**1. Introduction**

Investors can potentially improve the risk-adjusted performance of their portfolios by investing internationally and thereby take advantage of the associated return and diversification benefits. The potential gains provided by emerging markets have attracted significant investor attention which, in turn, has led to substantial capital inflows to these economies. Local currency-denominated sovereign bonds have been the fastest growing market in emerging market space in the past few years. More recently, the global quest for yield in a context of accommodative monetary policies in the advanced economies has created a positive external environment for emerging market bonds. In addition, the secondary market for emerging market local currency debt has also been supported by high interest and amortisation payments. Emerging market bonds are also benefiting from a track record of strong risk-adjusted returns and low correlations with other asset classes. Such characteristics are attractive from a portfolio optimisation perspective. These attributes have also seen considerable attention devoted to analysis of the various risk-return attributes of these markets. In particular, recent empirical literature has focused on the characterisation of the volatility profile of emerging market bond returns. Indeed, the accurate estimation of volatility plays a central role in many applications in finance, including optimal portfolio selection (e.g., diversification strategies), valuation of derivatives (e.g., option pricing) and risk management (e.g., value-at-risk calculation). These applications have motivated an extensive empirical literature on volatility modelling.

While recent empirical literature has focused on the characterisation of the volatility profile of a variety of asset classes; in particular, the long memory properties of these assets, surprisingly little attention has been devoted to the analysis of fixed income markets,

© 2012 Thupayagale, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Thupayagale, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

especially in emerging markets. The empirical literature on long memory dynamics of fixed income volatility and its implications for portfolio and risk management appears to be limited. Most of the extant literature appears concentrated in the advanced economies. For example, several authors have examined the various aspects of long memory behaviour in interest rates and yield spreads. [1-3] The purpose of this chapter is to augment this line of analysis concerning the long memory attributes of fixed income volatility in emerging market local currency debt market in light of investor interest in the potential alpha generation of these markets, amid wider capital inflows into emerging market bonds as investors search for yield.

Long Memory in the Volatility of Local Currency Bond Markets:

study attempts to help fill this gap by addressing a range of issues relating to the estimation

Against this background, this paper has three objectives. First, evidence of long memory in volatility within leading emerging fixed income markets is investigated. In particular, the existence of long memory behaviour in the volatility of returns from Hong Kong, Mexico and South Africa are examined which appears to have little or no previous research establishing the existence of its long memory properties. In order to estimate the long memory parameter *d*, this study makes use of methods based on wavelets, which have been recently used to capture the fractal structure of high frequency data.[6] Second, the existence of long memory dynamics in bond volatility data (i.e., a fractal structure in the data) will be further investigated in order to test if the extraction of this long-run component can be exploited for purposes of generating improved volatility forecasts especially over long horizons. The long memory property is examined in order to determine if it helps deliver more accurate forecasts over a long(er) horizon. Third, an important and topical area of research concerns the calculation of value-at-risk (VaR) in financial markets. This methodology is widely used by financial institutions and regulatory agencies to measure, monitor and manage market risk. This analysis compares whether long memory volatility estimates can help deliver more accurate VaR estimates relative to standard models (i.e.,

In total, the findings of this investigation will provide a range of volatility estimates and forecasts which could potentially inform portfolio management strategies and guide policymaking. In particular, while most empirical studies focus on the United States and other developed markets, recent research has begun to look at emerging markets, however, limited evidence exists with respect to these markets. This analysis contributes to the empirical literature by focusing on various aspects of long memory behaviour in local currency debt markets. The findings from this research complement those in previous

The rest of the chapter is structured as follows. Section II presents a description of long memory in time series. Section III introduces the data. Section IV presents the empirical methodology and associated results. In particular, this starts with a presentation of the standard GARCH model which is often used to present initial evidence of long memory behaviour. Then the wavelet method and the estimator employed is introduced, along with the relevant findings. This includes a discussion on wavelet analysis, the discrete wavelet transform, the estimator employed and the relevant findings. Section V provides the forecast evaluation techniques used and the out-of-sample forecast results. Section VI considers the evaluation of value-at-risk in the context of the Basle adequacy criteria. Section VII

Interest in long memory (or long range dependent) processes can be traced to the examination of data in the physical sciences. Formal models with long memory initially pertained to hydrological studies investigating how to regularise the flow of the Nile river

studies and may provide an interesting comparison to existing studies.

concludes and identifies topics for further research.

