Pfm in Crisis

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DYNAMIC CORRELATIONS, ESTIMATION RISK, AND

PORTFOLIO MANAGEMENT DURING

THE FINANCIAL CRISIS

Luis Garca-lvarez and Richard Luger

CEMFI Working Paper No. 1103

April 2011

CEMFICasado del Alisal 5; 28014 Madrid

Tel. (34) 914 290 551 Fax (34) 914 291 056Internet: www.cemfi.es


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CEMFI Working Paper 1103April 2011

DYNAMIC CORRELATIONS, ESTIMATION RISK, AND

PORFOLIO MANAGEMENT DURING

THE FINANCIAL CRISIS

Abstract

We evaluate alternative multivariate models of dynamic correlations in terms of realizedout-of-sample Sharpe ratios for an active portfolio manager who rebalances a portfolioof international equities on a daily basis. The evaluation period covers the recentfinancial crisis which was marked by increased volatility and correlations across

international stock markets. Our results show that international correlations fluctuateconsiderably from day to day, but we find no evidence of decoupling between emergingand developed stock markets. We also find that the recursively updated dynamiccorrelation models display remarkably stable parameter estimates over time, but thatnone yields statistically better portfolio performances than the naive diversificationbenchmark strategy. The results clearly show the erosive effects of model estimationrisk and transactions costs, the benefits of limiting short sales, and the far greaterimportance of including a risk-free security in the asset mix whether or not marketturbulence is high.

.

Keywords: Portfolio selection, DC model, international diversification, decouplinghypothesis, estimation risk short-sale constraints.

JEL Codes: C53, C58,G11, G15.

Luis Garca-lvarezCEMFI

[email protected]

Richard LugerJ. Mack Robinson College of Business

[email protected]


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1 Introduction

The classical theory of portfolio selection pioneered by Markowitz (1952) remains to this

day an immensely prominent approach to asset allocation and active portfolio manage-

ment. The key inputs to this approach are the expected returns and the covariance

matrix of the assets under consideration in the portfolio selection problem. According

to the classical theory, optimal portfolio weights can be found by minimizing the vari-

ance of the portfolios returns subject to the constraint that the expected portfolio return

achieves a specified target value. In practical applications of mean-variance portfolio the-

ory, the expected returns and the covariance matrix of asset returns obviously need to be

estimated from the historical data. As with any model with unknown parameters, this

immediately gives rise to the well-known problem of estimation risk; i.e., the estimated

optimal portfolio rule is subject to parameter uncertainty and can thus be substantially

different from the true optimal rule.

The implementation of mean-variance portfolios with inputs estimated via their sample

analogues is notorious for producing extreme portfolio weights that fluctuate substantially

over time and perform poorly out of sample; see Hodges and Brealey (1972), Michaud

(1989), Best and Grauer (1991), and Litterman (2003). A recent study by DeMiguel, Gar-

lappi, and Uppal (2009) casts further doubt on the usefulness of estimated mean-variance

portfolio rules when compared to a naive diversification rule. The naive strategy, or 1/N

rule, simply invests equally across the N assets under consideration, relying neither on

any model nor on any data. Those authors consider various static asset-allocation models

at the monthly frequency and find that the asset misallocation errors of the suboptimal

(from the mean-variance perspective) 1/N rule are smaller than those of the optimizing

models in the presence of estimation risk. See also Jobson and Korkie (1980), Michaud

and Michaud (2008), and Duchin and Levy (2009) for more on the issue of estimation

errors in the implementation of Markowitz portfolios.

Kirby and Ostdiek (2011) show that the research design in DeMiguel, Garlappi, andUppal (2009) places the mean-variance model at an inherent disadvantage relative to naive

diversification because it focuses on the tangency portfolio which targets a conditional

expected return that greatly exceeds the conditional expected return of the 1/N strategy.

The result is that estimation risk is magnified which in turn leads to excessive portfolio

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turnover and hence poor out-of-sample performance. Kirby and Ostdiek (2011) argue that

if the mean-variance model is implemented by targeting the conditional expected return

of the 1/N portfolio, then the resulting static mean-variance efficient strategies can out-

perform naive diversification for most of the monthly data sets considered by DeMiguel,

Garlappi, and Uppal(2009). Kirby and Ostdiek (2011) note, however, that this finding is

not robust to the presence of transactions costs.

The correlation structure across assets is a key feature of the portfolio allocation

problem since it determines the riskiness of the investment position. It is well known

that these correlations vary over time and the econometrics literature has proposed many

specifications to model the dynamic movements and co-movements among financial asset

returns, which are quite pronounced at the daily frequency. The importance of dynamic

correlation modeling is obvious for risk management because the risk of a portfolio dependsnot on what the correlations were in the past, but on what they will be in the future.

Engle and Colacito (2006) take the asset allocation perspective to measure the value

of modeling dynamic correlation structures. Just like DeMiguel, Garlappi, and Uppal

(2009) and Kirby and Ostdiek (2011) (and others), they too study the classical asset

allocation problem, but with forward-looking correlation forecasts obtained from dynamic

correlation models. They consider the realized volatility of optimized portfolios and find

that it is smallest when the dynamic correlation model is correctly specified. Their focus,

however, is primarily on a setting with only two assetsa stock and a bondand hencewith relatively little estimation risk since few parameters need to be estimated.

In this paper, we further contribute to the literature on portfolio management in the

presence of estimation risk by considering an active portfolio manager who uses forecasts

from dynamic correlation models to rebalance an international equity portfolio on a daily

basis. The asset mix consists of dollar-denominated Morgan Stanley Capital International

(MSCI) stock market indices for five geographical areas: North America, Europe, Pacific,

Latin America, and Asia. The first three of those are developed market indices, whereas

the last two represent emerging market indices. The MSCI indices represent a broad

aggregation of national equity markets and are the leading benchmarks for international

portfolio managers. This choice of indices allows us to examine whether there is any evi-

dence supporting the decoupling hypothesis in international finance, namely that recently

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the evolution of emerging stock markets has decoupled itself from the evolution of more

developed stock markets.

In addition to the usual plug-in method which simply replaces the covariance matrix

by its sample counterpart, we also consider four popular models to forecast the inputs to

the portfolio selection problem. These include the constant conditional correlation (CCC)

model of Bollerslev (1990), the dynamic conditional correlation (DCC) model of Engle

(2002), the time-varying correlation (TVC) model of Tse and Tsui (2002), and the recent

dynamic equi-correlation (DECO) model of Engle and Kelly (2009). Each of these models

builds on a decomposition of the conditional covariance matrix into a product involving the

conditional correlation matrix and a diagonal matrix of conditional standard deviations.

The conditional variances (standard deviations) are specified according to standard gen-

eralized autoregressive conditional heteroskedasticity (GARCH) models (Bollerslev 1986)and the return innovations are modeled according to a generalized version of the standard

multivariate Student-t distribution proposed by Bauwens and Laurent (2005) that allows

for asymmetries in each of the marginal distributions.

As its name implies, the CCC model restricts the conditional correlations between

each pair of assets to be time invariant. Of all the considered models, the CCC model is

the most tractable one. The DCC model follows the same structure as the CCC model but

allows the correlations to vary over time. The new information in each period about the

volatility-adjusted returns is used to update the correlation estimates in a GARCH-likerecursion. An important alternative to the DCC specification is the TVC model, which

updates the estimates using a rolling-window correlation estimator of the standardized

returns. The DECO model is a middle ground between the CCC model on one hand and

the DCC and TVC models on the other in that all pairwise correlations are restricted to

be equal. Engle and Kelly (2009) propose the DECO model recognizing the need to reduce

the number of parameters and hence estimation risk in dynamic correlation models. The

most sophisticated correlation models we consider have a total of 18 parameters that need

to be estimated, even though we follow Engle and Colacito (2006) and make use of the

correlation targeting method of Engle and Mezrich (1996) to reduce the dimensionality

problem; see Engle (2009) for a discussion of variance and correlation targeting.

Following DeMiguel, Garlappi, and Uppal (2009) and Kirby and Ostdiek (2011), the

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1/N rule serves as our benchmark strategy for comparison purposes and, as in Kirby

and Ostdiek, we level the playing field by using the conditional expected return of the

1/N rule as the target expected portfolio return. We hasten to emphasize that our goal

is not to advocate the use of the 1/N rule as an asset allocation strategy, but merely

to use it as a benchmark to assess the performance of the more sophisticated data- and

model-dependent strategies. The evaluation period covers the recent financial crisis which

was marked by increased volatility and correlations across international stock markets.

We consider portfolios that comprise risky assets only and we also examine the effects of

introducing a risk-free security in the asset mix. As in Frost and Savarino (1988), Chopra

(1993), and Jagannathan and Ma (2003), we examine whether the imposition of short-sale

constraints leads to improved portfolio performance.

We focus on the realized out-of-sample Sharpe ratio as the measure of portfolio per-formance, which is defined as the ratio of the realized excess return of an investment

to its standard deviation. This choice is motivated by the fact that the Sharpe ratio is

the most ubiquitous risk-adjusted measure used by financial market practitioners to rank

fund managers and to evaluate the attractiveness of investment strategies in general. For

example, Chicago-based Morningstar calculates the Sharpe ratio for mutual funds in its

Principia investment research and planning software. In order to gauge the statistical

significance of the differences between Sharpe ratios we use the block bootstrap inference

methodology proposed by Ledoit and Wolf (2008), which is designed specifically for thatpurpose.

At the same time, though, the Sharpe ratio is not without its limitations when it comes

to performance evaluation. Fundamentally, the Sharpe ratio is a measure that attempts

to calculate the amount of reward (excess return) per unit of risk (standard deviation)

of an investment. When excess returns are positive, a higher Sharpe ratio indicates a

better performance because it means a higher excess return or a smaller standard devia-

tion, or both. When excess returns are negative, however, the Sharpe ratio can yield an

unreasonable performance ranking. For example, consider two managed portfolios that

achieve the same negative excess return, but with different standard deviations. In that

case, the usual Sharpe ratio would indicate that the portfolio with the highest standard

deviation is the better one, even though it achieved the same negative expected return

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by taking riskier positions. Israelsen (2003, 2005) proposes a simple modification to the

Sharpe ratio that overcomes this shortcoming and yields a correct perfomance ranking

during periods of negative excess returns. We compute the values of this modified Sharpe

ratio and again test for statistical differences vis-a-vis the 1/N strategy using the Ledoit

and Wolf (2008) bootstrap methodology.

