dynamically evolve during a downturn is often neglected in the debate about diversification and contagion. A small change for the worse, like a minor decline in the realized trend or volatility, may cause correlations and volatilities to shift dramatically thereby affecting the efficiency of diversification, compare also, Christoffersen et al. [2012] and You and Daigler

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[2010]. For a US investor we investigate international diversification in China resp. the UK. In a crisis we find that diversification reduces portfolio volatility by 12% resp. 7%. The influence of longer term market movements is often assessed by defining characteristic market environments (typically, normal market versus downturn) according to historical

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consensus and to compare sample correlations that are estimated for each market environment. Closely related is the idea to compute correlations for stock returns that are below

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a pre-specified, negative threshold; such correlations are known as exceedance correlations, see, for example, Longin and Solnik [2001], Ang and Bekaert [2002], Ang and Chen [2002]. Both approaches are statistically challenging, for example, Campbell et al. [2008] show that a naive estimation of exceedance correlations may ‘show’ a correlation increase where correlation is actually constant. This problem is known as conditioning bias, see also Boyer et al. [1997] and Corsetti et al. [2005]. Dealing with conditioning bias is not straightforward and an important source of disagreement in the debate1 on diversification and financial contagion 1

For example, conditioning bias is the crux in Forbes and Rigobon [2002] who claim that financial contagion is a statistical artefact, and in Campbell et al. [2008] who question whether exceedance correlations increase in a downturn; for a recent discussion also see Forbes [2012].

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ACCEPTED MANUSCRIPT that is ongoing until today. Our approach provides a statistically sound alternative to determine correlations in different market environments. Estimated correlations from our model are not prone to conditioning bias; but in a downturn they have an interpretation similar to exceedance correlations.

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Popular models that incorporate a time dynamics for volatilities and correlations are regime shift models and GARCH-type models. Conventional models with regime-shifts, like Pelletier [2006], ignore the influence of past prices because they assume that volatilities and correlations are determined by a Markov process. These models investigate volatilities and

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correlations in different market regimes, usually in a normal market versus a downturn. But in each market regime volatilities and correlations are assumed to be constant, providing only a first order approximation to the non-linear dependence on the market trend that we find.

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Bauwens and Otranto [2013] depart from a Markov dynamics and propose a regime shift model for correlations where the transition probabilities depend on the actual volatility. Various authors extended GARCH-type models to incorporate the influence of returns or volatilities on conditional volatilities or correlations. For example, multivariate GARCH models like Cappiello et al. [2006] assume that conditional volatilities and correlations increase more

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strongly if the current return of stocks is negative. Univariate GARCH models that are based on this idea are, for example, Nelson [1991], Rabemananjara and Zakoïan [1993] and Glosten et al. [1993]. A large collection of alternative GARCH-type models is compared by Hansen et al. [2003] in the framework of model confidence sets. Bauwens and Otranto [2013] extend

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the Dynamic Conditional Correlation (DCC) model of Engle [2002] by including the volatility as driving factor of conditional correlations. GARCH-type models by design take into account

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individual returns from the more distant past but it is not straightforward to incorporate a nonlinear dependence of volatilities and correlations on the whole trend or realized volatility. We find that by allowing for a flexible and non-linear dependence on the realized trend or volatility of the market we obtain an improved model for the dynamics of stock returns. CAPM with dynamic conditional betas also attempt to provide a better explanation of the dynamics of returns, see, e.g. Jagannathan and Wang [1996], Adrian and Franzoni [2009], Engle [2014]. In general, conditional beta is a function of past observations, which has to be made explicit. For example, Bali et al. [2012] calculate conditional betas from a DCC model, Reeves and Wu [2013] use autoregressive models of time-varying realized beta, Jagannathan

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ACCEPTED MANUSCRIPT and Wang [1996] consider conditional betas that are correlated with conditional market risk premiums. Knowing conditional betas and conditional volatilities of market premiums implies knowledge of conditional correlations and conditional volatilities. As soon as conditional betas are functions of the realized market trend or market volatility, the conditional CAPM is related

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to our approach. However the factor structure of conditional CAPM restricts the flexibility (rank of matrix) of implied correlations, whereas our approach would yield an explicit and flexible functional dependence.

How well does our model capture the dynamics of stock returns in a downturn? Engle and

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Colacito [2006] propose an approach to compare the performance of competing models. The idea is to allocate minimum variance portfolios with volatilities and correlations as determined by the models. The model that yields the portfolio with the smallest empirical variance

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captures the behavior of volatilities and correlations best. It turns out that our models perform better than DCC-GARCH in the majority of cases and are better than 30 or 90 days moving average models, particularly in periods of market stress. The paper is organized as follows. In Section 2 we introduce our modeling approach in its general form and provide an example model where correlations and volatilities are driven by

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the trend or volatility of a market index. We apply the model to stocks from the S&P 500 in Section 3. Economic implications of our findings are investigated in Section 4. In Section 5 we discuss the relation of our modeling approach to multivariate GARCH models. Section 6

A class of asset price models with state-dependent correlation

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concludes.

and volatility

The vector of asset prices S(t) = S 1 (t), . . . , S N (t)), t = 0, 1, . . . , is a discrete time RN -valued stochastic process defined on a probability space (Ω, F, P ). The stochastic factors in the prices dynamics are shocks (t) = (1 (t), . . . , N (t)) that are standard normally distributed, independent over time, but whose coordinates i (t), i = 1, . . . , N are correlated. In our asset price model volatilities and correlations are allowed to depend on past price realizations. The history of observed asset prices looking back n periods from t, that is, over the time window n

from t − n to t, is denoted by St← = (S(t − n), . . . , S(t)).

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ACCEPTED MANUSCRIPT Let X i (t + 1) = log(S i (t + 1)) − log(S i (t)) denote the log-return process of asset S i . Our class of models is defined as follows,

(1) (2)

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X i (t + 1) = µi (θ, t) + σi θ, F (t) · i (t + 1)
Corr(i (t + 1), j (t + 1)) = ρi,j θ, F (t) , i, j = 1, . . . , N ,

with volatilities σi and correlations ρi,j being parameterized functions of a driving factor F (t) that is determined from past price observations and reflects the market environment at time t. Throughout the paper the factor process F (t) is a function of a rolling window of past asset nF

nF

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prices; it is a function F (t) = F (St← ) of the past segment St← with “memory” nF and with some state mapping F . Natural specifications for the factor F are the trend or the realized

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volatility in the market over the time window t − nF , . . . , t, which will be made precise in formulas (3), (4) below. The parameter θ takes values in a set Θ, which will be detailed later. The correlation ρi,j is nothing else but the conditional correlation between the returns of nF

assets S i and S j given the state St← :

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ρi,j

n   F
i j θ, F (t) ≈ Corr X (t + 1), X (t + 1) St← . nF

Next, we specify the state function F (t) = F (St← ) and parameterizations for volatilities σi and correlations ρi,j . While there are various useful definitions for the state function F to

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reflect relevant market situations, in this paper we concentrate on the realized trend, the realized volatility of prices or the realized volatility of the VIX over the past time window

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t − nF , . . . , t. All three are natural statistics to assess the actual state of the market. There is strong evidence in the literature that changes in the trend of market indices influence correlations, see Karolyi and Stulz [1996]. nF

In case of the trend, we define the state F (St← ) as weighted average of log-returns based nF

on discrete observations St← = (S(t − nF ), . . . , S(t)),  F

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N X j=1

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nF
1 X X j (t − k + 1) , nF k=1

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(3)

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with weights ωj ≥ 0. For the realized volatility as state variable2 , we define F (t) = F (St← ) by  n  X q N F c2 (X j (t − k + 1) : k = 1, . . . , nF ) F St← = ωj σ

(4)

j=1

with

!2 .

