Modeling conditional covariance for mixed-asset portfolios

Economic Modelling 40 (2014) 242–249

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Modeling conditional covariance for mixed-asset portfolios Jian Zhou ⁎ Real Estate and Housing, Department of Marketing and Consumer Studies, College of Management and Economics, University of Guelph, Guelph, ON N1G 2W1, Canada

a r t i c l e

i n f o

Article history: Accepted 15 April 2014 Available online 11 May 2014 Keywords: Mixed-asset portfolio Conditional covariance Forecast Portfolio diversification Risk management

a b s t r a c t This paper studies the issue of modeling conditional covariance for a mixed-asset portfolio consisting of stock, bond, and REITs. We examine the performances of six commonly used covariance estimators. We find that no single estimator delivers the best performance when a wide range of statistical and economic criteria are considered. The optimal estimator to use is found to depend on the evaluation criterion under consideration. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Understanding and predicting the temporal dependence of asset returns is important for many issues in financial economics. For instance, in the classical asset allocation context, a risk-averse investor following the mean-variance criterion of Markowitz (1952) would seek to adequately model the interdependence among asset returns. A measure which captures the (second-order moment) interdependence structure of asset returns is known as covariance. As such, many financial tasks (e.g. portfolio diversification, risk management and asset pricing) require accurate modeling and estimation of the covariance matrix. Essentially speaking, covariance modeling is a multivariate extension of the simpler problem of modeling the conditional variances of individual assets. A number of models have been developed for modeling covariance, most of which belong to the family of multivariate GARCH models. Refer to Bauwens et al. (2006) for a comprehensive review. The popularity of the multivariate GARCH family is also reflected in the real estate literature, as they have often been employed to study either the co-movements among real estate assets (e.g. Cotter and Stevenson, 2006; Michayluk et al., 2006) or across different assets (e.g. Case et al., 2012; Yang et al., 2012). Given the importance of covariance, this paper aims to tackle the issue of how to best model the conditional covariance for a mixedasset portfolio consisting of stock, bond, and REITs. Three points are worth noting: one, we include REITs into the portfolio. The reason is that REITs, as a distinctive investment alternative to the mainstream stocks and bonds, allow easy access to real estate investments without directly owning or managing the underlying assets. Because of their significant market capitalization growth, improved liquidity and imperfect ⁎ Tel.: +1 519 824 4120x56634; fax: +1 519 823 1964. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.econmod.2014.04.010 0264-9993/© 2014 Elsevier B.V. All rights reserved.

covariance with equities and bonds, REITs have received increasing attention from security analysts and pension fund managers in the execution of their investment plans (Chandrashekaran, 1999). Two, even though a variety of multivariate GARCH models have been applied in the real estate literature, no one has yet examined which one is the most appropriate for modeling portfolios with real estate as a component. This paper attempts to fill this hole; three, our focus is on forecasting covariance. Given that financial decision making is typically of forward-looking nature (i.e. making a decision for the next investment period given all currently available information), forecasting is more practically relevant than simply estimating covariance using in-sample analysis. A resultant complication, however, is that forecasting is more econometrically challenging than in-sample estimation (or fitting), as will be shown later. To carry out the investigation, we compile a group of six widely used models, which cover a varying degree of complexity. More specifically, our study includes the simple rolling estimator, the RiskMetrics estimator of J.P. Morgan (1996), and four sophisticated multivariate GARCH models (i.e. VECH-GARCH of Bollerslev et al., 1988, BEKK-GARCH of Engle and Kroner, 1995; CCC-GARCH of Bollerslev, 1990; and the DCC-GARCH of Engle, 2002). We attempt to determine which of them yields the best forecasts of conditional covariance. To ensure robustness of our findings, we resort to a wide range of evaluation metrics, including statistical and more importantly economic criteria. The statistical criteria to be utilized include the multivariate version of root mean square error (RMSE), mean absolute error (MAE) and Thiel's Inequality Coefficient (also known as Thiel's U). As can be imagined, focusing solely on statistical comparison is not particularly informative to investors as it falls short of measuring which competing model can bring the most tangible economic benefits. As a result, following the statistical assessments, we cast all six models into two practical scenarios in an attempt to perform economic assessments. The practical scenarios we