The remainder of the paper is organized as follows. Section 2 presents the modeling

framework. It begins with a description of the multivariate distribution of returns, and

then moves on to present the volatility specification and the various correlation specifica-

tions that we consider. Details about how the model parameters are estimated are also

given in that section. Section 3 presents the in-sample estimation results for each model

specification. Section 4 considers the portfolio management problem and presents the

out-of-sample portfolio performance results. Section 5 concludes.

2 Modeling framework

2.1 Multivariate distribution

Consider a collection ofN assets whose day-t returns are stacked in the N 1 vector yt.We assume that the daily returns can be represented as

yt = + t, (1)

t = H1/2t zt, (2)

where is the vector of expected returns, t is the vector of unexpected returns, and

zt is the vector of innovation terms with conditional moments E(zt|It1) = 0 a n dV ar(zt|It1) = IN, the identity matrix. Here It represents the information set avail-able up to time t. The conditional covariance matrix of yt given It1 is Ht and thematrix H

1/2t is its Cholesky factorization. In our empirical application we have N = 5

risky assets and, as will become clear, the number of parameters to be estimated in each

of our model specifications is fairly large. So to make the estimation simpler, we follow

the common practice and use auxiliary estimators for unconditional moments. The first

instance of this is that we replace in (1) by the vector of sample means, y.

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It is well known that financial asset returns exhibit excess kurtosis which is at odds

with models that assume a multivariate normal distribution of returns. While the com-

monly used multivariate Student-t distribution can capture heavy tails, it still restricts

returns to be symmetrically distributed around their means. In order to allow for pos-

sible return asymmetries, the zt innovation terms in (2) are assumed to be independent

and identically distributed random vectors following the multivariate skewed Student-t

distribution of Bauwens and Laurent (2005), which generalizes the standard multivariate

Student-t distribution by allowing each marginal distribution to have its own asymmetry

parameter. Specifically, the density of zt, given the shape parameters = (1,...,N),

i > 0, and the degrees-of-freedom parameter v > 2, is given by

f(zt; , v) = 2

N Ni=1

isi

1 + 2i ( v+N2 )

( v2 )(v 2)N2

1 +

zt zt

v 2 v+N

2

, (3)

where

zt = (z1,t,...,z

N,t)

,

zi,t = (sizi,t + mi)Iii ,

mi =( v12 )

v 2

( v2 )

i 1

i

,

s2i =

2i +12i 1 m2i ,

Ii =

1 if zi misi ,

1 if zi < misi .

Note that mi and s2i are functions of i, so they do not represent additional parameters.

Here 2i determines the skewness in the marginal distribution of zi,t. With this specifi-

cation, the marginal distribution of zi,t is symmetric when log i = 0 and it is skewed to

the right (left) when log i > 0 (< 0). See Bauwens and Laurent (2005) for the derivationand further discussion of this distribution. It should be clear that this specification also

implies the same degree of tail heaviness across marginals, controlled by the parameter

v. In a Bayesian setting, Jondeau and Rockinger (2008) consider a more general alterna-

tive where each marginal distribution has its own kurtosis parameter. Further discussion

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of these generalized multivariate Student-t distributions is found in Jondeau, Poon, and

Rockinger (2007).

2.2 Volatility specificationThe conditional covariance matrix ofyt is written in the familiar form

Ht = DtCtDt, (4)

where Dt = diag(h1/211,t,...,h

1/2NN,t) is a diagonal matrix of conditional standard deviations

and Ct is the conditional correlation matrix. The elements ofDt are the square roots

of the expected return variances based on the past information set, specified here as

GARCH(1,1) models of the form

hii,t = i + i2i,t1 + ihii,t1, (5)

where the restrictions i > 0, i, i 0, and i + i < 1 ensure that the conditionalvariance processes are stationary with unconditional (long-run) variances given by 2i =

i/i, where i = 1 i i (Bollerslev 1986). In order to reduce the dimension of theparameter space and the computational complexity of the estimation problem, we use the

variance targeting method proposed by Engle and Mezrich (1996). The method takes the

models in (5) and rewrittes them as

hii,t = i2i + i

2i,t1 + ihii,t1, (6)

with i + i + i = 1, and where the positivity and stationarity constraints become

i, 2i > 0, i 0, and i + i 1. From (6) we see that todays volatility, hii,t, is

a weighted average of the long-run variance, 2i , the square of the lagged unexpected

return, 2i,t1, and the lagged volatility, hii,t1. The parameter i is the weight on the

long-run variance in that average. Variance targeting here consists of replacing 2i in (6)

by the sample variance 2i = T

1Tt=1(yi,t yi)

2

and then estimating the is and isby maximum likelihood. We also follow the common practice and set the initial value

as hii,1 = 2i . See Francq, Horvath, and Zakoian (2009) for further discussion about the

method of variance targeting. With the volatilities in hand we can then define the vector

of standardized (or de-GARCHed) returns ut = (u1,t,...,uN,t) whose typical element

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is ui,t = i,t/

hii,t. If a portfolio of the assets comprising yt is formed with weights ,

then its conditional return variance is V ar(yt|It1) = Ht. In light of (4), thisexpression makes clear that the correlation matrix Ct is a key feature of the portfolio

management problem. We turn next to a description of the various correlation structures

that we consider in our empirical analysis.

2.3 CCC specification

Our starting point is the constant conditional correlation (CCC) model proposed by

Bollerslev (1990) in which the conditional correlations between each pair of asset returns

are restricted to be constant over time. So in the CCC model, the conditional covariance

matrix is defined as

Ht = DtCDt, (7)

where C = [ij ] is a well-defined correlation matrix; i.e., symmetric, positive definite,

and with ii = 1 for all i. As Engle (2009) discusses, it is particularly simple to obtain a

well-defined CCC matrix estimate if we use correlation targetinga direct analogue of the

variance targeting methodwhich consists of replacing C by its empirical counterpart:

C= diag

1

T

T

t=1uu

12

1

T

T

t=tuu

diag

1

T

T

t=1uu

12

, (8)

the sample correlation matrix of standardized returns.

2.4 DCC specification

The CCC model has a clear advantage of computational simplicity, but the assumption

of constant conditional correlations in (7) may be too restrictive for practical portfolio

management. The dynamic conditional correlation (DCC) model of Engle (2002) gener-

alizes the CCC model by allowing the correlations to vary over time. It is defined as in

(4) with the following correlation matrix:

CDC Ct = diag [Qt]12 Qtdiag[Qt]

12 , (9)

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where diag[Qt] is a matrix with the same diagonal as Qt and zero off-diagonal entries.

The matrix of quasi-correlations Qt evolves according to the GARCH-like recursion

Qt = (1 a b)Q + a(ut1ut1) + bQt1, (10)

where the persistence parameters a and b are non-negative scalars satisfying a + b j

qij,tqii,tqjj,t

, (13)

where is an N-vector of ones and qij,t is the (i, j)-th element of the Qt matrix in (10).The DCC-based equicorrelation (DECO) matrix is then

CDECOt = (1 t)IN + tJN, (14)

where JN is an N N matrix of ones. Engle and Kelly show that the transformationofCDC Ct in (9) to the equicorrelation structure via (13) and (14) results in a positive

definite CDECOt matrix.

2.7 Model estimation

Let denote a generic parameter vector. In the benchmark CCC specification, that

vector comprises 16 parameters (1,...,5, v , 1, 1,...,5, 5). The DCC, TVC, and DECO

specifications add two more parameters (a, b) for a total of 18. Given the sample of

asset returns y1,...,yT, we estimate each model by maximizing its sample log-likelihood

function L() = Tt=1 log f(yt|,It1) with respect to , wheref(yt

|,

It1) =

|Ht

|1/2fH1/2t (yt

)

|,

It1, (15)

subject to the models positivity and stationarity constraints. The functional form of the

multivariate density is given in (3) and the term |Ht|1/2 in (15) is the Jacobian factorthat arises in the transformation from zt to yt. We should emphasize that the non-normal

distribution used here does not allow the log-likelihood function to be decomposed as in

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Engle (2002), Engle and Sheppard (2005), and Engle and Kelly (2009). This means that

we cannot employ Engles two-step estimation approach, but instead we estimate the

parameters of the multivariate skewed Student-t distribution together with those of the

conditional variances and covariances processes in one step.1

3 Estimation results

Our empirical assessment uses daily data on dollar-denominated MSCI stock market in-

dices for five geographical areas: North America, Europe, Pacific, Latin America, and

Asia. The first three of those are developed market indices, whereas the last two rep-

resent emerging market indices. In international finance, the traditional argument for

reaping the benefits of international diversification has relied on the presence of low cross-country correlations. While the early literature studied developed markets, here we follow

more recent studies and also examine the benefits offered by emerging markets.2 The

MSCI indices represent a broad aggregation of national equity markets and are the lead-

ing benchmarks for international portfolio managers. The point of view adopted here is

that of a U.S. investor managing an international portfolio, but who does not hedge any

currency risk. The indices data were retrieved from Datastream for the sample period

covering July 16, 1990 to July 16, 2010. For each index, we computed a series of corre-

sponding log-returns and multiplied by 100 so they can be read as percentage returns.Figure 1 shows the five return series, each comprising 5219 observations. The presence of

volatility clustering effects is evident in each series and the increased volatility during the

financial crisis of 2008 is truly astounding.