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nF nF X 1 X 1 c2 (x1 , . . . , xn ) = x − xk σ l F (nF − 1) nF l=1

(5)

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In our empirical studies below, one of the securities is a market index, for example, the first security S 1 is the S&P 500 index. In this case we base our empirical analysis on the trend or

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volatility of this index, that is, the weights are ω1 = 1, ωj = 0 for j > 1.

When analyzing the correlation between two selected securities, we use the parametrization
2 arctan h(ξ,θ0 ) (f ) . π

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ρ(θ0 , f ) =

(6)

The function h(ξ,θ0 ) is a cubic spline characterized by points ξ = (ξl )l=1,...,nρ and values
θ0 = θl0 l=1,...,nρ : h(ξ,θ0 ) (ξl ) = θl0 ,

l = 1, . . . , nρ .

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We choose (quantile-) equidistant discretization points ξl , l = 1, . . . , nρ such that, for observed stock prices S, they cover the range of observed values of F ; ξ1 is the one percent smallest and ξnρ is the 99%-largest observation of F . While it is certainly also of interest to analyse

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the full N -variate correlation matrix3 , in this paper we concentrate our empirical analysis on pair-wise correlations.

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The volatilities are parameterized as σi (θi , f ) = g(i,ξ,θ (f ) , ˜ i)

i = 1, . . . , N.

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˜ The functions g(i,ξ,θ ˜ i ) are defined analogously to the function h(ξ,θ0 ) , with a vector ξ =

ξ˜l l=1,...,nσ of nσ discretization points and values θi = θli l=1,...,nσ . The parameterizations (6),(7) are flexible and able to capture nonlinear relationships. We find that spline functions with nρ = nσ = 4 discretization points yield a good description of the dependence of volatilities and correlations on our state F ; our empirical findings are robust with respect to the 2 3

The volatility of the VIX is based on levels instead of changes of the index. For a parametrization of a full correlation matrix see Becker and Schmidt [2013].

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ACCEPTED MANUSCRIPT choice of nρ , nσ . For the drift µi we use a constant µi (θ, t) = θiµ . Alternatively, the drift could be modeled also as a function of the market state F , however, it turns out that our empirical findings are

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quite insensitive to the choice of the drift, see Appendix A.3, Tables 4 - 6.
To sum up, the model parameters are θ = θ0 , θ1 , . . . , θn , θµ , nF . We provide an estimation method for our model parameters in Appendix A.

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Empirical results

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In a preliminary analysis we motivate that there is a strong, nonlinear influence of the observed nF

market trend or volatility F (St← ) on both correlations and volatilities when there are extreme

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market movements. In Figure 1 we plot4 estimated correlations and volatilities against the trend of the S&P 500, as defined in (3) with S 1 =S&P 500 and ω1 = 1. The estimated correlations and volatilities have to be interpreted with care, since at this stage we merely use pragmatic estimators that are inspired by Barndorff-Nielsen and Shephard [2004], see Appendix A.1 for details. To identify potential functional relationships ρ(F ) and σi (F ) we use

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non-parametric regressions5 . Figure 2 shows the corresponding relationships for the volatility

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We use the function smoothScatter in R. We apply the Nadaraya-Watson estimator ’ksmooth’ from the ’stats’-package in R.

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Figure 2: Typical plots of volatilities and correlations versus the realized volatility of the S&P 500 over a rolling window of nF = 15 business days. Plots are for Exxon - Microsoft, Jan. 2004- Sept. 2011. Figures 1 and 2 suggest that both, in strong market downturns and in market upturns,

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correlations and volatilities stabilize around a nonlinear function of the realized market trend or market volatility. In a normal market environment factors other than the market trend or volatility prevail. This preliminary analysis suggests that the market trend or volatility captures a substantial portion of the behavior of correlations and volatilities at least in market downturns and upturns.

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We estimate our model (1),(2) for pairs of large cap stocks from the S&P 500 using daily data from Jan. 2004 - Sept. 2011. For the dependence of volatilities and correlations on the market state F we use parameterization (6),(7). All analyzed pairs of stocks share a common pattern for the dependence of correlations and volatilities on the market state F 6 . Figure 3

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shows a typical example with the realized trend as market state variable: both correlations and volatilities increase in market downturns and respond to the market trend in a strongly

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nonlinear way. While volatilities and correlations mildly increase in a bull market, their increase in a downturn can be explosive. The severity of the downturn strongly influences the level of volatilities and correlations. 6 Estimates of volatility and correlation functions and the corresponding maximum of the likelihood function turn out to be insensitive to the choice of the drift in our model. This justifies that we assume constant drifts in our estimates, see also Appendix A.3, Tables 4 - 6.

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Figure 3: Typical dependency structure of correlation and volatilities on the realized market trend (3) of the S&P 500, with market memory nF = 95 business days. Dashed lines are 95% confidence bands, dotted lines are the non-parametric estimators from Figure 1. Data from Jan. 2004 - Sept. 2011.

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In Figure 4 we show the corresponding results for the realized volatility of the S&P 500

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Figure 4: Typical dependency structure of correlation and volatilities on the realized market volatility (4) of the S&P 500, with market memory nF = 15 business days. Dashed lines are 95% confidence bands, dotted lines are the non-parametric estimators from Figure 2. Data from Jan. 2004 - Sept. 2011. Finally, Figure 5 shows estimated relationships in case the driving state variable F is the

realized volatility of the VIX. In both, Figures 4 and 5, volatilities and correlation increase if the realized index volatility is high.

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Figure 5: Typical dependency structure of correlation and volatilities on the realized volatility (4) of the VIX index, with market memory nF = 20 business days. Dashed lines are 95% confidence bands, dotted lines are the non-parametric estimators. Data from Jan. 2004 Sept. 2011.

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We report detailed parameter estimates for Figures 3-5 in Appendix A.3, Tables 4, 7 and 8. How quickly the market state F responds to market developments depends on the market memory, that is, the number of days nF the observed trend or volatility looks into the past. Estimates for the market memory are about nF = 95 business days for the trend, about nF = 15 business days for the realized S&P 500 volatility and about nF = 20 days for the

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volatility of the VIX, see Appendix A.2 for details.

