# Volatility term structure and estimation of yield curve ... Volatility term structure and estimation

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Volatility term structure and estimation of yield curve: Inferring their connections and movements

Ana Maria A, ChirinosLeaez

Miriam Maita Bolvar

This version: October 2012

Abstract:

This paper estimates yield curve under forward interest rates and its conditional variance for two types of markets; domestic and external debt market, using monthly data from January 2004 to December 2011 and from April 2005 to December 2011, respectively. Yield curve is estimated using the parametric model proposed by Nelson Siegel (1987) and the conditional variance is computed under stochastic volatility model characterization (EGARCH). We find the vast majority of estimations displayed an upward sloping yield curve in each market, excluding recession periods in which yield curve showed downward sloping pattern. At the end of the sample humped shaped was exhibited. Other findings reveal yield curve parameters of long term component (level) could be mainly connected to economic fundamentals and market risk expectations at the same time (CDS and EMBI). While short term component of the yield curve (slope) might be affecting by price variables (nominal depreciations of non - official exchange rate, expansions in oil prices and inflation). Additionally, short term bonds are more volatile compare to other maturities horizons, especially those instruments issued in external debt market. Finally, positive economic performance could reduce conditional variance of bond returns in both markets. Key words: yield curve, volatility term structure, conditional variance, Venezuelan bond market, local debt market, foreign debt market, Nelson-Siegel model, investors expectations, level, slope and curvature JEL classification code: C13, C21, G12, N26

Economic Analyst of the Research Department at Central Bank of Venezuela and Professor of Universidad Catlica Andres Bello, achirino@bcv.org.ve Economic Analyst of Economic Analysis Department at Central Bank of Venezuela, mmaita@bcv.org.ve

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Introduction hilarious

Which are the main factors driving shape of yield curve and volatility term structure

(VTS)? Are these representing market expectations? Is there any difference between

VTS and term structure of interest rates (TSIR) depending on the type of market in

which bonds are issued (local market or foreign market)?

An extensive body of the literature has analyzed common factors that affect yield curve

(Litterman and Scheinkman 1991, Dielbod-Li 2005, Prignon and Villa 2006).

Nevertheless, a new trend of financial researches have focused on the second moment of

this financial indicator (Benito and Novales 2005, Diaz et al. 2010b and Jareo and

Tolentino 2011), as a new mechanism of extracting information for risk portfolio

management, and predicting future movements of interest rates.

For Venezuelan economy, levels of government bonds issued either local or foreign

markets have increased from 2006 onwards (an average growth of 39% and 12%

respectively), motivating empirical research regarding fixed income securities. In this

context, Chirinos and Moreno (2009), and Maita (2011) build the TSIR using

parametric approaches for a small group of debt securities. They find that different

shapes of the yield curve are attributed to variations in three elements: level, slope and

curvature (Litterman and Scheinkman (1991)). However, to the best of our knowledge,

there is no previous empirical research that has addressed volatility term structure for

Venezuelan debt market (domestic-foreign) or associated the possible factors that

generate its movements, and yield curve variations.

Purposes of this research are threefold: First we estimate yield curve under parametric

characterization (Nelson Siegel, 1987) including a wide spectrum of bonds for both

market during a recent period (2004-2011). Second, using EGARCH model, we

compute conditional variance of term structure of interest rates under instantaneous

forward rates. After that, and with the intention to reduce dimensionality of volatility

across maturities of bonds selected, we apply principal components analysis (PCA) to

obtain the main representative volatility factors. Third, we connect a group of

macroeconomic variables with the main unobservable factors of the yield curve and

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volatility, with the purpose to establish conjectures concerning possible variables driven

movements on yield curve and VTS.

Determining changes in volatility term structure and its interrelations with yield curve

are relevant to comprehend investors expectations, which are a useful tool for policy

makers. Nowadays, this tool of analysis has become in an alternative way, especially

after financial crisis, of understanding a fraction of the intrinsic movements of financial

market.

This paper is divided into five sections: First section describes theoretical framework

used to estimate yield curve and compute conditional variance. Second section explains

the estimation methodology. Third one characterizes features of the data, while fourth

section shows empirical results. Finally last section summarizes main concluding

remarks.

1. Theoretical Framework 1.1 TSIR estimation

Extensive approaches of the estimation of term structure of interest rates have been

applied by financial literature1, including stochastic term structure models and affine

term models (Vasicek 1977, Cox, Ingersol and Ross 1985, Hull and White 1990, among

others) and the parametric or parsimonious representations. These characterizations

summarize the key hypothesis behind fixed income analysis2. First group of models are

built under the main assumption that interest rates follow a stochastic process. However,

issues arise to fit observed yield data and in terms of computational tractability for

empirical scenarios. Such problems are dealt under parametric models, since they

provide good fit with a minimum level of requirements for empirical applications.

Having said that, we decide to estimate Nelson Siegel (NS) model, which is the typical

characterization of parsimonious modelling of the interest rates. Specifically, we

compute yield curve using instantaneous forward interest rates (rates at which contract

1See Chirinos and Moreno (2010) for an extensive description 2Expectation hypothesis theory ( Fisher, 1930), market segmentation theory ( Culbertson, 1953), preferred habitat theory (see Modigliani and Sutch, 1966) and liquidity preference theory ( Hicks, 1946).

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are negotiated at future dates)3. Since doing so, it is possible to separate short, medium

and, more important, long expectations of investors that are imbedded in the yield

curve.

Under NS model instantaneous forward rate is given by following function:

where T denotes maturity, f(t,T) is the forward rate for period [t,T], and ,,, 210

are the coefficients to be estimated4. Terms of equation (1) can be interpreted in the

following fashion: 0 measures long-term component (positive constant) and it is

frequently associated to indicators of economic fundamentals. The second term,

Texp1 , is related to the component of the short term. This coefficient can be

monotonically decreasing or increasing depending on the sign of 1 . The third

component

TT exp2 is the medium term component, responsible of generating

the hump shapes and U shapes that the term structure can exhibit. The parameter

depends on the rate at which the forward rate achieves its asymptotic value ( 0 ) and

must be positive since it is a time constant variable.

The main advantage of this model is to capture all the possible shapes that yield curve

usually can exhibit over the time (monotonic, hump and even S shapes).

According to Dai and Singleton (2000) a considerable fraction of movements and

shapes of yield curve are mainly attributable to unobservable factors called level, slope 3From a theoretical perspective, spot rates are often used to construct the yield curve. This concept tends to be understood as the yield to maturity. Nevertheless both concepts differ. The yield to maturityis the internal rate of return at time t on a debt security with maturity s = t + T. The rates r(t,T) considered as function on T will be referred as the continuously compounded spot rates. It can be shown that

),(log1),( TttPT

Ttr T>0, where P is the corresponding bond price for the period [t,T]. 4In their original version Nelson-Siegel (1987) implement ordinary least-squares to estimate equation (1)

since the parameter is settled in a range of values. As mentioned, this paper estimates the time constant parameters.

These authors do estimate and compute equation (1) for a reasonable range of values for this parameter. In contrast to this, and following Svensson (1997), we determine all the parameters of the model.

(1)

TTTTtf expexp),( 210

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and curvature. Diebold and Li (2002) find that parameters of Nelson Siegel model can

be interpreted as such latent factors, where 0 is the level, 1 the slope and 2 is

curvature. From now on, we use this result to refer to these parameters

1.2 Volatility Estimation Understanding the way and the reasons why fixed income returns change, is crucial to

comprehend movements of yield curve and somehow investors strategies as well.

During decade

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