J. Zhou / Economic Modelling 40 (2014) 242–249

consider are portfolio diversifications and risk measurements. Regarding portfolio diversifications, we want to see which method leads to the best performance for the global minimum-variance portfolio (GMV), which is the leftmost point of the mean-variance efficient frontier. There are two reasons why we consider GMV: one, it allows the variation in the portfolio weights to be driven purely by changes in the conditional covariance matrix, which is the major focus of this study. This way GMV is immune to estimation error of the instable mean returns (Jorion, 1985; Lee and Stevenson, 2005). Two, many empirical studies show that an investment in GMV often yields better outof-sample results than does an investment in the regular mean-variance portfolio and therefore suggest investing in GMV (e.g. Jagannathan and Ma, 2003; Jorion, 1991; Ledoit and Wolf, 2003). Regarding risk measurement, we utilize the concept of Value-at-Risk (VaR). Originally proposed by J.P. Morgan in 1994, VaR estimates the maximal loss that a financial position could incur within a given time period with a prespecified probability (Tsay, 2005; p288). Its main advantages are simplicity and wide applicability (e.g. Jorion, 2006). In a single statistic it provides an estimate of the potential loss faced by a financial position, and can be used by a large variety of investors. These features have made VaR being increasingly used to quantify downside risks. We will see which method leads to the most adequate measure of VaR. We conduct the study using U.S. data. Our data sample ranges from January 1999 to December 2011 with a total of 677 weekly data points. Our major findings are summarized as follows: (1) statistical assessment results show that DCC-GARCH and CCC-GARCH provide the most statistically precise covariance forecast while the rolling estimator provides the worst. VECH-GARCH, BEKK-GARCH, and RiskMetrics fall in between. (2) From the perspective of portfolio diversification, BEKKGARCH and VECH-GARCH lead to overall better GMV performances than the other candidates do. (3) In terms of risk management, only the rolling estimator fails to yield adequate VaR forecasts. When it comes to having the lowest average VaR forecast — a property especially welcomed by investors, as it implies the lowest level of risk capital reserve, VECH is found to prevail over other competitors. Our findings have important implications. First and most importantly, there does not exist a universally best method to model the conditional covariance for a mixed-asset portfolio. Rather, the optimal method to use depends on the evaluation criterion. For instance, using BEKK-GARCH leads to the best portfolio performance, indicating that one can enhance the portfolio performance by explicitly taking into account the dynamic correlations among assets through the use of a model like BEKK. However, when it comes to risk management, VECH becomes the choice. Second, our findings imply that there are inconsistencies between the assessments using statistical criteria and economic criteria. The good statistical performance of certain models (e.g. DCC-GARCH & CCC-GARCH) does not necessarily translate into good economic performances. This finding highlights the importance of considering the economic criteria — a point that our paper emphasizes. The reminder of this paper is organized as follows. Section 2 provides a brief review of the competing covariance estimators. Section 3 discusses the data. Section 4 presents the empirical findings and discusses the implications. Section 5 concludes. 2. Econometric models Consider a N × 1 vector return process { yt}, which has the following dynamics yt ¼ μ t þ εt

ð1Þ

where μ t = E(yt | It − 1) is the conditional mean vector (It − 1 is the information set up to time t − 1), εt = Ht1/2zt. Ht1/2 is a N × N positive definite matrix, and zt is a N × 1 vector of return innovations with zero mean and unit variance. Note that N is the number of financial assets.

243

The N × N conditional covariance matrix of yt can be shown as Co varðyt jI t−1 Þ ¼ Co
vart ðyt Þ ¼ Co var
t ðεt Þ  1=2 1=2 0 Co varðzt Þ H t ¼ Ht ¼ Ht :

ð2Þ

Various specifications are formulated to estimate Ht. Bauwens et al. (2006) have done a comprehensive survey. In this paper, we restrict our attention to those most commonly used estimators, even though we are aware that there exist many alternatives. Specifically, we include the following six competitors: 2.1. The rolling estimator The rolling estimator simply estimates Ht as the sample covariance matrix of the returns. At time t, the one-step-ahead forecast of H, i.e. Ht + 1 is written as H tþ1 ¼ Co varðΣÞ

ð3Þ

where Σ = [ε t …ε t − k] is a N × k matrix of demeaned returns. k indicates how many periods of past returns are being used in the forecast. Among the six competitors, this estimator is the easiest to apply. As seen from Eq. (3), it does not have an explicit parameterization of the covariance dynamics. So the rolling estimator is purely data-driven. This may backfire because Eq. (3) also implies that history will repeat itself the next day. 2.2. RiskMetrics This estimator is generally considered an improvement upon the rolling estimator. First proposed by J.P.Morgan (1996), this method estimates Ht as follows: 0

