domain: math stat

text:
Mohamed Saidane
email: m.saidane@qu.edu.sa

A New Viterbi-Based Decoding Strategy for Market Risk Tracking: An application to the Tunisian Foreign Debt Portfolio during 2010-2012

Keywords: C38, C53, G17, G32 Factor analysis, Volatility Clustering, Hidden Markov Models, Viterbi-EM Algorithm, Portfolio's Value-at-Risk

In this paper, a novel market risk tracking and prediction strategy is introduced. Our approach takes volatility clustering into account and allows for the possibility of regime shifts in the intra-portfolio's latent correlation structure. The proposed specification combines hidden Markov models (HMM) with latent factor models that takes into account the presence of both the conditional skewness and leverage effects in stock returns.

A computationally efficient expectation-maximization (EM) algorithm based on the Viterbi decoder is developed to estimate the model parameters. Using daily exchange rate data of the Tunisian dinar versus the currencies of the main Tunisian government's creditors, during the 2011 revolution period, the model parameters are estimated. Then, the suitable model is used in conjunction with a Monte Carlo simulation strategy to predict the Value-at-Risk (VaR) of the Tunisian government's foreign debt portfolio. The backtesting results indicate that the new approach appears to give a good fit to the data and can improve the VaR predictions, particularly during financial instability periods.

INTRODUCTION

According to [START_REF] Saidane | A Monte-Carlo-based Latent Factor Modeling Approach with Time-Varying Volatility for Value-at-Risk Estimation: Case of the Tunisian Foreign Exchange Market[END_REF] and [START_REF] Mosbahi | Mixture of Probabilistic Factor Analyzers for Market Risk Measurement: Empirical Evidence from the Tunisian Foreign Exchange Market[END_REF], the understanding of co-movements among asset returns is a central element in the portfolio risk management process. The authors advocate the use of a mixture of probabilistic factor analyzers and the conditionally heteroskedastic latent factor model to handle co-movements, heterogeneity and time-varying volatility embedded in financial data.

They demonstrate how their proposed strategies can be applied to the estimation of the portfolio's Value-at-Risk (VaR). However, an assumption of these models is that the correlation structure of the portfolio is assumed to be constant over time, but recent empirical works (e.g. [START_REF] Saidane | Forecasting Portfolio-Value-at-Risk with Mixed Factorial Hidden Markov Models[END_REF][START_REF] Tsang | Regime Change Detection Using Directional Change Indicators in the Foreign Exchange Market to Chart Brexit[END_REF][START_REF] Hamilton | Macroeconomic Regimes and Regime Shifts[END_REF][START_REF] Ang | Regime Changes and Financial Markets[END_REF] have shown that this assumption of structural stability is invalid for financial returns, especially during crisis periods. For example, when the economy is hit by a permanent or temporary exogenous unpredictable shock, the crosscorrelation behavior among several financial assets and the inter-relationship between volatilities can be expected to shift simultaneously. In light of this, we propose a novel market risk prediction strategy considering the possibility of regime switching in the interrelationships among several asset classes.

The new approach presented in this paper allows for the possibility of regime shifts in the intraportfolio's latent correlation structure and takes volatility clustering into account. The proposed specification combines latent factor models that takes into account the presence of both the conditional skewness and leverage effects with hidden Markov models (HMM). To capture the volatility clustering and the leverage effect patterns of the return series, we assume that the common variances are modeled separately using quadratic generalized autoregressive conditionally heteroskedastic (GQARCH) processes. This provides a more tractable way to handle the time-varying volatility, co-movements and the latent heterogeneity in financial data.

For the maximum likelihood estimation we proceed in two steps. In the first step, we use the Viterbi decoding algorithm to find the most probable path through the HMM, given the observed data, which we take as an estimate of the true path. In the second step, we implement the Expectation-Maximization (EM) algorithm introduced by [START_REF] Dempster | Maximum Likelihood from Incomplete Data via the EM Algorithm[END_REF], to estimate the model parameters. Our proposed estimation strategy overcomes the complexity and limitations of the exact learning algorithm, especially when the number of hidden states and the length of the time sequence become larger.

The remainder of this paper is organized as follows. In the next section, we provide further background on the factorial hidden Markov volatility model. In section 2, we discuss the inference procedure for the latent factors structure. We then present our iterative maximum-likelihood expectation-maximization (EM) algorithm in section 3. We describe the portfolio's VaR simulationbased Viterbi tracking strategy in section 4 and report on the backtesting results in section 5. In this paper, the currency risk of the Tunisian government's foreign debt portfolio during the revolution period of 14 January 2011 is considered as the basis for an application to our novel prediction strategy. Our portfolio includes the main debt currencies against the Tunisian dinar, such as the European euro, the American dollar, the Japanese yen, the Swiss franc and the British pound. Finally, we conclude the paper by summarizing our contributions and discussing the future research directions.

THE FACTORIAL HIDDEN MARKOV VOLATILITY MODEL

Throughout this paper, we consider a multivariate discrete-time model. The closing price of the k-th asset in the portfolio at the t-th trading day is denoted by 𝑝 𝑘,𝑡 , and the opening price at the first trading day by 𝑝 𝑘,0 .

For each 𝑡 ≥ 1, let 𝑟 𝑘,𝑡 = log(𝑝 𝑘,𝑡 /𝑝 𝑘,𝑡-1 ) be the log-return of the k-th asset. Our model assumes a Markov switching relationship between the observed variables (the log-returns) and a set of q latent factors, which depend on the market regime. This new framework, called factorial hidden Markov volatility model (FHMV), is defined by:

𝒓 𝑡 = 𝚽 𝑗 𝒛 𝑡 + 𝝐 𝑡 , (1)
where, ∀ 𝑡 = 1, … , 𝑇, 𝒓 𝑡 is a (𝑝 × 1) vector of log-returns. The transition probabilities of the first order homogenous hidden Markov process from state 𝑖 to state 𝑗 (∀ 𝑖, 𝑗 = 1, … , 𝑛) are represented by 𝑝(𝑆 𝑡 = 𝑗|𝑆 𝑡-1 = 𝑖), where 𝑗 is the actual market regime at time 𝑡, given the previous regime 𝑖 at time 𝑡 -1. In a specified regime 𝑆 𝑡 = 𝑗, 𝚽 𝑗 is the (𝑝 × 𝑞) factor loadings matrix.

The common latent factors 𝒛 𝑡 are generated from the multivariate normal distributions:

𝒛 𝑡 ~𝒩(𝟎, 𝛀 𝑗 ), (2)
where 0 and 𝛀 𝑗 denote, respectively, the (𝑞 × 1) mean vectors and (𝑞 × 𝑞) diagonal covariance matrices of the latent vectors 𝒛 𝑡 . The diagonal elements of 𝛀 𝑗 (common variances) are described by switching univariate quadratic GARCH [START_REF] Bastianin | Modelling Asymmetric Dependence Using Copula Functions: An application to Value-at-Risk in the Energy Sector[END_REF][START_REF] Bastianin | Modelling Asymmetric Dependence Using Copula Functions: An application to Value-at-Risk in the Energy Sector[END_REF] processes. Under a particular regime 𝑆 𝑡 = 𝑗 since 𝑆 𝑡-1 = 𝑖, the l-th common factor variance is given by:

𝜔 𝑙,𝑡 𝑗 = 𝛽 0𝑙 𝑗 + 𝛽 1𝑙 𝑗 𝑧 𝑙,𝑡-1 𝑖 + 𝛽 2𝑙 𝑗 𝑧 𝑙,𝑡-1 𝑖2 + 𝛽 3𝑙 𝑗 𝜔 𝑙,𝑡-1 𝑖 . ( 3
)
Assuming that 𝛽 1𝑙 𝑗 , 𝛽 2𝑙 𝑗 > 0, if 𝑧 𝑙,𝑡-1 < 0, its impact on the variance 𝜔 𝑙,𝑡 is lower than in the case where 𝑧 𝑙,𝑡-1 > 0.