**2. Long memory in time series** 

and forecasting of fixed income return volatility especially over long horizons.

GARCH and RiskMetrics).

Evidence from Hong Kong, Mexico and South Africa 99

This study will focus on government bond markets from Hong Kong, Mexico and South Africa. According to the most recent survey from the Emerging Markets Trading Association (EMTA) these three local currency debt markets are among the most vibrant and actively traded in emerging markets. [4] As a result of their (comparatively high) liquidity, developed institutional frameworks and credible monetary policies, these markets are therefore of interest to investors. Indeed, local currency bond markets have emerged as an important asset class in many emerging markets; a point which becomes salient in the current low-yield environment, where investors targeting high returns and diversification benefits have channelled capital to emerging bond markets such as these.

As a result of regulatory initiatives and various reforms, emerging market local currency sovereign debt markets have has grown rapidly in size and sophistication. According to [4], emerging market debt trading volumes were at USD1.8 trillion in the third quarter of 2011. This represents a 3 percent increase from the USD1.7 trillion reported for the second quarter of 2011. Turnover in local market instruments was at USD1.3 trillion in the third quarter of 2011, (i.e., 76 percent of total reported volume). Mexican and Hong Kong securities were the first and second most frequently traded emerging market debt in the third quarter of 2011 at USD282 billion and USD176 billion, respectively compared to USD136 billion and USD201 billion a year earlier. The next most frequently traded local markets debt were those from Brazil (USD160 billion), South African (USD113 billion) and Turkey (USD69 billion).1

Domestic institutional investors are typically the largest investors in local currency bond markets. For example, pension funds tend to have long-term liabilities which are typically funded by investments in long-term investment grade securities that provide a prudent riskreturn profile. As a result, the examination of long memory in volatility would appear to be of interest to institutional and other long-term investors. Furthermore, it has been shown that it is important to model the long memory volatility structure when pricing derivative contracts with long maturity. [5] In addition, in order to assess future returns from both active and passive investment strategies or the need for policy intervention, especially over long horizons, it is important to forecast volatility. These applications have motivated an extensive empirical literature on modelling long memory dynamics in asset return data. Analysis of the long-term volatility dynamics of emerging market local currency bonds appear limited. Therefore, this

<sup>1</sup> Analysis is on Hong Kong, Mexico and South Africa due to the availability of data.

study attempts to help fill this gap by addressing a range of issues relating to the estimation and forecasting of fixed income return volatility especially over long horizons.

Against this background, this paper has three objectives. First, evidence of long memory in volatility within leading emerging fixed income markets is investigated. In particular, the existence of long memory behaviour in the volatility of returns from Hong Kong, Mexico and South Africa are examined which appears to have little or no previous research establishing the existence of its long memory properties. In order to estimate the long memory parameter *d*, this study makes use of methods based on wavelets, which have been recently used to capture the fractal structure of high frequency data.[6] Second, the existence of long memory dynamics in bond volatility data (i.e., a fractal structure in the data) will be further investigated in order to test if the extraction of this long-run component can be exploited for purposes of generating improved volatility forecasts especially over long horizons. The long memory property is examined in order to determine if it helps deliver more accurate forecasts over a long(er) horizon. Third, an important and topical area of research concerns the calculation of value-at-risk (VaR) in financial markets. This methodology is widely used by financial institutions and regulatory agencies to measure, monitor and manage market risk. This analysis compares whether long memory volatility estimates can help deliver more accurate VaR estimates relative to standard models (i.e., GARCH and RiskMetrics).

In total, the findings of this investigation will provide a range of volatility estimates and forecasts which could potentially inform portfolio management strategies and guide policymaking. In particular, while most empirical studies focus on the United States and other developed markets, recent research has begun to look at emerging markets, however, limited evidence exists with respect to these markets. This analysis contributes to the empirical literature by focusing on various aspects of long memory behaviour in local currency debt markets. The findings from this research complement those in previous studies and may provide an interesting comparison to existing studies.

The rest of the chapter is structured as follows. Section II presents a description of long memory in time series. Section III introduces the data. Section IV presents the empirical methodology and associated results. In particular, this starts with a presentation of the standard GARCH model which is often used to present initial evidence of long memory behaviour. Then the wavelet method and the estimator employed is introduced, along with the relevant findings. This includes a discussion on wavelet analysis, the discrete wavelet transform, the estimator employed and the relevant findings. Section V provides the forecast evaluation techniques used and the out-of-sample forecast results. Section VI considers the evaluation of value-at-risk in the context of the Basle adequacy criteria. Section VII concludes and identifies topics for further research.