H t ¼ ð1−λÞεt−1 εt−1 þ λHt−1

ð4Þ

or alternatively, t X i−1 0 λ εt−i εt−i H t ¼ ð1−λÞ

ð5Þ

i¼1

where λ is a smoothing factor (0 b λ b 1). It is easy to see that Ht is now formed as a sum of weighted averages of the past return covariance. Given that 0 b λ b 1, this method places greater weights on more recent returns. The rationale is that recent observations tend to have larger impacts on Ht than distant ones. The value of λ determines how fast the weight decays back into the past. A commonly used value for λ is 0.94. As such, Eq. (5) can be considered an augmented version of the rolling estimator (Eq. (3)), which treats all past returns equally important. We label the estimator of Eq. (5) as RiskMetrics, since it forms the basis of the risk management system called RiskMetrics. It is also referred to as the exponentially weighted moving average (EWMA) model in the literature. RiskMetrics has been widely used. Like the rolling estimator, it is easy to apply. But it imposes restrictive dynamics for covariance by using a common smoothing factor to every time series under consideration. This may not hold in practical applications. As a final note, implementing Eq. (5) for forecasting Ht + 1 requires imposing some starting conditions, for which we set ε0 = 0. In contrast with RiskMertics, multivariate GARCH models allow for flexible dynamics for covariance. In what follows, we present four popular multivariate GARCH models. We treat them as a group and will discuss their pros and cons in the end.

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J. Zhou / Economic Modelling 40 (2014) 242–249

2.3. VECH-GARCH Bollerslev et al. (1988) proposed the following multivariate GARCH model to estimate Ht: H t ¼ A0 þ

q X

p X  0 Ai ∘ εt−i εt−i þ B j ∘Ht− j

i¼1

ð6Þ

2.4. BEKK-GARCH Engle and Kroner (1995) proposed a model called BEKK-GARCH to estimate Ht: 0

2.6. DCC-GARCH

j¼1

where p and q are non-negative integers, Ai and Bj are symmetric matrices, and ∘ denotes Hadamard product (i.e. (C ∘ D)ij = (C)ij ∘ (D)ij for two matrices Cm × n and Dm × n). Eq. (6) is referred to as diagonal vectorized GARCH model (i.e. VECH-GARCH). Estimation of Eq. (6) may incur two problems: one, it involves a very large number of parameters so that the computational burden is daunting; two, it may not produce a positivedefinite covariance matrix — a characteristic expected from a true covariance matrix. To overcome these problems, Ledoit et al. (2003) developed a two-step estimation procedure. We follow this procedure here. The technical details are not presented here to save space. Interested readers may refer to their paper. Using Eq. (6) to forecast covariance can be solved through a recursive procedure, which ultimately requires imposing some initial conditions, for which we set ε 0 = 0 & H 0 = 0. Note that we also set p = q = 1, hence VECH-GARCH(1,1).

Ht ¼ C C þ

and then based on the estimated hii,t use a maximum likelihood cri
 ^¼ ρ ^ ij is obtained, to foreterion to estimate R. Once the estimated R cast ht + 1 we first use Eq. (10) to have forecasts of hii,t + 1, i = 1,…, N,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ij hii;tþ1 hjj;tþ1 . and then Htþ1 ¼ ρ

q X

0

0

Ai εt−i εt−i Ai þ

i¼1

p X

0

B j H t− j B j

ð7Þ

j¼1

where C, A, and B are all N × N matrices with C being upper triangular. Generally speaking, Eq. (7) involves too many parameters to be of practical interest. Even for BEKK-GARCH(1,1) (p = q = 1), there are N(5 N + 1)/2 parameters. To reduce the generality, one needs to impose some restrictions on Ai & Bj. One common way is to restrict them to be a scalar times the identity matrix of order N (i.e. IN), leading to the name of scalar BEKK-GARCH model. In this paper, we estimate the following scalar BEKK-GARCH(1,1) model. 0

0

0

0

H t ¼ C C þ ðaIN Þ εt−1 εt−1 ðaIN Þ þ ðbI N Þ H t−1 ðbIN Þ:

ð8Þ

Forecasting using Eq. (8) can be solved through a similar recursive procedure as used for Eq. (6). 2.5. CCC-GARCH Bollerslev (1990) developed this new type of multivariate GARCH models. The constant conditional correlation (CCC) GARCH model assumes that the conditional correlations are constant over time, which makes the conditional covariance proportional to the product of the corresponding conditional standard deviations of individual return series. Specially, CCC-GARCH estimates Ht as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H t ¼ Dt RDt ¼ ρij hii;t hjj;t

is a diagonal matrix whose elements are where Dt = the conditional standard deviation of individual asset returns estimated from some appropriate univariate GARCH model, and R = (ρij) is the matrix containing the constant conditional correlation correlations ρij. To estimate Ht according to Eq. (9), it takes two steps: first estimate a separate GARCH(1,1) model for each return series to have hii,t: 2