The strong increase in volatilities and correlations in a downturn has important economic implications, for example, it influences the benefits of portfolio diversification. Even more importantly, if volatilities and correlations of international stock market indices increase in a

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downturn in a similar way like individual stocks, there would be negative implications for the stability of the international financial system. We investigate whether there exists contagion

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in international financial markets and explore consequences for the potential of international diversification to stabilize portfolio risk in Section 4. Finally, let us compare the behavior in periods of market stress defined either via the

market trend, the market volatility or the volatility of the VIX as driving state variable F . By how much do volatilities and correlations change if the driving state F shifts from its median (50%-quantile, normal market) to its 95%-quantile (crisis)? We compute percentage changes in volatility and correlations relative to the volatility and correlation taken at the median and then average over all pairs of stock out of 14 large cap US stocks7 . Figure 6 shows, 7

We use AT&T, Apple, Exxon, Pfizer, Walt Disney, Colgate, Microsoft, Citigroup, Bank of America, Coca Cola, Walmart, General Electric, Halliburton, Johnson & Johnson.

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ACCEPTED MANUSCRIPT that for a shift of the market trend from its median to its 95% quantile volatilities increase from the level taken at the median by 200% and correlations by 100% on average. Volatilities respond most strongly to changes in the market trend, slightly less to changes in the market volatility or to changes in the volatility of the VIX. Correlations, on the other hand, respond

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stronger to changes in the volatility of the VIX. The sample standard deviations in observed increases in both volatilities and correlations are large. For the volatilities it is 92% in case of shift in the trend, 79% for shifted market volatility, and 37% in case of the volatility of the

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Figure 6: Average relative increase in volatilities and correlations as market trend, market volatility, and volatility of VIX shift from their median to the 95% quantile. Numbers are percentage increases relative to the level at the median of the market state, numbers in brackets are sample standard deviations. Data from 2004-2011.

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Figure 7 shows that the large variation in the increase of volatilities and correlations under market stress can be explained by the stocks’ factor loading on the HML-factor from Fama and French [1993], suggesting that the large spread is due to the different behavior of value and growth stocks. For each stock, we plot the increase in volatility against its factor loading8 . Regressions using a heteroscedasticity consistent estimator by MacKinnon and White [1985] yield that the slope is positive and statistically significant at the 1%-level for all market states, and the adjusted R2 is between 78% and 94%. 8

We regress daily excess returns of each stock on the Fama and French three factor model, using data from Jan 2004-Sep 2011. The estimated coefficient on the HML factor is displayed on the x-axis in Figure 7. Data on the Fama and French factors are from Kenneth French’s website.

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Figure 7: Relative increase in volatility for changes in market trend, market volatility, and volatility of VIX as market state, plotted against stocks’ sensitivity to the Fama/French factor HML. To compute the sensitivity we regress daily excess returns of stocks on the three Fama French factors, see Fama and French [1993]. We plot the increase in a stock’s volatility against the regression coefficient estimate for the HML factor. Numbers are percentage increases relative to the level at the median of the market state. Data from 2004-2011. For the variation of correlations, Table 1 shows that for shifts in market trend the correlation increase is particularly large for stocks with low factor loading on HML, and small for stocks with high factor loading. The differences in the correlation increase are statistically significant9 at the 5%-level. For changes in the market volatility the correlation increase is also larger for stocks with low factor loading on HML, but the differences are not statistically

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significant. For changes in the volatility of the VIX the correlation increase is similar for the different groups of stocks.

(a) Market Trend

hml > q50% 106% 72%

hml ≤ q50% hml > q50%

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hml ≤ q50% hml > q50%

hml ≤ q50% 115% 106%

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hml ≤ q50% 104% 77%

hml > q50% 77% 62%

(c) Volatility of VIX

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hml ≤ q50% 115% 109%

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hml > q50% 109% 107%

Table 1: Average increase in correlation for market trend, market volatility, and volatility of VIX as market state, for shifts from the median to the 95% quantile. Averages are over all pairs of stocks with regression coefficient of stock returns on the HML factor in the Fama and French three factor model that are either below or above the median q50% of observed betas. Our findings contribute to answering questions discussed in Bauwens and Otranto [2013], who studied the influence of volatility on correlation in various VDCC models. They find a statistically significant influence, but the marginal impact on correlation appears small and 9

We use one-sided t-tests.

13

ACCEPTED MANUSCRIPT depends on the model. Our analysis confirms that realized volatility influences correlation significantly, but, contrary to Bauwens and Otranto [2013], the marginal sensitivity of correlations to changes in volatility is substantial. Furthermore, we find evidence that the sensitivity of volatilities and correlations to changes in the market state is influenced strongly by stocks’

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sensitivity to the value/growth factor HML. Value stocks exhibit a much larger relative increase in volatility than growth stocks for all considered market states, and during market downturns the increase in correlation among value stocks is significantly larger than for growth

4

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stocks.

Is there contagion in international financial markets? How

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effective is international diversification?

We emphasize that in this section we understand contagion in a narrow sense as increase in correlation between indices of international financial markets in a downturn. This is one of the mainstream definitions of contagion in the literature, but various other definitions exist; for an overview see, for example, Pericoli and Sbracia [2003]. Estimating contagion is

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technically involved, because it is not straightforward to obtain valid estimates for correlations in specific market environments like a normal market versus a downturn. A popular approach is to compute the sample Pearson correlation coefficient for returns in each environment. But such conditional sample correlations depend on the volatility in the conditioning environment.

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As shown in Boyer et al. [1997], even if correlation is constant, by conditioning on an event where volatility is high, the corresponding conditional correlation increases, which suggests an

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increase in correlation that is actually not there. Therefore Boyer et al. [1997] adjust sample correlations for volatilities in different market environments; their approach is also applied by Loretan and English [2000]. Using a similar adjustment, Forbes and Rigobon [2002] claim that correlations do not increase in periods of market turmoil. But when we adjust sample correlations for changes in volatility a second problem appears: we need reliable estimates for volatilities in characteristic market environments. In particular, overstating volatility in a downturn produces adjusted correlations that are too low, causing a bias towards accepting the hypothesis that correlations do not increase. Corsetti et al. [2005] challenge the claim that correlations remain stable in a crisis. They argue that correlations should be adjusted

14

ACCEPTED MANUSCRIPT for the volatility of the common factor that drives contagion effects in both markets instead of the larger volatilities observed in individual markets. Our approach provides a method to adjust correlation for the common factor. As an example, we analyze whether there exists contagion between the US and China,

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and between the US and the UK. The first pair represents contagion between a developed and an emerging market, the second pair comprises contagion among two developed markets. We investigate the S&P 500, the FTSE 100, and the Hang Seng Index10 and estimate the volatilities and correlation in our model as functions of the state variable F . For data, we use

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weekly instead of daily market prices to take into account that international financial markets operate in different time zones.

