Journal of Empirical Finance · portfolios calculated on the basis of different multi-factor...

Journal of Empirical Finance 45 (2018) 243–268

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Journal of Empirical Finance

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A factor-based approach of bond portfolio value-at-risk: Theinformational roles of macroeconomic and financial stress factorsAnthony H. Tu a, Cathy Yi-Hsuan Chen b,*a Newhuadu Business School, Minjiang University, No. 1 Wenxiao Road, University town, Fuzhou city, Fujian province, 350108, PR Chinab Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. - Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Unter denLinden 6, 10099 Berlin, Germany

a r t i c l e i n f o

JEL classification:G12G14C51C52

Keywords:Nelson–Siegel factor-augmented modelValue-at-riskBacktestsConditional predictability

a b s t r a c t

Based on the Nelson–Siegel term structure framework, we develop a new factor-augmented modelfor the computation of the value-at-risk (VaR) of bond portfolios, and examine whether theinclusion of information contained within macroeconomic variables and financial stress shockscan enhance the accuracy of VaR forecasts. We examine three Citi US bond indices and theempirical results reveal that: (1) based upon the geometric-VaR backtest, proposed by Pelletierand Wei (2016), the new factor-augmented approach provides reasonably accurate VaR forecasts;(2) there is a clear tendency toward better VaR forecasting performance as a result of the inclusionof the macroeconomic variables and financial stress shocks in the Nelson–Siegel factor model; (3)the impact of the inclusion of financial stress shocks appears to be stronger than the impact ofthe inclusion of the macroeconomic variables.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

The 1996 Market Risk Amendment to the Basel Accord established value at risk (VaR) as the basis for determining market riskcapital requirements. Since then, the use of VaR as a risk measure has become standard in the specialized literature, and also popularlyimplemented by most financial institutions and mutual funds.

Although VaR has attracted a considerable amount of theoretical and applied research, the vast majority of the existing studieson VaR modeling have tended to follow three basic methodologies – Monte Carlo simulation, the variance–covariance approach,and the historical simulation approach – and focus on measuring the risk of equity portfolios.1 However, serious restrictions areencountered when applying these techniques to a bond portfolio with a large number of assets. For example, it is well known thatthe implementation of multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models with more than afew dimensions is extremely difficult, essentially because the model has many parameters and the likelihood function becomes veryflat; consequently, the optimization of the likelihood becomes practicably impossible. In other words, there is no way that the fullmultivariate GARCH models can be used to estimate directly the large covariance matrices that are required to capture all of the riskvariables in a large bond portfolio.

Factor models have become increasingly popular over recent decades, and indeed are now widely applied as a means of solvingthe problem mentioned above. VaR was first computed by Golub and Tilman (1997) and Singh (1997), who used principal componentanalysis (PCA) to extract the yield curve risk factors from a series of bond returns. Alexander (2002) subsequently employed a principal

* Corresponding author.E-mail addresses: [email protected] (A.H. Tu), [email protected] (C.Y.-H. Chen).

1 Jorion (2006) and McNeil et al. (2005) provide excellent introductions to these estimation techniques.

https://doi.org/10.1016/j.jempfin.2017.11.010Received 17 August 2016; Received in revised form 18 August 2017; Accepted 19 November 2017Available online 6 December 20170927-5398/© 2017 Elsevier B.V. All rights reserved.

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A.H. Tu, C.Y.-H. Chen Journal of Empirical Finance 45 (2018) 243–268

component GARCH (PC-GARCH) model to generate large GARCH covariance matrices and found that the model had many practicaladvantages beyond VaR model estimations.

Fiori and Iannotti (2007) went on to develop a principal component VaR (PC-VaR) methodology to assess the interest rate riskexposure of Italian banks; based upon their use of daily data covering a five-year period, the risk was evaluated through a VaR measureusing a principal component-based Monte Carlo simulation of interest rate changes, with the interest rate changes being modeled asa function of three underlying risk factors, ‘‘shift’’, ‘‘tilt’’, and ‘‘twist’’, derived from the principal components decomposition of theEuropean Union (EU) yield curve.2 Abad and Benito (2009) compare the accuracy of three different VaR measures of fixed-incomeportfolios calculated on the basis of different multi-factor empirical models of the term structure of interest rates. The three modelsare: (1) regression models, (2) principal component models, and (3) parametric (Nelson–Siegel; hereinafter NS) models. In addition,the cartography system used by Riskmetrics is also included in the comparison. At the 99% (95%) confidence level, the NS model(Riskmetrics model) is the one that yields the most accurate VaR estimates. Date and Bustreo (2015) also propose a way of measuringthe risk of a sovereign debt portfolio using a simple two-factor short rate model. The model is calibrated from data and then thechanges in the bond prices are simulated using a Kalman filter (KF). The KF-based method is computationally cheaper compared tothe principal component-based models. Recently, Caldeira et al. (2015) estimated the VaR for bonds employing the factor approach,which differs significantly from the existing methods as it is built on a well-established term structure factor model. They use thedynamic version of the NS three-factor (level, slope, and curvature) model, as proposed by Diebold and Li (2006) and Diebold et al.(2006).

Although our study relates closely to the factor VaR approach adopted in Caldeira et al. (2015), the focus here is that we take theexisting evidence a step further, expanding the NS three-factor model by including macroeconomic and financial stress factors. Weare motivated to include these additional factors by several recent findings, described below.

Yu and Zivot (2011) find that the monthly forecasts of United States (US) yield curves related to Treasury, investment-grade, andspeculative-grade bonds are greatly improved by introducing macroeconomic variables into the yield level in the NS three-factormodel. Dewachter and Iania (2012) have subsequently extended the benchmark macro-finance model of Dewachter and Lyrio (2006)by introducing financial stress factors in addition to the standard macroeconomic factors; these include liquidity-related (or moneymarket spread) and return-forecasting (or risk-premium) factors. They find that in terms of the cross-sectional fit to the yield curve,the factor-augmented model significantly outperforms most macro-finance yield curve models.

Financial stress shocks (both liquidity related and return forecasting) are found to have statistically and economically significantimpacts on the yield curve, accounting for a substantial proportion of the variation in the curve.3 In their very recent study, Frickeand Menkhoff (2015) also find that the expected element of the excess returns of bonds is driven by macroeconomic factors, whereasthe innovation element seems to be influenced primarily by financial stress conditions.

Guided by the above findings, in this study we expand the NS three-factor model to include three macroeconomic variables, theinflation rate, the Standard and Poors (S&P) 500 index, and the federal funds rate as the macroeconomic (macro) factor, and fourfinancial shocks, the London Interbank Offered Rate (LIBOR) spread, the T-bill spread, default probability, and the Chicago BoardOptions Exchange (CBOE) Volatility Index (VIX), as the financial stress (financial) factor.4

In an attempt to provide a firm understanding of the importance of the three types of factors (NS, macro, and financial), weemploy nested and non-nested regressions to facilitate an examination of (1) whether the macroeconomic and financial stress factorscan provide incremental information capable of predicting and explaining the variations in the bond portfolio yields, and (2) whichtype of factors (or factor-combinations) offers the best predictive ability or explanatory power for the variations in bond portfolioyields. Turning to the derivation in the closed-form solution for the conditional expected bond portfolio return, we use the dynamicversions of the NS factor-augmented models and employ the dynamic conditional correlation GARCH (DCC-GARCH) specificationto model conditional variance and covariance. As a consequence, the one-day-ahead VaR estimates obtained from the first twoconditional moments are based on the information revealed from three types of factors. Finally, we apply VaR backtesting and VaRperformance ranking techniques to examine the impacts of the various types of factors (or factor-combinations) on the VaRs in thebond portfolios.

We first examine the accuracy of the VaR estimates through backtesting, which is based on the coverage/independence criteriaproposed by Kupiec (1995), Christoffersen (1998), Berkowitz et al. (2011), and Pelletier and Wei (2016). We then compare and rankthe VaR predictive performance between the factors and the factor-combined models by applying the conditional predictive ability(CPA) test proposed by Giacomini and White (2006) to identify (3) whether the macroeconomic or financial stress factors can improvethe forecasting performance of bond VaR, and (4) which factor (macro or financial) makes the greater contribution.

To the best of our knowledge, this study represents the first attempt to identify specifically the effect of the factors driving the VaRof bond portfolios. We provide empirical evidence concerning the applicability of the proposed approach by considering three bondindices: the Citi US Treasury 10Y–20Y Index, the Citi US Broad Investment-Grade Bond Index, and the Citi US High-Yield MarketIndex; these respectively comprise Treasury securities, investment-grade, and speculative-grade (high-yield) bonds.

2 Prior related studies have shown that the three-factor model can successfully explain the main variations in government bond yields (Christensen et al., 2011;de Rezende and Ferreira, 2013), as well as in investment-grade and speculative-grade corporate bond yields (Yu and Salyards, 2009; Yu and Zivot, 2011), with thethree factors being regarded as the fundamental (or yield-curve) factors of the bond yields.

3 Numerous studies have investigated the determinants of corporate yield spreads and have linked such spreads to credit risk and liquidity (Collin-Dufresne et al.,2001; Elton et al., 2001; Ericsson and Renault, 2006; Friewald et al., 2012; Helwege et al., 2014; Huang and Huang, 2012), as well as macroeconomic risk (Duffie etal., 2007; Jarrow and Turnbull, 2000; Yu and Zivot, 2011).

4 Other studies that take the financial stress factor into consideration include Liu et al. (2006), Feldhütter and Lando (2008), and Dewachter and Iania (2012).

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The results show that our study makes a three-fold contribution to the extant literature. First, the extended five-factor (three NSfactors plus macro and financial factors) model provides reasonable accurate VaR forecasting. Second, the empirical results show thatthe three NS factors are rarely sufficient to provide reasonably accurate VaR forecasts, and macroeconomic variables and financialstress shocks are also important driving factors; indeed, the inclusion of macroeconomic variables and financial stress shocks inthe NS term structure model significantly enhances VaR forecasting performance. Third, interestingly our results also reveal thatgreater impact is discernible as a result of the inclusion of financial stress shocks compared to the impact from the inclusion of themacroeconomic variables.

The remainder of this paper is organized as follows. An introduction to the NS factor-augmented models used for modeling thejoint framework of the term structure, the macroeconomic variables, and the financial stress shocks is provided in Section 2. This isfollowed in Section 3 by a description of the procedure for computing the VaRs, the econometric specification for the construction andestimation of the factor-augmented models, and the closed-form solution for the first two conditional moments of the bond portfolioyields. Section 4 presents the results of the VaR estimates and describes the informational roles of the various factor combinations,followed in Section 5 by the evaluation tests of the VaR estimates. Finally, the conclusions drawn from this study are presented inSection 6.

2. Factor models

2.1. Dynamic Nelson–Siegel (NS) three-factor model

Nelson and Siegel (1987) introduced a parsimonious and influential three-factor model for zero coupon bond yields, which isexpressed as:

𝑦𝑡(

𝜏𝑖)

= 𝛽1𝑡 + 𝛽2𝑡

(

1 − 𝑒−𝜂𝜏𝑖𝜂𝜏𝑖

)

+ 𝛽3𝑡

(

1 − 𝑒−𝜂𝜏𝑖𝜂𝜏𝑖

− 𝑒−𝜂𝜏𝑖)

+ 𝜀𝑡 (1)

with 𝑦𝑡 = [𝑦𝑡 (𝜏1), 𝑦𝑡 (𝜏2) . . .𝑦𝑡 (𝜏𝑁 )]′ as N ×1 vector of yields at time t, and disturbance vector 𝜀𝑡 =(

𝜀1𝑡,… , 𝜀𝑁𝑡)′; 𝜏𝑖 refers to the

maturity of bond i, 𝑖 = 1, 2,…𝑁 , which usually ranges from 3 months to 30 years.The NS specification in Eq. (1) can generate several yield curve shapes, including upward sloping, downward sloping, and (inverse)

hump-shaped curves, with the parameter 𝜂 determining the rate of exponential decay. The three factors are 𝛽1𝑡, 𝛽2𝑡, and 𝛽3𝑡. Thefactor loading on 𝛽1𝑡 is unity for all maturities, which implies that a change in 𝛽1 will lead to a parallel shift in the yield curve bythe same amount of change as in 𝛽1; thus, it is referred to as the ‘‘level factor’’. As 𝜏𝑖 becomes greater, 𝛽1𝑡 begins to play a moreimportant role in forming the yields, compared to the smaller factor loadings on 𝛽2𝑡 and 𝛽3𝑡. At the limit 𝑦𝑡 (∞) = 𝛽1𝑡, therefore, 𝛽1𝑡 isalso referred to as the ‘‘long-term factor’’. The factor loading on 𝛽2𝑡 is (1 − 𝑒−𝜂𝜏𝑖 )/𝜂𝜏𝑖, which is close to unity (zero) for the short-term(long-term) maturity yield, and since it loads short rates more heavily than long rates, it changes the slope of the yield curve. Thus,𝛽2𝑡 is regarded as a ‘‘short-term factor’’, also referred to as the ‘‘slope factor’’.