Table 1 presents some summary statistics of the returns data. The top portion reports

the means, standard deviations, maximum and minimum values, skewness, and kurtosis

of each return series. Each series exhibits negative skewness, except the MSCI Pacific

1The computations were done in Fortran using IMSL. A quasi-Newton method with finite-difference

gradients was used for the maximization of the sample likelihood function of each model.2Early studies on the benefits of international diversification include Solnik (1974) for developed

markets and Errunza (1977) for emerging markets. More recent evidence is found in DeSantis and

Gerard (1997), Errunza, Hogan, and Hung (1999), Bekaert and Harvey (2000), and Christoffersen et al.

(2010).

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returns which have a slight positive skew. Moreover, we see that the kurtosis in every

case is much higher than it would be if the returns were normally distributed. The bottom

portion of Table 1 displays the sample correlation matrix for these five return series. It is

interesting to note how the pairwise correlations are related to geographic proximity. For

instance, Latin American returns have a 58% correlation with North American returns

and a 54% correlation with European ones, whereas the correlations with the farther

Pacific and Asian markets are much smaller (21% and 32%, respectively). This is further

illustrated by the 54% correlation between Asian and Pacific region returns.

Tables 25 show the maximum-likelihood parameter estimates of the CCC, DCC,

TVC, and DECO models, respectively, using the entire sample of returns data. We began

by estimating the CCC model and retained as final point estimates those that attained

the highest value of the log-likelihood function over a grid of initial values. We then usedthe CCC estimates as starting values for the estimation of the DCC, TVC, and DECO

models. In Tables 25, the numbers in parentheses are the standard errors associated

with each point estimate.

Each model reveals that the marginal distributions of return innovations are skewed to

the left (since log i < 0), except for the Pacific region returns under the CCC and DECO

models which appear more symmetric than those of the other regions. The degrees-of-

freedom parameter is around 7.7 in each case, implying tails far thicker than those of

a normal distribution. The GARCH parameter estimates are quite typical of what isusually found with daily returns; i.e., the weights on the long-run variances i and the

ARCH parameters i are quite small, while the GARCH parameters i are much closer

to one in magnitude.

The estimated a and b parameters in Tables 35 determine the persistence over time of

the respective model-implied conditional correlations. The DCC, TVC, and DECO models

each imply a great deal of persistence, as evidenced by the values of a + b all close to

one, which in turn implies very slow rates of mean-reversion in correlations. To illustrate

these effects, Figures 24 shows the model-implied correlations between the standardized

returns for the five regions. The horizontal lines in each plot are the correlation implied by

the estimated CCC model, which are seen to be slightly below the corresponding sample

correlations in Table 1. The solid lines in Figures 24 are associated with the DCC model,

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the dashed lines with the TVC model, and the dotted line (which is the same in each plot)

with the DECO model. The time-varying correlations are unmistakable in each case. Note

that the DCC and TVC models in each case imply very similar patterns over time, while

the DECO model exhibits a different pattern since it captures the average correlation

across all pairs of assets. It is interesting to observe the relative increases in correlations

during the financial crisis. For instance, the North America-Europe conditional correlation

(in the upper left plot) in Figure 2 rises from about 0.4 to 0.7 during that period. This

finding of increasing correlations between stock market indices during volatile periods

(typically bear markets) is in line with the inverse relationship between market value and

correlations documented in Longin and Solnik (1995, 2001), Ang and Chen (2002), and

Engle and Kelly (2009).

Recall that the decoupling hypothesis is the idea that recently the evolution of stockmarkets in emerging markets has started decoupling itself from the evolution of more

developed stock markets. This notion is clearly not supported by the evidence presented

here. Indeed, the overall picture that emerges from Figures 24 is that the correlations

between all pairs of indicesdeveloped and emergingfluctuate considerably over time,

either around the values implied by the CCC model or have been on an upward trend in the

last decade. Nowhere do we see evidence of recent downward trends in daily correlations,

which clearly contradicts the decoupling hypothesis. Christoffersen et al. (2010) reach

the same conclusion with weekly returns during the 19732009 period.3

Our focus, however, is not on the in-sample comparison of these models, but rather on

their out-of-sample performance in portfolio management. In the application that follows,

the models are used to produce forecasts for time t + 1 using only information available

up to time t, as would be done in real-time forecasting. For the forecasting exercise we

set aside the last 1000 observationsabout 4 years. So at time t each model is used to

produce the one-day-ahead forecasts t+1|t and Ht+1|t. The first of these one-day-ahead

forecasts is made on 15 September 2006, and every day the returns data available up to

that point in time are used to update the forecasts of the next day. Following Bauwens

3Despite the evidence of recent upward tends in correlations, Christoffersen et al. (2010) argue that

their findings of very low tail dependence at the end of their sample suggests that there are still benefits

from adding emerging markets to a portfolio.

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and Laurent (2005), the model parameters are reestimated every 50 trading daysabout 2

monthsusing the previous estimates as initial values for the numerical optimization. The

Appendix shows the full set of recursively updated parameter estimates for each model.

It is noteworthy that the parameters estimates change by very little between consecutive

updates, which is perhaps not surprising since each time they merely incorporate the most

recent 50 days worth of observations. What is remarkable though is that the recursively

updated parameter estimates for each of the four model specifications appear very stable

by any standard over the entire sample period, from the relatively tranquil days of 2006

07 through the tumultuous days of 200810 (see Figure 1). Indeed, the averages of the

standard deviations of each parameter series for the CCC, DCC, TVC, and DECO models

are 0.003, 0.002, 0.002, and 0.002, respectively. Note that Brownlees, Engle, and Kelly

(2009) have a similar finding with volatility models. It will be important to bear in mindthis remark about parameter stability when interpreting the portfolio performance results.

4 Application to portfolio management

4.1 Management strategies

We consider the problem faced by an active portfolio manager who rebalances a portfolio

of assets on a daily basis. Here we begin by assuming the absence of a risk-free security, sothe portfolio comprises only risky assets. The naive approach to this problem is simply to

invest the initial wealth each day equally across the N risky assets under consideration, so

the fraction of wealth in each asset is 1/N. This is referred to as the naive diversification

(or 1/N) rule. It should be noted that the 1/N rule doesnt entail any estimation risk

since it relies neither on the data nor on any model. And nonetheless DeMiguel, Garlappi,

and Uppal (2009) find that this portfolio strategy performs remarkably well in out-of-

sample comparisons against more sophisticated approaches based on classical portfolio

optimization using monthly returns data. Here we use the naive 1/N rule as our firstobvious benchmark for comparison purposes.

According to the classical theory of optimal portfolio selection by Markowitz (1952),

the manager allocates the wealth across the N assets so as to minimize the portfolios

variance subject to the constraint that the expected portfolio return attains a specified

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target, t . Without loss of generality, the initial wealth can be normalized to one so the

portfolio managers problem is to find the optimal normalized portfolio weights, t, as

the solution to:min

t

tHt+1|tt

s.t. tt+1|t = t ,

t1 = 1,

t 0,

(16)

where 1 is a vector of ones. The non-negativity constraints t 0 in this formulation ofher problem mean that the portfolio manager is prohibited from making short sales. We

shall also consider a version of (16) without those short-sale constraints so that the optimal

solution, t, may contain negative weights (short positions). The portfolio optimization

problem in (16) is a standard quadratic programming problem that is readily solvednumerically. In this formulation of the portfolio problem, the manager only allocates

wealth to a set of risky assets. We also consider the case where she has access to a risk-

free asset with a zero rate of return, which seems quite realistic given that the portfolio

is rebalanced on a daily basis. In the presence of a risk-free asset, the constraint t1 = 1

is dropped from (16) so that the portfolio weights need not sum to one; i.e., 1 t1 isthe share in the risk-free asset.

The quantities that the manager needs to input into (16) are the forecasts t+1|t

and Ht+1|t. The so-called plug-in approach simply computes the sample mean yt andcovariance matrix of asset returns up to time t and uses those as the inputs to (16). That

approach obviously uses the data, but just like the 1/N rule it remains model-free. The

common plug-in approach is included in our comparisons. The model-based approaches

that we consider are those that use the multivariate model of returns presented in Section

2 with either a CCC, DCC, TVC, or DECO correlation structure. Note that the implied

forecast t+1|t in each case is the same as that of the plug-in approach: yt. So the plug-in,

CCC, DCC, TVC, and DECO approaches differ only by their forecasts of the conditional

covariance matrix, Ht+1|t.

The fact that Ht+1|t is an estimated quantity gives rise to estimation risk (owing to

the uncertainty about the data-generating process) and that risk becomes particularly

important when the cost of rebalancing the portfolio is taken into account. In order

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to get a sense of the amount of trading required by each portfolio approach, we follow

DeMiguel, Garlappi, and Uppal (2009) and Kirby and Ostdiek (2011) and compute the

portfolio turnover at time t + 1, defined as the sum of the absolute values of the trades

across the N assets. More specifically, notice that if we let Ri,t+1

denote the simple return

between times t and t + 1 on asset i, then for each dollar invested in the portfolio at time

t there is i,t+1(1 + Ri,t+1) dollars invested in asset i at time t + 1. Note also that at

the daily frequency, log-returns are virtually identical to simple returns. So the share of

wealth in asset i before the portfolio is rebalanced at time t + 1 is

i,t+ =i,t(1 + Ri,t+1)N

i=1 i,t(1 + Ri,t+1)

and when the portfolio is rebalanced it gives rise to a trade in asset i of magnitude

|i,t+1 i,t+|, where i,t+1 is the optimal portfolio weight on asset i at time t + 1 (afterrebalancing). So the total amount of turnover in the portfolio is

t+1 =N

i=1

|i,t+1 i,t+|. (17)

Roughly speaking, the turnover is the average percentage of wealth traded each period.

It is important to remark that with the naive strategy, i,t = i,t+1 = 1/N, but i,t+ may

be different owing to changes in asset prices between times t and t + 1. Ifc denotes the

proportional transaction cost, then the total cost to rebalance the portfolio is ct+1. LetRp =

Ni=1 Ri,t+1i,t denote the portfolio return from a given strategy before rebalancing

occurs. The evolution of wealth invested according to a given strategy is then given by

Wt+1 = Wt(1 + Rp)(1 c t+1) (18)

and the simple return net of rebalancing costs is rt+1 = Wt+1/Wt 1. Since the portfoliot is formed using only information available at time t and held for one day before being

rebalanced at time t + 1, the return rt+1 represents the one-day out-of-sample return.