Finally, the (𝑝 × 1) vector of specific factors can be written as follows:

𝝐 𝑡 ~𝒩(𝝁 𝑗 , 𝚲 𝑗 ), (4)
where 𝝁 𝑗 and 𝚲 𝑗 are, respectively, the (𝑝 × 1) mean vectors and (𝑝 × 𝑝) diagonal covariance matrices of the specific factors.

Assumption 1:

In order to insure the positivity of the common variances and the stationarity of the covariance structure of the studied series, we introduce some constraints on the parameters of the quadratic GARCH specification, such as:

𝛽 2𝑙 𝑗 + 𝛽 3𝑙 𝑗 < 1, 𝛽 0𝑙 𝑗 , 𝛽 2𝑙 𝑗 , 𝛽 3𝑙 𝑗 > 0 and 𝛽 1𝑙 𝑗 2 ≤ 4𝛽 0𝑙 𝑗 𝛽 2𝑙 𝑗 , ∀ 𝑗 = 1, … , 𝑛, 𝑙 = 1, … , 𝑞.
Assumption 2: To guarantee the model identification in [START_REF] Bastianin | Modelling Asymmetric Dependence Using Copula Functions: An application to Value-at-Risk in the Energy Sector[END_REF], we assume that ∀ 𝑗, 𝑟𝑎𝑛𝑘(𝚽 𝑗 ) = 𝑞 and 𝑝 ≥ 𝑞. The factors 𝒛 𝑡 and 𝝐 𝑡 are also assumed to be uncorrelated and mutually independent. For more detailed discussions of the identification problem, the reader can refer to Saidane and[START_REF] Saidane | Can the GQARCH Latent Factor Model Improve the Prediction Performance of Multivariate Financial Time Series[END_REF][START_REF] Carnero | Persistence and Kurtosis in GARCH and Stochastic Volatility Models[END_REF].

INFERENCE OF THE LATENT FACTORS STRUCTURES

Our model can be expressed as a switching state-space system with a measurement equation:

𝒓 𝑡 = 𝝁 𝑗 + 𝚽 𝑗 𝒛 𝑡 + 𝝐 𝑡 , (5)
and a transition equation:

𝒛 𝑡 = 𝟎 • 𝒛 𝑡-1 + 𝒛 𝑡 , (6)
In order to find the optimal sequences of hidden states 𝑆 𝑡 and latent factors 𝒛 𝑡 , we can use the Viterbi decoding algorithm based on the minimization of the Hamiltonian cost function given by the following equation: 𝛿 𝑡-1,𝑖 is the "optimal" HMM sequence up to time 𝑡 -1 when the market state is in regime 𝑖 at time 𝑡 -1.

ℋ(𝑟 1:𝑇 , 𝑍 1:𝑇 , 𝑆 1:𝑇 ) ≃ 𝑐 + 𝓢 1 ′(-log 𝝅) + ∑ 𝓢 𝑡 ′(-log 𝑷)𝓢 𝑡-1 𝑇 𝑡=2 + 1 2 ∑ ∑[log|𝚲 𝑗 | + (𝒓 𝑡 -𝚽 𝑗 𝒛 𝑡 -𝝁 𝑗 )′𝚲 𝑗 -1
Firstly we define the "optimal" partial Hamiltonian cost up to time 𝑡 of the observed log-return sequence 𝑟 1:𝑡 when the market state is in regime 𝑗 at time 𝑡:

𝛿 𝑡,𝑗 = min

𝑆 1:𝑡-1 ,𝑍 1:𝑡 ℋ(𝑍 1:𝑡 , {𝑆 1:𝑡-1 , 𝑆 𝑡 = 𝑗}, 𝑟 1:𝑡 ). ( 10
)
To calculate this cost correctly, we need the optimal filtered estimates of the common latent factors We need also the predicted and filtered common variances: , for 𝑙 = 1, … , 𝑞.

𝛀 𝑡|𝑡-1 𝑖(𝑗) = 𝔼 [(𝒛 𝑡 -𝒛 𝑡|𝑡-1 𝑖(𝑗) ) (𝒛 𝑡 -𝒛 𝑡|𝑡-1 𝑖(𝑗) ) ′ |𝑟 1:𝑡-1 , 𝑆 𝑡 = 𝑗, 𝑆 𝑡-1 = 𝑖] , (11)
Given the information set 𝒟 1:𝑡-1 = {𝑟 1:𝑡-1 , 𝑍 1:𝑡-1 , 𝑆 1:𝑡-1 }, the predicted variances are calculated as the conditional expectations of the predicted volatilities, 𝔼(𝜔 𝑙,𝑡 |𝒟 1:𝑡-1 ), and from the total variance

formula 𝔼(𝑧 𝑙,𝑡-1 𝑖2 |𝒟 1:𝑡-1 ) = 𝑉𝑎𝑟(𝑧 𝑙,𝑡-1 𝑖 |𝒟 1:𝑡-1 ) + 𝔼(𝑧 𝑙,𝑡-1 𝑖 |𝒟 1:𝑡-1 ) 2 = 𝜔 𝑙,𝑡-1|𝑡-1 𝑖 + 𝑧 𝑙,𝑡-1|𝑡-1 𝑖2
, we obtain the filtered variances 𝜔 𝑙,𝑡-1|𝑡-1

𝑖

. When a novel observation 𝒓 𝑡 becomes available, all the prediction estimates can be updated recursively via the Kalman filtering equations:

𝒛 𝑡|𝑡 𝑖(𝑗) = 𝒛 𝑡|𝑡-1 𝑖(𝑗) + 𝑲 𝑡 (𝑖, 𝑗) [𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡|𝑡-1 𝑖(𝑗) ] , (15)
𝛀 𝑡|𝑡 𝑖(𝑗) = [𝑰 𝑘 -𝑲 𝑡 (𝑖, 𝑗)𝚽 𝑗 ]𝛀 𝑡|𝑡-1 𝑖(𝑗) = 𝛀 𝑡|𝑡-1 𝑖(𝑗) -𝑲 𝑡 (𝑖, 𝑗)𝚪 𝑡|𝑡-1 𝑖(𝑗) 𝑲 𝑡 (𝑖, 𝑗) ′ , ( 16
)
with 𝚪 𝑡|𝑡-1 𝑖(𝑗) = 𝚲 𝑗 + 𝚽 𝑗 𝛀 𝑡|𝑡-1 𝑖(𝑗) 𝚽 𝑗 ′ and 𝑲 𝑡 (𝑖, 𝑗) = 𝛀 𝑡|𝑡-1 𝑖(𝑗) 𝚽 𝑗 ′ 𝚪 𝑡|𝑡-1 𝑖(𝑗)-1 . The innovation cost 𝛿 𝑡,𝑡-1,𝑖,𝑗 related to each transition from state 𝑖 to state 𝑗, is given by:

𝛿 𝑡,𝑡-1,𝑖,𝑗 = 1 2 log |𝚪 𝑡|𝑡-1 𝑖(𝑗) | + 1 2 [𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡|𝑡-1 𝑖(𝑗) ] ′ 𝚪 𝑡|𝑡-1 𝑖(𝑗)-1 [𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡|𝑡-1 𝑖(𝑗) ] -log 𝑝 𝑖𝑗 . (17)
A substantial part of this cost is exclusively due to the transition of the latent factors, as illustrated by the innovation component in equation ( 17 As a result, we obtain for each time 𝑡 the "optimal" filtered latent factors 𝒛 𝑡|𝑡 𝑗 = 𝒛 𝑡|𝑡 𝜆 𝑡-1,𝑗 (𝑗) and their corresponding variances 𝛀 𝑡|𝑡 𝑗 = 𝛀 𝑡|𝑡 𝜆 𝑡-1,𝑗 (𝑗) = 𝑑𝑖𝑎𝑔 [ω 𝑙,𝑡|𝑡-1 𝜆 𝑡-1,𝑗 (𝑗) ].

When all the log-returns 𝑟 1:𝑇 become available, we obtain the optimal global cost 𝛿 𝑇 * = min 𝑗 {𝛿 𝑇,𝑗 }.

Then, we use the index of the optimal final state in order to decode the optimal sequence of HMM states: 𝑗 𝑇 * = arg min 𝑗 {𝛿 𝑇,𝑗 }. To get the best regime for all time steps, we trace back through the market regime switching record: 𝑗 𝑡 * = 𝜆 𝑡,𝑗 𝑡+1 * .

We note here that the smoothing gain matrix 𝑳 𝑡 (𝑗)𝑘 = 𝛀 𝑡|𝑡 𝑗 𝟎 𝑘 ′𝛀 𝑡+1|𝑡 (𝑗)𝑘-1 = 𝟎 and the smoothing equations are simply given by:

𝒛 𝑡|𝑇 (𝑗)𝑘 = 𝒛 𝑡|𝑡 𝑗 + 𝑳 𝑡 (𝑗)𝑘 [𝒛 𝑡+1|𝑇 𝑘 -𝒛 𝑡+1|𝑡 𝑗(𝑘) ] = 𝒛 𝑡|𝑡 𝑗 , ( 18
) 𝛀 𝑡|𝑇 (𝑗)𝑘 = 𝛀 𝑡|𝑡 𝑗 + 𝑳 𝑡 (𝑗)𝑘 [𝛀 𝑡+1|𝑇 𝑘 -𝛀 𝑡+1|𝑡 (𝑗)𝑘 ] 𝑳 𝑡 (𝑗)𝑘 ′ = 𝛀 𝑡|𝑡 𝑗 . ( 19
)
Following the smoothing procedure developed by [START_REF] Saidane | An EM-Based Viterbi Approximation Algorithm for Mixed-State Latent Factor Models[END_REF], the sufficient statistics for our estimation problem will be given by:

𝔼(𝒮 𝑡 | •) = 𝒮 𝑡 (𝑗 * ), 𝔼(𝒮 𝑡 𝒮 𝑡-1 ′| •) = 𝒮 𝑡 (𝑗 * )𝒮 𝑡-1 (𝑗 * )′ and 𝔼(𝒛 𝑡 𝒮 𝑡 (𝑗)| •) = 𝒛 𝑡|𝑇 𝑗 𝑡 *
, if 𝑗 = 𝑗 𝑡 * and 0 otherwise. In this case, the operator 𝔼(| •)

denotes the expectation with respect to the distribution 𝑝(𝑍, 𝑆|𝑟).

MAXIMUM LIKELIHOOD ESTIMATION

We propose a two-step learning algorithm combining the expectation maximization (EM) algorithm [START_REF] Dempster | Maximum Likelihood from Incomplete Data via the EM Algorithm[END_REF] and the Viterbi decoding algorithm in order to estimate the parameters Θ of our model. The E-step subsists in calculating the expected value of the complete data log-likelihood function with respect to the conditional distribution of the unobserved variables (𝑍, 𝑆) given the observed returns 𝑟 and Θ (𝑒) , the value of the parameter at the current iteration (𝑒). The conditional expectation is then, maximized with respect to Θ at the M-step. In this case, the auxiliary function that will be maximized can be approximated as follows:

𝒬(Θ, Θ (𝑒) ) ≃ ∑ 𝒮 1 (𝑗) log 𝑝(𝑆 1 ) -∑ ∑ ∑ 𝒮 𝑡 (𝑗)𝒮 𝑡-1 (𝑖) log 𝑝 𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 𝑇 𝑡=2 𝑛 𝑗=1 - 1 2 ∑ ∑ 𝒮 𝑡 (𝑗)[log|𝚲 𝑗 | + 𝔼{(𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡 𝑗 )′𝚲 𝑗 -1 (𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡 𝑗 )|𝑟 1:𝑇 , Θ (𝑒) }] 𝑛 𝑗=1 𝑛 𝑖=1 - 1 2 ∑ ∑ ∑ 𝒮 𝑡 (𝑗)𝔼 [log(𝜔 𝑙,𝑡 𝑗 ) + 𝑧 𝑙,𝑡 2 𝜔 𝑙,𝑡 𝑗 |𝑟 1:𝑇 , Θ (𝑒) ] 𝑇 𝑡=1 𝑞 𝑙=1 𝑛 𝑗=1 , ( 20
)
and the conditional expectations can be derived using the sufficient statistics obtained by the Viterbi algorithm in section 3.

The basic idea behind our algorithm is summarized as follows: At the end of each iteration (𝑒) we find Θ (𝑒+1) , the optimal value of the parameter Θ that maximizes the function in equation ( 20) over all possible values of Θ. Then Θ (𝑒+1) replaces Θ (𝑒) in the E-step and Θ (𝑒+2) is chosen to maximize 𝒬(Θ, Θ (𝑒+1) ), and so on until convergence. However, given the nonlinear dependency of the common variance parameters in the last summation of equation ( 20), we can maximize in a first time this function with respect to the probabilities of the initial state 𝜋 𝑗 , the transition probabilities 𝑝 𝑖𝑗 , the specific means 𝝁 𝑗 , the factor loadings 𝚽 𝑗 and the specific variances 𝚲 𝑗 . In a second time, the parameters of the common variances can be determined numerically.