Figure 8 shows the estimated volatility and correlation functions for the S&P 500 and the

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Hang Seng Index plotted against the observed trend of the MSCI World Index. For comparison

1.0 −0.5

0.0

market state F

0.5

0.2

−0.5

0.0

market state F

(b) Volatility of Hang Seng

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(a) Volatility of S&P 500

−1.0

0.0

0.2 0.0

−1.0

0.4

correlation

0.6 0.4

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volatility

0.6 0.4 0.0

0.2

volatility

0.6

0.8

0.8

1.0

and correlation.

0.8

we also show volatilities and correlation estimated in a model that assumes constant volatilities

0.5

−1.0

−0.5

0.0

0.5

market state F

(c) Correlation S&P 500 - Hang Seng

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Figure 8: Volatilities and correlation of the S&P 500 and the Hang Seng Index converted to US Dollar plotted against the trend of the MSCI World Index. Dotted lines are 95%-confidence bands. Horizontal dotted lines are estimated constant quantities. Weekly data from Jan. 2004 - Sept. 2011. The increase of state dependent correlation from 51% to 68% in a downturn indicates

contagion effects between the US and China. However, as the confidence bands for the correlation estimate are quite wide, a definite statement cannot be made. On the other hand, Figure 8 (a)(b) shows that, in a downturn, volatilities increase from 11% to 64%, resp. 16% to 51%, which has implications for the effectiveness of diversification that are investigated 10

The FTSE 100 and the Hang Seng Index are converted to US Dollar to exclude the influence of currency fluctuations.

15

ACCEPTED MANUSCRIPT in more detail below. Figure 9 shows the corresponding analysis of contagion between the US and the UK. We observe a significantly higher level of correlation compared to US-China, indicating a closer dependence between the two economies. But looking at correlation in a downturn we find no

0.8

1.0 0.0

0.5

market state F

−0.5

0.0

0.5

market state F

(a) Volatility of S&P 500

0.4 0.2

−1.0

−1.0

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−0.5

0.0

0.2 0.0 −1.0

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correlation

0.6

volatility

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0.6 0.4 0.0

0.2

volatility

0.6

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0.8

1.0

significantly different from its long term average.

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evidence for contagion between the US and the UK. The state dependent correlation is not

(b) Volatility of FTSE 100

−0.5

0.0

0.5

market state F

(c) Correlation S&P 500 - FTSE 100

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Figure 9: Volatilities and correlation of the S&P 500 and the FTSE 100 converted to US Dollar plotted against the trend of the MSCI World Index. Dotted lines are 95%-confidence bands. Horizontal dotted lines are estimated constant quantities. Weekly data from Jan. 2004 - Sept. 2011. We also investigate alternative choices of the driving factor F : the trend or volatility of the S&P 500 alone, or the average trend or volatility of the S&P 500 and the Hang Seng Index resp. FTSE 100, see Appendix A.3, Tables 9 and 10. The level of correlations and volatilities

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under market stress is similar for different choices of F . What do our findings imply for the effectiveness of international diversification? Studies

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that emphasize the benefits of diversification like Solnik [1995] and Grubel [1968] assume that volatilities and correlations are constant. But Figures 8 and 9 indicate that a model with constant volatilities (and correlations) underestimates the true portfolio risk in a market downturn.

To assess the risk-reducing potential of international diversification we analyze the variance of an internationally diversified minimum variance portfolio

ωS 1 + (1 − ω)S 2 ,

with S 1 either the Hang Seng Index or the FTSE 100 and S 2 the S&P 500; all indices are 16

ACCEPTED MANUSCRIPT converted to US Dollar. So, we assess the benefits of international diversification from the viewpoint of a US investor in either an emerging market (China) or a developed market (UK). The minimum variance portfolio weight ω is 2 σ2,const − σ1,const σ2,const ρconst 2 2 σ1,const + σ2,const − 2σ1,const σ2,const ρconst

,

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ω=

and is based on a simple model with constant volatilities and correlations, see Table 2.

FTSE - S&P 500

σ1,const 23.2 (21.6, 24.8) 21.9 (20.4, 23.4)

σ2,const 19.3 (18.0, 20.7) 19.3 (18.0, 20.7)

ρconst 58.2 (50.7, 63.9) 72.5 (67.0, 76.5)

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Assets Hang Seng - S&P 500

ω 28.7 27.6

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Table 2: Constant volatilities and correlations and the corresponding optimal exposure ω to different international indices in a minimum variance portfolio. All numbers in percent, with 95%-confidence intervals in brackets. Weekly data from Jan. 2004 - Sept. 2011. We compare the state dependent portfolio volatility

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q F 7→ ω 2 σ12 (F ) + (1 − ω)2 σ22 (F ) + 2ω(1 − ω)σ1 (F )σ2 (F )ρ(F )

(8)

with the volatility σ2 (F ) of an investment in the S&P 500 alone by computing the ratio of equation (8) over σ2 (F ). International diversification is beneficial if the ratio is smaller than one, investing only locally in the US stock market is beneficial if the ratio is greater than one.

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Figure 10 shows the ratio in crisis scenarios for different choices of the driving factor F : the

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trend of the S&P 500, the volatility of the S&P 500, and the volatility of the VIX index.

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0.15

0.20

0.25

0.95 0.90

Trend of S&P 500 Volatility of S&P 500 Volatility of VIX

0.05

sample quantile of market state

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0.10

1.00

1.05 0.05

0.85

ratio of portfolio volatilities

0.80

Trend of S&P 500 Volatility of S&P 500 Volatility of VIX

0.80

1.00 0.95 0.90 0.85

ratio of portfolio volatilities

1.05

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0.10

0.15

0.20

0.25

sample quantile of market state

(a) FTSE 100, S&P 500

(b) Hang Seng Index, S&P 500

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Figure 10: Ratio of portfolio volatilities (8) over σ2 (F ) plotted against the observed market state F . The state F is either the trend of the S&P 500, the volatility of the S&P 500, or the volatility of the VIX index. For the latter two the x-axis refers to 100% minus sample quantile. Weekly data from Jan. 2004 - Sept. 2011. Figure 10(a) shows that in periods of market stress (say the range below the 5%-quantile of the market state) for a US investor diversification in a developed market like the UK reduces volatility by about 7% compared to investing in the US market alone. Diversification in an emerging market like China reduces volatility by about 12%. Observe that the results are

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similar for the different state variables defining a crisis. Why is diversification under market stress still beneficial, even though the correlations increase or stay close to constant? The diversification gain, that is, the ratio of (8) over σ2 (F ) is also driven by the ratio of volatilities σ1 (F )/σ2 (F ), where, in our examples, under stress σ2 (F ) increases much faster than σ1 (F ).

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Diversification benefits are determined not only by the behavior of correlations under stress but also by the relative behavior of volatilities.