 −1=2 −1=2 −1=2 −1=2 Rt ¼ diag q11;t …qNN;t Q t diag q11;t …qNN;t

ð10Þ

ð11Þ

where Q t = (qij,t) is a N × N symmetric positive definite matrix and meets 0

Q t ¼ ð1−a−bÞQ þ azt−1 zt−1 þ bQ t−1

ð12Þ

where a and b are nonnegative scalar parameters satisfying a + b b 1, and Q is the unconditional covariance matrix of zt. ^ are estimated and by imposing Q = 0, we use Eq. (12) ^ and b Once a 0 to recursively solve for the forecast of Qt + 1. Following Eq. (11), we then –1/2 –1/2 –1/2 have Rt + 1 = diag(q–1/2 11,t + 1…qNN,t + 1)Q t + 1diag(q11,t + 1…qNN,t + 1). Finally, the one-step-ahead forecast of the conditional covariance matrix Ht + 1 can be obtained as H tþ1 ¼ Dtþ1 Rtþ1 Dtþ1

ð13Þ

1/2 where Dt + 1 = diag(h1/2 11,t + 1…hNN,t + 1) is a diagonal matrix consisting of the forecasted conditional standard deviation of individual asset returns based on Eq. (10). To wrap up our discussions of the four multivariate GARCH models, first notice that they share some common ground: all of them allow a separate GARCH process for each element of the covariance matrix Ht. This way they can accommodate more flexible dynamics for covariance than RiskMetrics. But the four models also have differences. On the one hand, VECH-GARCH and BEKK-GARCH are considered the same subgroup of models as both attempt to jointly estimate the entire covariance matrix which is known to consist of two parts (i.e. variance of individual time series and the correlations). They differ only in their way to guarantee the positive-definite constraint for the covariance matrix (see Bauwens et al., 2006). On the other hand, CCC-GARCH and DCC-GARCH belong to another subgroup as both estimate covariance through separated estimations of variances and correlations. DCC-GARCH has additional flexibilities by allowing correlations to be time varying. Thanks to its multiple-step estimation strategy, this subgroup, in comparison with the first subgroup, is particularly suitable when a large number of assets are under consideration. However, the

Table 1 Summary statistics.

ð9Þ

1/2 diag(h1/2 11,t…hNN,t)

hii;t ¼ wi þ αεi;t−1 þ βhii;t−1 ; i ¼ 1; …; N;

Engle (2002) extends the CCC model by making the conditional correlation matrix time dependent. The resulting model is called dynamic conditional correlation model (DCC) GARCH. The conditional correlation matrix (Rt) is assumed to be determined as follows:

Stock Bond REIT

Mean

Std. Dev.

Skewness

Kurtosis

Normality

Unit-root

−0.003 0.110 0.057

2.596 0.536 3.525

−0.535⁎ −0.147 −0.665⁎

4.131⁎ 0.518⁎ 7.775⁎

0.513 ⁎ 0.010 ⁎ 1.755 ⁎

−14.829⁎ −13.399⁎ −14.241⁎

Notes: This table reports the summary statistics of weekly returns for the three assets from Jan 1999 to Dec 2011. Stock is represented by S&P 500 Index, bond by Merrill Lynch U.S. Domestic Master Index, and REIT by the FTSE NAREIT All-REITs Index. Mean and Standard deviation are expressed in the percentage form. Normality is tested for using the Jarque– Bera test and the test statistics are expressed in the unit of 1000. Unit-root is tested for using the ADF (Augmented Dickey–Fuller) test, under which the null hypothesis H0 is that the series of interest is I(1). ⁎ Significant at 5% level.

J. Zhou / Economic Modelling 40 (2014) 242–249

245

Stock 10 0 -10

1999 2

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

Bond

1 0 -1

1999 20

2000

REIT

0

-20 1999

2000

Fig. 1. Time series plots of weekly returns. The sample period is from Jan 1999 to Dec 2011.

strategy also has a drawback: it may lead to inconsistent estimates of the covariance matrix (Aielli, 2013). With the above being said, what estimator has the best performance is ultimately an empirical question.

3. Data We focus on a mixed-asset portfolio consisting of U.S. stock, bond and REITs. Stock is represented by S&P 500 Index, bond by Merrill Lynch U.S. Domestic Master Index, and REITs by the FTSE NAREIT AllREIT Index. The data obtained are weekly price indices (Wednesday to Wednesday)1, ranging from January 1999 to December 2011. This leads to a total of 677 data points.2 The obtained price indices are then transformed into returns by taking the log first difference. Table 1 presents the summary statistics of the returns. As shown in the table, the mean stock return is negative while its counterparts for bond and REITs are both positive. Bond returns demonstrate the lowest variability among the three assets over the sample period. All returns except those of bond exhibit skewness (i.e. asymmetry between negative and positive returns). Moreover, all returns display significant kurtosis, which is consistent with the stylized fact of fat-tailness for financial returns. As such, all return series reject the notion of following normal distribution. Finally, all return series appear to be stationary, as indicated by the unit-root test. Fig. 1 presents the time-series plots of returns. Note that pronounced swings in volatility occurred during the recent financial crisis.