For the initial state probabilities 𝜋 𝑗 , we use the Lagrange multipliers approach subject to the condition that the sum ∑ 𝜋 𝑗 𝑛 𝑗=1

= 1, and we obtain the updated estimation:

𝜋 ̂𝑗 = 𝒮 1 (𝑗) ∑ 𝒮 1 (𝑖) 𝑛 𝑖=1 . ( 21
)
We use also the Lagrange formalism, subject to the unity constraint ∑ 𝑝 𝑖𝑗 𝑛 𝑗=1

= 1, to obtain the updated transition probabilities:

𝑝î 𝑗 = ∑ 𝒮 𝑡 (𝑗)𝒮 𝑡-1 (𝑖) 𝑇 𝑡=2 ∑ 𝒮 𝑡-1 (𝑖) 𝑇 𝑡=2 . ( 22
)
The maximization of the auxiliary function with respect to the specific means yields the updated estimates:

𝝁 ̂𝑗 = 1 ∑ 𝒮 𝑡 (𝑗) 𝑇 𝑡=1 ∑ 𝒮 𝑡 (𝑗)(𝒓 𝑡 -𝚽 𝑗 𝒛 𝑡|𝑇 𝑗 ) 𝑇 𝑡=1 . ( 23
)
The updated l-th row of the factor loadings matrix 𝚽 ̂𝑗can be expressed as follows:

𝝓 ̂𝑙,𝑗 = [∑ 𝒮 𝑡 (𝑗)(𝑟 𝑙,𝑡 -𝜇 𝑙,𝑗 )𝒛 𝑡|𝑇 𝑗 𝑇 𝑡=1 ] ′ [∑ 𝒮 𝑡 (𝑗)[𝛀 𝑡|𝑇 𝑗 + 𝒛 𝑡|𝑇 𝑗 𝒛 𝑡|𝑇 𝑗′ ] 𝑇 𝑡=1 ] -1 , ( 24
)
where 𝜇 𝑙,𝑗 is the specific mean of the l-th asset return 𝑟 𝑙,𝑡 under the market regime 𝑗. Then, given these updated parameters, we can update the specific variances according to the following rule:

𝚲 ̂𝑗 = 1 ∑ 𝒮 𝑡 (𝑗) 𝑇 𝑡=1 ∑ 𝒮 𝑡 (𝑗)𝑑𝑖𝑎𝑔 [𝚽 𝑗 𝛀 𝑡|𝑇 𝑗 𝚽 𝑗 ′ + (𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡|𝑇 𝑗 )(𝒓 𝑡 -𝝁 𝑗 -𝚽 𝑗 𝒛 𝑡|𝑇 𝑗 ) ′ ] 𝑇 𝑡=1 . ( 25
)
In a second time, given the new values of 𝜋 𝑗 , 𝑝 𝑖𝑗 , 𝝁 𝑗 , 𝚽 𝑗 and 𝚲 𝑗 , we can approximate the conditional distribution of the log-returns by the normal distribution:

𝒓 𝑡 |𝑟 1:𝑡-1 , 𝑆 𝑡 = 𝑗, 𝑆 1:𝑡-1 ∼ 𝒩[𝝁 𝑗 , 𝚪 𝑡|𝑡-1 𝑗 ]
(e.g. [START_REF] Harvey | Unobserved Component Time Series Models with ARCH Disturbances[END_REF]. In this case, 𝚪 𝑡|𝑡-1 (𝑗) ] is the conditional expectation of 𝛀 𝑡 , given the sequences 𝑟 1:𝑡-1 and 𝑆 1:𝑡-1 , obtained via the modified Kalman filter approach based on the Viterbi decoder developed in section 3.

𝑗 = 𝚲 𝑗 + 𝚽 𝑗 𝛀 𝑡|𝑡-1 𝑗 𝚽 𝑗 ′ and 𝛀 𝑡|𝑡-1 𝑗 = 𝑑𝑖𝑎𝑔 [𝜔 𝑙,𝑡|𝑡-1 𝜆 𝑡-1,𝑗
Using these approximations and ignoring the initial conditions, we obtain the following pseudo loglikelihood function:

ℒ * = 𝑐 - 1 2 ∑ ∑ 𝒮 𝑡 (𝑗) [log|𝚪 𝑡|𝑡-1 𝑗 -1 | + (𝒓 𝑡 -𝝁 𝑗 )′𝚪 𝑡|𝑡-1 𝑗 -1 (𝒓 𝑡 -𝝁 𝑗 )] 𝑛 𝑗=1 𝑇 𝑡=1 . ( 26
)
In a first stage, we ignore the elements in the last summation of equation ( 20) and then we maximize the remaining terms with respect to (𝜋 𝑗 , 𝑝 𝑖𝑗 , 𝝁 𝑗 , 𝚽 𝑗 and 𝚲 𝑗 ) using the EM algorithm. During this step, the parameters of the quadratic GARCH processes 𝛽 = {𝛽 0 , 𝛽 1 , 𝛽 2 , 𝛽 3 } are kept fixed to their values obtained in the previous iteration. In a second stage, we optimize the pseudo log-likelihood in equation ( 26) with respect to 𝛽, using the values of 𝜋 𝑗 , 𝑝 𝑖𝑗 , 𝝁 𝑗 , 𝚽 𝑗 and 𝚲 𝑗 found in the first step. The R package NlcOptim, developed by [START_REF] Chen | Solve Nonlinear Optimization with Nonlinear Constraints[END_REF], can be used in this step to find quickly and most accurately the parameters of the conditionally heteroskedastic component 𝛽.

THE FHMV APPROACH FOR VALUE-AT-RISK PREDCTION

Formally put, Value-at-Risk is a financial metric that measures the worst expected loss that could happen in an investment portfolio over a given horizon for a given confidence level. In this section a general Monte Carlo simulation FHMV-based framework for value-at-risk prediction, under regime switching dynamics, will be proposed. This approach will then be used, in section 5, for the evaluation of the currency risk associated with the Tunisian government's foreign debt portfolio during the revolution period of 14 January 2011.

Forecasting future market regime changes

Given the information set available at time 𝑡, 𝒟 1:𝑡 , and the actual market regime 𝑖, the conditional mean of the multivariate predictive distribution given by our FHMV model is as follows:

𝔼(𝒓 𝑡+1 |𝒟 1:𝑡 ) = 𝝁 𝑗 , ( 27
)
and the conditional variance-covariance matrix is given by:

𝚪 𝑡+1|𝑡 𝑗 = 𝚲 𝑗 + 𝚽 𝑗 𝛀 𝑡+1|𝑡 𝑗 𝚽 𝑗 ′ . ( 28
)
Within this framework, the forecasts of the future market regime jumps and the model parameters updating process are implemented simultaneously. Thus, by the end of each transaction day the closing prices will be included in the database. Thereafter, the parameters of our model will be updated using the newer information set available at this point in time, and the updated one-stepahead forecasts of the common latent factor variances will be derived via the relation: 𝜔 ̃𝑙,𝑡+1|𝑡

The simulation strategy

Our simulation strategy consists of the following steps:

1. Firstly, we define the coverage rate 𝛼 of the VaR. 5. Finally, to compute the portfolio's VaR for the period 𝑡 + 1, we sort the simulated values in ascending order and we exclude the 𝛼% lowest returns ℛ 𝑠,𝑡+1|𝑡 . In this case, the predicted VaR is the minimum of the remaining returns.