The factor loading on 𝛽3𝑡 is (1 − 𝑒−𝜂𝜏𝑖 )/𝜂𝜏𝑖 − 𝑒−𝜂𝜏𝑖 , which is close to zero for both the short-term and long-term maturities of theyield curve, and higher for the middle-term maturities. As it loads medium rates more heavily, 𝛽3𝑡 is regarded as a ‘‘medium-termfactor’’, also referred to as the ‘‘curvature factor’’, essentially because an increase in 𝛽3𝑡 will lead to a corresponding increase in thecurvature of the yield curve. These three factors can be regarded as the ‘‘fundamental’’ or ‘‘yield-curve’’ factors of bond yields.5

2.2. The Nelson–Siegel (NS) factor-augmented model

Despite its obvious importance, the NS factor model has gone through several modifications and extensions over the yearsto include additional variables or factors. The improvements in this study derive primarily from the incorporation of additionalmacroeconomic variables and financial stress shocks.

5 As opposed to the arbitrage free dynamic NS (AFNS) framework developed by Christensen et al. (2011), we choose to work within the dynamic NS (DNS) model.To examine whether the no-arbitrage constraints are relevant, we follow Coroneo et al. (2011) to conduct a test for the equality of the NS factor loadings on theno-arbitrage ones derived from the Gaussian affine arbitrage-free model defined by Joslin et al. (2011). Joslin et al. (2011) show that any no-arbitrage NS modelcan be normalized to a model in which the factors are observed bond portfolios. As the NS factors are just linear combination of yields, any no-arbitrage NS modelconsidered so far falls in the class of Gaussian no-arbitrage yield curve models as defined by Joslin et al. (2011). The corresponding null hypothesis created in Equation(13) of Coroneo et al. (2011) is therefore applied to test whether the NS model is statistically different from the AFNS model. The F-test statistics (with p-value inparentheses) for the null hypothesis, as reported in the following table, show the statistical insignificance of factor loadings between these two models.

Intercept Level Slope Curvature Joint

F statistic 0.14(0.99)

0.05(1.00)

0.10(1.00)

0.12(0.99)

0.41(1.00)

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2.2.1. Macroeconomic variablesIn earlier related studies, it has been established that macroeconomic variables are related to the dynamics of the yield curves, and

that their inclusion in yield-only models obviously improves forecasting performance. Ang and Piazzesi (2003), for example, find thatthe one-month-ahead out-of-sample vector autoregressive forecasting performance of Treasury yields improves when macroeconomicvariables are included.

Dewachter et al. (2006) present a methodology for estimating the term structure model of interest rates that incorporates bothobservable and unobservable factors with macroeconomic interpretations. As such, the model is well suited for tackling questionsrelating to the interactions between financial markets and the macroeconomy, and is also better at describing the joint dynamicsfor the macroeconomy and the yield curve. Ludvigson and Ng (2009) find that real and inflated macroeconomic variables havepredictive power for future government bond yields, and the macroeconomic variables are also able to explain a larger proportionof the variations in corporate credit spreads over time.6 Prior related studies have also found that macroeconomic variables tendto explain a significant portion of their variations. Jarrow and Turnbull (2000), for example, suggest that a reduced-form model ofcredit spreads might be improved by incorporating macroeconomic variables, while Duffie et al. (2007) use macroeconomic variablesto improve predictive performance with regard to corporate defaults.7

Using Treasury yields and nine different ratings of corporate bonds, Yu and Zivot (2011) examine comprehensive short- andlong-term forecasting evaluation of the two-step and one-step approaches of Diebold and Li (2006) and Diebold et al. (2006), andfind that forecasting using the NS factor model could be improved through the incorporation of macroeconomic variables. In a morerecent study based on Japanese government bond data, Ullah et al. (2013) also find that the yield-macro model with the incorporationof macroeconomic factors leads to a better in-sample fit and out-of-sample forecasting of the NS term structure than the yield-onlymodel. In the following empirical model, we follow Diebold et al. (2006) and Yu and Zivot (2011) in this study to consider threemacroeconomic variables: the inflation rate (Infl), S&P500 index returns (SP), and the federal funds rate (FFR).

2.2.2. Financial shocksDewachter and Iania (2012) extend the macro-finance yield-curve model by introducing liquidity-related and return-forecasting

(risk premium) shocks in addition to the standard macroeconomic variables. From an examination of US data, they find that in termsof the cross-sectional fit of the yield curve, the extended model significantly outperforms the macro-finance yield curve models.In their study, liquidity-related shocks are obtained from the decomposition of the TED spread in the money market, while thereturn-forecasting (risk premium) shock is extracted by imposing a single-factor structure on the one-period expected excess holdingreturn.

The TED spread, which is defined as the difference between the three-month Treasury bill (T-bill) and the relevant unsecuredmoney market rate (i.e., the LIBOR), is often considered to be a key indicator of financial strain (market liquidity and credit risk)within the money markets, with an increase being associated with an increase in counterparty and/or funding liquidity risk. We followseveral prior studies in decomposing this money market spread into two distinct spread shocks (LIBOR spread and T-bill spread),8 asfollows:

𝑇𝐸𝐷𝑡 ≡ 𝑖𝐿𝐼𝐵𝑂𝑅𝑡 − 𝑖𝑇 𝑏𝑖𝑙𝑙𝑡 =(

𝑖𝐿𝐼𝐵𝑂𝑅𝑡 − 𝑖𝑟𝑒𝑝𝑜𝑡)

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟LIBOR spread

+(

𝑖𝑟𝑒𝑝𝑜𝑡 − 𝑖𝑇 𝑏𝑖𝑙𝑙𝑡)

⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟T-bill spread

where 𝑖𝐿𝐼𝐵𝑂𝑅, 𝑖𝑇 𝑏𝑖𝑙𝑙, and 𝑖𝑟𝑒𝑝𝑜 denote the three-month LIBOR rate, the three-month T-bill rate, and the general collateral-secured reporate respectively.

By comparing unsecured money market rates with their secured counterparts, the LIBOR spread (LIBORS) provides an indicatorof counterparty or more general credit risk within the money markets. A widening of the LIBOR spread typically indicates increasedcredit risk exposure in the money markets. The T-bill spread (TbillS), which measures the convenience yield of holding governmentbonds, is generally considered to be a proxy for ‘‘flight to quality’’ (or ‘‘flight to liquidity’’).9 Typically, a widening of this spread isoften associated with frequent flight to quality.

A considerable number of prior studies have provided comprehensive empirical analysis of the effects of liquidity and credit riskon corporate yield spreads,10 with the vast majority suggesting that liquidity and credit risk are the two most important determinantsof expected corporate bond returns. In particular, several related studies have suggested that liquidity effects are more pronouncedin periods of financial crisis (such as the subprime mortgage crisis), particularly for bonds with high credit risk (or inferior credit

6 Examples of related studies examining the joint dynamics between the macroeconomy and the Treasury yield curve include Piazzesi (2005), Wu (2006), Dieboldet al. (2006), Rudebusch and Wu (2007, 2008), and Wu and Zhang (2008), among others.

7 Other related works include Davies (2008) and Castagnetti and Rossi (2013), among others.8 See, for example, Liu et al. (2006), Feldhütter and Lando (2008), and Dewachter and Iania (2012).9 In fixed-income securities markets, investors are often observed to rebalance their portfolios toward less risky and more liquid securities during periods of

economic distress. These respective phenomena are commonly referred to as flight to quality and flight to liquidity. Although the economic motives for these twophenomena clearly differ from each other, empirically disentangling a flight to quality from a flight to liquidity is difficult. In the context of the corporate bond marketin the US, these two attributes of a fixed-income security (credit quality and liquidity) are usually positively correlated (Ericsson and Renault, 2006). Using data onthe Euro-area government bond markets, Beber et al. (2009) find that credit quality matters for bond valuation, but that in times of market stress, investors chaseliquidity, not credit quality.

10 See, for example, Elton et al. (2001), Collin-Dufresne et al. (2001), Ericsson and Renault (2006), Friewald et al. (2012), Dick-Nielsen et al. (2012), Huang andHuang (2012), and Helwege et al. (2014), among others.

246


ratings).11 In other words, the spread contribution from Illiquidity is shown to have increased dramatically with the onset of thesubprime crisis.

In addition to the two money market spread shocks, we consider two other financial shocks, default probability (DP) and theVIX. Elton et al. (2001) find that expected defaults account for a surprisingly small proportion of the spread between corporate andgovernment bond rates. Dionne et al. (2010), subsequently revisiting the estimation of the proportions of default risk in corporate yieldspreads, find that the estimated proportions of default in credit spreads are sensitive to changes in recovery rates, the sample period,and the type of data filtration method used. To obtain an effective measure of the default probability variable of speculative-gradecorporate bonds, we take the difference between the Moody’s AAA and BBB bond yields as our ‘‘default probability’’ variable measure,with our empirical results revealing that the default probability variable employed in this study has more significant explanatorypower than the spread between corporate and government bond rates. The VIX measures the implied volatility of S&P 500 indexoptions over the subsequent 30-day period. It has also come to be known generally as the ‘‘investor fear gauge’’ (Low, 2004; Whaley,2000) or the ‘‘sentiment index’’ (Wall Street Journal). The VIX index has become widely accepted as a measure of uncertainty andinstability in financial markets (Hakkio and Keeton, 2009), and is also regarded as a financial stress indicator.

Fricke and Menkhoff (2015) decompose the excess returns of bonds into the expected proportion (the risk premium) and theinnovative proportion of the excess returns, and find that the expected proportion of the excess returns of the bonds is driven bymacroeconomic factors, whereas the innovation proportion appears to be influenced primarily by financial stress conditions. Giventhat the four financial shocks examined above are all highly correlated with market stress conditions or sentiment, they can also beregarded as ‘‘stress’’ or ‘‘sentiment’’ factors, and as investor sentiment is usually found to have a negative correlation with market-widerisk aversion, as well as uncertainty with regard to future economic conditions, the specification is consistent with the notion thatcredit spreads are dependent on investors’ attitude toward risk, as well as their uncertainty with regard to future economic prospects.As a consequence, the four financial shocks are included in the empirical model in the following section.

3. Model estimation

In this section, we go on to apply the dynamic NS factor model to the yield curve, together with the addition of the macroeconomicvariables and financial shocks as previously discussed, to obtain the closed-form solution for the conditional expected bond yields, aswell as their conditional covariance matrix. From these two moments, we are able to derive the distribution of the bond prices andreturns, and then use this to compute the VaR of a bond portfolio. The achievement of our proposed target depends upon four steps,as described in the following sub-sections.

3.1. Step one: estimation of the Nelson–Siegel (NS) state-space model

The NS model, Eq. (1), can be rewritten as an alternative three-factor version, as follows:

𝑦𝑡 (𝜏) = 𝐿𝑡 + 𝑆𝑡

(

1 − 𝑒−𝜂𝜏𝜂𝜏

)

+ 𝐶𝑡

(

1 − 𝑒−𝜂𝜏𝜂𝜏

− 𝑒−𝜂𝜏)

+ 𝜀𝑡 (2)

where the latent factors 𝐿𝑡, 𝑆𝑡, and 𝐶𝑡 refer to the level, slope, and curvature factors respectively. We follow the dynamic frameworkof Diebold et al. (2006) in specifying the first-order vector autoregressive processes for the factors, and propose a linear Gaussianstate-space approach involving the use of a one-step KF12 to simultaneously carry out parameter estimation and signal extraction inthe dynamic NS factor model.13 The ‘‘measurement (observation) equation’’ of the state-space form for the dynamic NS three-factormodel is expressed as:

⎛

⎜

⎜

⎜

⎜

⎝

𝑦𝑡(

𝜏1)

𝑦𝑡(

𝜏2)

⋮𝑦𝑡(

𝜏𝑁)

⎞

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

1 1 − 𝑒−𝜂𝜏1𝜂𝜏1

1 − 𝑒−𝜂𝜏1𝜂𝜏1

− 𝑒−𝜂𝜏1

1 1 − 𝑒−𝜂𝜏2𝜂𝜏2

1 − 𝑒−𝜂𝜏2𝜂𝜏2

− 𝑒−𝜂𝜏2

⋮ ⋮ ⋮

1 1 − 𝑒−𝜂𝜏𝑁𝜂𝜏𝑁

1 − 𝑒−𝜂𝜏𝑁𝜂𝜏𝑁

− 𝑒−𝜂𝜏𝑁

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎝

𝐿𝑡𝑆𝑡𝐶𝑡

⎞

⎟

⎟

⎠

+

⎛

⎜

⎜

⎜

⎜

⎝

𝜀𝑡(

𝜏1)

𝜀𝑡(

𝜏2)

⋮𝜀𝑡(

𝜏𝑁)

⎞

⎟

⎟

⎟

⎟

⎠

= 𝛬𝛽 (𝜂) 𝛽𝑡 + 𝜀𝑡 (3)

where 𝛽𝑡 =(

𝐿𝑡, 𝑆𝑡, 𝐶𝑡)′, and 𝑁 × 3 factor loading matrix 𝛬𝛽 (𝜂). Eq. (3) relates the observed yields to the three latent NS factors (state

variables) and measurement errors. The ‘‘transition equation’’, which describes the evolution of the state variables as a first-order

11 See Gefang et al. (2011), Dick-Nielsen et al. (2012), and Friewald et al. (2012).12 The KF is a recursive procedure for computing the optimal estimator of the state vector at time t given the information available at time 𝑡 − 1.13 Diebold and Li (2006) earlier proposed a two-step procedure for the estimation of the NS factor yield curve; however, Diebold et al. (2006) argue that the second

step of the two-step procedure used in Diebold and Li (2006) suffered from a lack of consideration of the parameter estimation and signal extraction uncertaintyassociated with the first step. Thus, the one-step approach is regarded as superior to the two-step approach, essentially because the simultaneous estimation of all theparameters produces correct inferences with respect to standard theory.