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From the perspective of a US investor, international diversification in China (emerging

market) is more beneficial than in the UK (developed market). Our findings confirm results in You and Daigler [2010] and Christoffersen et al. [2012].

5

Comparison with multivariate GARCH models

Multivariate GARCH models are standard for modeling the dynamics of volatilities and correlations; see, for example, Silvennoinen and Teräsvirta [2009].

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ACCEPTED MANUSCRIPT For a vector of log returns

X(t) = X 1 (t), . . . , X N (t)




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with discrete time t = 1, 2, . . . multivariate GARCH models describe the dynamics11 of X via

X(t + 1) = C(t)(t + 1).

(9)

The matrix C(t) is the Cholesky decomposition of a covariance matrix H(t) ∈ RN,N , which

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by assumption depends on the observations up to time t. The RN -valued random process

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 has independent, identically distributed realizations (t), t = 1, 2, . . . . The random vector
(t) has components 1 (t), . . . , N (t) with covariances


Cov i (t), j (t) =

   1,

i=j

  0,

i 6= j

.


Under the assumption E i (t) = 0 the returns are centered. For normally distributed (t)

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the dynamics (9) yields a multivariate normal distribution conditional on the observations up to time t for the returns,

X(t + 1) ∼ N (0, H(t)) .

(10)

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The core element of multivariate GARCH models is the updating rule for the covariance matrix H(t) that is based on past observations of the covariance matrix H and past returns

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X. One group of models defines linear updating rules, for example, the VEC-model by Bollerslev et al. [1988], a second group introduces nonlinear updating rules, for example, the Dynamic Conditional Correlation GARCH model by Engle [2002]. The difference between our model and multivariate GARCH models is the explicit and

flexible dependence of the covariance matrix H(t) on the realized market trend or market volatility. Instead of updating the covariance matrix based directly on past observations of individual returns and covariance matrices, in our model first the state F of the market is updated, which in turn determines the covariance matrix. So far this has not been investigated 11

We use a standard definition for multivariate GARCH models, see for example McNeil et al. [2005], Section 4.6.1. Our Cholesky factor C of the covariance matrix H is a lower triangular matrix that is commonly denoted by H 1/2 , see for example McNeil et al. [2005], Section 3.1.1.

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ACCEPTED MANUSCRIPT in a systematic way. How well does our model (1),(2) capture the real-world dynamics of volatilities and correlations in comparison to GARCH models? To answer this question we follow an approach suggested by Engle and Colacito [2006], which has been applied in Colacito et al. [2011] in a

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similar context. They compare different models by optimizing a portfolio based on volatilities and correlations that are predicted by the corresponding model. The model that yields the smallest portfolio variance is then best able to capture the real dynamics of volatilities and correlations.

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For the period from time t to t + 1 we denote the asset weights in the portfolio by w(t) = (w1 (t), w2 (t)). The return of the asset portfolio is then w1 (t)X 1 (t+1)+w2 (t)X 2 (t+1).

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The covariance matrix H(t) = (hi,j (t)) is predicted by the model at time t, so the predicted portfolio volatility is q

w12 (t)h1,1 (t) + 2w1 (t)w2 (t)h1,2 (t) + w22 (t)h2,2 (t).

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Denote by µ = (µ1 , µ2 ) the vector of expected excess returns, µi = E(X i (t) − rf ),

i = 1, 2,

where we assume that these expectations are constant in t. For the period from t to t + 1,

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the asset weights w(t) minimizing the portfolio variance are the solution of the problem

min

w: wT (t)µ=µ0

wT (t)H(t)w(t),

(11)

where µ0 is the required excess return of our portfolio. The solution to problem (11) is

w(t) =

H −1 (t)µ µ0 . µT H −1 (t)µ

(12)

The conditional portfolio variance for the period from t to t + 1 is 
2 σ(t + 1) = Et wT (t) X(t + 1) − X ,

(13)

where X = Et X(t + 1) is the conditional mean of return X(t + 1) given the observations up 20

ACCEPTED MANUSCRIPT to time t. By Theorem 2 in Engle and Colacito [2006], the unknown conditional mean X in (13) can be safely approximated by the sample mean. The portfolio weights w(t) in (12) depend on the model predicted conditional covariance matrix H(t) and the vector µ. The corresponding weights using the true (however, unknown)

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conditional covariance matrix would lead to a different portfolio. Comparing the variances of these two portfolios Engle and Colacito [2006], Theorem 1, show that the latter variance is always smaller, no matter the choice of µ. This justifies that the model generating the smallest portfolio variance is considered to be superior.

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As in Colacito et al. [2011] we assume that µ = (µ0 , µ0 ). Since no asset outperforms, the portfolio can be interpreted as a global minimum variance portfolio. For the portfolio weights

w1 (t) =

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we then obtain h2,2 (t) − h1,2 (t) , h1,1 (t) + h2,2 (t) − 2h1,2 (t)

w2 (t) = 1 − w1 (t).

We compare the following models: the market state-dependent model (1), (2), with the market trend over 95 business days as market state (denoted by ‘StateTrend’ in the following

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tables); model (1), (2) with the market volatility over 15 business days as market state (‘StateVol’); model (1), (2) with the volatility of the VIX index over 20 business days as market state (‘StateVix’); the Dynamic Conditional Correlation GARCH model by Engle [2002] (denoted by ‘DCC’); and moving averages over the past 30 days (‘Avg30’) or over the

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past 90 days (‘Avg90’). We estimate the models for data from 2004 − 2011 for all pairs of 14 large cap US-stocks12 . Recall that we want to investigate how well the models capture

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the dynamics of volatilities and correlations on a given set of data. To this end we estimate the models and optimize the portfolios period by period in the time from 2004 − 2011; for all models, estimates are based on one and the same set of daily (trading day) data from Thomson Reuters. In Table 3 we list the percentage of asset pairs for which the model in the respective row yields a smaller portfolio variance than the model in the respective column. Knowledge about asset correlations can be particularly valuable in periods of market stress, that is, when correlations are high; see Engle and Colacito [2006]. To assess how well our model describes the dynamics of correlations under market stress we also compute the portfolio volatility on 12

We use AT & T, Apple, Exxon, Pfizer, Walt Disney, Colgate, Microsoft, Citigroup, Bank of America, Coca Cola, Walmart, General Electric, Halliburton, Johnson & Johnson.