4. Empirical findings Table 2 reports the unconditional covariance and correlation of the returns over the entire sample period, offering us a first glimpse of the data. Consistent with the summary statistics, bond returns shows overall the lowest level of volatility whereas REIT returns have the highest. Stock returns fall in between. Furthermore, bond returns appear to be negatively correlated with both stock returns and REIT returns, indicating that diversification benefits may exist for a mixedasset portfolio. As expected, stock returns and REIT returns are positively correlated. Our focus is on covariance forecasting. We adopt a moving window framework. This framework uses a fixed-length window. As we roll the window forward by adding one new observation and dropping the most distant one, we re-estimate the model parameters and make a sequential one-step-ahead forecast. In the end we generate a sequence of forecasts that do not overlap. We set the size (k) of the moving window to be 156 (i.e. k = 156), which corresponds to a 3-year period.3 Also to simplify our analysis, we model μt of Eq. (1) as the mean in-sample return for each moving window. We apply the above framework to each estimator under consideration. The resultant out-of-sample forecasts will form the basis for the subsequent analysis. Note that out-of-sample period, forecast period and asset allocation period have the same meaning. They will be used interchangeably throughout the remainder of the paper. 4.1. Statistical evaluations

1

If there is no index for a Wednesday, we choose to use the next available index of the same week. 2 We also tried the SNL U.S. REIT indices which cover a longer data period from Jan 1990 to Dec 2011. The results, reported in Appendix, are comparable to those obtained using the NAREIT indices.

A tricky problem in comparing the covariance forecasts is that the true matrix is unobservable. A proxy needs to be constructed. A 3 Experimenting with different window sizes such as k = 104 leads to qualitatively similar results.

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J. Zhou / Economic Modelling 40 (2014) 242–249

Table 2 Full-sample unconditional covariance and correlation of returns. Stock

Table 4 Comparison of ex post portfolio performances.

Bond.

REIT

Covariance Stock Bond REIT

6.749 −0.215 6.395

−0.215 0.287 −0.075

6.395 −0.075 12.446

Correlations Stock Bond REIT

1.000 −0.154 0.698

−0.154 1.000 −0.040

0.698 −0.040 1.000

Notes: This table presents the estimates of unconditional covariance and correlations of returns over the full sample period (Jan 1999–Dec 2011).

common and successful approach, termed integrated volatility, is to use the cumulative cross-product of intraday return residuals over the forecast horizon (e.g. Andersen et al., 1999; Andersen et al., 2001). Unfortunately, we only have daily returns, but the same methodology still applies. Denoted by Ht the covariance forecast, based on the information available at time t − 1. Let Σt be the cumulative cross-products of daily demeaned returns during that period. The typical elements of these two matrices are denoted by ℏij,t and σij,t, respectively. Then we can use the following multivariate version of root mean square error (RMSE), mean absolute error (MAE) and Thiel's Inequality Coefficient (also known as Thiel's U) to assess the statistical precision of the forecasts:

Rolling RiskMetrics VECH BEKK CCC DCC

Out-of-sample (2006–2011)

Out-of-sample (2008–2011)

μ

σ

SR

μ

σ

SR

5.216 4.648 4.494 5.681 4.782 4.841

2.665 2.666 2.332 2.893 2.564 2.598

1.957 1.743 1.927 1.964 1.865 1.863

4.997 4.310 4.493 5.153 4.421 4.512

3.294 3.326 2.962 3.074 3.169 3.212

1.517 1.296 1.669 1.676 1.395 1.405

Notes: This table compares the ex post performance of portfolios constructed using the covariance forecasts from different estimators. The portfolio under consideration is the global minimum-variance portfolio. μ is the net annualized mean realized return of the portfolio, σ is the annualized realized standard deviation of portfolio returns, and SR is the realized Sharp ratio (μ/σ).

of the multivariate GARCH model. Following this subgroup is the duo of BEKK and VECH from the other multivariate GARCH subclass we consider. RiskMetrics ranks further behind. Rolling ranks last, as its scores on the three criteria are way higher than those of the other five. Given these statistical evaluation results, a more practically relevant question is how the six estimators would fare in economic evaluations. In what follows, we will apply each estimator to the two most commonly considered financial activities in which covariance forecasting is used. 4.2. Portfolio diversifications