247


Markov process, is given by14 :

⎛

⎜

⎜

⎝

𝐿𝑡 − 𝜇1𝑆𝑡 − 𝜇2𝐶𝑡 − 𝜇3

⎞

⎟

⎟

⎠

=⎛

⎜

⎜

⎝

𝑎11 𝑎12 𝑎13𝑎21 𝑎22 𝑎23𝑎31 𝑎32 𝑎33

⎞

⎟

⎟

⎠

⎛

⎜

⎜

⎝

𝐿𝑡−1 − 𝜇1𝑆𝑡−1 − 𝜇2𝐶𝑡−1 − 𝜇3

⎞

⎟

⎟

⎠

+⎛

⎜

⎜

⎝

𝜖1𝑡𝜖2𝑡𝜖3𝑡

⎞

⎟

⎟

⎠

(4)

where the decay parameter, 𝜂, here is 0.0609 (Diebold et al., 2006).We assume that the measurement and transition disturbances are Gaussian white noise, diagonal, and orthogonal to each other,

as is the standard treatment of the state-space model (Durbin and Koopman, 2001):(

𝜀𝑡𝜖𝑡

)

∼ 𝑖.𝑖.𝑑. 𝑁[(

00

)

,(

𝛴𝜀𝑡 00 𝛴𝜖𝑡

)]

(5)

where 𝛴𝜖𝑡 is the (3 × 3) covariance matrix of the innovations of the transition system and is assumed to be unrestricted, while thecovariance matrix (𝛴𝜀𝑡) of the innovations to the measurement system, of (N×N) dimension, is assumed to be diagonal. The latterassumption means that the deviations in the observed yields from those implied by the fitted yield curve are uncorrelated acrossboth time and maturity. Given the large number of observed yields used, the assumption of the diagonality of the measurement errorcovariance matrix is necessary for computational tractability.

Given that both macroeconomic and financial stress factors are taken into consideration,15 the measurement and transitionequations can be extended and expressed respectively as follows:

𝑦𝑡 = 𝛬 (𝜂) 𝛿𝑡 + 𝜀𝑡 (6)

𝛬 =(

𝛬𝛽 (𝜂) , 𝜆𝑀 , 𝜆𝐹)

, 𝛿′𝑡 =(

𝛽′𝑡 ,𝑀𝑡, 𝐹𝑡)

(7)

⎛

⎜

⎜

⎜

⎜

⎜

⎝

𝐿𝑡 − 𝜇1𝑆𝑡 − 𝜇2𝐶𝑡 − 𝜇3𝑀𝑡 − 𝜇4𝐹𝑡 − 𝜇5

⎞

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

𝑎11 𝑎12 𝑎13 𝑎14 𝑎15𝑎21 𝑎22 𝑎23 𝑎24 𝑎25𝑎31 𝑎32 𝑎33 𝑎34 𝑎35𝑎41 𝑎42 𝑎43 𝑎44 𝑎45𝑎51 𝑎52 𝑎53 𝑎54 𝑎55

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎜

⎜

⎝

𝐿𝑡−1 − 𝜇1𝑆𝑡−1 − 𝜇2𝐶𝑡−1 − 𝜇3𝑀𝑡−1 − 𝜇4𝐹𝑡−1 − 𝜇5

⎞

⎟

⎟

⎟

⎟

⎟

⎠

+

⎛

⎜

⎜

⎜

⎜

⎜

⎝

𝜖1𝑡𝜖2𝑡𝜖3𝑡𝜖4𝑡𝜖5𝑡

⎞

⎟

⎟

⎟

⎟

⎟

⎠

(8)

where 𝑀𝑡 denotes the common macroeconomic factor, which is the first principal component of the three macroeconomic variables(FFR, Infl, and SP), and which explains 80% of the total sample variance. Similarly, 𝐹𝑡 represents the common financial stress factor,which is the first principal component of the four financial shock variables (LIBORS, TbillS, DP, and VIX), and which dramaticallyexplains 93% of the total sample variance.16 We then apply the one-step KF, explained in Appendix A, to the above state-spacerepresentation, using Eqs. (6), (7), and (8) to estimate the three NS factors, �̂�𝑡, �̂�𝑡, and �̂�𝑡.

3.2. Step two: Conditional expected bond portfolio yield and conditional variance

Under a multivariate setting involving a bond portfolio, the computation of VaR requires two main ingredients, namely, thevector expected returns and their covariance matrix. Although the factor models for the interest rate term structure discussed aboveare designed to model only bond yields, it is possible to obtain expressions for the expected bond portfolio returns, along with theircorresponding conditional covariance matrices, based on the distribution of the expected yields.17

Following Caldeira et al. (2015, 2016), given the system of equations in (3) and (4), the conditional expectation of the yield-only(or NS) factor model is:

𝜇(𝑦𝑝,𝑡|𝑡−1) = 𝛬′𝛽𝛽𝑡|𝑡−1 (9)

where 𝜇(𝑦𝑝,𝑡|𝑡 − 1) denotes the expected bond portfolio yield at time t, conditional on the available information set 𝛺𝑡 − 1 at time𝑡 − 1; 𝛽𝑡|𝑡−1 is a vector of the one-day-ahead forecast of the three NS factors. Given the system of equations in (6), (7), and (8), withthe inclusion of the macroeconomic and financial stress factors, the conditional expected bond portfolio yield becomes:

𝜇(𝑦𝑝,𝑡|𝑡−1) = 𝛬′𝛽𝛽𝑡|𝑡−1 + 𝜆𝑀𝑀𝑡|𝑡−1 + 𝜆𝐹𝐹𝑡|𝑡−1. (10)

14 To identify the model and simplify the computations, following Yu and Zivot (2011), we assume that the coefficient matrix in Eq. (4) is diagonal for two reasons:first, Diebold et al. (2006) report that most off-diagonal elements of this matrix are statistically insignificant, with small magnitudes; second, the main objective ofthe approach is to estimate yield curve factors, rather than identifying the best fitting model. This restriction therefore simplifies the estimation of the model withoutaffecting our results.

15 The factor-augmented model has been widely applied in the term structure literature (see Diebold et al., 2006; Exterkate et al., 2013; Ullah et al., 2013; Yu andZivot, 2011), among others.

16 The principal component method is a dimension reduction technique that transforms a number of correlated macroeconomic variables (or financial shockvariables) into a smaller number of uncorrelated variables based on explaining the majority of the information in the sample covariance matrix of variables. Theprimary aims of applying the principal component technique in this paper are: (i) to reduce the number of factors to a manageable dimension, and (ii) to solve themulticollinearity problem when all interrelated macroeconomic variables and financial shocks are used as explanatory variables in a linear regression model. Theresults of the PCA are not reported here for reasons of space; however, they can be provided upon request.

17 Ferreira and Lopez (2005) provide empirical support for the interest-rate portfolio VaR models based on simple covariance matrix forecasts and distributionalassumptions.

248


The corresponding conditional variances of Eqs. (9) and (10) are, respectively, given by:

𝜎2(𝑦𝑝,𝑡|𝑡−1) = 𝛬′𝛽(

𝛴𝑡|𝑡−1(𝛽𝛽′))

𝛬𝛽 + 𝜎2𝜀,𝑡|𝑡−1 (11)

and

𝜎2(𝑦𝑝,𝑡|𝑡−1) = 𝛬′ (𝛴𝑡|𝑡−1(𝛿𝛿′))

𝛬 + 𝜎2𝜀,𝑡|𝑡−1 (12)

where 𝛴𝑡|𝑡−1 (⋅) represents the factor variance–covariance matrix at time t, conditional on the available information set 𝛺𝑡−1 at time𝑡 − 1; 𝜎2𝜀, 𝑡|𝑡 − 1 is the conditional variance of the error term.18

3.3. Step three: Calculation of bond portfolio expected return

In this sub-section, we begin by deriving the distribution of the expected fixed-maturity bond prices. Let us consider the price ofa bond at time t. Given that 𝑃𝑡(𝜏) is the present value at time t of the $1 receivable 𝜏 periods ahead, the vector of the expected bondprice, 𝑃𝑡|𝑡−1, for all maturities can be obtained by:

P𝑡|𝑡−1 = exp(

−𝜏 ⊗ 𝑦𝑡|𝑡−1)

(13)

where 𝑦𝑡|𝑡−1 denotes the one-step ahead forecast of the continuously compounded zero-coupon nominal yield to maturity; ⊗ is theHadamard (or element by element) multiplication; 𝜏 is the vector of the maturities. Thus, the log-return of the bond portfolio can beexpressed as:

𝑟𝑝,𝑡 = log P𝑡 − log P𝑡−1 = −𝜏 ⊗ (𝑦𝑝,𝑡 − 𝑦𝑝,𝑡−1). (14)

It is possible to find a closed-form solution for the conditional expected bond portfolio return as well as for its conditionalcovariance using Eq. (14). With a simple derivation, similar to that of Caldeira et al. (2015), the conditional expected log return forthe bond portfolio is:

𝜇(𝑟𝑝,𝑡|𝑡−1) = −𝜏 ⊗ 𝜇𝑝,𝑡 + 𝜏 ⊗ 𝑦𝑝,𝑡−1. (15)

With the inclusion of the macroeconomic and financial stress factors, and inserting Eq. (10) into (15), the conditional expectedlog return becomes:

𝜇(𝑟𝑝,𝑡|𝑡−1) = 𝜏 ⊗

(

𝛬′𝛽𝛽𝑡|𝑡−1 + 𝜆𝑀𝑀𝑡|𝑡−1 + 𝜆𝐹𝐹𝑡|𝑡−1

)

+ 𝜏 ⊗ 𝑦𝑝,𝑡−1 (16)

with respective corresponding conditional variances of19 :

𝜎2(𝑟𝑝,𝑡|𝑡−1

) = 𝜏′𝜏 ⊗[

𝛬′𝛽(

𝛴𝑡|𝑡−1(

𝛽𝛽′))

𝛬𝛽 + 𝜎2𝜀,𝑡|𝑡−1]

(17)

and

𝜎2(𝑟𝑝,𝑡|𝑡−1

) = 𝜏′𝜏 ⊗[

𝛬′ (𝛴𝑡|𝑡−1(𝛿𝛿′))

𝛬 + 𝜎2𝜀,𝑡|𝑡−1]

. (18)

The above results show that it is possible to obtain a closed form solution for the conditional expected bond portfolio log return,and its corresponding conditional variance and covariance based on the NS model and its extensions.

To facilitate the modeling of the conditional covariance matrices, 𝛴𝑡|𝑡−1(𝛽𝛽′) and 𝛴𝑡|𝑡−1(𝛿𝛿′), we consider the DCC-GARCH modelproposed by Engle (2002), which is given by20 :

𝛴𝑡|𝑡−1(

𝛿𝛿′)

= 𝐷𝑡𝛹𝑡𝐷𝑡, (19)

where 𝐷𝑡 is a (5 × 5) diagonal matrix with the elements given by ℎ𝑓𝑘,𝑡 (the conditional variance of the 𝑘th factor), and 𝛹𝑡 is asymmetric correlation matrix with elements 𝜌𝑖𝑗,𝑡, where i, 𝑗 = 1,…5. The conditional correlation, 𝜌𝑖𝑗,𝑡, in the DCC model is given by:

𝜌𝑖𝑗,𝑡 =𝑞𝑖𝑗,𝑡

√

𝑞𝑖𝑖,𝑡𝑞𝑗𝑗,𝑡, (20)

where 𝑞𝑖𝑗,𝑡, i, 𝑗 = 1,…5, are the elements of the (5 × 5) matrix 𝑄𝑡, which follows GARCH-type dynamics, expressed as:

𝑄𝑡 = (1 − 𝛼 − 𝛽)𝑄 + 𝛼𝑍𝑡−1𝑍′𝑡−1 + 𝛽𝑄𝑡−1, (21)

where𝑍𝑡 are the (5 × 1) standardized vectors of the factor returns, the elements of which are𝑍𝑓𝑘𝑡 = 𝑓𝑘𝑡∕√

ℎ𝑓𝑘𝑡;𝑄 is the unconditionalcovariance matrix of 𝑍𝑡; 𝛼 and 𝛽 are the non-negative scalar parameters satisfying 𝛼 + 𝛽 < 1.

18 The derivation of Eqs. (10) and (12) is similar to Proposition 1 in Caldeira et al. (2015).19 The conditional covariance matrix was shown to be positive definite by Caldeira et al. (2015).20 Caldeira et al. (2015) note that among all the GARCH-type specifications considered, the most accurate VaR estimates are those obtained from the NS yield curve

models, with the conditional covariance matrix given by a DCC-GARCH model.