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ACCEPTED MANUSCRIPT those days only when the market state is below its empirical 10%-quantile. DCC 55.0%(8/21/62) 71%(6/0/85) 57%(10/9/72) 78%(7/0/84) 64%(10/10/71) 90%(16/0/75)

Avg30 69%(22/7/62) 90%(9/0/82) 85%(26/1/64) 97%(8/0/83) 84%(34/3/54) 100%(25/0/66)

Avg90 57%(4/19/68) 84%(15/0/76) 63%(6/6/79) 91%(10/0/81) 73%(10/10/71) 97%(32/0/59)

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MinVar StateTrend bottom 10% StateVol bottom 10% StateVix bottom 10%

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Table 3: Percentage of asset pairs for which the model in the row yields a smaller portfolio variance than the model in the column. In brackets we state how often the portfolio variance is significantly smaller at the 5%-level for the model in the row (first number), for the model in the column (second number), and how often the difference in portfolio variance is not statistically significant. In total we employ 91 comparisons. ’Bottom 10%’ states the portfolio volatility on those days only when the market state is below its empirical 10%-quantile. Data from Jan 2004 - Sep 2011. For each market state – the trend of the market, the volatility of the market, and the volatility of the VIX – our model performs better than the other models in the majority of cases. In particular, our model performs significantly better in periods of stress, making the model especially useful for investigating the dynamics of correlations and volatilities in downturns. In terms of significance at the 5% level our model yields better results in almost

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all cases where significance can be assessed, in particular under stress. However, there is a large number of cases where none of the models is significantly different. The explanatory power of the considered market states differs: volatility of the VIX is more

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valuable than market volatility, which in turn is better than the market trend. Our findings suggest that realized volatility of the VIX provides most information about correlations and

6

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volatilities.

Conclusion

This paper contributes to understanding the determinants of correlations and volatilities of stock returns. Knowledge about these determinants is vital to identify international contagion effects and benefits of diversification in market downturns. We propose a new model where volatilities and correlations are functions of the realized trend, or the volatility of a market index, or the realized volatility of the VIX, all calculated on a rolling time window whose length we interpret as the memory of the market. Each of these drivers quantifies the severity of

22

ACCEPTED MANUSCRIPT market stress in a natural way, which makes the model particularly well suited for investigating correlations and volatilities during periods of market stress. Our model yields a more realistic dynamics of correlations and volatilities than conventional approaches. Volatilities and correlations exhibit a strong nonlinear dependence on each

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of the determinants, in particular, volatilities and correlations strongly increase during market stress. For the sensitivity of correlations and volatilities to changes in their determinants, we find that if the realized market trend moves from its empirical 50% to its 95% quantile, correlation increases by about 100% and volatilities by about 200% on average; variations in

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the stocks’ sensitivity can be explained by their factor loadings on the HML-factor, suggesting that this variation is due to the different behavior of value and growth stocks. The sensitivity of correlations to changes in the volatility of the VIX is strongest, whereas stock volatilities

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are most sensitive to changes in the market trend.

Estimates for the market memory are about 95 business days for the market trend, about 15 business days for the realized market volatility and about 20 days for the volatility of the VIX.

We apply our approach to analyzing contagion and the benefits of international diversi-

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fication. We find some indication for contagion between US and China, that is, correlations between the corresponding market indices increase in a crisis, but the increase is not statistically significant. For US and UK correlation remains stable. For a US investor, international diversification in China or the UK reduces volatility by 12% resp. 7% in a crisis, which is due

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to the heterogenous increase in volatilities. Our approach provides a natural framework to deal with the conditioning bias for corre-

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lations that is caused by the way volatilities are estimated in a crisis. To compare how well our models capture the dynamics of correlations and volatilities,

we follow Engle and Colacito [2006] and compare minimum variance portfolios based on correlations and volatilities that are computed by different models. Our approach performs better than DCC-Garch in the majority of cases and is better than 30 or 90 days moving averages, particularly in periods of market stress. Using information about the volatility of the market index or the volatility of the VIX provides superior portfolios than using information about the market trend.

23

ACCEPTED MANUSCRIPT

A

Estimation methods

A.1

Covariation based estimators

We first employ a pragmatic estimator based on the discrete covariation or variation of the underlying price processes to investigate the general relationship between the realized market nF

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trend or volatility F (St← ) on one hand and correlations and volatilities on the other hand. The motivation for our analysis in Figures 1 and 2 is first, to show that there is a relationship between realized market trend or volatility on one side and correlations/volatilities between stocks on the other side, and, second, to get an indication on appropriate parameterizations

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of the model.

Given realized time series data xi (t − lρ ), . . . , xi (t) of returns X i of assets S i , i = 1, . . . , N

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that follow our model (1), (2) the correlation between asset returns is approximately estimated by the quotient of the realized covariation and the square root of realized quadratic variations of the processes log S i ,

Plρ

=

− k + 1)xj (t − k + 1) . Plρ j (t − k + 1))2 i (t − k + 1))2 (x (x k=1 k=1 k=1 x

q Plρ

i (t

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ρˆij

(14)

In a similar way, we estimate volatilities σi from realized quadratic variations of the process log S i ,

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σbi

v u lσ u1 X t = (xi (t − k + 1))2 . lσ

(15)

k=1

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The estimates for correlation and volatility are based on the past lρ and lσ asset returns, which generally differ from the number nF of past asset returns to determine the state of the nF

market F (St← )13 .

The rationale for the estimators (14), (15) comes from the continuous time limit of our

model (1), (2), which is

d log S i (t) = µi (θ, t) dt + σi (θ, F (t)) dWt dWti dWtj 13

= ρij (θ, F (t)) dt,

We consider all combinations with nF , lρ , lσ ∈ {5, 10, 15, 20, 30, 40, 60, 90, 200, 250}.

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ACCEPTED MANUSCRIPT with Wiener processes W i , W j . In this model we have Rt

≈

σi (θ, F (u)) σj (θ, F (u)) ρij (θ, F (u)) du Rt 2 2 t−r (σi (θ, F (u))) du t−r (σj (θ, F (u))) du

qRt−r t 1 r

Z

t


ρij θ, F (u) du,

t−r

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(14) ≈

where r = lρ ∆ with time step size ∆. The last approximation is an equality if the volatilities are constant. In the same way

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s Z

2 1 r σi θ, F (u) du, (15) ≈ r t−r with r = lσ ∆.

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The asymptotic properties of this type of estimators are analyzed in Barndorff-Nielsen and Shephard [2004] for a slightly different model setting.

We emphasize that, at this stage, (14), (15) are just pragmatic estimators aimed at detecting the general relationship between the realized market state F and correlations or volatilities.

Maximum likelihood estimators

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A.2

We provide a maximum likelihood estimator for the parameter θ ∈ Θ in our model (1),(2). For this estimator we study consistency and the asymptotic distribution.

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Consider discrete observations

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s(t) t=0,...,K = s1 (t), . . . , sN (t) t=0,...,K

of the asset price processes S together with the observed log-returns


x(t) = x1 (t), . . . , xN (t) , xi (t) = log(si (t)) − log(si (t − 1)), t = 1, . . . K.