The target t in (16) needs to be specified. Kirby and Ostdiek (2010) argue that

the amount of turnover in the portfolio is very sensitive to the selected target value.

So here we follow those authors and set t equal to the estimated conditional expected

return of the benchmark 1/N portfolio; i.e., t = yt1/N. The 1/N portfolio is expected

to have relatively low turnover, so this choice levels the playing field between the naive

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approach, the plug-in approach, and the four model-based approaches. Figure 5 shows

the time-series plot of the daily portfolio target returns (in basis points) on each day from

September 15, 2006 to July 15, 2010 that the portfolio is rebalanced. We see that the

target return values vary between about 1 and 3 basis points (or between 2.5 and 7.5%

in annual terms).

It remains to discuss how we evaluate portfolio performance. For that we choose

the out-of-sample realized Sharpe ratio because it is the most ubiquitous risk-adjusted

measure used by financial market practitioners to rank fund managers and to evaluate the

attractiveness of investment strategies in general. We use an expanding-window procedure

to compare the portfolio performances. Let T denote the total number of returns under

consideration in the data set and let t1 be the first day of portfolio formation. With daily

rebalancing, we obtain a time-series of out-of-sample returns rt for t = t1 + 1,...,T. Inour empirical assessments, we consider three out-of-sample periods: (i) September 18,

2006 to July 16, 2010, (ii) September 18, 2006 to August 15, 2008, and (iii) August 18,

2008 to July 16, 2010. The first of those represents the entire out-of-sample evaluation

period with 999 returns, and the second and third periods each comprise 499 return

observations.4 As in Brownless, Engle, and Kelly (2009), we consider September 2008

the month in which Lehman Brothers filed for Chapter 11 bankruptcy protectionas the

beginning of the financial crisis, so the period from September 18, 2006 to August 15,

2008 represents our pre-crisis subsample. The Sharpe ratio of strategy i is then computedas SRi = ri/

2i , where ri is the average of the out-of-sample returns to strategy i and

2i is the corresponding sample variance. We also report the average portfolio turnover

i, computed as the out-of-sample average of (17) for each strategy.

The Sharpe ratio works well as a performance gauge when excess returns are posi-

tive. In that case, higher the Sharpe ratio, the better; and at a given level of excess

return, lower the standard deviation of return, the better. During periods of negative

excess returns, however, the Sharpe ratio yields unreasonable performance rankings. For

instance, consider two portfolios that achieve the same negative excess return, but with

different standard deviations. According to the usual Sharpe ratio, the one with the larger

standard deviation would be ranked higher! In recognition of this shortcoming, Israelsen

4Note that one observation is lost when computing rt, the returns net of rebalancing costs.

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(2003, 2005) proposes to modify the Sharpe ratio as follows:

SR-m =

r/

2, if r 0,r

2, if r < 0.

So when returns are negative, a portfolio with a smaller standard deviation gets ranked

higher than one that took more risk. Whereas when returns are positive, the Sharpe ratio

and its modified version yield the same performance ranking. The modified Sharpe ratio

is thus entirely consistent with the basic principle of risk-adjusted returns, which is that

higher risk is only preferable if accompanied by higher return.5

To assess the statistical significance of the differences between the Sharpe ratio of the

benchmark 1/N strategy (SR1/N) and those of the plug-in and model-based strategies,

we use a bootstrap inference method. Specifically, we test the equality of Sharpe ratiosaccording to the block bootstrap proposed in Ledoit and Wolf (2008) which is designed

to accommodate serially correlated and heteroskedastic time series of returns. The null

hypothesis is H0 : SRi SR1/N = 0 for which we compute a two-sided p-value usingLedoit and Wolfs studentized circular block bootstrap with block size equal to 5 and

1000 bootstrap replications. Following the suggestion in Ledoit and Wolf (2008), we

use the same methodology to test for statistical differences between the modified Sharpe

ratios.

4.2 Portfolio performance results

Tables 6 and 7 report the portfolio performance results over the entire out-of-sample

period (September 18, 2006 to July 16, 2010) when the portfolio comprises only risky

assets (Table 6) and when a risk-free security is part of the asset mix (Table 7). Tables 8

and 9 show the corresponding results over the pre-crisis period from September 18, 2006

to August 15, 2008, and Tables 9 and 10 pertain to the later period from August 18, 2008

to July 16, 2010. The results for the benchmark 1/Nstrategy are given in the leading row

of each table. The subsequent rows refer to the plug-in and model-based strategies, with

and without short selling. For the portfolios of risky assets only (in Tables 6, 8, and 10),

5As Israelsen (2005) notes, there is an odd feature of the modified Sharpe ratio: its magnitude can be

quite large. So its interest here is mainly as a ranking criterion.

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the short selling constraints are not binding for the plug-in strategy. The reported out-

of-sample portfolio results include the annualized mean return, the annualized standard

deviation of returns, the annualized Sharpe ratio (SR), the annualized modified Sharpe

ratio (SR-m), and the average turnover (Turnover).6 The entries in the columns labeled

p-value are the two-sided bootstrap p-values for the null hypothesis of equal Sharpe

ratios vis-a-vis the 1/N strategy. Following Kirby and Ostdiek (2011), each table shows

the results when there are no transactions costs and assuming proportional transactions

costs of 50 basis points (bps) for c in (18).

Table 6 shows that when the portfolio comprises risky assets only over the entire

out-of-sample period, the plug-in and the model-based strategies yield significantly worse

Sharpe and modified Sharpe ratios than the 1/N rule. The only two noticeable exceptions

occur with 50 bps transactions costs for the plug-in method and the TVC model whenshort sales are allowed. In those two instances the p-values are 0.75 and 0.15 for the

Sharpe ratio and 0.62 and 0.52 for the modified Sharpe ratio, respectively, indicating

no significant differences with the 1/N rule. In every other case, however, the p-values

never exceed 9% at best, indicating that portfolio performances with negative Sharpe

(or modified Sharpe) ratios are indeed significantly worse than what is achieved through

naive diversification.

The portfolio performance results change dramatically as soon as the manager has

access to a risk-free asset, even if it yields a zero return. Comparing Tables 6 and 7, weclearly see how the introduction of a risk-free asset leads to increased mean returns and

much smaller standard deviations. So in the presence of a risk-free asset the Sharpe ratios

delivered by the plug-in and model-based strategies continue to be lower than that of the

1/N rule, but the differences are unmistakably nowhere statistically significant. The only

apparent exceptions are for the CCC, DCC, and TVC Sharpe ratios when short sales

are allowed and with 50 bps transactions costs, but even those differences cease to be

significant under the modified Sharpe ratio performance measure. The following quote

from Amenc and Martellini (2011) summarizes well this finding:

6The annualized mean return is computed here as the daily mean return times 252 and the annualized

standard deviation of returns, SR, and SR-m are obtained by multiplying their daily counterparts by the

square root of 252.

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Any attempt at improving portfolio diversification techniques, either by

introducing sophisticated time- and state-dependent risk models, or by ex-

tending them to higher-order moments, is also equally misleading if the goal

is again to hope for protection in 2008-like market conditions. When there is

simply no place to hide, even the most sophisticated portfolio diversification

techniques are expected to fail.

Table 6 shows what happens when there is no place to hide (i.e. when the portfolio

manager can only take risky positions) and Table 7 shows that even if the risk-free rate

of return is zero, having access to a risk-free investment vehicle provides valuable pro-

tection against downside risk. Indeed, the fact that the statistically significant negative

performance measures in Table 6 cease to be significantly different vis-a-vis the 1/N rule

(in Table 7) once the constraint t1 = 1 is dropped from (16) is quite remarkable.7 It

is also interesting to note the effects of short-sale constraints in Tables 6 and 7. When

that constraint binds, the reduction in the amount of portfolio turnover can be quite

important. For instance, in Table 7 we see that prohibiting short sales with the CCC

model reduces portfolio turnover from 13.28 to 1.93, lower than the turnover value of 3.2

achieved by the 1/N rule. Similar reductions in turnover can be seen for the other models

as well. These results are in line with the findings of Jagannathan and Ma (2003) who

show that imposing a short-selling constraint amounts to shrinking the extreme elements

of the covariance matrix and thereby stabilizes the portfolio weights. As in DeMiguel,

Garlappi, and Uppal (2009) though, we find that even restraining short sales is not suf-

ficient to completely mitigate the error in estimating the covariance matrix and thus to

provide better portfolio Sharpe ratios than the naive 1/N rule, which ignores the data

altogether.

Tables 8 and 9 show the corresponding analysis during our pre-crisis period from

September 18, 2006 to August 15, 2008. During this period the performance of the 1/N

rule is even better, with an annualized Sharpe ratio of 0.37 in the absence of transactions7Note that relaxing the constraint is the main effect here, since the two series of portfolio target

returns (in Figure 5) are very close. We further confirmed this by performing the portfolio exercises with

a risk-free asset in the mix (so N=6) but targeting the return implied by the 1/5 rule. The results were

virtually identical to those reported in Tables 7, 9, and 11.

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it it nonetheless interesting to observe from Table 11 that the the modified Sharpe ratio

ranks the performances of the model-based strategies above that of naive diversification,

except in the case of the DCC model with 50 bps transactions costs. The reversal of

rankings between the model-based strategies and the naive one, when moving from Table

10 to Table 11, further illustrates the important role played by the risk-free asset in

portfolio risk management.