2

31=2

2 1 X
E  hij;t −σ ij;t 5 RMSE ¼ 4 2 N i; j

MAD ¼

ð14Þ

 1 X   E hij;t −σ ij;t Þ 2 N i; j 2 4 12 N

U¼2 4 12 N

ð15Þ

31=2 2 X
E  hij;t −σ ij;t 5 i; j

31=2 2 31=2 : 2 X
2 X
1 E  hij;t 5 þ 4N2 E σ ij;t 5 i; j

ð16Þ

i; j

Table 3 presents the results. For notional convenience, we use Rolling, RiskMetrics, VECH, BEKK, CCC, and DCC to represent the estimators. To ensure robustness of our results, we consider two asset allocation periods: 2006–2011 and 2008–2011. The first thing to notice is that among all candidates DCC scores the lowest across the three criteria. So it is safe to say that DCC generates the most statistically precise forecasts of conditional covariance matrix. CCC closely follows DCC. As noted before, these two models belong to the same subgroup Table 3 Statistical evaluation of the forecasts of conditional covariance. Out-of-sample (2006–2011)

Rolling RiskMetrics VECH BEKK CCC DCC

Out-of-sample (2008–2011)

RMSE

MAE

U

RMSE

MAE

U

33.748 28.766 27.765 26.546 24.949 24.769

9.089 7.130 7.068 7.160 6.492 6.464

0.731 0.583 0.560 0.510 0.480 0.474

41.189 35.127 34.445 31.414 30.419 30.198

12.733 9.898 9.813 9.350 8.926 8.888

0.732 0.585 0.561 0.510 0.481 0.475

Notes: This table reports the evaluation results of covariance forecasting using the multivariate version of root mean square error (RMSE) of Eq. (14), mean absolute error (MAE) of Eq. (15), and Thiel's U of Eq. (16).

The arguably most important application of the covariance matrix is as an input to the mean-variance portfolio optimization model of Markowitz (1952). Implementing the model, in general, requires forecasts of both the expected conditional return and expected conditional covariance. However, it is widely regarded as a tough task to estimate the expected conditional returns (Fleming et al., 2001; Merton, 1980) because the mean returns are very unstable over time. The instability of mean returns would cause substantial estimation errors for portfolio performances (e.g. Jorion, 1985; Lee and Stevenson, 2005). To avoid this problem, we focus on the global minimum-variance portfolio (GMV), which is the leftmost point of the mean-variance efficient frontier. Doing so also allows the variation in the portfolio weights (wt) to be driven purely by changes in the conditional covariance matrix. This way wt is not affected by the errors from estimating the uncertain mean returns. Another reason to use GMV is that many empirical studies suggest investing into GMV instead of the classic mean-variance portfolio due to the former's superior out-of-sample results (e.g. Jagannathan and Ma, 2003; Jorion, 1991; Ledoit and Wolf, 2003). The optimization problem of the GMV portfolio can be written as: 0

min wt H t wt wt

0

s:t:wt l ¼ 1

ð17Þ

where Ht is the forecasted conditional covariance matrix, and l is a N × 1 vector of ones. Note that we do not put any further constraints on the problem. In particular, short sales are allowed. Given a series of covariance forecasts, the solution (wt) to Eq. (17) constitutes a portfolio strategy that specifies the optimal asset weights as a function of time. Applying the estimated wt to the actual asset returns in the out-ofsample period yields ex post portfolio returns. Table 4 reports the net annualized mean realized portfolio return (μ), and the annualized realized standard deviation of portfolio returns (σ), and the realized Sharp ratio (SR).4 Our findings suggest that BEKK leads to best portfolio performances. The reason is that BEKK generates the highest SR over the two out-of-sample periods. Note that SR, being risk-adjusted, is a more 4 To simplify our calculations of the Sharpe ratio, we assume the risk-free rate is zero. This does not affect the ranking of the estimators.

J. Zhou / Economic Modelling 40 (2014) 242–249

meaningful performance measure than μ. Using the same measure, VECH is found to have a very comparable performance as BEKK. Furthermore, we find that Rolling matches up with the top duo over 2006–11 while it fades away when a shorter forecast (i.e. 2008–11) period is considered. Moving further down the ranking list is the duo of CCC and DCC. Their Sharpe ratios are 10 basis points below the one from the top performer in BEKK over 2006–11. The gap seems to increase (more than 25 basis points) over the shorter forecast period (i.e. 2008–11). RiskMetrics is found to lie at the bottom. As can be seen here, the ranking of the models is vastly at odds with the one obtained based on statistical criteria. This finding echoes those of Caporin and McAleer (2011) which also reports inconsistencies between statistical and economic evaluation results. Furthermore, our findings here lend credence to the conclusion of Zumbach (2013) that shows evidence of a preference for the covariance subgroup models (BEKK & VECH) with respect to the subgroup of variance and correlation specifications (CCC & DCC). As explained earlier, the distinctive subclass of CCC and DCC models has the advantage of estimating covariance for a large system but is exposed to estimation inconsistency problems. Our results show that the disadvantage outweighs the advantage. Finally, the naïve Rolling estimator, given its data-driven nature, seems to enjoy some success for mixed-asset portfolio diversifications.