The conditional density of the return vector X(t + 1) = (X 1 (t + 1), . . . , X N (t + 1)),

Pθ



  nF n←F n←F ← X(t + 1) ∈ dx St = st ) = pt+1 x θ, st dx, x ∈ RN ,

is a N -variate normal density with expectation vector (µ1 (θ, t), . . . , µN (θ, t)) standard devi-

25

ACCEPTED MANUSCRIPT nF

nF

ations σi (θ, F (st← )), i = 1, . . . , N and correlations ρij (θ, F (st← )), i, j = 1, . . . , N . The loglikelihood function is given by

L(θ, nF ) =

K−1 X

 nF  log pt+1 x(t + 1) θ, st← .

(16)

The maximum likelihood estimator for the parameter θ is then

θbK = argmaxθ∈Θ L(θ, nF ).

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t=nF

(17)

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The estimation problem (17) is solved numerically. Remember from Section 2 that θ consists of parameters to describe volatilities and correlations by spline functions of the state variable nF

nF

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F (St← ). At this stage, the market memory nF in the market state F (St← ) (cf. (3),(4)) is assumed to be known in (17). In a subsequent step the market memory nF is estimated from

nc F = argmaxnF ∈{5,10,15,...,250} max L(θ, nF ).

(18)

θ∈Θ

Figure 11 shows that the peak of the maxima of the likelihood function and hence the es-

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timated market memory is about 90 (lower quartile) to 130 (upper quartile) business days

0

50

100

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200

market memory

(a) Microsoft - S&P 500

250

12350 12250

max. likelihood

12150

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max. likelihood

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12350 12250 12150

max. likelihood

12450

days14 .

12250 12300 12350 12400 12450

for the observed trend as state variable; the median estimated market memory is 95 business

0

50

100

150

200

market memory

(b) Walt Disney - S&P 500

250

0

50

100

150

200

250

market memory

(c) Pfizer - S&P 500

Figure 11: Estimated maxima of the likelihood function plotted against market memory nF for pairs of stock and S&P 500. The state F is the trend of the S&P 500, with market memory nF . Data from Jan. 2004 - Sept. 2011. 14 We consider AT&T, Apple, Exxon, Pfizer, Walt Disney, Colgate, Microsoft, Citigroup, Bank of America, Coca Cola, Walmart, General Electric, Halliburton, Johnson & Johnson.

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ACCEPTED MANUSCRIPT For the observed market volatility as state F , Figure 12 shows that the peak of the maxima

100

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250

12450 12400 12350

max. likelihood 0

50

market memory

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market memory

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market memory

(b) Walt Disney - S&P 500

(c) Pfizer - S&P 500

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(a) Microsoft - S&P 500

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50

12300

max. likelihood 0

12350 12400 12450 12500 12550 12600

12500 12400 12300

max. likelihood

of the likelihood function; the estimated market memory is 15 business days.

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Figure 12: Estimated maxima of the likelihood function plotted against market memory nF for pairs of stock and S&P 500. The state F is the volatility of the S&P 500, with market memory nF . Data from Jan. 2004 - Sept. 2011. For the observed volatility of the VIX as state F , Figure 13 shows that the peak of the

50

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150

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250

0

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market memory

(a) Microsoft - S&P 500

12350 12250

12300

0

12300

max. likelihood

12450 12350

12400

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max. likelihood

12400 12300 12200

max. likelihood

12500

12400

maxima of the likelihood function; the estimated market memory is 20 business days.

100

150

200

250

market memory

(b) Walt Disney - S&P 500

0

50

100

150

200

250

market memory

(c) Pfizer - S&P 500

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Figure 13: Estimated maxima of the likelihood function plotted against market memory nF for pairs of stock and S&P 500. The state F is the volatility of the VIX, with market memory nF . Data from Jan. 2004 - Sept. 2011. Using well-known martingale limit techniques, e.g., from Hall and Heyde [1980], Chapter 6,

under technical conditions we can derive the following asymptotic distribution of the estimator θbK as the number K of observations tends to infinity, d

1/2

(hG(θ)iK )(θˆK − θ) −→ N(0, Ip ).

(19)

Here GK (θ) is the score function, that is, the derivative of the log-likelihood function with 27

ACCEPTED MANUSCRIPT respect to the unknown p-dimensional parameter θ. It is a p-dimensional function with coordinates

GiK (θ)

=

K−1 X

uit+1 (θ)

(20)

t=nF

=

  nF ∂ ← , i = 1, . . . , p. log pt+1 X(t + 1) θ, St ∂θi

Their matrix of covariations hG(θ)iN is given by

hG(θ)iij K

=

K X

(21)

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uit+1 (θ)

  Et uit+1 (θ)ujt+1 (θ) ,

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t=nF

which we compute by Monte-Carlo simulation. Result (19) is the basis for our estimates of

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confidence intervals.

Results of parameter estimations

We report parameter estimates for Figures 3–5. The numbers in brackets are the 95% confidence intervals.

µ2

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σ1 (θ1 , f )

θ2i

θ3i

θ4i

21.5 (20.6, 22.5) 23.8 (22.7, 24.9) 35.2 (29.5, 40.4)

18.9 (18.0, 19.8) 17.2 (16.3, 18.1) 35.3 (29.0, 40.9)

18.3 (15.6, 21.0) 25.9 (22.3, 29.5) 36.2 (14.9, 51.2)

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µ1

θ1i 8.9 (−6.0, 23.7) 0.7 (−14.3, 15.7) 79.2 (68.9, 89.6) 72.6 (62.7, 82.5) 83.7 (75.9, 87.8)

σ2 (θ2 , f )

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ρ(θ0 , f )

Table 4: Parameter estimates for correlation and (annualized) volatility functions in Figure 3 assuming a constant drift. Model uses market trend over 95 business days as market state. Confidence intervals at 95% in brackets. Data for Exxon - Microsoft, from Jan. 2004Sept. 2011. All numbers in percent.

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µ2 σ1 (θ1 , f ) σ2 (θ2 , f ) ρ(θ0 , f )

θ2i 3.8 (−18.0, 25.6) 10.2 (−13.9, 34.2) 21.5 (20.6, 22.5) 23.8 (22.7, 24.9) 35.3 (29.5, 40.5)

θ3i 11.7 (−9.1, 32.6) 1.0 (−19.1, 21.1) 18.9 (18.0, 19.8) 17.2 (16.3, 18.1) 35.3 (29.0, 40.8)

θ4i 25.6 (−43.6, 94.9) 24.2 (−62.6, 110.9) 18.2 (15.5, 20.9) 25.9 (22.3, 29.5) 36.7 (15.3, 51.7)

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µ1

θ1i 69.8 (−169.7, 309.2) 1.7 (−225.7, 229.1) 79.2 (68.6, 89.8) 72.8 (62.8, 82.8) 83.8 (75.9, 87.8)

µ1 µ2 σ1 (θ1 , f ) σ2 (θ2 , f )

θ3i

θ4i

21.5 (20.6, 22.5) 23.8 (22.7, 24.9) 35.3 (29.5, 40.5)

18.9 (18.0, 19.8) 17.2 (16.3, 18.1) 35.2 (29.0, 40.8)

18.4 (15.6, 21.1) 25.9 (22.4, 29.5) 36.3 (15.1, 51.3)

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ρ(θ0 , f )

θ2i

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θ1i 0 0 79.2 (68.8, 89.6) 72.6 (62.7, 82.5) 83.7 (75.7, 87.8)

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Table 5: Parameter estimates for drift, correlation and (annualized) volatility functions in Figure 3. Model uses market trend over 95 business days as market state. Confidence intervals at 95% in brackets. Data for Exxon - Microsoft, from Jan. 2004-Sept. 2011. All numbers in percent.