### **2. Long memory in time series**

98 Risk Management – Current Issues and Challenges

investors search for yield.

especially in emerging markets. The empirical literature on long memory dynamics of fixed income volatility and its implications for portfolio and risk management appears to be limited. Most of the extant literature appears concentrated in the advanced economies. For example, several authors have examined the various aspects of long memory behaviour in interest rates and yield spreads. [1-3] The purpose of this chapter is to augment this line of analysis concerning the long memory attributes of fixed income volatility in emerging market local currency debt market in light of investor interest in the potential alpha generation of these markets, amid wider capital inflows into emerging market bonds as

This study will focus on government bond markets from Hong Kong, Mexico and South Africa. According to the most recent survey from the Emerging Markets Trading Association (EMTA) these three local currency debt markets are among the most vibrant and actively traded in emerging markets. [4] As a result of their (comparatively high) liquidity, developed institutional frameworks and credible monetary policies, these markets are therefore of interest to investors. Indeed, local currency bond markets have emerged as an important asset class in many emerging markets; a point which becomes salient in the current low-yield environment, where investors targeting high returns and diversification

As a result of regulatory initiatives and various reforms, emerging market local currency sovereign debt markets have has grown rapidly in size and sophistication. According to [4], emerging market debt trading volumes were at USD1.8 trillion in the third quarter of 2011. This represents a 3 percent increase from the USD1.7 trillion reported for the second quarter of 2011. Turnover in local market instruments was at USD1.3 trillion in the third quarter of 2011, (i.e., 76 percent of total reported volume). Mexican and Hong Kong securities were the first and second most frequently traded emerging market debt in the third quarter of 2011 at USD282 billion and USD176 billion, respectively compared to USD136 billion and USD201 billion a year earlier. The next most frequently traded local markets debt were those from

Brazil (USD160 billion), South African (USD113 billion) and Turkey (USD69 billion).1

1 Analysis is on Hong Kong, Mexico and South Africa due to the availability of data.

Domestic institutional investors are typically the largest investors in local currency bond markets. For example, pension funds tend to have long-term liabilities which are typically funded by investments in long-term investment grade securities that provide a prudent riskreturn profile. As a result, the examination of long memory in volatility would appear to be of interest to institutional and other long-term investors. Furthermore, it has been shown that it is important to model the long memory volatility structure when pricing derivative contracts with long maturity. [5] In addition, in order to assess future returns from both active and passive investment strategies or the need for policy intervention, especially over long horizons, it is important to forecast volatility. These applications have motivated an extensive empirical literature on modelling long memory dynamics in asset return data. Analysis of the long-term volatility dynamics of emerging market local currency bonds appear limited. Therefore, this

benefits have channelled capital to emerging bond markets such as these.

Interest in long memory (or long range dependent) processes can be traced to the examination of data in the physical sciences. Formal models with long memory initially pertained to hydrological studies investigating how to regularise the flow of the Nile river

in view of its nonperiodic (flooding) cycles. [7] This feature was described as the "Joseph effect" alluding to the biblical reference in which seven years of plenty where to be followed by seven years of famine. [8] In this sense, long memory process concern observations in the remote past that are highly correlated with observations in the distant future. The implications of long memory in financial markets was related to the use of Hurst's 'rescaled range' statistic to detect long memory behaviour in asset return data. [9] It was observed that if security prices display long memory then the arrival of new market information cannot be arbitraged way, which in turn means that martingale models for security prices cannot be derived through arbitrage. As such, long memory processes can be characterised as having fractal dimensions, in the form of non-linear behaviour marked by distinct but nonperiodic cyclical patterns and long-term dependence between distant observations. [10]

Long Memory in the Volatility of Local Currency Bond Markets:

(1)

More formally, the change in the local bond index ˆ can be expressed as:

<sup>1</sup> 01 *t tt t <sup>t</sup> yield yield DV coupon return* and <sup>1</sup> 1 1 *t t tt*

markets. In addition, the volatility series is standardised prior to further analysis.

GARCH(1,1) model using the assumption of the Student *t* distribution.