249


The estimation of the DCC model can be conveniently divided into two univariate parts, conditional volatility and correlation.The univariate conditional volatility of the factors can be modeled using a GARCH-type specification, with the parameters beingestimated by quasi-maximum likelihood (QML) with the assumption of Gaussian innovations. To estimate the parameters of thecorrelation elements of Eqs. (20) and (21), we employ the composite likelihood (CL) method proposed by Engle et al. (2008).21

3.4. Step four: computation of the value-at-risk and the expected shortfall

𝑉 𝑎𝑅𝑝,𝑡 is defined as the loss of a bond portfolio that, within a given day, will be exceeded with a certain probability 𝛼 (1% and5%); this is expressed as:

prob[

𝑃𝐿𝑝,𝑡 < −𝑉 𝑎𝑅𝑝,𝑡|𝛺𝑡−1]

= 𝛼

where 𝑃𝐿𝑝,𝑡 denotes the profit/loss of the bond portfolio at time t.Using the conditional variance–covariance approach, the one-day-ahead VaR estimate of the bond portfolio at time t, is 𝑉 𝑎𝑅𝑝,𝑡+1

(1 − 𝛼) = �̃�𝛼𝑊 𝜎(𝑟𝑝,𝑡|𝑡), where 𝛼 = 1% or 5%, �̃�𝛼 is the value of the 𝛼 percentile approximated by the Cornish–Fisher expansion,22 andW is the initial portfolio value. 𝜎(𝑟𝑝,𝑡|𝑡) is represented by either Eq. (17) or (18) and its dynamics are estimated through DCC-GARCH.We then compare 𝑉 𝑎𝑅𝑡+1|𝑡 to the actual profit/loss of the portfolio on day 𝑡 + 1, denoted as 𝑃𝐿𝑝,𝑡+1. If 𝑃𝐿𝑝,𝑡+1 < 𝑉 𝑎𝑅𝑝,𝑡+1|𝑡, we havean exception (or violation). For backtesting purposes, the violation indicator variable is defined as:

𝐼𝑡+1 ={

1 𝑖𝑓𝑃𝐿𝑝,𝑡+1 < 𝑉 𝑎𝑅𝑝,𝑡+1|𝑡0 𝑖𝑓𝑃𝐿𝑝,𝑡+1 ≥ 𝑉 𝑎𝑅𝑝,𝑡+1|𝑡.

(22)

As the VaR is not in general coherent, we also estimate the bond portfolio’s expected shortfall (ES), given by23 :

𝐸𝑆𝑝,𝑡+1 (1 − 𝛼) = 𝐸[

𝐿𝑜𝑠𝑠|𝐿𝑜𝑠𝑠 ≤ 𝑉 𝑎𝑅𝑝,𝑡+1(1 − 𝛼)]

(23)

where 𝛼 = 1% or 5%.

4. Data and empirical results

4.1. Data

The data examined in this study are the US spot rates for Treasury zero-coupon and coupon-bearing AA- and BBB-rated corporatebonds covering the period January 2011 to December 2014; these data, obtained from Bloomberg, provide us with a total of 924daily observations. The summary statistics on the yield data across different maturities for Treasury zero-coupon (zero), AA-rated,and BBB-rated bonds are presented in Table 1. For each maturity period, we report the mean, standard deviation (SD), maximum,minimum, skewness (skew), kurtosis (kurt), and autocorrelation coefficients at various displacements for Treasury zero, AA-rated,and BBB-rated yield data.

A cross-section of the three types of yields over time is illustrated in Fig. 1. The summary statistics and figures reveal that theaverage yield curves for the three types of yields are all upward sloping. Skewness also has a downward trend, with both the maturityand kurtosis of the short rates being higher than those of the long rates.

We examine three bond indices, the Citi US Treasury 10Y–20Y Index, the Citi US Broad Investment-Grade Bond Index, and the CitiUS High-Yield Market Index, which are, respectively, composed of Treasury, investment-grade, and high-yield bonds. The three bondindices are all composed of a large number of bonds, which are weighted by their market capitalizations. The main characteristicsof the three bond indices are provided in Table 2. For the three macroeconomic variables and the four financial shocks, we use dailydata for the same sample period used in the above yield analysis for the inflation rates, the S&P 500 index returns, and the federalfund rates, as well as the LIBOR spreads, the T-bill spreads, the default probabilities, and the VIX values.

Details on the three macroeconomic variables, comprising S&P 500 returns, federal fund rates, and inflation rates, are respectivelyobtained from Datastream, the ‘‘FRED’’ website (http://research.stlouisfed.org/fred2/series/DFF), and the Federal Reserve Bank ofSt. Louis.24 Details on the financial stress shocks, LIBOR, repo rates, and T-bill rates are collected from Datastream, data on theVIX index are obtained from the CBOE website, and the Moody’s AA- and BBB-rated bond yields are available from the FederalReserve Website (http://www.federalreserve.gov/releases/h15/data.htm). The descriptive statistics (mean, SD, maximum, minimum,skewness, kurtosis, and autocorrelation coefficients) are summarized in Table 3.

21 Compared to the two-step procedure proposed by Engle and Sheppard (2001) and Sheppard (2003), the CL estimator proposed by Engle et al. (2008) providesmore accurate parameter estimates, particularly for large-dimensional problems.

22 To take into account non-normality in factor distribution, we employ the Cornish–Fisher approximation to estimate the 𝛼 percentile, as this takes into accountthe skewness and kurtosis of a distribution. See Appendix B for a description of the Cornish–Fisher approximation of the 𝛼 percentile.

23 The VaR fails to be a coherent measure because it does not satisfy the sub-additivity property, so that diversification does not necessarily result in a reduction ofrisk as measured by VaR. To cope with this shortcoming, Artzner et al. (1999) and Basak and Shapiro (2001) propose the use of the so-called expected shortfall as analternative measure of risk.

24 The daily inflation rates used in this study are the five-year break-even inflation rates from the Federal Reserve Bank of St. Louis. The break-even inflationrate, which represents a measure of expected inflation derived from five-year Treasury Constant Maturity Securities, provides an assessment of the average expectedinflation rate by market participant over the subsequent five-year period.

250

http://research.stlouisfed.org/fred2/series/DFF

http://www.federalreserve.gov/releases/h15/data.htm


Table 1Descriptive statistics of yield data across maturities (months).

Maturity Mean SD Max Min Skew Kurt �̂� (1) �̂� (6) �̂� (12)

Treasury zero yield (100 bp)

3 1.168 1.799 5.676 0.020 1.446 3.491 0.997 0.986 0.9736 1.241 1.802 5.646 0.028 1.432 3.461 0.998 0.987 0.973

12 1.327 1.690 5.483 0.125 1.383 3.383 0.998 0.986 0.97224 1.596 1.543 5.428 0.271 1.209 3.083 0.997 0.985 0.97136 1.932 1.482 5.544 0.410 0.968 2.653 0.997 0.986 0.97248 2.276 1.403 5.577 0.605 0.756 2.379 0.997 0.985 0.97260 2.598 1.317 5.633 0.838 0.551 2.190 0.997 0.985 0.97072 2.884 1.252 5.671 1.102 0.389 2.049 0.997 0.984 0.97084 3.178 1.194 5.711 1.348 0.204 1.919 0.997 0.983 0.96896 3.403 1.127 5.748 1.614 0.149 1.861 0.997 0.981 0.965

108 3.598 1.043 5.782 1.855 0.094 1.890 0.996 0.979 0.960240 3.790 0.959 5.810 2.146 0.062 1.946 0.996 0.976 0.954360 4.482 0.790 5.936 2.863 −0.323 1.934 0.994 0.968 0.942

AA-rated yield (100 bp)

12 2.074 2.163 6.452 0.047 0.904 2.151 0.998 0.990 0.97924 2.257 2.044 6.337 0.111 0.789 1.948 0.998 0.991 0.98136 2.558 2.083 6.543 0.242 0.734 1.844 0.999 0.992 0.98248 2.970 1.988 6.881 0.495 0.578 1.703 0.999 0.992 0.98460 3.329 1.994 7.597 0.735 0.520 1.726 0.999 0.992 0.98472 3.748 1.900 7.985 1.053 0.387 1.745 0.998 0.992 0.98484 4.173 1.829 8.454 1.375 0.262 1.783 0.998 0.991 0.98396 4.451 1.776 8.654 1.699 0.223 1.820 0.998 0.992 0.983

108 4.719 1.698 8.727 2.013 0.178 1.808 0.998 0.991 0.983120 5.059 1.845 9.472 2.287 0.319 1.790 0.999 0.992 0.983180 5.367 1.700 9.862 2.787 0.375 1.969 0.998 0.991 0.981240 5.472 1.529 9.568 3.088 0.372 2.057 0.998 0.990 0.978360 5.998 1.390 9.349 3.579 0.154 1.926 0.998 0.989 0.977

BBB-rated yield (100 bp)

12 3.113 2.094 6.889 0.682 0.569 1.577 0.999 0.991 0.98224 3.253 1.976 6.989 0.986 0.555 1.540 0.999 0.992 0.98436 3.620 1.960 7.548 1.230 0.536 1.568 0.999 0.993 0.98548 4.048 1.916 8.412 1.570 0.461 1.617 0.999 0.994 0.98760 4.445 1.918 9.042 1.871 0.402 1.648 0.999 0.994 0.98772 4.848 1.851 9.241 2.176 0.333 1.731 0.999 0.994 0.98784 5.250 1.799 9.385 2.501 0.300 1.830 0.999 0.993 0.98696 5.537 1.769 9.971 2.825 0.324 1.935 0.999 0.993 0.986

108 5.797 1.748 10.231 3.086 0.323 2.012 0.999 0.993 0.986120 6.145 1.862 10.719 3.366 0.329 1.804 0.999 0.993 0.986180 6.486 1.752 11.270 3.907 0.399 1.963 0.999 0.992 0.984240 6.504 1.633 10.660 4.016 0.340 1.945 0.998 0.991 0.982360 7.009 1.509 10.706 4.557 0.395 2.213 0.998 0.991 0.982

This table presents the basic statistics of yield data on Treasury zero and bonds with AA- and BBB-rated across different maturities (months). The sample period rangesfrom Jan. 2011 to Dec. 2014 with 924 daily observations. �̂� (⋅) denotes the sample autocorrelation at lag 1, 6, and 12 days.

4.2. Empirical results

4.2.1. Basic characteristics of the factorsIn step one, we use the full-sample observed yields of the Treasury zero coupon, together with the AA- and BBB-rated coupon

bonds, and apply the one-step KF to the state–space representation, shown in Eqs. (6), (7), and (8), to estimate the three NS factors(�̂�𝑡, �̂�𝑡, and �̂�𝑡) for the Treasury zero, AA- and BBB-rated yield curves (Table 3). The results of the augmented Dickey–Fuller (ADF)unit root test indicate that the three estimated NS factors (level, slope, and curvature) and the two additional factors (macro andfinancial) all exhibit stationary time series. The time series of the three estimated NS factors, together with the two additional macroand financial stress factors, are illustrated in Fig. 2.

4.2.2. Explanatory power of the NS factor-augmented modelsIn this subsection, we take the actual bond index yields as dependent variables, and run a linear regression on the NS, macro,

and financial factors. Our aim is to determine how much of the yield variations the three NS factors are capable of explaining, andwhether the macro and financial stress factors can add additional explanatory power. The five-factor predictive regression can bespecified as follows:

𝑦𝑝,𝑡+1 = 𝑦0 + 𝛬′𝛽𝛽𝑡 + 𝜆𝑀𝑀𝑡 + 𝜆𝐹𝐹𝑡 + 𝜀𝑝,𝑡+1 (24)

251


Fig. 1. Time series plot of Treasury zero, AA- and BBB-rated yield curves. Note: The sample comprises of daily Treasury zero, AA- and BBB-rated yield data acrossvarious maturities from January 2011 to December 2014.

where 𝑦𝑝,𝑡+1 is the bond index yield at time 𝑡+1; 𝑦0 denotes the intercept term; 𝛽𝑡 = [�̂�𝑡, �̂�𝑡, �̂�𝑡]′ is the (3 × 1) vector of factors estimatedfrom the NS model in step one; 𝑀𝑡 is a principal component macroeconomic factor; 𝐹𝑡 is a principal component financial stress factor;𝛬𝛽 = [𝜆𝛽1, 𝜆𝛽2, 𝜆𝛽3] ′, 𝜆𝑀 and 𝜆𝐹 are the corresponding factor coefficients; 𝜀𝑝,𝑡+1 is the error term, with zero mean, which representsthe proportions of the yields that are not explained by the factors.

252


Table 2Characteristics of three bond indices used in the sample.

Name of the bond index Citi US Treasury 10Y–20Y Bond Index Citi US Broad Investment Grade BondIndex

Citi US High Yield Market Index

Basic description The US Treasury 10Y–20Y Bond Indextracks the performance of US 10Y–20YTreasury notes and bonds.

The US Broad Investment-Grade BondIndex (USBIG) tracks the performance ofUS Dollar-denominated bonds issued inthe US investment-grade bond market.

The US High-Yield Market Index is a USDollar-denominated index whichmeasures the performance of high-yielddebt issued by corporations domiciled inthe US or Canada.