NUMERICAL EXPERIMENTS USING EXCHANGE RATE DATA

In this empirical experiment, we use the FHMV model to analyze the dynamic latent correlation structure of the five dominant currencies in the Tunisian government's foreign debt portfolio. The optimal specification obtained with the Viterbi algorithm will then be used to evaluate, through a backtesting exercise, the performance of the new methodology in detecting the foreign exchange risk associated with this portfolio. All the numerical results and the graphs in this section are obtained using the R statistical freeware, version 4.1.

Data presentation and summary

We focus in this section on the main five currencies forming the basis for the Tunisian government's foreign debt portfolio, namely the European euro (EUR), the American dollar (USD), the Japanese yen (JPY), the Swiss franc (CHF) and the British pound (GBP). Our dataset, downloaded from the Yahoo Finance website, spread over the period between 02/01/2007 and 30/12/2012, consists of 1500 daily exchange rates for the different currencies expressed in terms of Tunisian dinar (TND). This dataset includes the period of social mobilization and political change in Tunisia (the revolution of 14 January 2011). In this case, taking into account the period of social instability we will be permitted to investigate the efficiency of our Jump-VaR methodology during crisis times.

In Table 1, we give a variety of descriptive statistics to study the distributional characteristics of the data and to test the empirical skewness and Kurtosis against the values of normal distributions (e.g. D 'Agostino, 1970 and[START_REF] Anscombe | Distribution of Kurtosis Statistic for Normal Statistics[END_REF]. We implemented also the normality test [START_REF] Jarque | Efficient Test for Normality, Homoscedasticity and Serial Independence of Residuals[END_REF]. From these results, we note that all the log-returns are non-normally distributed, they are still skewed (positive for EUR, USD, JPY and GBP and negative for CHF). We note also a positive excess kurtosis for all the currencies. The results of the [START_REF] Ljung | On a Measure of Lack of Fit in Time Series Models[END_REF] statistic show the presence of volatility clustering. This imply that we have a non-constant conditional volatility, and the use of a

Markov-switching specification with a time-varying co-movement structure for the log-return series, is more realistic in this situation.

insert Table 1 about here.

A preliminary latent structure analysis of the data

In order to select the most appropriate model that fits better our dataset, we used the Akaike (AIC)

and the Bayesian (BIC) information criteria. To this end, we trained standard and conditionally heteroskedastic models using one or two common factors and a number of hidden states varying between one and three, on the period from 02/01/2010 to 30/12/2012. Then, we used the selection criteria to identify the best model with the minimum AIC and BIC values.

insert Table 2 about here.

The results reported in Table 2 show that the FHMV model with two common factors and two HMM states is the best one fitting our dataset. For this optimal specification, the initial state probability vector and the transition probability matrix are as follows:

𝝅 = [ 1 0
] and 𝑷 = [ 0.9491 0.0509 0.2311 0.7689 ].

In Figure 1, we depict the percentage of the variability of the different currencies expressed in terms of specific and common factors. During the crisis period, we can see that, on average, 90% and 95% of the variances of the EUR and the CHF are explained by the first common factor. During the normal period (before and after the 2011-2012), we can see that the first common factor explains on average 50% and 90% of the variability of the EUR and CHF.

The contribution of the second common factor to the variability of the EUR and CHF, during the instability period, is almost insignificant. During the revolution period, this factor explains more than 85% and 65% of the variability of the USD and JPY. However, the contribution of the first factor to the JPY variability is around 45% during the normal period. For the GBP, the contribution of this factor is around 35% over the whole period.

insert Figure 1 about here.

From these results, we can conclude that the first common factor is associated with the volatility dynamics of the European currencies. During the social mobilization period, the second factor is associated with the volatility dynamics of the American and Japanese currencies. We can conclude also that the first common latent factor expresses the relative value of the TND against the major trading partner's currencies (the European community countries). The second factor reproduces the relative value of the TND against a basket of global currencies in which the American and Japanese currencies are dominant.

insert Table 3 about here.

From the estimation results presented in Table 3, we note that the first common factor can be regarded as a European factor: it represents a basket of currencies, where the EUR dominates with relatively high loadings (50% in the first regime and 76% in the second regime). The weight of the GBP is relatively reduced in this basket. We note also that the second common factor represents a basket of currencies, where the USD dominates with relatively high loadings (52% in the first regime and nearly 60% in the second regime). In order to satisfy the identification constraints (e.g. Saidane and Lavergne 2011), we have taken 𝜙 1,2,𝑗 = 0, ∀𝑗 = 1,2, which imply that the European currency EUR is entirely absent from the second factor. The relative weight of the CHF is also reduced in this factor. Hence, we can consider the second common factor as an American factor.

insert Table 4 about here.

From Table 4, it appears that the excess of volatility during the political instability period (the second hidden regime) is relatively due to the significant increase in volatility persistence (e.g. [START_REF] Klaassen | Improving GARCH Volatility Forecasts with Regime-Switching GARCH[END_REF]. We observe from this table that the sum of the volatility parameters, 𝛽 2 and 𝛽 3 , of the two common factors in the second regime is nearly close to 1.

All the previous conclusions are strongly confirmed by the estimated values of the specific variances, given in Table 3. Hence, during the social mobilization period, the specific variance of the British pound is relatively high, which indicates its aberration from its latent factorial class. On the other hand, the specific variances of the European euro and the American dollar are the smallest ones, which indicate their determinant role in their latent factorial class.

Finally, in Figure 2, we depict the correlation structure of the different log-returns during the period 02/01/2010 to 30/12/2012. The graph picks up co-movement increases between all the log-returns from the beginning of 2011 until near the end of the study period. This result confirms the financial contagion that affected the Tunisian economy during the revolution period.

insert Figure 2 about here.

Selection of the most appropriate VaR model

In order to assess the currency risk associated with the Tunisian government's foreign debt portfolio during the social mobilization period of 14 January 2011, we divided in a first time our dataset into calibration set and test set. The calibration, called also training, set contains the log-returns of the different exchange rates during the period 02/01/2007-30/12/2009 (750 observations). The test, called also backtesting, set contains the remaining 750 observations covering the period 02/01/2010-30/12/2012. Then, we used the Monte Carlo simulation strategy (section 4.2.) to evaluate the VaR of our portfolio. For each coverage rate 𝛼, we used the portfolio weights given in Table 5. Here, the weight of each exchange rate 𝛾 𝑘 is determined by the relative share of currency 𝑘 in the payment of the total foreign debt. For example, in 2010 Tunisia settled 61.3% of its foreign loans in Euro, 14.3% in American dollar, 16.1% in Japanese yen, 2.4% in Swiss franc and 5.9% in British pounds. Hence, in 2010, 𝛾 1 = 0.613, 𝛾 2 = 0.143, 𝛾 3 = 0.161, 𝛾 4 = 0.024 and 𝛾 5 = 0.059. For 2011 and 2012, the weights are determined in the same way.

insert Table 5 about here.