0.040496 0.339043 0.538071 20.74158 33829\*\* 64.97\*\* 42.23\*\*


2/ Normality test follows a Chi-squared distribution 3/ ARCH (x) test follows an F-statistic with parameters (x, n-x)

1/ '\*\*' and '\*' indicate statistical significant at the 1% and 5% levels, respectively.

where is the local currency return and

**4. Empirical methodology and results** 

**4.1. Preliminary observations** 

Mean

Skewness Kurtosis Normality test ARCH (5) test ARCH (10) test ADF unit root test:

Constant

Note:

Constant & Trend

**Table 1.** Description of the Data

Standard deviation

<sup>1</sup> ˆ ˆ <sup>1</sup> *t tt*

coupon return is the return derived from the interest payment made on the fixed income product. Therefore, when an investor buys a local market index, equation (1) suggests that fixed income returns can be decomposed into its predictable coupon return, FX changes, and changes in local yields. Furthermore, in order to compute return volatility, this analysis focuses on squared daily returns, as a proxy for the volatility of the selected emerging

Table 1 presents the time series properties of the data using some basic methods. The results of the Augmented Dickey-Fuller (ADF) unit root test offer evidence in favour of stationary fixed income returns. While this test may be deficient in terms of its ability to capture an order of integration that may not be an integer, the finding of stationary bond returns is consistent with those of many previous studies. [15] However, based on the standard normality and Lagrange Multiplier ARCH tests, fixed income return data exhibit non-normality and ARCH effects. [16-17] These non-white noise characteristics of the data motivate estimation of

Mexico South Africa Hong Kong

0.040533 0.381478 -0.220239 8.738281 3546\*\* 92.71\*\* 53.92\*\*


Evidence from Hong Kong, Mexico and South Africa 101

is the currency return.

*FX FX* and the

0.017929 0.186277 -0.008431 6.603305 1390\*\* 72.38\*\* 47.92\*\*


A variety of measures have been used to detect long memory in time series. For example, in the time domain, long memory is associated with a hyperbolically decaying autocovariance function. Meanwhile, in the frequency domain, the presence of long memory is indicated by a spectral density function that approaches infinity near the zero frequency; in other words, such series display power at low frequencies. [11] Finally, a pattern of self-similarity in the aggregated sequences of a time series is an indicator of long memory (this refers to the property of a self-similar process, in which, different time aggregates display the same autocorrelation structure). These notions have led several authors to develop stochastic models that capture long memory behaviour, such as the fractionally-integrated I(*d*) time series models. [12-13] In particular, fractional integration theory asserts that the fractional difference parameter which indicates the order of integration, is not an integer value (0 or 1) but a fractional value. Fractionally integrated processes are distinct from both stationary and unit-root processes in that they are persistent (i.e., they reflect long memory) but are also mean reverting and as a consequence provide a flexible alternative to standard I(1) and I(0) processes. [14] Specifically, the long memory parameter is given by *d* 0, 0.5 while when *d* > 0.5 the series is nonstationary and when *d* 0.5, 0 the series is antipersistent.

Since, non-zero values of the fractional differencing parameter imply dependence between distant observations, considerable attention has been directed to the analysis of fractional dynamics in financial time series data. Indeed, long memory behaviour has been reported in the returns of various asset classes. [15] Against this background, a rapidly expanding set of models has been developed to capture long memory dynamics is asset return data.

### **3. Data description**

The data analysed in this study are obtained from the global bond index (GBI) series for emerging markets (EM) compiled by JP Morgan. In particular, the fixed income data used comprise of daily total returns for Hong Kong, Mexico, and South Africa from December 31, 2001 to April 9, 2012, representing 2571 observations.

More formally, the change in the local bond index ˆ can be expressed as:

$$
\hat{\mathbf{B}} \Big/ \hat{\mathbf{B}}\_{t-1} - \mathbf{1} = \mathfrak{R}\_t + \mathcal{Y}\_t \tag{1}
$$

where is the local currency return and is the currency return. <sup>1</sup> 01 *t tt t <sup>t</sup> yield yield DV coupon return* and <sup>1</sup> 1 1 *t t tt FX FX* and the coupon return is the return derived from the interest payment made on the fixed income product. Therefore, when an investor buys a local market index, equation (1) suggests that fixed income returns can be decomposed into its predictable coupon return, FX changes, and changes in local yields. Furthermore, in order to compute return volatility, this analysis focuses on squared daily returns, as a proxy for the volatility of the selected emerging markets. In addition, the volatility series is standardised prior to further analysis.