Coupon Fixed-rate Fixed-rate Fixed rateCurrency USD USD USDMinimum maturity Between 10 and 20 years At least one year At least one yearWeighting Market capitalization Market capitalization Market capitalization

Credit quality Minimum quality: AAA by S&P or Aaaby Moody’s

Minimum quality: BBB- by S&P or Baa3by Moody’s

Maximum Quality: BB+ by S&P andBa1 by Moody’s; Minimum Quality: C byS&P and Ca by Moody’s (excludesdefaulted bonds)

Composition US Treasuries (excluding FederalReserve purchases, inflation-indexedsecurities and STRIPS)

US Treasuries (excluding FederalReserve purchases, inflation-indexedsecurities and STRIPS); US agencies(excluding callable zeros and bondscallable less than one year from issuedate); supranationals; mortgagepass-throughs; asset-backed; credit(excluding bonds callable less than oneyear from issue date); Yankees, globals,and corporate securities issued underRule 144A with registration rights

Cash-pay, Zero-to-Full (ZTF),Pay-in-Kind (PIK), step-coupon bonds,and Rule 144A bonds issued bycorporations domiciled in the UnitedStates or Canada only

Table 3Descriptive statistics of NS three factors, macro and financial stress factors.

Mean SD Max Min Skew Kurt �̂� (1) �̂� (6) �̂� (12) ADF p-values

Treasury zero yield curve

𝐿𝑡 2.493 0.673 4.822 0.485 −0.083 3.093 0.486 0.432 0.456 0.000𝑆𝑡 −2.183 0.651 −0.719 −4.329 −0.417 2.866 0.518 0.481 0.499 0.000𝐶𝑡 −6.262 1.419 −1.550 −9.545 1.087 3.712 0.890 0.838 0.792 0.090𝑀𝑡 −0.040 0.403 1.339 −1.109 0.554 3.877 −0.052 −0.036 0.020 0.000𝐹𝑡 1.913 0.324 2.621 0.552 0.065 2.625 0.839 0.796 0.740 0.095

AA-rated yield curve

𝐿𝑡 3.222 0.746 4.919 0.524 −0.189 2.787 0.459 0.396 0.439 0.000𝑆𝑡 −1.346 0.886 0.373 −3.732 −0.105 1.862 0.672 0.648 0.659 0.000𝐶𝑡 −7.540 2.173 −1.788 −11.265 0.368 2.030 0.936 0.887 0.844 0.090𝑀𝑡 −0.044 0.455 1.474 −1.293 0.557 4.183 −0.051 −0.053 0.036 0.000𝐹𝑡 2.091 0.329 2.746 0.590 −0.529 2.688 0.803 0.763 0.709 0.095

BBB-rated yield curve

𝐿𝑡 3.839 0.860 6.022 0.585 −0.157 3.047 0.429 0.363 0.411 0.000𝑆𝑡 −1.723 1.217 0.825 −4.928 −0.242 2.100 0.760 0.734 0.750 0.000𝐶𝑡 −5.715 2.696 1.032 −10.749 0.370 1.974 0.940 0.912 0.885 0.090𝑀𝑡 −0.053 0.536 1.753 −1.649 0.542 4.095 −0.050 −0.058 0.038 0.000𝐹𝑡 2.491 0.361 3.193 0.661 −0.583 3.353 0.779 0.738 0.681 0.091

Macroeconomic variables

FFR 1.084 1.762 5.410 0.040 1.588 3.863 0.997 0.985 0.972 0.008Infl 1.769 0.644 2.720 −2.240 −2.459 11.329 0.994 0.954 0.918 0.294SP 0.002 0.635 4.450 −4.113 −0.437 9.940 −0.122 0.076 0.020 0.000

Financial stress shocks

LIBORS 0.337 0.439 3.400 −0.018 2.828 14.331 0.990 0.934 0.857 0.027TbillS −0.045 0.291 0.630 −2.017 −2.691 11.516 0.943 0.846 0.798 0.000DP 1.280 0.586 3.500 0.720 2.185 7.226 0.999 0.987 0.965 0.506VIX 0.230 0.106 0.809 0.099 2.064 8.349 0.978 0.921 0.867 0.080

This table presents the basic statistics of estimated NS three factors (level (𝐿𝑡), slope (𝑆𝑡) and curvature (𝐶𝑡) and two additional factors (macro (𝑀) and financial (𝐹 )).The sample period ranges from Jan. 2011 to Dec. 2014 with 924 daily observations. �̂� (⋅) denotes the sample autocorrelation at lag 1, 6, and 12 days. The last columnreports the p-value of augmented Dickey–Fuller (ADF) unit root test. The VIX index of CBOE has been multiplied by 0.01. Federal fund rate (FFR), annual inflationrate (Infl), S&P 500 return (SP), LIBOR spread (LIBORS), T-bill spread (TbillS), default probability (DP) are all presented as percentage.

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Fig. 2. Time series plot of the three estimated NS factors, macro and financial factors. Note: This figure illustrates the time variations in the three estimated NS factors(�̂�, �̂�, �̂�), along with the macro (M) and financial stress (F ) factors, respectively derived from the Treasury zero, AA- and BBB-rated yield curves. The estimation ofthe three NS factors is driven by M and F as designed in the factor-augmented model shown in Eqs. (6), (7) and (8).

For the full-sample period, the regression results of the various factor and factor-combined models for variations in the yields inthe three selected bond indices are presented in Table 4, with Panel A reporting the regression results for the yield variations in thenested models of the NS (Treasury zero, AA- and BBB-rated yield curves), 𝑁𝑆 +𝑀 , 𝑁𝑆 + 𝐹 , and 𝑁𝑆 +𝑀 + 𝐹 .

We find that the three NS factors (the ‘‘fundamental factors’’), based on the Treasury zero, AA- and BBB-rated yield curves, havequite high explanatory capabilities in terms of yield variations in the Treasury 10Y–20Y index, the Broad Investment-Grade BondIndex, and the High-Yield Market Index, with respective adjusted 𝑅2 values of 0.937, 0.455, and 0.369; however, the explanatoryabilities of the NS three factors based on the Treasury zero yield curve, yield variations in the Broad Investment-Grade Bond Index,and the High-yield Market Index are quite limited, with respective 𝑅2 values of only 0.356 and 0.081.

The inclusion of the macro and/or financial stress factors improves the explanatory power of all of the regression models, resultingin increased adjusted 𝑅2 values. All of the coefficients for the financial stress factor for all of the indices are found to be significantlydifferent from zero; conversely, whilst the coefficients for the macro factor are found to be significantly different from zero for mostof the indices, this is not the case for the High-Yield Market Index (based on the Treasury zero-coupon yield curve) or the BroadInvestment-Grade Bond Index (based on the AA-rated yield curve).

254

A.H.Tu,C.Y.-H

.ChenJournalofEm

piricalFinance45

(2018)243–268

Table 4The explanatory power of the NS three factors, macro factor and financial stress factor regressed on three bond indices.

Panel A: nested model

Treasury zero-coupon yield curve AA-rated yield curve BBB-rated yield curve

Treasury 10Y–20Y Index Broad Investment Grade Bond Index High Yield Market Index Broad Investment Grade BondIndex

High Yield Market Index

Intercept 0.041 0.143* −0.591* −0.559* 1.993* 1.941* 2.484* 2.873* 6.926* 6.714* 8.699* 9.753* 1.168* 0.970* 1.791* 1.423* 7.601* 6.609* 10.829* 9.440*

(1.31) (5.48) (−18.36) (−10.98) (30.93) (29.83) (28.12) (20.70) (37.34) (37.86) (36.12) (26.13) (14.72) (12.54) (18.81) (9.68) (38.80) (43.45) (59.73) (29.98)

𝐿𝑡0.173* 0.045* 0.168* 0.161* 0.080* 0.145* 0.084* −0.007 −0.017 0.247* −0.003 −0.250* 0.116* 0.258* 0.126* 0.197* 0.012 0.599* 0.053* 0.263*

(58.50) (6.78) (77.04) (16.85) (13.00) (8.68) (14.06) (−0.27) (−1.00) (5.42) (0.16) (−3.58) (19.68) (17.60) (22.34) (8.84) (0.85) (24.11) (5.02) (6.48)

𝑆𝑡−0.613* −0.653* −0.640* −0.642* −0.238* −0.218* −0.217* −0.233* 0.244* 0.327* 0.322* 0.278* −0.236* −0.203* −0.221* −0.210* 0.256* 0.327* 0.266* 0.290*

(−108.3) (−128.7) (−150.0) (−140.5) (−20.28) (−17.22) (−18.54) (−18.73) (7.52) (9.49) (10.21) (8.32) (−26.41) (−22.41) (−25.75) (−22.82) (12.64) (21.18) (18.19) (19.23)

𝐶𝑡0.093* 0.141* 0.129* 0.131* 0.030* 0.005 0.018 0.019* −0.156* −0.255* −0.258* −0.211* −0.058* −0.104* −0.091* −0.101* −0.097* −0.255* −0.210* −0.235*

(27.6) (38.71) (46.22) (39.59) (4.30) (0.61) (0.23) (2.12) (−8.08) (−10.32) (−12.52) (−8.75) (−13.23) (−17.21) (−17.56) (−16.80) (−9.77) (−26.54) (−25.52) (−25.17)

𝑀𝑡0.567* 0.034 0.290* 0.416* −1.174* 1.129* −0.661* −0.348* −2.803* −1.067*

(20.44) (0.81) (4.19) (3.61) (−6.22) (3.64) (−10.44) (−3.27) (−26.27) (−5.35)

𝐹𝑡0.455* 0.438* −0.355* −0.581* −1.281* −1.894* −0.505* −0.292* −2.361* −1.621*

(27.73) (15.48) (−7.85) (−7.54) (−10.54) (−9.14) (−10.57) (−3.62) (−28.57) (−10.11)𝑅-square 0.937 0.957 0.966 0.967 0.356 0.368 0.397 0.406 0.081 0.119 0.184 0.194 0.455 0.513 0.515 0.521 0.369 0.643 0.670 0.680

Panel B: nonnested model

Treasury 10Y–20Y Index Broad Investment Grade Bond Index High Yield Market IndexIntercept 0.041 2.743* 2.278* 1.993* 3.097* 3.641* 6.926* 7.032* 8.072*

(1.31) (152) (15.87) (30.93) (266) (39.76) (37.34) (262) (37.94)

𝐿𝑡 0.173* 0.080* −0.017(58.50) (13.00) (−1.00)

𝑆𝑡 −0.613* −0.238* 0.244*

(−108.3) (−20.28) (7.52)

𝐶𝑡 0.093* 0.030* −0.156*

(27.6) (4.30) (−8.08)𝑀𝑡 0.000 −0.005 −0.053

(0.00) (−0.21) (−0.85)𝐹𝑡 0.255* −0.299* −0.573*

(3.26) (−5.98) (−4.94)𝑅-square 0.937 0.001 0.011 0.356 0.011 0.038 0.382 0.008 0.026

This table reports the nested and nonnested regression results of NS three factors (𝐿, 𝑆, 𝐶), macro factor (𝑀), and financial stress factor (𝐹 ) on yields of three bond indices used in the study. The numbers in parentheses are tstatistics.

* Indicates that the coefficient is significantly different from zero at the 5% level.

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We re-run the regressions of the three bond index yields on the three types of factors alone (non-nested model), based on theTreasury zero yield curve, with the results reported in Panel B of Table 4. For the three bond indices, the three NS term structurefactors dominate the explanatory abilities of their yield variations and the explanatory powers of the financial stress factor are allstronger than those of the macro factor. All of the coefficients for the financial stress factor for the three bond indices are found tobe significantly different from zero, while no significance is found for any of the coefficients for the macro factor. The respective 𝑅2

values for the financial stress (macro) factor are 0.011 (0.001), 0.038 (0.011), and 0.026 (0.008) for the Treasury 10Y–20Y index,the Broad Investment-Grade Bond Index, and the High-Yield Market Index.

In this section, we use the entire sample period to estimate the three NS factors as in Section 4.2.1, then we run a predictiveregression to determine how much of the yield variations can be explained by the NS factors and the macroeconomic and financialstress factors additionally as in Section 4.2.2. Data snooping bias arises as these two processes use the same sample period, and hencesome of the regression results in Table 4 may be misleading.

To alleviate the data snooping bias, we re-run the predictive nested regression in Eq. (24) by using a 500-day rolling window.25

Given 924 daily yields and starting from the first 500 (from 2011∕1∕4 to 2013∕3∕11) yields, we (i) estimate the five factors by one-stepKF; (ii) make forecasts of the yields; (iii) regress the actual against the predicted yields. We iterate this process by discarding theoldest observation and including a new one until the last sample. We therefore obtain a series of 424 adjusted 𝑅-square values. Fig. 3shows the adjusted 𝑅-square values for 3-factor, 4-factor (𝑁𝑆 +𝑀 and 𝑁𝑆 + 𝐹 ) and 5-factor (𝑁𝑆 +𝑀 + 𝐹 ) regressions.

As shown in Fig. 3, the adjusted 𝑅-square values of 5-factor (or 4-factor) model are all slightly higher than those of 3-factor modelin all bond indices. This result confirms the previous analysis that the incremental information is generated by the macroeconomicand financial stress factors.

4.3. VaR estimation

The statistics for the estimated VaRs across the various factor models during the sample period, summarized in Table 6, includethe mean, standard deviation, and 75% and 25% quantiles. Our focus in this study is on the estimation of the 95% coverage ratesand one-day-ahead VaR in Panel A, and on the 99% estimations in Panel B, which provide the relevant risk levels for all financialinstitutions.26

To estimate the one-day-ahead VaR for the three bond indices, we consider a 500-day rolling window strategy. Starting from thefirst 500 yields and the corresponding five factors estimated by one-step KF approach, we estimate the VaR parameters in Eqs. (10) and(12). Following the same rolling process as mentioned previously, we ultimately obtain a series of 424 one-day-ahead VaR forecasts(from 2013∕3∕12 to 2014∕12∕31), as shown in Figs. 4a and 4b.