So what do we learn from these portfolio performance comparisons? For starters,

the significantly poorer performance of the model-based strategies over the entire out-of-

sample period (from September 18, 2006 to July 16, 2010) in Table 6 is driven by the

pre-crisis subsample ending on August 15, 2008 (Table 8). Indeed from August 18, 2008

to July 16, 2010, the model-based strategies are not statistically different from the 1/N

rule (Tables 10 and 11). Recall also that the recursively updated parameter estimates ofeach considered model are remarkably stable over the entire period, but that the portfolio

strategies are extremely sensitive to whether a risk-free asset is available. Comparing

Table 6 versus 7, and 8 versus 9, shows that this constraint plays a far more important

role than the short-sale constraints. Indeed, the effects of limiting short sales on the

statistical significance of the portfolio performances is tiny compared to the effects of

relaxing the constraint that the portfolio weights (of the risky assets) sum to one, which

changes the performance results from much worse to not statistically different from the

1/N rule. This is not to say that limiting the amount of short selling is not important. InTable 10 for instance, prohibiting short sales reduces the turnover of the TVC portfolio

from 43 to 2.9, and that value is further reduced to 0.75 in Table 11 by the introduction

of a risk-free asset.

5 Conclusion

In a recent study, DeMiguel, Garlappi, and Uppal (2009) cast serious doubt on the value

of mean-variance optimization as a method of portfolio selection. They consider various

static asset-allocation models at the monthly frequency and find that there is no single

model that delivers an out-of-sample Sharpe ratio that is higher than that of the naive 1/N

portfolio. Kirby and Ostdiek (2011) argue that the research design in DeMiguel, Gralppi,

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and Uppal casts mean-variance optimization in an unfavorable light because it targets a

conditional expected return that greatly exceeds the conditional expected return of the

1/N strategy. This has the effect of magnifying estimation risk and leads to excessive

portfolio turnover. Kirby and Ostdiek argue further that if the mean-variance model

is instead implemented by targeting the conditional expected return of the 1/N rule, it

generally performs better than the naive strategy at least in the absence of transactions

costs. They find nevertheless that the 1/N strategy remains a statistically challenging

benchmark to match when transactions cost are considered.

For the purpose of modeling the dynamic movements and co-movements among finan-

cial asset returns, which are quite pronounced at the daily frequency, the econometrics

literature has proposed many specifications. In order to measure the value of modeling

such dynamic variances and correlations, Engle and Colacito (2006) propose an asset al-location perspective. Their main focus, however, is on a setting with only two assetsa

stock and a bondand hence with relatively few parameters that need to be estimated. In

the same spirit as Engle and Colacito, we conduct and empirical assessment of alternative

time-series models for variances and correlations for the purpose of portfolio management

at the daily frequency.

In addition to the usual plug-in method which simply replaces the covariance matrix

by its sample counterpart, we also consider four popular models to forecast the inputs

to the portfolio selection problem. These include the CCC model of Bollerslev (1990),the DCC model of Engle (2002), the TVC model of Tse and Tsui (2002), and the recent

DECO model of Engle and Kelly (2009). Here we examine a portfolio of five international

MSCI equity indices, which results in a total of 18 parameters that need to be estimated

for the most sophisticated correlation models we consider, even though we make use

of correlation targeting to reduce the dimensionality problem, as in Engle and Colacito

(2006). Following DeMiguel, Gralppi, and Uppal (2009) and Kirby and Ostdiek (2011),

the 1/N rule serves as our benchmark strategy for comparison purposes and, as in Kirby

and Ostdiek, the conditional expected return of the 1/N rule is used as the target expected

portfolio return.

Our empirical assessment reveals that in the two years leading up to the recent fi-

nancial crisis, the 1/N strategy delivered positive out-of-sample Sharpe ratios while the

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model-based strategies were statistically worse with negative average returns when the

portfolio manager could only take risky positions. During the financial crisis, the plug-in

strategy and the model-based strategies are surprisingly not statistically different from

the deteriorated 1/N portfolio strategy. These results provide further evidence that the

desire to elaborate multivariate conditional heteroskedasticity and correlation models is

necessarily accompanied by greater estimation risk. And that parameter uncertainty can

easily translate into quite poor out-of-sample risk-adjusted portfolio performances, even

during relatively tranquil markets. A very important finding here is that the recursively

updated parameter estimates of each considered model specification appear remarkably

stable over the entire evaluation period. This sheds more light on the fragility of mean-

variance optimizing portfolios. Our results clearly show the benefits of limiting short sales

and the far greater importance for the portfolio manager of including a risk-free securityin the asset mix, even if it yields a zero return.

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References

Amenc, N. and L. Martellini. 2011. In diversification we trust? Journal of Portfolio

Management 37: 12.

Ang, A. and J. Chen. 2002. Asymmetric correlations of equity portfolios. Journal of

Financial Economics 63: 443494.

Bauwens, L. and S. Laurent. 2005 . A new class of multivariate skew densities, with ap-

plication to generalized autoregressive conditional heteroscedasticity models. Jour-

nal of Business and Economic Statistics 3: 346354.

Bekaert, G. and C. Harvey. 2000. Foreign speculators and emerging equity markets.

Journal of Finance 55: 565613.

Best, M.J. and R.R. Grauer. 1991. On the sensitivity of mean-variance-efficient portfo-

lios to changes in asset means: some analytical and computational results. Review

of Financial Studies 4: 315342.

Bollerslev, T. 1986. Generalized autoregressive conditional heteroskedasticity. Journal

of Econometrics 31: 307327.

Bollerslev, T. 1990. Modeling the coherence in short-run nominal exchange rates: a

multivariate generalized ARCH model. Review of Economics and Statistics 72:

498505.

Brownlees, C., Engle, R. and B. Kelly. 2009. A practical guide to volatility forecasting

through calm and storm. New York University Working Paper.

Chopra, V.K. 1993. Improving optimization. Journal of Investing 8: 5159.

Christoffersen, P., Errunza, V., Jacobs, K. and C. Jin. 2010. Is the potential for

international diversification disappearing? McGill University Working paper.

DeMiguel, V., Garlappi, L. and R. Uppal. 2009. Optimal versus naive diversification:

how inefficient is the 1/N portfolio strategy? Review of Financial Studies 22:

191553.

25


28/50

DeSantis, G. and B. Gerard. 1997. International asset pricing and portfolio diversifi-

cation with time-varying risk. Journal of Finance 52: 18811912.

Duchin, R. and H. Levy. 2009. Markowitz versus the Talmudic portfolio diversification

strategies. Journal of Portfolio Management 35: 7174.

Engle, R. 2002. Dynamic conditional correlation: a simple class of multivariate gen-

eralized autoregressive conditional heteroskedasticity models. Journal of Business

and Economic Statistics 20: 339350.

Engle, R. 2009. Anticipating correlations: a new paradigm for risk management. Prince-

ton University Press, Princeton, New Jersey.

Engle, R. and R. Colacito. 2006. Testing and valuing dynamic correlations for assetallocation. Journal of Business and Economic Statistics 24: 238253.

Engle, R. and B. Kelly. 2009. Dynamic equicorrelation. New York University Working

Paper.

Engle, R. and J. Mezrich. 1996. GARCH for groups. Risk 9: 3640.

Engle, R. and K. Sheppard. 2005. Theoretical properties of dynamic conditional corre-

lation multivariate GARCH. University of California, San Diego Working Paper.

Errunza, V. 1977. Gains from portfolio diversification into less developed countries

securities. Journal of International Business Studies 55: 8399.

Errunza, V., Hogan, K., and M.W. Hung. 1999. Can the gains from international

diversification be achieved without trading abroad? Journal of Finance 54: 2075

2107.

Francq, C., Horvath, L. and J.-M. Zakoian. 2009. Merits and drawbacks of variance

targeting in GARCH models. CREST Working Paper 2009-17.

Frost, P.A. and J.E. Savarino. 1988. For better performance constrain portfolio

weights. Journal of Portfolio Management 15: 2934.

26


29/50

Hodges, S.D. and R.A. Brealey. 1972. Portfolio selection in a dynamic and uncertain

world. Financial Analysts Journal 28: 5869.

Israelsen, C.L. 2003. Sharpening the Sharpe ratio. Financial Planning 33: 4951.

Israelsen, C.L. 2005. A refinement to the Sharpe ratio and information ratio. Journal

of Asset Management 5: 423427.

Jagannathan, R. and T. Ma. 2003. Risk reduction in large portfolios: why imposing

the wrong constraints helps. Journal of Finance 58: 16511884.

Jobson, D.J. and B.M. Korkie. 1980. Estimation for Markowitz efficient portfolios.

Journal of the American Statistical Association 75: 544554.

Jondeau, E., Poon, S.-H. and M. Rockinger. 2007. Financial Modeling Under Non-

Gaussian Distributions. Springer-Verlag, London.

Jondeau, E. and M. Rockinger. 2008. The economic value of distributional timing.

University of Lausanne Working Paper.

Kirby, C. and B. Ostdiek. 2011. Its all in the timing: simple active portfolio strate-

gies that outperform naive diversification. Journal of Financial and Quantitative

Analysis, forthcoming.

Ledoit, O. and M. Wolf. 2008. Robust performance hypothesis testing with the Sharpe

ratio. Journal of Empirical Finance 15: 850859

Litterman, B. 2003. Modern Investment Management: An Equilibrium Approach. Wi-

ley, New York.

Longin, F. and B. Solnik. 1995. Is the correlation in international equity returns

constant: 19601990? Journal of International Money and Finance 14: 326.

Longin, F. and B. Solnik. 2001. Extreme correlation of international equity markets.

Journal of Finance 56: 649676.

Markowitz , H.M. 1952. Portfolio selection Journal of Finance 7: 7791.

27


30/50

Michaud, R.O. 1989. The Markowitz optimization enigma: is optimized optimal?

Financial Analysts Journal 45: 3142.

Michaud, R.O. and R.O. Michaud. 2008. Efficient Asset Management: A Practical

Guide to Stock Portfolio Optimization and Asset Allocation, Second Edition. Ox-

ford University Press, New York.

Solnik, B. 1974. The international pricing of risk: an empirical investigation of the

world capital market structure. Journal of Finance 29: 365378.

Tse, Y.K. and A.K.C. Tsui. 2002. A multivariate generalized autoregressive conditional

heteoscedasticity model with time-varying correlations. Journal of Business and

Economic Statistics 20: 351362.