In a second time, the effectiveness of our methodology is justified by some experiments, using unconditional [START_REF] Kupiec | Techniques for Verifying the Accuracy of Risk Measurement Models[END_REF] and conditional [START_REF] Christoffersen | Elements of Financial Risk Management[END_REF] tests and the rolling sample method based on a one-day moving window scheme with the coverage rates from the level of 0.005 to 0.1 by 0.005. All these calculations have been carried out by simulations from our FHMV model, the mixed factorial hidden Markov model (MFHMM) by [START_REF] Saidane | Forecasting Portfolio-Value-at-Risk with Mixed Factorial Hidden Markov Models[END_REF], the latent factor model with time varying volatility (FM) by [START_REF] Saidane | A Monte-Carlo-based Latent Factor Modeling Approach with Time-Varying Volatility for Value-at-Risk Estimation: Case of the Tunisian Foreign Exchange Market[END_REF] and the classical Monte Carlo simulation method (CMC) by [START_REF] Mosbahi | Mixture of Probabilistic Factor Analyzers for Market Risk Measurement: Empirical Evidence from the Tunisian Foreign Exchange Market[END_REF].

In order to compromise between precision and efficiency, we generated 𝑚 = 25.000 scenarios from each competing model (e.g. [START_REF] Saidane | Switching latent factor value-at-risk models for conditionally heteroskedastic portfolios: A comparative approach[END_REF][START_REF] Lu | Portfolio Value-at-Risk Estimation in Energy Futures Markets with Time-Varying Copula-GARCH Model[END_REF][START_REF] Bastianin | Modelling Asymmetric Dependence Using Copula Functions: An application to Value-at-Risk in the Energy Sector[END_REF][START_REF] Fantazzini | Dynamic Copula Modelling for Value at Risk[END_REF].

Then, we calculated the VaR, the exception rates and the likelihood ratios for the proportion of failure test (LR-pof), the independence test (LR-ind) and the conditional coverage test (LR-cc).

All the results of the backtesting experiments are given in table 6 and figures 3-4. The Kupiec and

Christoffersen backtesting results show that the optimal FHMV model, with 2 latent factors and 2

hidden states, provides good results and gives exception rates very close to the target (the true coverage rates 𝛼). The likelihood ratios associated with the unconditional and independence tests, for our proposed model, are always lower than the critical values, which imply a significant conditional coverage tests for all the confidence levels. 1insert Table 6 about here.

For the coverage rates from 0.5% to 2%, Figure 3 shows promising results for the optimal FHMV model compared to those given by the best MFHMM (with 2 mixture components and 2 latent factors). For the significance level 2%, our FHMV model gives, for example, an exception rate equal to 2.17%, versus 2.45% obtained by the MFHMM. From this figure, we can see also that the optimal MFHMM looks better than the FM and CMC, especially at low confidence levels. Hence, we can argue that the FHMV model is the more precise and yields higher-quality predictions, as compared to the other competing models.

insert Figure 3 about here.

In order to compare the results given by the different models, we used the squared relative prediction

error criteria 𝑆 = ∑ [(𝐸 𝑖 -𝛼 𝑖 )/𝛼 𝑖 ] 2 20 𝑖=1
, where 𝐸 𝑖 are the estimated exception rates obtained with the different specifications, and 𝛼 𝑖 the coverage rates. It appears from the results that the most adequate model to evaluate the VaR of our portfolio, during this period, is the FHMV framework. This specification gives the estimated exception rates closest to all the true significance levels with 𝑆 = 0.4561. The second ranked model is the MFHMM (𝑆 = 2.5824), the third is the FM with (𝑆 = 4.9783), and the CMC is the worst one with 𝑆 = 6.0673.

insert Figure 4 about here.

Finally, we can see from Figure 4 the significant effect of the volatility shocks on the predicted VaR given by the optimal FHMV model. Hence, we can argue that the major reason for the bad results

given by the CMC, FM, and to a lesser degree the MFHMM, is that they do not take into account the abnormal switching behaviors, which can affect the volatility and the co-movement dynamics in financial markets during crisis periods. the new approach appears to give a good fit to the data, allows to more close forecasts to the market changes and can improve the VaR predictions and offer more accurate VaR estimates than the other competing models for all coverage rates from 0.5% to 10%.

CONCLUSION

We conclude that our Viterbi-based decoding strategy using the factorial hidden Markov volatility model seems to be a useful tool for portfolio risk management and control, especially during periods of financial market stress. These results support our argument for integrating time-varying volatility and regime jumps into the risk measurement framework. In the forthcoming works, we intend to reflect the interaction between the common latent factors with a dynamic structure for the idiosyncratic variances. We will address also nonlinear behaviors, non-homogeneous transition probabilities and other areas of application, like options, or credit derivatives.

  [𝒛 𝑡 |𝑟 1:𝑡 , 𝑆 𝑡 = 𝑗], the one-step ahead predictions of the common latent factors 𝒛 𝑡|𝑡-1 𝑖(𝑗) = 𝔼[𝒛 𝑡 |𝑟 1:𝑡-1 , 𝑆 𝑡 = 𝑗, 𝑆 𝑡-1 = 𝑖] and their optimal filtered estimates, 𝒛 𝑡|𝑡 𝑖(𝑗) = 𝔼[𝒛 𝑡 |𝑟 1:𝑡 , 𝑆 𝑡 = 𝑗, 𝑆 𝑡-1 = 𝑖].

  ). The remaining part (-log 𝑝 𝑖𝑗 ) reflects the transition of the market state from regime 𝑖 to regime 𝑗. In this case, the minimization of the global cost at time 𝑡 requires the selection of the optimal previous market state 𝑖: 𝛿 𝑡,𝑗 = min 𝑖 {𝛿 𝑡,𝑡-1,𝑖,𝑗 + 𝛿 𝑡-1,𝑖 }. The resulting index is then recorded in the regime switching record, 𝜆 𝑡-1,𝑗 = arg min 𝑖 {𝛿 𝑡,𝑡-1,𝑖,𝑗 + 𝛿 𝑡-1,𝑖 }.

2 . 3 .

 23 Then, taking into account the presence of leverage effects and conditional skewness in financial time series, we simulate different return scenarios from the conditional distribution of the common latent factors 𝒛 𝑡+1|𝑡 𝑠 , using the optimal specification obtained by the Viterbi algorithm at time 𝑡 (section 5.1). a. We use in a first time the normal distribution 𝒩(𝟎, 𝑰 𝑞 ) to generate the standardized factors After that, we simulate different return scenarios from the conditional distribution of the specific factors 𝝐 𝑡+1|𝑡 𝑠 , using also the optimal specification obtained by the Viterbi algorithm at time 𝑡. a. We generate in a first time from the normal distribution 𝒩(𝟎, 𝑰 𝑝 ) the standardized specificities 𝝐 𝑡+1 * . b. Then, we compute the lower triangular Cholesky factor 𝚲 𝑗 * of the variance-covariance matrix 𝚲 𝑗 , and we obtain: the fourth step, we compute 𝑚 different portfolio's returns for the period 𝑡 + 1 as, ℛ 𝑠,1 , 𝛾 2 , … , 𝛾 𝑝 denote the portfolio weights of the 𝑝 risk factors and 𝒓 𝑡+1|𝑡 𝑠 = 𝝁 𝑗 + 𝚽 𝑗 𝒛 𝑡+1|𝑡 𝑠 + 𝝐 𝑡+1|𝑡 𝑠 (∀ 𝑠 = 1, … , 𝑚).