5. VaR evaluation tests

In this section, we propose two methods for evaluating the accuracy of VaR estimates: (i) backtesting based upon unconditionaland conditional coverage, duration independence, geometric, and geometric-VaR tests, and (ii) ranking comparison based upon theconditional loss function. Our focus is on the examination of whether the macroeconomic and financial stress factors can improvethe forecasting performance of bond VaR, and which factor contributes more.

5.1. VaR and ES backtests27

Assuming that the set of VaR estimates and the underlying model are accurate, violations can be modeled as independent drawsfrom a binomial distribution with a probability of occurrence equal to 𝛼. Accurate VaR estimates should have an unconditionalcoverage (�̂� = x/T ) value equal to 𝛼, where x is the number of violations and T is the number of VaR observations. Kupiec (1995)derives a likelihood ratio (LR) statistic, 𝐿𝑅𝑢𝑐 , to test the hypothesis �̂� = 𝛼, with an asymptotic 𝜒2(1) distribution.

The 𝐿𝑅𝑢𝑐 test is an unconditional test of the coverage of VaR estimates as it simply counts violations over the entire period withno reference to the information available at each point in time. However, if the underlying portfolio returns exhibit time-dependentheteroskedasticity, the conditional accuracy of the VaR estimates is probably a more important issue. In such cases, VaR modelsignoring such variance dynamics will generate VaR estimates that while having correct unconditional coverage at any given time,will have incorrect conditional coverage. To address this issue, Christoffersen (1998) proposes a likelihood ratio statistic, 𝐿𝑅𝑐𝑐 , forthe conditional test of VaR estimates based on interval forecasts.

As accurate VaR estimates have correct conditional coverage, the violation indicator variable, 𝐼𝑡+1, must exhibit both correctunconditional coverage and serial independence. The 𝐿𝑅𝑐𝑐 test is a joint test of these properties; the relevant test statistic is𝐿𝑅𝑐𝑐 = 𝐿𝑅𝑢𝑐 + 𝐿𝑅𝑖𝑛𝑑 , which is asymptotically distributed as 𝜒2(2). The 𝐿𝑅𝑖𝑛𝑑 statistic is the likelihood ratio statistic for the nullhypothesis of serial independence against the alternative of first-order Markov dependence.

One of the drawbacks of Christoffersen’s (1998) 𝐿𝑅𝑐𝑐 test is that it is not capable of capturing dependence in all forms as it onlyconsiders the dependence of observations between two successive days. To address this, Christoffersen and Pelletier (2004) introducethe duration-based test of independence, which is an improved test for both independence and coverage. Berkowitz et al. (2011)

25 Using other shorter or longer window is possible, but the conclusion remains unchanged. We appreciate the valuable comments from the reviewer.26 All financial institutions are required to report this level as a means of measuring their market risk exposure in accordance with the Basel Accords.27 For the purposes of this study, only first-order Markov dependence is used. The likelihood ratio statistics for 𝐿𝑅𝑢𝑐 , 𝐿𝑅𝑐𝑐 , 𝐿𝑅𝑖𝑛𝑑 , 𝐿𝑅𝐷𝑖𝑛𝑑 , and 𝐿𝑅𝐺𝑉 are standard,

in line with the approach of Christoffersen (1998), Christoffersen and Pelletier (2004), and Pelletier and Wei (2016).

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Fig. 3. Time variation of adjusted 𝑅-square values estimated by the predictive regression. Note: Based upon a rolling horizon of 500 daily observations, this figurereports the adjusted 𝑅-square values of the predictive regressions in Eq. (24). The entire-sample regression results can be found in Table 4.

further implement a discrete duration-based test under the likelihood ratio framework. Under the null hypothesis that durations haveno memory, discrete durations follow the geometric distribution, which they refer to as a ‘‘geometric test’’. The geometric test canbe decomposed into a test of unconditional coverage and a test of duration independence. Specifically, the likelihood ratio of the

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Fig. 4a. Time series plot of 95% VaR estimates. Note: Based upon a rolling horizon of 500 daily observations, this figure illustrates the daily profit or loss (in indexpoints) in the 95% VaR estimates of the Treasury 10Y–20Y Bond Index, the Broad Investment-grade Bond Index and the High-yield Market Index over our sampleperiod which runs from January 2011 to December 2014; these are derived from models containing the dynamic NS three factors (NS), the macro factor (M), thefinancial stress factor (F ), NS with the macro factor (𝑁𝑆 +𝑀), NS with the financial stress factor (𝑁𝑆 + 𝐹 ) and NS with both the macro and financial stress factors(𝑁𝑆 +𝑀 + 𝐹 ). As illustrated in the figure, a violation occurs when the PL (profit or loss) is negative and falls below the solid line.

geometric test, 𝐿𝑅𝐺, is equal to the sum of the two test statistics. That is, 𝐿𝑅𝐺 = 𝐿𝑅𝑢𝑐 + 𝐿𝑅𝐷𝑖𝑛𝑑 , where 𝐿𝑅𝐷𝑖𝑛𝑑 is the test statisticof duration independence. Recently, Pelletier and Wei (2016) have proposed the geometric-VaR test to evaluate the performanceof VaR forecasts. This test draws strength from a duration-based approach, which captures general forms of time dependence inviolations, as well as the strength of a regression-based approach, capturing the dependence between VaR forecasts and violations.It can be decomposed into three individual tests: a test of unconditional coverage, a test of duration independence, and a test of VaR

258


Fig. 4b. Time series plot of 99% VaR estimates. Note: Based upon a rolling horizon of 500 daily observations, this figure illustrates the daily profit or loss (in indexpoints) in the 99% VaR estimates of the Treasury 10Y–20Y Bond Index, the Broad Investment-grade Bond Index and the High-yield Market Index over our sampleperiod which runs from January 2011 to December 2014; these are derived from models containing the dynamic NS three factors (NS), the macro factor (M), thefinancial stress factor (F ), NS with the macro factor (𝑁𝑆 +𝑀), NS with the financial stress factor (𝑁𝑆 + 𝐹 ) and NS with both the macro and financial stress factors(𝑁𝑆 +𝑀 + 𝐹 ). As illustrated in the figure, a violation occurs when the PL (profit or loss) is negative and falls below the solid line.

independence. In other words, the likelihood ratio of the geometric-VaR test, 𝐿𝑅𝐺𝑉 , can be written as 𝐿𝑅𝐺𝑉 = 𝐿𝑅𝑢𝑐+𝐿𝑅𝑖𝑛𝑑+𝐿𝑅𝐷𝑖𝑛𝑑 ,and has an asymptotic 𝜒2(3) distribution.

Although there are many alternative backtesting methods from which to choose, we adopt a series of LR tests for a couple ofreasons. First, the unified framework of LR tests allows us not only to test whether VaR forecasts are misspecified overall, but alsoto understand how they are misspecified by looking into which individual hypothesis is rejected. Second, Pelletier and Wei (2016)

259


conduct a Monte Carlo study and show that the geometric-VaR test has better power than other duration-based tests or regression-based tests, and it has power against various forms of misspecifications.

In addition to the measurement of the VaR, we also evaluate the accuracy of the expected shortfall (ES) forecasts. We evaluatethe accuracy of the ES on the basis of the deviation of the realized shortfall against the ES in the case of a VaR violation. Similar toMcNeil and Frey (2000), we apply the bootstrap method to perform the backtest. The null hypothesis of a correct ES corresponds tothe zero deviation between the ES and the realized shortfall.

5.2. Conditional loss function

Although independence/coverage tests may be appropriate for the evaluation of the accuracy of a single model, they may not beappropriate for ranking alternative estimates of the VaR and may well provide an ambiguous decision with regard to which candidatemodel is better. Thus, it is important to enhance the backtesting analysis using statistical tests designed to evaluate the comparativeperformance of the candidate models. We follow Santos et al. (2013), using the equal conditional predictive ability (CPA) test ofGiacomini and White (2006),28 specifically, for a given asymmetric loss function at the (1 − 𝑞)% quantile, which is defined as:

𝐿𝑞𝑡+𝜏(

𝑟𝑝𝑡+𝜏 , 𝑓𝑡)

= (𝑞 − 𝐼((

𝑓𝑡 − 𝑟𝑝𝑡+𝜏)

< 0)

)(𝑓𝑡 − 𝑟𝑝𝑡+𝜏 ), (25)

The null hypothesis of equal conditional predictive ability of forecast function f and g for the target date t + 𝜏 can be written asfollows:

𝐻0 ∶ 𝐸[

𝐿𝑞𝑡+𝜏(

𝑟𝑝𝑡+𝜏 , 𝑓𝑡)

− 𝐿𝑞𝑡+𝜏(

𝑟𝑝𝑡+𝜏 , �̂�𝑡)

|𝐼𝑡]

≡ 𝐸[

𝛥𝐿𝑡+𝜏 |𝐼𝑡]

= 0, (26)

where 𝑟𝑝𝑡+𝜏 are the actual bond portfolio returns on day t + 𝜏, and 𝑓𝑡 and �̂�𝑡 can be any one of the VaR estimates.For a given chosen test function 𝐻𝑡 = (1, 𝛥𝐿𝑡+1), which is a q ×1 vector, a Wald-type test statistic corresponding to the null

hypothesis is given by:

𝐶𝑃𝐴𝑞 = 𝑛

(

𝑛−1𝑇−1∑

𝑡=1𝐻𝑡𝛥𝐿𝑡+1

)

�̂�−1𝑛

(

𝑛−1𝑇−1∑

𝑡=1𝐻𝑡𝛥𝐿𝑡+1

)

= 𝑛𝑍′�̂�−1𝑛 𝑍 (27)

where 𝑍 = 𝑛−1∑𝑇−1𝑡=1 𝑍𝑡+1, 𝑍𝑡+1 = 𝐻𝑡 𝛥𝐿𝑡+1, �̂�𝑛 = 𝑛−1

∑𝑇−1𝑡=1 𝑍𝑡+1 ×𝑍

′𝑡+1 is a q ×q matrix that consistently estimates the variance in 𝑍𝑡+1,

and n is the number of out-of-sample forecasts. A test for the level of 𝛼 can be carried out by rejecting the null hypothesis of equalconditional predictive ability in cases in which 𝐶𝑃𝐴𝑞 > 𝜒2

𝑞 ,1− 𝛼 , where 𝜒2𝑞 ,1− 𝛼 is the 1 − 𝛼 quantile of a 𝜒2

𝑞 distribution.

5.3. Evaluation results

5.3.1. VaR and ES backtestsThe results of the unconditional, independence, and conditional coverage, geometric, and geometric-VaR tests, the violation ratios,

ES and bootstrap test for the three bond indices are presented in Table 6. A violation ratio is defined as the backtesting violationnumber divided by the number of VaR estimates. We compare the VaR performance of various factor models along two dimensions:the number of backtesting violations and the average size of the violations (or ES).

The number of backtesting violations is the primary indicator of VaR performance. If the VaR model works well, we would expectto find the VaR estimates passing the geometric VaR test. The results in Panel A of Table 6, based on 95% VaR estimates, show thatsome of the NS, M, F, NS + M, and NS+F models pass the conditional coverage and geometric tests, but none of them pass thegeometric VaR test. The arbitrage-free NS (AFNS) model performs no better than the NS model. However, the NS factor-augmentedmodel (𝑁𝑆 +𝑀 + 𝐹 ) successfully passes the geometric VaR tests for all three bond indices, which provides a clear answer to ourthird concern of whether the macro and financial stress factors provide any improvement in VaR forecasting performance.

The overall performance of the 99% VaRs is better than that of the 95% VaRs. In particular, the factor-combined models (NS +M, NS + F, and NS + M + F) pass the geometric VaR tests and perform better than the individual factor models (NS, M, and F).

In terms of the second indicator of VaR performance, the results for the ES bootstrap backtests are similar to those for the VaRbacktests. Overall, the factor-combined models (NS + M, NS + F, and NS + M + F) perform better than the individual factor models(NS, M, and F) for both 95% (Panel A) and 99% (Panel B) ESs, which implies that the macro and financial factors do indeed providevaluable information and improvements on the VaR performance of the NS three-factor model.

To further illustrate the results presented in Tables 5 and 6, we present the 95% (99%) VaR estimates of the daily returns on theTreasury 10Y–20Y Bond Index, Broad Investment-Grade Bond Index, and High-Yield Bond Index over the sample period in Fig. 3a(Fig. 3b), where we show the estimates provided by the NS, M, F, NS + M, NS + F, and NS + M + F models. A violation occurs if thenegative return (or loss) falls below the solid line in the figures.

28 The Giacomini and White (2006) test essentially provides improvements, in several respects, on the Diebold and Mariano (1995) approach, which has beenin widespread use for predictive evaluation. First, the test can exist in an environment in which the sample is finite; second, and more importantly, the modelaccommodates conditional predictive evaluation in such a way that we can predict which forecast will be more accurate on a specific future day.