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Table 1: Summary statistics of daily MSCI index log-returns

Developed markets Emerging markets

North America Europe Pacific Latin America Asia

Mean 0.021 0.015 -0.002 0.051 0.009

Std Dev 1.148 1.230 1.348 1.767 1.372Max. 10.427 10.697 10.823 15.363 12.651Min. -9.504 -10.178 -9.182 -15.060 -8.620Skewness -0.285 -0.159 0.084 -0.332 -0.227Kurtosis 12.312 12.088 7.840 12.943 8.822

Correlation matrix

North America 1Europe 0.484 1Pacific 0.105 0.356 1Latin America 0.588 0.543 0.211 1Asia 0.175 0.374 0.547 0.319 1

Notes: The data consists of daily percentage log-returns on dollar-denominated MSCI stockmarket indices for five geographical areas: North America, Europe, Pacific, Latin America, andAsia. The first two of those are developed market indices, whereas the last two are emergingmarket indices. The entire sample period comprises 5219 return observations from July 17,1990 to July 16, 2010.

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Table 2: Parameter estimates of the CCC model



Shape parameters

i 0.9385 0.9251 1.0072 0.9352 0.9484(0.0176) (0.0211) (0.0191) (0.0171) (0.0206)

Degrees-of-freedom parameter

v 7.7551(0.4463)

GARCH parameters

i 0.0087 0.0118 0.0150 0.0213 0.0098(0.0032) (0.0031) (0.0051) (0.0048) (0.0034)

i 0.0616 0.0625 0.0559 0.0861 0.0646(0.0057) (0.0014) (0.0043) (0.0051) (0.0059)

i 0.9295 0.9256 0.9289 0.8925 0.9254

(0.0048) (0.0052) (0.0061) (0.0063) (0.0053)Log-likelihood -36764.4

Notes: This table shows the maximum-likelihood parameter estimates of the constant conditionalcorrelation model based on the entire sample of returns data. The numbers in parentheses arestandard errors.

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Table 3: Parameter estimates of the DCC model



Shape parameters

i 0.9266 0.9153 0.9866 0.9245 0.9522(0.0128) (0.0160) (0.0144) (0.0125) (0.0162)


v 7.7873(0.3446)

GARCH parameters

i 0.0061 0.0085 0.0123 0.0154 0.0068(0.0018) (0.0021) (0.0048) (0.0041) (0.0021)

i 0.0606 0.0579 0.0586 0.0778 0.0590(0.0022) (0.0026) (0.0062) (0.0043) (0.0024)

i 0.9331 0.9334 0.9290 0.9066 0.9341(0.0034) (0.0036) (0.0049) (0.0047) (0.0037)

DCC persistence parameters

a 0.0076(0.0221)

b 0.9905(0.0243)

Log-likelihood -36386.1

Notes: This table shows the maximum-likelihood parameter estimates of the dynamic conditionalcorrelation model based on the entire sample of returns data. The numbers in parentheses arestandard errors.

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Table 4: Parameter estimates of the TVC model



Shape parameters

i 0.9257 0.9296 0.9973 0.9240 0.9474(0.0162) (0.0180) (0.0191) (0.0185) (0.0196)


v 7.7544(0.4332)

GARCH parameters

i 0.0051 0.0076 0.0123 0.0124 0.0062(0.0019) (0.0020) (0.0037) (0.0031) (0.0021)

i 0.0578 0.0594 0.0591 0.0757 0.0607(0.0039) (0.0038) (0.0030) (0.0031) (0.0041)

i 0.9370 0.9329 0.9284 0.9117 0.9329(0.0063) (0.0068) (0.0076) (0.0074) (0.0070)

TVC persistence parameters

a 0.0077(0.0441)

b 0.9909(0.0357)


Notes: This table shows the maximum-likelihood parameter estimates of the time-varyingcorrelation model based on the entire sample of returns data. The numbers in parentheses arestandard errors.

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Table 5: Parameter estimates of the DECO model



Shape parameters

i 0.9289 0.9258 1.0071 0.9089 0.9582(0.0164) (0.0202) (0.0180) (0.0181) (0.0179)

Degrees-of-freedom parameterv 7.7529

(0.5975)

GARCH parameters

i 0.0057 0.0069 0.0131 0.0130 0.0067(0.0015) (0.0018) (0.0044) (0.0036) (0.0021)

i 0.0577 0.0598 0.0595 0.0813 0.0623(0.0023) (0.0024) (0.0039) (0.0035) (0.0024)

i 0.9365 0.9331 0.9272 0.9056 0.9309(0.0050) (0.0054) (0.0061) (0.0064) (0.0055)

DECO persistence parameters

a 0.0123(0.0221)

b 0.9862(0.0425)


Notes: This table shows the maximum-likelihood parameter estimates of the dynamic equicorrelationmodel based on the entire sample of returns data. The numbers in parentheses are standard errors.

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Table 6: Portfolio of risky assets only: September 18, 2006 to July 16, 2010

No transactions costs Transactions costs = 50 bps

Strategy Mean Std Dev SR p-value SR-m p-value Turnover Mean Std Dev SR p-value SR-m p-value

1/N 0.29 23.75 0.01 1.00 0.01 1.00 3.20 -3.74 23.71 -0.15 1.00 -0.35 1.00Short sales allowed

Plug-in -2.93 22.77 -0.12 0.09 -0.26 0.08 23.63 -32.73 54.98 -0.59 0.75 -7.14 0.62CCC -5.96 22.60 -0.26 0.00 -0.53 0.01 2.84 -9.47 22.56 -0.42 0.00 -0.84 0.03DCC -8.69 22.19 -0.39 0.00 -0.76 0.01 5.04 -14.98 22.45 -0.66 0.00 -1.33 0.01TVC -8.47 22.22 -0.38 0.00 -0.74 0.01 23.40 -37.92 54.09 -0.70 0.15 -8.14 0.52DECO -6.77 22.77 -0.29 0.00 -0.61 0.01 3.11 -10.66 22.72 -0.46 0.00 -0.96 0.01

Short sales prohibited

CCC -5.66 22.64 -0.25 0.00 -0.50 0.01 2.90 -9.26 22.59 -0.40 0.01 -0.83 0.03DCC -7.67 22.31 -0.34 0.00 -0.67 0.01 3.03 -11.43 22.25 -0.51 0.00 -1.00 0.02TVC -7.51 22.33 -0.33 0.00 -0.66 0.00 5.25 -14.06 22.82 -0.61 0.00 -1.27 0.03DECO -6.19 22.80 -0.27 0.01 -0.56 0.02 2.92 -9.82 22.75 -0.43 0.01 -0.88 0.02

Notes: This table reports the daily out-of-sample portfolio performances of the 1/N strategy, the plug-in strategy, and fourmodel-based strategies. The portfolio return statistics are the annualized mean, the annualized standard deviation, the annualizedSharpe ratio, the annualized modified Sharpe ratio, and the average turnover. The p-values are for the null hypothesis that the

modified) Sharpe ratio of a given strategy equals the (modified) Sharpe ratio of the 1/N strategy. Values less than 0.01 arereported as zero.

Table 7: Portfolio of risky assets and a risk-free asset: September 18, 2006 to July 16, 2010



1/N 0.24 19.79 0.01 1.00 0.01 1.00 3.20 -3.80 19.76 -0.19 1.00 -0.29 1.00

Short sales allowed



Plug-in -0.33 12.44 -0.02 0.83 -0.02 0.91 2.01 -2.86 12.42 -0.23 0.85 -0.14 0.72CCC -2.91 12.59 -0.23 0.21 -0.14 0.62 1.93 -5.32 12.58 -0.42 0.27 -0.26 0.94

DCC -2.37 12.36 -0.19 0.34 -0.11 0.71 1.15 -3.80 12.32 -0.31 0.59 -0.18 0.80TVC -2.44 12.43 -0.20 0.28 -0.12 0.67 1.31 -4.07 12.39 -0.32 0.52 -0.20 0.82DECO -2.33 12.61 -0.18 0.32 -0.11 0.65 2.22 -5.12 12.65 -0.40 0.30 -0.25 0.92

Notes: See footnote of Table 6.

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Table 8: Portfolio of risky assets only: September 18, 2006 to August 15, 2008



1/N 5.79 15.52 0.37 1.00 0.37 1.00 3.02 1.98 15.50 0.12 1.00 0.12 1.00

Short sales allowed

Plug-in 2.24 14.07 0.16 0.11 0.16 0.20 4.46 -3.38 14.23 -0.23 0.06 -0.19 0.14CCC -5.11 14.06 -0.36 0.00 -0.28 0.00 3.66 -9.66 14.09 -0.68 0.00 -0.54 0.00DCC -8.97 13.97 -0.64 0.00 -0.49 0.00 2.99 -12.67 13.94 -0.91 0.00 -0.70 0.00TVC -8.73 14.02 -0.62 0.00 -0.48 0.00 3.01 -12.44 13.99 -0.88 0.00 -0.69 0.00DECO -6.09 14.06 -0.43 0.00 -0.34 0.00 2.87 -9.66 14.02 -0.68 0.00 -0.53 0.01


CCC -4.62 14.08 -0.32 0.00 -0.26 0.00 3.75 -9.29 14.12 -0.65 0.00 -0.52 0.01DCC -6.80 14.04 -0.48 0.00 -0.38 0.00 2.72 -10.16 14.02 -0.72 0.00 -0.56 0.01TVC -6.66 14.08 -0.47 0.00 -0.37 0.00 7.54 -16.10 15.69 -1.02 0.00 -1.00 0.03

DECO -5.11 14.09 -0.36 0.00 -0.28 0.00 2.52 -8.24 14.05 -0.58 0.00 -0.45 0.02


Table 9: Portfolio of risky assets and a risk-free asset: September 18, 2006 to August 15, 2008