  This paper develops a new multivariate approach for Value-at-Risk (VaR) prediction. Our strategy considers the possibility of regime jumps in the intra-portfolio's latent correlation structure and allows for time-varying volatility in the factor variances. The proposed framework combines factor analysis models with GQARCH processes and hidden Markov models. During financial crisis periods, this specification provides a more tractable way to capture simultaneously the switching interrelations between assets and the time-varying volatility of each individual asset.The accuracy of the new prediction approach in comparison with other existing models (such as the mixed factorial hidden Markov model, the latent factor model with time varying volatility and the classical Monte Carlo method) is evaluated through a real dataset example from the Tunisian foreign exchange market for the period 02/01/2010 to 30/12/2012. Our strategy aims to select the best model that could predict the VaR of the Tunisian government's foreign debt portfolio during the social mobilization period of 14 January 2011. In that period the Tunisian economy has experienced the longest, deepest and most broad-based recession in its history since the 1978. The main results of the empirical example and the backtesting experiments, based on the rolling sample method, show that

Figure 1 Source

 1 Figure 1 The percentage of variability of the different log-returns associated to the common and specific factors over the period from 02/01/2010 till 30/12/2012

Figure 2 Figure 3

 23 Figure 2 Plot of the estimated correlations from the best FHMV model, during the period 02/01/2010 -30/12/2012

  (𝒓 𝑡 -𝚽 𝑗 𝒛 𝑡 -𝝁 𝑗 )]𝒮 𝑡(𝑗) where 𝑟 1:𝜏 = {𝒓 1 , 𝒓 2 , … , 𝒓 𝜏 }, 𝑍 1:𝜏 = {𝒛 1 , 𝒛 2 , … , 𝒛 𝜏 }, 𝑆 1:𝜏 = {𝑺 1 , 𝑺 2 , … , 𝑺 𝜏 } are, respectively, the sequences of observed returns, latent common factors and HMM states up to time 𝜏; 𝝅 the vector of initial state probabilities, of length n-states and summing to 1; 𝑷 the matrix of transition probabilities of the hidden Markov chain, the sum of all elements in the i-th row [𝑝 𝑖1 … 𝑝 𝑖𝑛 ] is 1, ∀ 𝑖 = 1, … , 𝑛 and

	𝓢 𝑡 = [𝒮 𝑡 (1), … , 𝒮 𝑡 (𝑛)]′, where 𝒮 𝑡 (𝑗) = 1, if 𝑆 𝑡 = 𝑗 and 0 otherwise.
	If we denote by 𝑆 1:𝑇 * the optimal sequence of HMM states, the posterior distribution 𝑝(𝑍 1:𝑇 , 𝑆 1:𝑇 |𝑟 1:𝑇 )
	can be approximated as:
		𝑝(𝑍 1:𝑇 , 𝑆 1:𝑇 |𝑟 1:𝑇 ) ≃ 𝜂(𝑆 1:𝑇 -𝑆 1:𝑇 * )𝑝(𝑍 1:𝑇 |𝑆 1:𝑇 , 𝑟 1:𝑇 ),	(8)
	i.e. the posterior distribution of the HMM state sequence 𝑝(𝑆 1:𝑇 |𝑟 1:𝑇 ) is approached by its mode,
	where 𝜂(𝑦) = 1 for 𝑦 = 𝜙 and zero otherwise. The optimal sequence of HMM states can formally be
	obtained by solving the dynamic optimization program: 𝑆 1:𝑇 * = arg max 𝑆 1:𝑇	𝑝(𝑆 1:𝑇 |𝑟 1:𝑇 ). In this case, an
	almost optimal solution can be reached by maximizing recursively the probability of the best HMM
	sequence up to time t:
	𝛿 𝑡,𝑗 = max 𝑆 1:𝑡-1	𝑝(𝑆 1:𝑡-1 , 𝑆 𝑡 = 𝑗, 𝑟 1:𝑡 )
			≃ max 𝑖	{𝑝(𝒓 𝑡 |𝑆 𝑡 = 𝑗, 𝑆 𝑡-1 = 𝑖, 𝑆 1:𝑡-2 *	(𝑖), 𝑟 1:𝑡-1 )𝑝(𝑆 𝑡 = 𝑗|𝑆 𝑡-1 = 𝑖)
				× max 𝑆 1:𝑡-2	𝑝(𝑆 1:𝑡-2 , 𝑆 𝑡-1 = 𝑖, 𝑟 1:𝑡-1 )},	(9)
	* where 𝑆 1:𝑡-2	(𝑖) = arg max 𝑆 1:𝑡-2
			𝑇	𝑛
			𝑡=1	𝑗=1
	+	1 2	𝑛 𝑗=1 ∑ ∑[log|𝛀 𝑗 | + 𝒛 𝑡 ′𝛀 𝑗 𝑇 𝑡=1 -1 𝒛 𝑡 ]𝒮 𝑡 (𝑗)	,	(7)

  𝑡 obtained by the Viterbi algorithm, through the state transition record 𝜆 𝑡-1,𝑖 at each time step. The FHMV model with the optimal future hidden state 𝑆 ̂𝑡+1|𝑡 , will be used in the simulation procedure as the data generating process, to calculate the VaR of our portfolio.

				𝑗	=
	𝛽 0𝑙 𝑗 + 𝛽 1𝑙 𝑗 𝑧 𝑙,𝑡|𝑡 𝑖	+ 𝛽 2𝑙 𝑗 𝑧 𝑙,𝑡|𝑡 𝑖 2 + 𝛽 3𝑙 𝑗 𝜔 𝑙,𝑡|𝑡 𝑖	. Then, the future market regime 𝑆 𝑡+1 = 𝑗 can be obtained as a
	solution of the optimization problem: 𝑆 ̂𝑡+1|𝑡 = arg max 𝑗	𝑝(𝑆 𝑡+1 = 𝑗|𝑆 𝑡 = 𝑖 𝑡 * ), where 𝑖 𝑡 * is the optimal
	market regime at time

Table 1

 1 Basic descriptive characteristics of the daily exchange rate returns from 02/01/2010 to

	30/12/2012
	Statistic	EUR/TND USD/TND	JPY/TND	CHF/TND	GBP/TND
	Mean	0.000327	0.000381	0.000179	0.000274	0.000363
	Max	0.0439	0.0423	0.0514	0.0424	0.1468
	Median	0.000198	0.000255	0.000163	0.000308	0.000212
	Min	-0.0538	-0.0619	-0.0718	-0.0653	-0.0083
	Std. Dev.	0.00425	0.00510	0.00674	0.00583	0.00620
	D'Agost. test	7.49623	2.72961	4.14286	-6.21753	9.57321
		(0.0000)	(0.0047)	(0.0000)	(0.0081)	(0.0000)
	A-Glyn. test	18.751	16.173	15.927	16.369	21.916
		(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)
	LB. test	117.67	108.21	41.962	62.114	123.813
		(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)
	J-Bera. test	65.321	29.655	14.533	26.123	34.259
		(0.0000)	(0.0000)	(0.0074)	(0.0000)	(0.0000)

Note:

The values into brackets represent the p-values of the corresponding tests.