247

Table 5 Comparison of portfolio VaR (Value-at-Risk) estimates. hit% p = 1%

LRuc p = 5%

Ave. VaR

p = 1%

p = 5%

p = 1%

p = 5%

Panel a: Out-of-sample (2006–2011) Rolling 0.032 0.070 RiskMetrics 0.016 0.061 VECH 0.019 0.073 BEKK 0.017 0.058 CCC 0.016 0.058 DCC 0.016 0.058

0.058 0.328 0.147 0.275 0.328 0.328

0.119 0.399 0.071 0.557 0.551 0.551

5.737 5.812 5.409 5.858 5.845 5.868

3.583 3.625 3.491 3.681 3.644 3.658

Panel b: Out-of-sample (2008–2011) Rolling 0.024 0.067 RiskMetrics 0.014 0.057 VECH 0.019 0.072 BEKK 0.014 0.048 CCC 0.014 0.053 DCC 0.014 0.053

0.086 0.553 0.238 0.553 0.553 0.553

0.283 0.631 0.174 0.886 0.863 0.863

6.959 6.874 6.507 6.845 6.991 7.007

4.387 4.335 4.097 4.333 4.403 4.413

Notes: This table compares the VaR estimated using the filtered historical simulation (FHS) method for different covariance estimators. p is the VaR level. hit% is the proportion of VaR violations predicted by the models. LRuc is the unconditional coverage test of Kupiec (1995), for which the p-values of test statistics are reported. Those significant test statistics (at the 10% level) are highlighted in bold, indicating that the corresponding model fails to generate adequate VaR forecast. Ave. VaR is the average of VaR estimates over the out-ofsample period.

4.3. Risk management Another important application we consider is risk management. To illustrate, we consider the role of covariance in the calculation of Value-at-Risk (VaR). As a measure of downside risk, VaR is defined as: VaR ¼ F

−1

ð1−pÞ

ð18Þ

where F–1 is the quantile function of the financial return distribution and p is the quantile level of interest. VaR thus measures the maximal loss occurring to a portfolio within a given time period with a prespecified probability (Tsay, 2005, p288). Originally proposed by J.P. Morgan in 1994, VaR has since been widely embraced by risk managers as an important tool to assess risks. Given the covariance forecast of Ht and assume the assets to be equally weighted, we first obtain the forecasted portfolio variance at ^w;t . Then conditioning on the past realized portfolio returns time t as h and realized conditional portfolio variances, we use the filtered historical simulation (FHS) method to find out the p-level quantiles of the standardized portfolio returns, namely zp (p = 1% and 5%).5 Finally, we compute the one-step-ahead VaR forecast as VaRt;p ¼ zp

qffiffiffiffiffiffiffiffi ^ : h w;t

(denoted by hit%) to equal p — the pre-specified VaR level. To test for whether hit% equals p, we utilize the likelihood ratio test of Kupiec (1995). A significant test statistic LRuc would suggest that hit% differs considerably from p. Table 5 presents the results. In most cases, the estimated hit% is reasonably close to the respective p, although hit% tends to be a bit larger on average. For a rigorous statistical check, we turn to the LRuc tests. Note that significant test statistics (5% level) are highlighted in bold. The testing results indicate that among the six estimators only Rolling fails to provide adequate risk forecasts. The other five (i.e. RiskMetrics, VECH, BEKK, CCC, & DCC) work well in both panels. Since VaR can be used to set the amount of risk capital reserves (e.g. the Basel Committee on Banking Supervision, 1996), obtaining adequate VaR forecasts is thus critical for risk management. Conservative forecasts (i.e. overestimating the risks) would lead to too much risk capital to be set aside and hence incurs large opportunity costs whereas aggressive forecasts (i.e. underestimating the risks) would imply insufficient capital reserve and could cause considerable losses or even bankruptcy. As a final note, Table 5 also reports the average size of VaR forecasts (Ave. VaR). The property of having the lowest VaR forecast is especially welcomed by investors, as it implies the lowest level of risk capital reserves, hence the lowest amount of opportunity costs. In this regard, VECH is found to perform better than the others.

ð19Þ 4.4. Implications of the findings

To gauge the accuracy of the risk forecasts, define the ‘hit’ sequence:
 0 hit t;p ¼ I wt yt bVaRt;p

ð20Þ

where I(·) is the indicator function. The hit sequence returns a 1 if the loss at time t is larger than the predicted VaR for that time. Otherwise, it returns a 0. The case of hitt,p = 1 is called a VaR violation. If VaR is accurately estimated, we would expect the proportion of VaR violations 5 As a nonparametric method, filtered historical simulation (FHS) uses the empirical distribution of standardized returns rather than assume a specific distribution function. It is easy to implement, and has been found in a number of studies to perform well (e.g. Barone-Adesi et al., 1999; Hull and White, 1998; Pritsker, 2001). In implementing this approach, we use returns of 104 past weeks — the same size as the moving window.