1/N 4.83 12.93 0.37 1.00 0.37 1.00 3.02 1.01 12.93 0.07 1.00 0.07 1.00

Short sales allowed

Plug-in 4.14 10.38 0.39 0.91 0.39 0.93 3.15 0.13 10.37 0.01 0.83 0.01 0.82CCC -0.21 10.31 -0.02 0.23 -0.01 0.32 22.58 -28.71 25.60 -1.12 0.14 -2.91 0.31DCC -1.65 10.33 -0.16 0.14 -0.07 0.27 5.73 -8.85 11.19 -0.79 0.06 -0.39 0.29TVC -1.47 10.34 -0.14 0.16 -0.06 0.28 7.45 -10.85 11.16 -0.97 0.01 -0.48 0.17DECO 1.39 11.18 0.12 0.66 0.12 0.63 12.76 -14.70 15.76 -0.93 0.16 -0.92 0.24


Plug-in 4.33 10.37 0.41 0.88 0.41 0.87 2.96 0.59 10.40 0.05 0.92 0.05 0.94CCC -0.36 10.36 -0.03 0.16 -0.01 0.26 1.77 -2.58 10.34 -0.24 0.25 -0.10 0.53DCC -0.46 10.38 -0.04 0.16 -0.02 0.28 1.56 -2.41 10.35 -0.23 0.31 -0.09 0.57

TVC -0.51 10.40 -0.05 0.17 -0.02 0.24 1.87 -2.85 10.38 -0.27 0.23 -0.11 0.56DECO 0.40 10.30 0.04 0.23 0.04 0.31 1.71 -1.74 10.27 -0.16 0.38 -0.07 0.60


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Table 10: Portfolio of risky assets only: August 18, 2008 to July 16, 2010



1/N -4.97 29.83 -0.16 1.00 -0.58 1.00 3.38 -9.26 29.77 -0.31 1.00 -1.09 1.00

Short sales allowed



CCC -6.64 28.80 -0.23 0.46 -0.75 0.50 2.04 -9.17 28.71 -0.32 0.93 -1.04 0.80DCC -8.48 28.30 -0.30 0.41 -0.95 0.45 3.34 -12.64 28.22 -0.44 0.41 -1.41 0.47TVC -8.30 28.32 -0.29 0.42 -0.93 0.41 2.96 -11.98 28.25 -0.42 0.45 -1.34 0.52

DECO -7.18 29.05 -0.24 0.44 -0.82 0.42 3.31 -11.33 28.99 -0.39 0.47 -1.30 0.50


Table 11: Portfolio of risky assets and a risk-free asset: August 18, 2008 to July 16, 2010



1/N -4.14 24.86 -0.16 1.00 -0.41 1.00 3.39 -8.44 24.82 -0.34 1.00 -0.83 1.00

Short sales allowed



Plug-in -4.96 14.22 -0.34 0.50 -0.28 0.89 1.06 -6.28 14.18 -0.44 0.71 -0.35 0.69CCC -5.42 14.50 -0.37 0.45 -0.31 0.91 2.08 -8.02 14.49 -0.55 0.46 -0.46 0.77DCC -4.21 14.08 -0.29 0.66 -0.23 0.85 0.75 -5.14 14.03 -0.36 0.92 -0.28 0.62

TVC -4.31 14.18 -0.30 0.59 -0.24 0.85 0.75 -5.24 14.14 -0.37 0.91 -0.29 0.63DECO -5.05 14.57 -0.34 0.51 -0.29 0.89 2.73 -8.48 14.66 -0.57 0.41 -0.49 0.80


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1

0

0

10

1990 1995 2000 2005 2010

North America

10

0

10

1990 1995 2000

Euro

5

5

1990 1995 2000 2005 2010

Pacific

15

0

15

1990 1995 2000

Latin Am

5

5

1990 1995 2000 2005 2010

Asia

Figure 1: Time-series plots of the daily log-returns (in percentages) on 3 MSCI developed markets indices (North America, Eur

and 2 MSCI emerging markets indices (Latin America, Asia) from July 17, 1990 to July 16, 2010.


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0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000 2005 2010

North AmericaEurope

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000

North Ameri

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000 2005 2010

North AmericaLatin America

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000

North Amer

Figure 2: Time-series plots of the model-implied conditional correlations between the standardized returns on the MSCI indice

and Europe (upper left), North America and the Pacific region (upper right), North America and Latin America (lower left), and

Asia (lower right). The horizontal lines correspond to the CCC model, the solid lines correspond to the DCC model, the dashed

the TVC model, and the dotted line (which is the same in each plot) corresponds to the DECO model.


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0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000 2005 2010

EuropePacific

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000

EuropeLati

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000 2005 2010

EuropeAsia

Figure 3: Time-series plots of the model-implied conditional correlations between the standardized returns on the MSCI indice

Pacific region (upper left), Europe and Latin America (upper right), and Europe and Asia (lower left). The horizontal lines corr

model, the solid lines correspond to the DCC model, the dashed lines correspond to the TVC model, and the dotted line (which

plot) corresponds to the DECO model.


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0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000 2005 2010

PacificLatin America

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000

Pacific

0.

0

0.

2

0.

4

0.

6

0.

8

1990 1995 2000 2005 2010

Latin AmericaAsia

Figure 4: Time-series plots of the model-implied conditional correlations between the standardized returns on the MSCI indices

and Latin America (upper left), the Pacific region and Asia (upper right), and Latin America and Asia (lower left). The horizon

the CCC model, the solid lines correspond to the DCC model, the dashed lines correspond to the TVC model, and the dotted li

in each plot) corresponds to the DECO model.


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1.

0

1.

5

2.

0

2.

5

3.

0

2007 2008 2009 2010

Figure 5: Time-series plot of the daily portfolio target returns (in basis points) from September

15, 2006 to July 15, 2010. The solid lines is the target return when the portfolio comprises risky

assets only and the dashed line is the target return when the asset mix includes a risk-free security.


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Appendix

The tables in this appendix show the maximum-likelihood parameter estimates of the fourmodel specifications based on the sample of returns data from July 17, 1990 up to the given

date. The dates are in the format Month/Day/Year. The updated estimates on a givendate are found using the previous ones as initial values for the numerical optimization ofthe likelihood function. The shape parameters are i, the degrees-of-freedom parameteris v, the GARCH parameters are i and i (i = 1 i i are not shown), and theparameters of the conditional correlation process are a and b. The parameter subscriptsrefer to: (1) North America, (2) Europe, (3) the Pacific region, (4) Latin America, and(5) Asia.

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Table A1 : Recursively updated parameter estimates of the CCC model

Date 1 2 3 4 5 v 1 2 3 4 5 1 2 309/15/2006 0.9457 0.9286 1.0058 0.9382 0.9595 7.7565 0.0083 0.0127 0.0180 0.0256 0.0096 0.0489 0.0494 0.0564 11/24/2006 0.9458 0.9283 1.0041 0.9359 0.9591 7.7568 0.0075 0.0131 0.0169 0.0270 0.0093 0.0468 0.0497 0.0560 02/02/2007 0.9459 0.9281 1.0041 0.9354 0.9585 7.7569 0.0072 0.0129 0.0156 0.0270 0.0099 0.0469 0.0495 0.0565 04/13/2007 0.9458 0.9280 1.0041 0.9353 0.9587 7.7569 0.0071 0.0129 0.0158 0.0271 0.0097 0.0468 0.0495 0.0565 06/22/2007 0.9467 0.9286 1.0035 0.9351 0.9587 7.7571 0.0072 0.0139 0.0159 0.0289 0.0099 0.0469 0.0506 0.0574 08/31/2007 0.9466 0.9288 1.0035 0.9353 0.9586 7.7572 0.0071 0.0135 0.0154 0.0284 0.0097 0.0467 0.0507 0.0573 11/09/2007 0.9461 0.9289 1.0032 0.9343 0.9584 7.7572 0.0075 0.0146 0.0154 0.0284 0.0099 0.0456 0.0514 0.0563 01/18/2008 0.9456 0.9283 1.0037 0.9337 0.9592 7.7573 0.0075 0.0145 0.0157 0.0288 0.0100 0.0460 0.0521 0.0559 03/28/2008 0.9457 0.9284 1.0040 0.9346 0.9595 7.7574 0.0073 0.0134 0.0161 0.0281 0.0094 0.0464 0.0527 0.0565 06/06/2008 0.9416 0.9248 1.0003 0.9318 0.9600 7.7582 0.0079 0.0161 0.0166 0.0258 0.0101 0.0491 0.0584 0.0562 08/15/2008 0.9392 0.9225 0.9997 0.9312 0.9610 7.7586 0.0075 0.0151 0.0160 0.0273 0.0106 0.0467 0.0558 0.0561 10/24/2008 0.9354 0.9213 0.9963 0.9272 0.9570 7.7607 0.0072 0.0139 0.0159 0.0233 0.0093 0.0504 0.0594 0.0601 01/02/2009 0.9334 0.9204 0.9928 0.9271 0.9571 7.7612 0.0070 0.0127 0.0161 0.0215 0.0093 0.0538 0.0625 0.0620 03/13/2009 0.9324 0.9204 0.9926 0.9265 0.9568 7.7614 0.0067 0.0120 0.0149 0.0206 0.0092 0.0552 0.0629 0.0616 05/22/2009 0.9324 0.9204 0.9926 0.9265 0.9568 7.7614 0.0066 0.0119 0.0149 0.0206 0.0092 0.0553 0.0629 0.0616 07/31/2009 0.9322 0.9204 0.9921 0.9270 0.9567 7.7614 0.0069 0.0117 0.0156 0.0213 0.0090 0.0555 0.0630 0.0612 10/09/2009 0.9323 0.9198 0.9914 0.9266 0.9568 7.7618 0.0074 0.0121 0.0151 0.0222 0.0091 0.0577 0.0634 0.0612 12/18/2009 0.9321 0.9200 0.9912 0.9267 0.9558 7.7618 0.0073 0.0117 0.0154 0.0216 0.0093 0.0574 0.0634 0.0607 02/26/2010 0.9324 0.9200 0.9921 0.9265 0.9569 7.7620 0.0079 0.0119 0.0155 0.0227 0.0093 0.0593 0.0625 0.0603 05/07/2010 0.9324 0.9201 0.9922 0.9266 0.9569 7.7620 0.0081 0.0122 0.0155 0.0229 0.0093 0.0596 0.0626 0.0605