260


Table 5The VaR estimates based on the NS factor-augmented model.

Panel A: 95% VaR

Mean Standard deviation 75% quantile 25% quantile

Treasury 10Y–20Y Index

NS 26.705(1.19%)

14.610(0.65%)

38.281(1.71%)

10.867(0.48%)

𝑀 24.689(1.10%)

3.144(0.14%)

26.935(1.20%)

22.248(0.99%)

𝐹 25.342(1.13%)

4.857(0.22%)

27.668(1.23%)

22.359(1.00%)

NS +𝑀 26.432(1.18%)

13.247(0.59%)

36.806(1.65%)

11.777(0.53%)

NS + 𝐹 26.367(1.18%)

12.602(0.56%)

36.603(1.64%)

13.991(0.63%)

NS +𝑀 + 𝐹 26.161(1.17%)

11.904(0.53%)

35.323(1.58%)

14.570(0.65%)

Broad Investment Grade Bond Index

NS (Treasury zero) 12.776(0.69%)

4.994(0.27%)

16.677(0.90%)

7.471(0.41%)

NS (AA-rated) 12.366(0.67%)

4.312(0.23%)

15.451(0.84%)

9.130(0.49%)

𝑀 14.763(0.80%)

2.903(0.16%)

18.023(0.98%)

11.928(0.65%)

𝐹 14.082(0.76%)

3.605(0.19%)

16.735(0.91%)

10.914(0.59%)

NS +𝑀 15.241(0.83%)

5.906(0.32%)

19.607(1.06%)

9.289(0.50%)

NS + 𝐹 11.784(0.64%)

6.895(0.37%)

15.433(0.84%)

5.676(0.31%)

NS +𝑀 + 𝐹 12.269(0.67%)

7.190(0.39%)

16.618(0.91%)

5.950(0.32%)


NS (Treasury zero) 5.482(0.70%)

1.413(0.18%)

6.833(0.88%)

4.242(0.54%)

NS (BBB-rated) 4.836(0.62%)

0.739(0.09%)

5.420(0.69%)

4.154(0.53%)

𝑀 3.949(0.51%)

1.044(0.13%)

5.029(0.64%)

3.390(0.43%)

𝐹 3.638(0.47%)

0.749(0.09%)

4.234(0.54%)

3.390(0.43%)

NS +𝑀 5.201(0.67%)

0.862(0.11%)

5.995(0.77%)

4.497(0.58%)

NS + 𝐹 4.367(0.56%)

1.131(0.14%)

5.041(0.65%)

3.617(0.46%)

NS +𝑀 + 𝐹 4.410(0.57%)

1.050(0.13%)

5.182(0.67%)

3.652(0.46%)

Panel B: 99% VaR



NS 30.812(1.38%)

16.650(0.74%)

44.221(1.98%)

12.753(0.57%)

𝑀 28.644(1.28%)

3.718(0.17%)

32.018(1.43%)

26.135(1.17%)

𝐹 29.411(1.32%)

5.762(0.26%)

31.827(1.43%)

25.700(1.15%)

NS +𝑀 30.573(1.37%)

15.050(0.67%)

42.208(1.89%)

13.833(0.62%)

(continued on next page)

261


Table 5 (continued)

Panel B: 99% VaR


NS + 𝐹 30.512(1.37%)

14.295(0.64%)

42.621(1.91%)

16.512(0.74%)

NS +𝑀 + 𝐹

30.294(1.36%)

13.512(0.61%)

41.425(1.86%)

17.272(0.77%)


NS(Trea-suryzero)

16.123(0.87%)

5.856(0.31%)

20.338(1.10%)

11.626(0.63%)

NS(AA-rated)

16.669(0.90%)

6.765(0.36%)

21.758(1.17%)

9.454(0.51%)

𝑀 19.210(1.04%)

4.182(0.22%)

23.897(1.29%)

15.165(0.82%)

𝐹 18.273(0.99%)

4.734(0.25%)

21.944(1.19%)

14.081(0.76%)

NS +𝑀 19.885(1.08%)

8.006(0.43%)

25.414(1.37%)

11.780(0.64%)

NS + 𝐹 15.399(0.83%)

9.283(0.50%)

20.320(1.10%)

7.300(0.39%)

NS +𝑀 + 𝐹

16.031(0.87%)

9.684(0.52%)

21.915(1.18%)

7.579(0.41%)


NS(Trea-suryzero)

10.597(1.36%)

2.457(0.31%)

12.684(1.63%)

8.066(1.03%)

NS(BBB-rated)

9.364(1.20%)

1.213(0.15%)

10.275(1.32%)

8.486(1.09%)

𝑀 7.639(0.98%)

1.844(0.23%)

9.289(1.19%)

6.685(0.86%)

𝐹 7.067(0.91%)

1.424(0.18%)

8.189(1.05%)

6.555(0.84%)

NS +𝑀 10.089(1.29%)

1.529(0.20%)

11.231(1.44%)

8.852(1.14%)

NS + 𝐹 8.486(1.09%)

2.147(0.27%)

9.780(1.25%)

6.895(0.88%)

NS +𝑀 + 𝐹

8.589(1.10%)

2.110(0.27%)

10.217(1.31%)

6.953(0.89%)

This table summarizes the mean, standard deviation, and 75% and 25% quantiles of the VaR estimates across sample period. The value in the table denotes the lossmeasured by index points. The value in parenthesis is the ratio of VaR estimate to the initial value of bond portfolio. Using data between t -500 and 𝑡−1, the 95% and99% VaR at time t are estimated. In total, we produce 424 VaR estimates from 2013∕3∕12 to 2014∕12∕31.

5.3.2. Conditional Predictive Ability (CPA) testThe Wald-type test statistics for the pairwise comparisons between the factor and factor-combined models are reported in Table 7,

in which the CPA test, proposed by Giacomini and White (2006), is used for each of the three bond indices examined in this study.The null hypothesis is that the models in the row will have equal conditional predictive ability to the models in the column. If thevalue of the Wald-type test statistic is greater than 𝜒2

𝑞 ,1− 𝛼 , which is the 1 − 𝛼 quantile of a 𝜒2 distribution with q degrees of freedom,the null hypothesis of equal conditional predictive ability is rejected. Given that the 5% significance level of a 𝜒2

𝑞 ,1− 𝑑 distributionwith 𝑞 = 2 degrees of freedom is 5.99, the results show that for both the 95% and 99% VaR estimates, all the models in the rowsoutperform those in the columns.

The results reported in Table 7 corroborate the backtesting results discussed in relation to Table 6. For both the 95% and 99% VaRestimates, the 𝑁𝑆 +𝑀 +𝐹 specification outperforms all other specifications for all three bond indices at the 5% level of significance.This result emphasizes the usefulness of additional macro and financial stress factors in deriving improvements in VaR performance.We also find that the 𝑁𝑆 + 𝐹 model performs better than the 𝑁𝑆 +𝑀 model in all cases, thereby implying that financial shockvariables have a stronger effect than macroeconomic variables. For the non-nested comparison, we find that the F model performsbetter than the M (or even the NS) model, which indicates that the financial stress factor contributes more to bond VaR forecastingperformance than the macro factor (or even the NS factors).

262


Table 6Backtest results.

Panel A: 95% VaR

LRuc LRind LRcc LRDind LRG LRGV Violation ratio Expected shortfall Bootstrap 𝑝-value


NS 0.204(0.65)

4.792*

(0.03)4.996(0.08)

12.757*

(0.00)12.929*

(0.00)12.815*

(0.00)0.055 6.793 0.121

AFNS 1.095(0.30)

3.962*

(0.05)5.057(0.08)

12.396*

(0.00)13.491*

(0.00)12.166*

(0.00)0.061 7.342 0.123

𝑀 9.714*

(0.00)0.000(1.00)

9.714*

(0.00)0.023(0.87)

9.737*

(0.00)11.698*

(0.00)0.020 7.157 0.047*

𝐹 11.742*

(0.00)0.000(1.00)

11.742*

(0.00)0.498(0.48)

12.240*

(0.00)14.488*

(0.00)0.017 8.424 0.043*

NS +𝑀 0.218(0.64)

3.089(0.08)

3.307(0.19)

10.078*

(0.00)10.296*

(0.00)10.560*

(0.00)0.050 6.859 0.281

NS + 𝐹 0.498(0.48)

3.364(0.07)

3.862(0.14)

6.122*

(0.01)6.620*

(0.03)7.002*

(0.03)0.042 6.345 0.147

NS +𝑀 + 𝐹 0.053(0.81)

2.838(0.09)

2.891(0.23)

2.808(0.09)

2.861(0.24)

5.015(0.08)

0.047 5.425 0.230


NS (Treasury zero) 3.907*

(0.05)3.262(0.07)

7.170*

(0.03)3.077(0.08)

6.984*

(0.03)8.089*

(0.02)0.030 2.436 0.048*

AFNS(Treasury zero) 6.172*

(0.01)1.174(0.27)

7.345*

(0.03)1.003(0.31)

7.175*

(0.03)20.709*

(0.00)0.026 1.839 0.044*

NS (AA-rated) 1.435(0.23)

2.501(0.11)

3.936(0.14)

7.899*

(0.00)9.344*

(0.01)9.975*

(0.00)0.037 1.925 0.155

𝑀 23.367*

(0.00)0.000(1.00)

23.367*

(0.00)6.360*

(0.01)29.727*

(0.00)33.896*

(0.00)0.008 1.763 0.031*

𝐹 23.367*

(0.00)0.000(1.00)

23.367*

(0.00)0.079(0.77)

23.466*

(0.00)27.616*

(0.00)0.008 2.439 0.035*

NS +𝑀 14.062*

(0.00)0.000(1.00)

14.062*

(0.00)0.002(0.96)

14.064*

(0.00)16.646*

(0.00)0.015 1.979 0.031*

NS + 𝐹 0.794(0.37)

3.563(0.06)

4.356(0.11)

8.059*

(0.00)8.853*

(0.01)8.528*

(0.00)0.059 1.837 0.129

NS +𝑀 + 𝐹 0.453(0.50)

1.246(0.26)

1.699(0.43)

2.756(0.10)

3.209(0.20)

2.971(0.22)

0.057 1.715 0.155


NS (Treasury zero) 6.398*

(0.01)5.495*

(0.02)11.893*

(0.00)13.524*

(0.01)19.922*

(0.01)21.410*

(0.01)0.025 1.564 0.041*

AFNS (Treasury zero) 10.094*

(0.00)0.000(1.00)

10.094*

(0.00)19.037*

(0.00)29.131*

(0.00)25.383*

(0.00)0.019 1.785 0.039*

NS (BBB-rated) 2.928(0.08)

9.432*

(0.00)12.360*

(0.00)22.261*

(0.01)25.189*

(0.00)26.127*

(0.01)0.032 1.368 0.083

𝑀 0.000(1.00)

17.791*

(0.00)17.791*

(0.00)46.206*

(0.01)46.206*

(0.01)46.254*

(0.01)0.050 1.926 0.190

𝐹 0.052(0.82)

17.258*

(0.00)17.310*

(0.00)45.092*

(0.00)45.144*

(0.00)45.092*

(0.00)0.052 2.056 0.408

NS +𝑀 7.942*

(0.01)5.927*

(0.02)13.870*

(0.00)20.624*

(0.00)28.566*

(0.00)30.276*

(0.00)0.022 2.039 0.157

NS + 𝐹 6.398*

(0.01)1.393(0.23)

7.790*

(0.02)2.943(0.08)

9.341*

(0.00)5.828*

(0.05)0.025 1.978 0.269

NS +𝑀 + 𝐹 2.107(0.87)

0.877(0.35)

2.984(0.22)

1.289(0.25)

3.396(0.18)

2.338(0.31)

0.040 1.408 0.613

Panel B: 99% VaR



NS 23.627*

(0.00)4.670*

(0.03)28.297*

(0.00)9.915*

(0.00)33.542*

(0.00)30.705*

(0.00)0.042 6.151 0.044*

AFNS 19.366*

(0.00)7.941*

(0.00)27.307*

(0.00)14.397*

(0.00)33.763*

(0.00)32.415*

(0.00)0.038 7.382 0.045*

𝑀 0.276(0.60)

0.000(1.00)

0.276(0.87)

0.241(0.62)

0.517(0.77)

1.468(0.48)

0.007 8.291 0.143

𝐹 0.234(0.63)

0.000(1.00)

0.234(0.89)

0.083(0.77)

0.317(0.85)

0.083(0.95)

0.013 5.465 0.520

(continued on next page)

263


Table 6 (continued)

Panel B: 99% VaR


NS +𝑀

10.529*

(0.00)0.403(0.52)

10.933*

(0.00)3.579(0.06)

14.108*

(0.00)11.994*

(0.00)0.029 6.252 0.047*

NS+𝐹 3.131(0.08)

0.924(0.33)

4.055(0.13)

2.553(0.11)

5.684(0.06)

4.426(0.11)

0.020 8.702 0.965

NS +𝑀+𝐹

3.131(0.08)

0.000(1.00)

3.131(0.21)

1.903(0.16)

5.034(0.08)

3.776(0.15)

0.020 8.419 0.836


NS(Trea-suryzero)

0.234(0.63)

0.000(1.00)

0.234(0.89)

1.620(0.20)

1.854(0.40)

2.766(0.25)

0.012 2.248 0.193

AFNS(Trea-suryzero)