First and most importantly, our study implies that there does not exist a universally best method to model the conditional covariance for a mixed-asset portfolio. The optimal method to use depends on the evaluation criterion under consideration. For instance, from the perspective of portfolio diversification, using BEKK leads to the best performance. This holds regardless of the asset allocation period under study. Such a finding indicates that one can enhance the portfolio performance by explicitly taking into account the dynamic correlations among assets through the use of a model like BEKK. On the other hand, when it comes to risk management, VECH appears to prevail over other competitors. Second, there could be inconsistencies between the statistical assessments and the economic assessments. For example, DCC and CCC are found among the top two in terms of statistical criteria. However, such a status does not hold up in either portfolio diversification or risk management. In both cases, neither DCC nor CCC makes it to the top

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J. Zhou / Economic Modelling 40 (2014) 242–249

two. As a matter of fact, both of them fail to outperform the less statistical precise estimator in Rolling when it comes to portfolio diversification. Such a finding is well echoed in the forecasting literature, which demonstrates that more complicated models often provide poorer performance than simpler models (e.g. Swanson and White, 1997; Stock and Watson, 1999). This finding also highlights the necessity of considering economic criteria, as the superior statistical performances of certain models do not necessarily translate into economic gains. 5. Conclusions This paper studies how to forecast covariance for a mixed-asset portfolio consisting of stock, bond and REITs. Such a topic has eluded the real estate literature. Covariance forecasting is important because the forecasted covariance is an essential element for many important financial applications such as portfolio diversification and risk management. We carry out the investigation by comparing the forecasting performance of six commonly used covariance estimators: Rolling, RiskMetrics, VECH, BEKK, CCC and DCC. We apply these estimators to a U.S. mixed-asset portfolio. We find that: (1) DCC-GARCH and CCC-GARCH provide the most statistically precise covariance forecast while the rolling estimator provides the worst. VECH-GARCH, BEKK-GARCH, and RiskMetrics fall in between. (2) BEKK-GARCH and VECH-GARCH lead to overall better portfolio performances than the other candidates do. (3) Only the rolling estimator fails to yield adequate VaR forecasts. Among the remaining five estimators, VECH is found to deliver the lowest average VaR forecasts over the two forecast periods. Our findings imply that there is no single best method to model the conditional covariance for a mixed-asset portfolio. Rather, the optimal method to use depends on the evaluation criterion. Moreover, our findings imply that there are inconsistencies between the assessments using statistical criteria and economic criteria. The good statistical performance of certain models (e.g. CCC-GARCH & DCC-GARCH) does not necessarily translate into good economic performances. Appendix We use SNL indices in place of NAREIT and obtain the following results for out-of-sample 2006–11. They are comparable to what we Table A-1 Statistical evaluations.

Rolling RiskMetrics VECH BEKK CCC DCC

RMSE

MAE

U

48.866 45.587 44.763 44.867 44.085 44.690

19.319 18.765 18.830 18.204 18.852 18.817

0.760 0.709 0.711 0.714 0.670 0.669

Summary: CCC & DCC overall are still the best while Rolling is the worst. VECH, BEKK, & RiskMetrics in between.

Table A-2 Comparison of ex post portfolio performances.

Rolling RiskMetrics VECH BEKK CCC DCC

μ

σ

SR

5.753 5.165 5.970 5.741 5.214 5.176

6.051 6.201 5.922 5.654 5.657 5.595

0.951 0.832 1.008 1.015 0.921 0.925

Summary: BEKK & VECH have best performances, followed by Rolling and then the duo of CCC & DCC. RiskMetrics is the worst.

Table A-3 Comparison of portfolio VaR (Value-at-Risk) estimates. hit%

Rolling RiskMetrics VECH BEKK CCC DCC

LRuc

Ave. VaR

p = 1%

p = 5%

p = 1%

p = 5%

p = 1%

p = 5%

0.023 0.015 0.019 0.019 0.015 0.015

0.081 0.050 0.065 0.058 0.058 0.058

0.070 0.418 0.184 0.184 0.418 0.418

0.036 1.000 0.276 0.578 0.578 0.578

10.236 10.153 9.701 9.889 9.861 9.896

4.575 4.461 4.293 4.378 4.400 4.415

Summary: Rolling is still the only one which fails to deliver adequate VaR forecasts. VECH has the lowest VaR estimate.

have shown in paper. Also, the results for out-of-sample 2008–11 are qualitatively similar (not reported).

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