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Table A2 : Recursively updated parameter estimates of the DCC model

Date 1 2 3 4 5 v 1 2 3 4 5 1 2 309/15/2006 0.9349 0.9170 0.9907 0.9229 0.9556 7.7888 0.0065 0.0116 0.0149 0.0250 0.0080 0.0465 0.0488 0.0562 11/24/2006 0.9356 0.9174 0.9908 0.9226 0.9559 7.7890 0.0060 0.0113 0.0155 0.0246 0.0079 0.0468 0.0494 0.0566 02/02/2007 0.9356 0.9174 0.9908 0.9225 0.9560 7.7891 0.0057 0.0114 0.0151 0.0247 0.0083 0.0468 0.0493 0.0567 04/13/2007 0.9363 0.9177 0.9909 0.9226 0.9565 7.7892 0.0056 0.0117 0.0145 0.0251 0.0078 0.0466 0.0496 0.0564 06/22/2007 0.9367 0.9186 0.9908 0.9222 0.9567 7.7893 0.0058 0.0120 0.0140 0.0262 0.0086 0.0469 0.0497 0.0571 08/31/2007 0.9367 0.9186 0.9908 0.9222 0.9567 7.7893 0.0058 0.0120 0.0140 0.0262 0.0086 0.0469 0.0497 0.0571 11/09/2007 0.9364 0.9197 0.9912 0.9218 0.9561 7.7897 0.0058 0.0122 0.0142 0.0248 0.0084 0.0466 0.0504 0.0565 01/18/2008 0.9364 0.9197 0.9912 0.9218 0.9561 7.7898 0.0057 0.0123 0.0142 0.0247 0.0084 0.0466 0.0504 0.0565 03/28/2008 0.9366 0.9197 0.9909 0.9215 0.9570 7.7900 0.0057 0.0111 0.0142 0.0240 0.0082 0.0471 0.0519 0.0566 06/06/2008 0.9366 0.9197 0.9909 0.9215 0.9570 7.7900 0.0057 0.0111 0.0142 0.0240 0.0082 0.0471 0.0519 0.0566 08/15/2008 0.9371 0.9191 0.9913 0.9201 0.9572 7.7908 0.0056 0.0129 0.0134 0.0228 0.0086 0.0473 0.0557 0.0546 10/24/2008 0.9350 0.9178 0.9909 0.9187 0.9566 7.7914 0.0053 0.0110 0.0124 0.0182 0.0076 0.0505 0.0565 0.0561 01/02/2009 0.9350 0.9175 0.9912 0.9186 0.9569 7.7917 0.0049 0.0099 0.0126 0.0157 0.0070 0.0510 0.0579 0.0579 03/13/2009 0.9346 0.9173 0.9916 0.9186 0.9569 7.7918 0.0048 0.0093 0.0120 0.0154 0.0069 0.0523 0.0584 0.0583 05/22/2009 0.9337 0.9178 0.9922 0.9185 0.9587 7.7919 0.0048 0.0093 0.0112 0.0143 0.0066 0.0536 0.0593 0.0583 07/31/2009 0.9339 0.9178 0.9919 0.9200 0.9588 7.7921 0.0052 0.0089 0.0120 0.0147 0.0064 0.0555 0.0591 0.0587 10/09/2009 0.9336 0.9177 0.9916 0.9199 0.9587 7.7921 0.0053 0.0084 0.0119 0.0142 0.0066 0.0558 0.0586 0.0586 12/18/2009 0.9336 0.9176 0.9915 0.9201 0.9587 7.7922 0.0053 0.0085 0.0121 0.0144 0.0065 0.0557 0.0585 0.0585 02/26/2010 0.9336 0.9176 0.9915 0.9201 0.9587 7.7922 0.0053 0.0085 0.0121 0.0144 0.0065 0.0557 0.0585 0.0585 05/07/2010 0.9330 0.9176 0.9908 0.9197 0.9588 7.7925 0.0058 0.0086 0.0119 0.0146 0.0067 0.0579 0.0584 0.0578

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Table A3 : Recursively updated parameter estimates of the TVC model

Date 1 2 3 4 5 v 1 2 3 4 5 1 2 309/15/2006 0.9395 0.9324 0.9988 0.9225 0.9572 7.7564 0.0056 0.0106 0.0150 0.0220 0.0075 0.0448 0.0503 0.0588 11/24/2006 0.9394 0.9324 0.9988 0.9224 0.9572 7.7564 0.0053 0.0106 0.0151 0.0220 0.0074 0.0449 0.0503 0.0587 02/02/2007 0.9396 0.9324 0.9990 0.9225 0.9574 7.7565 0.0049 0.0106 0.0149 0.0221 0.0078 0.0450 0.0503 0.0587 04/13/2007 0.9398 0.9323 0.9987 0.9220 0.9575 7.7565 0.0049 0.0109 0.0147 0.0219 0.0078 0.0449 0.0503 0.0585 06/22/2007 0.9398 0.9323 0.9987 0.9220 0.9575 7.7565 0.0049 0.0109 0.0147 0.0219 0.0078 0.0449 0.0503 0.0585 08/31/2007 0.9393 0.9317 0.9977 0.9225 0.9582 7.7569 0.0050 0.0108 0.0137 0.0219 0.0079 0.0454 0.0505 0.0577 11/09/2007 0.9397 0.9316 0.9979 0.9223 0.9581 7.7570 0.0050 0.0113 0.0136 0.0219 0.0078 0.0452 0.0506 0.0573 01/18/2008 0.9397 0.9316 0.9979 0.9223 0.9581 7.7570 0.0050 0.0113 0.0136 0.0219 0.0078 0.0452 0.0506 0.0573 03/28/2008 0.9400 0.9325 0.9977 0.9225 0.9584 7.7571 0.0048 0.0099 0.0140 0.0217 0.0078 0.0461 0.0514 0.0574 06/06/2008 0.9399 0.9307 0.9969 0.9216 0.9589 7.7576 0.0050 0.0114 0.0129 0.0205 0.0078 0.0462 0.0544 0.0559 08/15/2008 0.9399 0.9306 0.9969 0.9217 0.9589 7.7576 0.0050 0.0114 0.0130 0.0205 0.0078 0.0462 0.0544 0.0559 10/24/2008 0.9395 0.9302 0.9955 0.9225 0.9586 7.7577 0.0044 0.0095 0.0122 0.0169 0.0076 0.0464 0.0548 0.0575 01/02/2009 0.9356 0.9279 0.9946 0.9213 0.9582 7.7584 0.0046 0.0088 0.0125 0.0142 0.0067 0.0524 0.0592 0.0603 03/13/2009 0.9354 0.9278 0.9947 0.9211 0.9583 7.7584 0.0043 0.0084 0.0121 0.0140 0.0067 0.0525 0.0592 0.0602 05/22/2009 0.9354 0.9275 0.9946 0.9211 0.9582 7.7585 0.0044 0.0078 0.0111 0.0132 0.0063 0.0530 0.0595 0.0600 07/31/2009 0.9356 0.9271 0.9945 0.9213 0.9584 7.7586 0.0045 0.0078 0.0113 0.0130 0.0063 0.0540 0.0590 0.0596 10/09/2009 0.9351 0.9273 0.9939 0.9211 0.9583 7.7587 0.0045 0.0077 0.0109 0.0130 0.0062 0.0547 0.0585 0.0587 12/18/2009 0.9351 0.9270 0.9939 0.9210 0.9582 7.7587 0.0046 0.0076 0.0111 0.0130 0.0062 0.0546 0.0585 0.0585 02/26/2010 0.9351 0.9270 0.9939 0.9210 0.9582 7.7587 0.0046 0.0076 0.0111 0.0130 0.0062 0.0546 0.0585 0.0585 05/07/2010 0.9351 0.9269 0.9938 0.9208 0.9581 7.7587 0.0047 0.0075 0.0111 0.0132 0.0063 0.0548 0.0585 0.0585

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Table A4 : Recursively updated parameter estimates of the DECO model

Date 1 2 3 4 5 v 1 2 3 4 5 1 2 309/15/2006 0.9324 0.9220 1.0101 0.9121 0.9697 7.7546 0.0058 0.0097 0.0154 0.0200 0.0078 0.0458 0.0499 0.0587 11/24/2006 0.9357 0.9199 1.0105 0.9149 0.9708 7.7552 0.0058 0.0099 0.0157 0.0203 0.0076 0.0467 0.0515 0.0579 02/02/2007 0.9360 0.9200 1.0105 0.9150 0.9708 7.7553 0.0059 0.0101 0.0158 0.0213 0.0079 0.0466 0.0519 0.0584 04/13/2007 0.9367 0.9195 1.0098 0.9151 0.9704 7.7554 0.0055 0.0096 0.0148 0.0221 0.0079 0.0445 0.0505 0.0571 06/22/2007 0.9369 0.9196 1.0098 0.9153 0.9703 7.7554 0.0057 0.0099 0.0147 0.0220 0.0082 0.0446 0.0503 0.0568 08/31/2007 0.9369 0.9196 1.0097 0.9153 0.9703 7.7554 0.0056 0.0099 0.0147 0.0219 0.0081 0.0446 0.0503 0.0568 11/09/2007 0.9367 0.9192 1.0097 0.9156 0.9703 7.7554 0.0056 0.0100 0.0147 0.0219 0.0081 0.0445 0.0503 0.0566 01/18/2008 0.9369 0.9187 1.0098 0.9160 0.9705 7.7555 0.0057 0.0103 0.0152 0.0213 0.0085 0.0454 0.0514 0.0565 03/28/2008 0.9368 0.9172 1.0095 0.9161 0.9699 7.7556 0.0057 0.0100 0.0151 0.0208 0.0076 0.0448 0.0530 0.0575 06/06/2008 0.9381 0.9171 1.0088 0.9166 0.9716 7.7557 0.0061 0.0116 0.0153 0.0211 0.0082 0.0467 0.0570 0.0573 08/15/2008 0.9381 0.9163 1.0092 0.9166 0.9717 7.7558 0.0058 0

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