0.832(0.26)

1.201(0.27)

2.033(0.36)

1.346(0.24)

2.178(0.33)

2.890(0.23)

0.008 2.129 0.189

NS(AA-rated)

0.000(1.00)

0.000(1.00)

0.000(1.00)

0.204(0.25)

0.204(0.90)

0.475(0.78)

0.010 1.733 0.215

𝑀 3.250(0.07)

0.000(1.00)

3.250(0.20)

0.204(0.25)

3.454(0.18)

5.824*

(0.05)0.003 2.087 0.045*

𝐹 3.250(0.07)

0.000(1.00)

3.250(0.20)

0.204(0.25)

3.454(0.18)

5.920*

(0.05)0.003 0.562 0.044*

NS +𝑀

1.237(0.27)

0.000(1.00)

1.237(0.54)

0.456(0.50)

1.693(0.43)

3.691(0.15)

0.005 1.496 0.111

NS+𝐹 1.857(0.17)

0.000(1.00)

1.857(0.40)

1.900(0.16)

3.757(0.15)

2.785(0.25)

0.017 2.436 0.966

NS +𝑀+𝐹

0.234(0.63)

0.000(1.00)

0.234(0.89)

0.676(0.41)

0.910(0.63)

0.676(0.71)

0.012 2.907 0.841


NS(Trea-suryzero)

3.250(0.07)

0.000(1.00)

3.250(0.19)

3.841(0.05)

7.091*

(0.03)8.389*

(0.00)0.003 0.657 0.038*

AFNS(Trea-suryzero)

0.013(0.91)

5.138*

(0.02)5.150(0.07)

3.210(0.07)

3.223(0.20)

7.054*

(0.03)0.010 1.292 0.147

NS(BBB-rated)

0.276(0.60)

0.000(1.00)

0.276(0.87)

3.741(0.06)

4.017(0.13)

5.002(0.08)

0.008 1.198 0.164

𝑀 3.131(0.08)

0.000(1.00)

3.131(0.21)

7.426*

(0.00)10.557*

(0.00)9.299*

(0.00)0.020 1.788 0.050

𝐹 1.857(0.17)

0.000(1.00)

1.857(0.40)

9.247*

(0.00)11.104*

(0.00)10.133*

(0.00)0.017 2.338 0.508

NS +𝑀

1.237(0.26)

0.000(1.00)

1.237(0.54)

4.693*

(0.03)5.930*

(0.05)7.928*

(0.02)0.005 1.344 0.400

NS+𝐹 0.234(0.63)

0.000(1.00)

0.234(0.89)

2.412(0.12)

2.646(0.26)

4.972(0.08)

0.012 1.368 0.547

NS +𝑀+𝐹

0.234(0.63)

0.000(1.00)

0.234(0.89)

1.052(0.30)

1.286(0.52)

3.182(0.20)

0.012 0.758 0.921

Note that the conditional coverage test (LRcc) is a joint test of the unconditional coverage (LRuc) and serial independence (LRind), that is LRcc = LRuc + LRind . Similarly,for geometric test (G), its likelihood ratio is LRG = LRuc + LRDind and, for geometric-VaR test (GV), its likelihood ratio is LRGV = LRuc + LRind + LRDind. The likelihoodratios LRcc, LRG and LRGV are, respectively, asymptotical distributed as 𝜒2(2), 𝜒2(2) and 𝜒2(3) . The null hypothesis of bootstrap test is that the difference between theexpected shortfall and the realized shortfall in the case of a VaR-violation is zero. The numbers in parentheses are p-values.

* Indicates statistically significance at the 5% level.

6. Conclusions

This study is motivated by the recent finding that variations in bond returns can be explained by macroeconomic variables andfinancial stress conditions, in addition to the spot-rate term structure model. We go beyond prior studies to develop a new factor-basedapproach, grounded in the Nelson–Siegel (NS) term structure factor-augmented model, to compute the VaR of bond portfolios, andthen use the model to investigate whether the information contained in macroeconomic variables and financial stress shocks can helpto improve the VaR performance of three bond indices.

In our extension of the factors affecting the yield variations, we consider several traditional macroeconomic variables (Federalfund rates, inflation rates, and S&P returns) and financial stress shocks (TED spread, default probability, and the VIX). Our findings

264

A.H.Tu,C.Y.-H

.ChenJournalofEm

piricalFinance45

(2018)243–268

Table 7Conditional predictive ability test.

Treasury 10Y–20Y Index Broad Investment Grade Bond Index High Yield Market Index

Panel A: 95%VaR

NS 𝑀 𝐹 NS + 𝑀 NS + 𝐹 NS + 𝑀 + 𝐹 NS 𝑀 𝐹 NS + 𝑀 NS + 𝐹 NS + 𝑀 + 𝐹 NS 𝑀 𝐹 NS + 𝑀 NS + 𝐹 NS + 𝑀 + 𝐹NS 246.21

(0.00)230.46(0.00)

159.92(0.00)

116.85(0.00)

165.72(0.00)

196.40(0.00)

190.49(0.00)

95.29(0.00)

175.46(0.00)

153.30(0.00)

330.21(0.00)

111.73(0.00)

165.09(0.00)

300.24(0.00)

131.96(0.00)

𝑀 99.44(0.00)

226.35(0.00)

205.86(0.00)

202.41(0.00)

183.90(0.00)

223.24(0.00)

147.16(0.00)

142.33(0.00)

362.54(0.00)

355.99(0.00)

71.51(0.00)

124.08(0.00)

𝐹 221.46(0.00)

206.82(0.00)

204.31(0.00)

218.74(0.00)

173.50(0.00)

211.06(0.00)

102.44(0.00)

361.53(0.00)

275.91(0.00)

NS+𝑀 116.25(0.00)

124.75(0.00)

206.86(0.00)

184.50(0.00)

355.01(0.00)

307.18(0.00)

NS+𝐹 107.48(0.00)

141.25(0.00)

107.85(0.00)

NS + 𝑀 + 𝐹

Panel B: 99%VaR

99%VaR NS 𝑀 𝐹 NS + 𝑀 NS + 𝐹 NS + 𝑀 + 𝐹 NS 𝑀 𝐹 NS + 𝑀 NS + 𝐹 NS + 𝑀 + 𝐹 NS 𝑀 𝐹 NS + 𝑀 NS + 𝐹 NS + 𝑀 + 𝐹NS 258.40

(0.00)236.94(0.00)

166.61(0.00)

118.02(0.00)

164.91(0.00)

206.74(0.00)

124.34(0.00)

366.45(0.00)

179.18(0.00)

187.09(0.00)

274.30(0.00)

310.38(0.00)

249.12(0.00)

184.07(0.00)

233.27(0.00)

𝑀 101.69(0.00)

238.21(0.00)

216.38(0.00)

212.58(0.00)

224.05(0.00)

176.98(0.00)

220.27(0.00)

215.49(0.00)

105.92(0.00)

382.63(0.00)

139.78(0.00)

194.70(0.00)

𝐹 229.86(0.00)

213.23(0.00)

210.53(0.00)

208.95(0.00)

186.41(0.00)

185.49(0.00)

381.25(0.00)

138.50(0.00)

170.65(0.00)

NS+𝑀 120.74(0.00)

128.40(0.00)

147.98(0.00)

157.02(0.00)

201.64(0.00)

188.40(0.00)

NS+𝐹 115.03(0.00)

127.83(0.00)

110.34(0.00)

NS + 𝑀 + 𝐹

This table reports the Wald-type test statistics for pairwise comparisons among factor models, using the conditional predictive ability (CPA) test of Giacomini and White (2006). The null hypothesis is that the models in the‘‘line’’ have the equal conditional predictive ability as the models in the ‘‘column’’. If the value of the Wald-type test statistic is greater than 𝜒2

𝑞,1−𝛼 , which is the 1−𝛼 quantile of a Chi-square distribution with q degree of freedom,then the null hypothesis is rejected. The test function ℎ𝑡 is chosen as ℎ𝑡 =

(

1, 𝛥𝐿𝑡)′. The 5% significance level of a 𝜒2

𝑞,1−𝛼 distribution with 𝑞 = 2 degree of freedom is 5.99 and the numbers in parentheses are 𝑝 values.

265


show that bond VaR forecasting performance is significantly improved when macroeconomic variables and financial stress shocks areincluded in the NS factor model.

Thus, our empirical evidence suggests that in addition to the NS term structure factors, macroeconomic variables and financialstress shocks may act as driving factors in the VaR of bond portfolios. Furthermore, the impact of the inclusion of financial stressshocks is found to be greater than that of macroeconomic variables. Our results may have important implications for risk managementand policy decisions oriented toward a framework of financial stability.

Acknowledgments

The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through SFB 649 ‘‘EconomicRisk’’ and IRTG 1792 ‘‘High Dimensional Non Stationary Time Series’’.

Appendix A. Estimation of Eqs. (6), (7), and (8) based on the Kalman Filter (KF)

The measurement (observation) Eq. (6) of the state-space form for the dynamic NS three-factor model is expressed as:

𝑦𝑡 = 𝛬 (𝜂) 𝛿𝑡 + 𝜀𝑡 𝜀𝑡 ∼ 𝑁𝐼𝐷(

0, 𝛴𝜀)

, 𝑡 = 1,… , 𝑇 . (A.1)

The transition Eq. (8), which describes the evolution of the state variables 𝛿𝑡 as a vector autoregressive process, is given as:

𝛿𝑡+1 = 𝜇 +𝛷𝛿𝑡 + 𝜖𝑡 𝜖𝑡 ∼ 𝑁𝐼𝐷(

0, 𝛴𝜖)

. (A.2)

Considering the NS model in the above two equations as a linear Gaussian state space model, the unobserved factor 𝛿𝑡 is the statevector, and can be estimated conditional on the past and concurrent observations via the KF. Let 𝑏𝑡|𝑡−1 denotes the minimum meansquare linear estimator of 𝛿𝑡 given the observations up to t − 1 with mean square error matrix 𝐵𝑡|𝑡−1. For given values of 𝑏𝑡|𝑡−1 and𝐵𝑡|𝑡−1, the KF first computes 𝑏𝑡|𝑡 and 𝐵𝑡|𝑡 as observation 𝑦𝑡 is available. The filtering step is:

𝑏𝑡|𝑡 = 𝑏𝑡|𝑡−1 + 𝐵𝑡|𝑡−1𝛬(𝜂)′F−1𝑡 𝑣𝑡 (A.3)

𝐵𝑡|𝑡 = 𝐵𝑡|𝑡−1 − 𝐵𝑡|𝑡−1𝛬(𝜂)′F−1𝑡 𝛬 (𝜂)𝐵𝑡|𝑡−1 (A.4)

where 𝑣𝑡 = 𝑦𝑡 − 𝛬 (𝜂) 𝑏𝑡|𝑡−1 is the prediction error vector and 𝐹𝑡 = 𝛬 (𝜂)𝐵𝑡|𝑡−1𝛬(𝜂)′ + 𝛴𝜀 is the prediction error variance matrix. Theminimum mean square linear estimator for the next period t + 1 conditional on the observations up to t is given by the predictionstep:

𝑏𝑡+1|𝑡 = 𝜇 +𝛷𝑏𝑡|𝑡 (A.5)

𝐵𝑡+1|𝑡 = 𝛷𝐵𝑡|𝑡𝛷′ + 𝛴𝜖 . (A.6)

Estimation of the unknown parameter set 𝜓 =(

𝜇,𝛷,𝛴𝜀, 𝛴𝜖)

is based on the numerical maximization of the Gaussian log-likelihoodfunction for the prediction error:

logL(

𝑦1,…, 𝑦𝑡;𝜓)

=𝑇∑

𝑡=1

(

−𝑁2

log (2𝜋) − 12log |

|

F𝑡|| −12𝑣′𝑡F

−1𝑡 𝑣𝑡

)

. (A.7)

Appendix B. Cornish–Fisher expansion

Using the first four moments (mean, variance, skewness, and kurtosis), the Cornish–Fisher expansion, approximating the 𝛼-percentile �̃�𝛼 of a standardized random variable, is calculated by:

�̃�𝛼 ≈ 𝑍𝛼 +16(

𝑍2𝛼 − 1

)

∗𝛾 + 124

(

𝑍3𝛼 − 3𝑍𝛼

)

∗𝜅 − 136

(

2𝑍3𝛼 − 5𝑍𝛼

)

∗𝛾2 (B.1)

where 𝑍𝛼 refers to the 𝛼 percentile of an N (0,1) distribution, 𝛾 represents the skewness, and 𝜅 is the excess kurtosis of the factorvariables. The skewness of the factor variables, f, where f = NS, M or F, is computed based upon historical data over n days (n = 500)as:

𝛾 = 1𝑛

𝑛∑

𝑡=1

(𝑓𝑡 − 𝑓 )3

�̂�3(B.2)

where 𝑓 is the expected value and �̂� is the volatility of f. The excess kurtosis of f is given by:

𝜅 = 1𝑛

𝑛∑

𝑡=1

(𝑓𝑡 − 𝑓 )4

�̂�4− 3. (B.3)

266


It should be noted that in the case of normal distribution, the skewness 𝛾 and the excess kurtosis 𝜅 will both be equal to zero,which thereby leads to �̃�𝛼 = 𝑍𝛼 .29

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