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Table 6: Parameter estimates for correlation and (annualized) volatility functions in Figure 3, and drift µi ≡ 0. Model uses market trend over 95 business days as market state. Confidence intervals at 95% in brackets. Data for Exxon - Microsoft, from Jan. 2004-Sept. 2011. All numbers in percent.

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µ1 µ2

σ1 (θ1 , f ) σ2 (θ2 , f ) ρ(θ0 , f )

θ1i 11.0 (−3.3, 25.3) 3.8 (−10.5, 18.0) 17.6 (15.7, 19.6) 14.8 (13.1, 16.5) 29.3 (14.0, 41.6)

θ2i

θ3i

θ4i

18.6 (17.7, 19.5) 17.8 (16.9, 18.6) 25.5 (19.2, 31.3)

21.5 (20.6, 22.5) 24.3 (23.3, 25.3) 37.8 (32.2, 42.8)

87.4 (72.8, 102.0) 75.7 (62.6, 88.9) 83.4 (70.6, 88.5)

Table 7: Parameter estimates for correlation function and (annualized) volatility functions in Figure 4 assuming a constant drift. Model uses market volatility over 15 business days as market state. Confidence intervals at 95% in brackets. Data for Exxon - Microsoft, from Jan. 2004-Sept. 2011. All numbers in percent.

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ACCEPTED MANUSCRIPT µ1 µ2 σ1 (θ1 , f ) σ2 (θ2 , f ) ρ(θ0 , f )

θ2i

θ3i

θ4i

19.6 (18.7, 20.6) 17.6 (16.7, 18.5) 28.8 (22.2, 34.9)

21.7 (20.7, 22.6) 24.4 (23.4, 25.5) 39.7 (34.2, 44.7)

73.3 (62.4, 84.2) 65.9 (56.1, 75.7) 86.0 (77.5, 89.9)

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θ1i 9.9 (−4.8, 24.5) 3.9 (−10.7, 18.6) 18.2 (16.4, 20.0) 18.1 (16.3, 19.8) 15.0 (0.6, 28.0)

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Table 8: Parameter estimates for correlation function and (annualized) volatility functions in Figure 5 assuming a constant drift. Model uses market volatility of the VIX index over 20 business days as market state. Confidence intervals at 95% in brackets. Data for Exxon Microsoft, from Jan. 2004-Sept. 2011. All numbers in percent.

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The contagion analysis in Section 4, Figures 8 and 9 was based on the trend of the MSCI index as driving market state variable. Tables 9 and 10 show estimates for volatilities and correlations assessed at the 1%-quantile of different market state factors, confirming that the choice of factor defining the crisis scenario does not matter. To complete the picture the lower parts in Tables 9 and 10 show the corresponding estimation results in case the driving market

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state is the realized volatility. Market state F Trend of MSCI

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Trend of SP

Avg. Trend of SP and HS

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Volatility of MSCI Volatility of SP

Avg. Vol of SP and HS

Vol. HS 51.4 (34.7, 68.0) 53.6 (36.8, 70.4) 60.2 (42.6, 77.7) 49.8 (31.0, 68.4) 50.4 (32.1, 68.7) 47.6 (28.0, 67.2)

Vol. SP 64.5 (45.8, 83.2) 74.4 (53.7, 95.2) 60.5 (43.2, 77.9) 71.8 (46.3, 97.3) 66.9 (43.8, 90.0) 70.8 (42.7, 98.9)

Correlation 67.9 (5.0, 82.5) 71.2 (12.7, 84.1) 73.6 (21.6, 85.1) 67.1 (−7.5, 82.9) 66.4 (−4.5, 82.3) 73.7 (−27.8, 87.5)

Table 9: Volatilities and correlation of the S&P 500 (SP) and the Hang Seng Index (HS) assessed at the 1%-quantile of the market state if defined as a trend resp. assessed at the 99%-quantile of the market state if defined as a volatility; 95% confidence bands in brackets. Weekly data from Jan. 2004 - Sept. 2011.

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Trend of SP Avg. Trend SP and FTSE Volatility of MSCI Volatility of SP Avg. Vol SP and FTSE

Vol. FTSE 63.8 (44.9, 82.7) 68.6 (48.5, 88.7) 68.0 (47.7, 88.3) 58.3 (36.7, 80.0) 59.0 (37.9, 80.0) 59.1 (37.2, 81.0)

Vol. SP 66.2 (47.7, 84.7) 73.9 (53.3, 94.4) 68.4 (48.9, 87.9) 72.3 (47.4, 97.1) 67.2 (44.4, 89.9) 70.1 (45.7, 94.6)

Correlation 68.4 (0.3, 83.2) 76.7 (17.5, 87.3) 72.9 (6.8, 85.5) 71.1 (3.5, 84.6) 70.8 (1.9, 84.5) 71.1 (−0.2, 84.8)

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Market state F Trend of MSCI

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Table 10: Volatilities and correlation of the S&P 500 (SP) and the FTSE 100 (FTSE) assessed at the 1%-quantile of the market state if defined as a trend resp. assessed at the 99%-quantile of the market state if defined as a volatility; 95% confidence bands in brackets. Weekly data from Jan. 2004 - Sept. 2011.

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*Title Page (including Author names)

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How past market movements affect correlation and volatility Christoph Becker∗

Wolfgang M. Schmidt†

Abstract

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July 9, 2014

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The influence of past stock price movements on correlations and volatilities is essential for understanding diversification and contagion in financial markets. We develop a model that makes the influence of past returns, aggregated into driving factors for correlations and volatilities, explicit. Employing information about recent market movements leads to a more realistic model for the behavior of stock returns in a downturn than conventional models. Our approach offers a fresh perspective on the behavior of stock markets, and provides an alternative to the concept of exceedance correlation. For a US investor we find that international diversification in China or the UK remains beneficial in a crisis.

Keywords: correlation, volatility, financial contagion, diversification, exceedance correlation, GARCH models

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JEL: C13, C32, C58, G11, G12

This paper is part of a research project that was funded by the Frankfurt Institute for Risk Management (FIRM). ∗ Frankfurt School of Finance & Management, Sonnemannstr. 9-11, 60314 Frankfurt am Main, e-mail: [email protected] † Corresponding author. Frankfurt School of Finance & Management, Sonnemannstr. 9-11, 60314 Frankfurt am Main, e-mail: [email protected], Tel. +4969154008707

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