Source: Own construction

Table 2

 2 AIC and BIC values for the different specifications over the period from 02/01/2010 till 30/12/2012 (750 observations)

Table 3

 3 Estimation results of the optimal FHMV model during the period 02/01/2010 -30/12/2012

	Model
	parameters
	(1e-04)

Note:

The selection criteria values for the standard models are given into brackets.

Source: Own construction

Table 4

 4 Estimated parameters of the conditionally heteroskedastic components of the optimal FHMV model during the period 02/01/2010 -30/12/2012

	Common	𝛽 0	𝛽 1	𝛽 2	𝛽 3
	Variance
	parameters
			Factor 1
	Regime 1	0.1224	0.0009	0.2411	0.3648
	Regime 2	0.1521	0.1382	0.2363	0.7601
			Factor 2
		0.0843	0.1774	0.1892	0.4667
		0.1102	0.1430	0.2711	0.7092

Source: Own construction

Table 5

 5 Structure of the Tunisian foreign debt by settlement currency for the period 2010-2012 Monetary and financial statistics of the Tunisian central bank

	Date			Indicators
		EUR	USD	JPY	GBP	CHF
	31/12/2010	61.3	14.3	16.1	5.9	2.4
	31/12/2011	56.8	20.1	15.3	5.6	2.2
	31/12/2012	59.6	18.9	13.8	5.1	2.6

Source:

Table 6

 6 Backtesting results for the FHMV model with two hidden states and two common factors

	Confidence	Exception rate 1st exception	LR-pof	LR-ind	LR-cc
	level
	0.950	0.050	18	1.7934	1.1152	2.9086
	0.960	0.040	22	1.8372	1.1256	2.9628
	0.970	0.029	38	2.0839	1.1398	3.2237
	0.980	0.022	38	2.1856	1.1458	3.3314
	0.990	0.013	56	2.3122	1.1593	3.4715

Source: Own construction

  The VaR exceptions obtained by the optimal FHMV model for different coverage rates and different portfolio weights

	0.90 0.90 02/01/2010 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.12 -0.005 Exception rates 0.000 0.005 0.010 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 02/01/2011 -0.005 0.000 0.005 Figure 4 Source: Own construction 0.94 FHMV MFHMM 0.94 MFHMM FM = 1% = 2% = 5% 02/01/2012	0.98 0.98	0.90 0.90 Confidence levels 0.94 0.98 0.00 0.02 0.04 0.06 0.08 0.10 0.12 FHMV FM 0.94 0.98 0.00 0.02 0.04 0.06 0.08 0.10 0.12 MFHMM CMC 01/05/2010 01/05/2011 01/05/2012	0.90 0.90 01/09/2010 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12 01/09/2011 01/09/2012	0.94 FHMV CMC 0.94 FM CMC	0.98 0.98	30/12/2010 Returns 30/12/2011 30/12/2012

The critical values for the Kupiec and Christoffersen tests are, respectively, 𝜒

(1) =

3.8414 and 𝜒 2 (2) = 5.9915 for 95% VaR.

ACKNOWLEDGMENT

We would like to thank the editor and the reviewers for their constructive, careful and helpful feedback.


Somto K Onukwuli

Charles Chikwendu Okpala

Fred N Okeagu

Review of Benefits and Limitations of Coir Fiber Filler Material in Composites

Keywords: fiber, composite, coir fiber, coir fiber reinforced composite, lignin, cellulose, history of coir, limitations of coir fiber, benefits of coir fiber

The applications of coir fiber reinforced composites have witnessed rapid growth in industrial applications as well as in fundamental research, due to their improved physical and mechanical characteristics. The paper started with the definitions of fiber and composites, before it provided a brief history of coir fiber starting from when it was observed from the Ramayana era around the third century B.C. The nature of coir fiber, it's processing and extraction, as well as its chemical properties, and physical and mechanical compositions were discussed in detail. Apart from being abundant in nature, cost effective, non-toxic, renewable, lowest thermal conductivity, and bulk density, the research pointed out that coir fiber is more durable than other natural fibers because of its high lignin content. Despite the remarkable properties of coir fiber, it was noted that untreated coir fiber composites have unwanted features such as dimensional instability and flammability which makes it unsuitable for high temperature applications. After explaining that the extremely hydrophilic character of coir, along with the hydrophobic nature of the polymers employed for matrix production makes the manufacture of coir polymer composites difficult, resulting in mechanical property loss following moisture uptake, the paper concluded that in order to reduce or eliminate the rate of moisture absorption, that coir fiber surface must be chemically modified.

I.INTRODUCTION

fiber is defined as a unit of matter distinguished by fineness, flexibility, and a high length-to-thickness ratio [START_REF] Farnfield | Textile terms and definitions[END_REF]. The first fibers used by humans were natural fibers such as wool, sisal, cotton, silk, flax, and hemp, while the first synthetic fiber was most likely glass, [START_REF] Grosberg | Fibre failure and wear of materials-an atlas of fracture, fatigue and durability[END_REF]. In a nutshell, natural fibers are fibers that are not organic or man-made. They can be obtained from either plants or animals, [START_REF] Ray | Thermoset biocomposites. Natural Fibers[END_REF]. Natural fibers are multicellular in nature, comprised of a number of continuous, primarily cylindrical honeycomb cells with variable sizes, forms, and structures [START_REF] Pothan | The role of fibre/matrix interactions on the dynamic mechanical properties of chemically modified banana fibre/polyester composites[END_REF].

The basic idea behind fiber composites is the use of fibers as reinforcement in a resin matrix. Fibers typically give the majority of the required strength, while resin provides binding to the fibers. This is because fibers cannot withstand actual loads on their own, and as a result, resin is utilized to bond and protect them.

Okpala, Chinwuko, and Ezeliora (2021), defined a composite as a material that comprise of two or more dissimilar constituents that when joined together are quite stronger than the individual components. They pointed out that a stronger material that is known as reinforcement is embedded in a weaker one known as matrix, to form a new material with enhanced properties like rigidity, long term durability, and improved strength. A composite material combines the most desired qualities of its parts while suppressing their least desirable properties, which in return gives better and unique mechanical and physical properties. Composites, plastics, and ceramics have been the dominating engineering materials during the past few decades.

Modern composite materials are made up of many different materials that are used in everyday life, as well as in sophisticated applications. While composites have already proven their worth as weight-saving materials, the current challenge is to make them durable enough to replace other materials while also remaining cost effective. This has led to the creation of several innovative procedures that are now in use in composite industry. At the moment, composite materials play an important part in the aerospace, car, and other technical applications due to their exceptional strengthto-weight and modulus-to-weight ratios [START_REF] Haruna | Prospects and challenges of composites in a developing country[END_REF]. Coir fiber composites are composite materials that are reinforced with fibers, particles, or platelets derived from coconut.

Composite material uses have expanded fast and have even found new markets, as numerous industries have utilized composites for diverse applications. Composite materials are divided into three types based on their matrix materials: polymer matrix composites, metal matrix composites, and ceramic matrix composites. Each form of these composites is appropriate for a variety of purposes. A matrix or binder material is the foundation material that binds or retains the filler material in structures, whereas filler material is present in the form of sheets, pieces, particles, fibers, or whiskers of natural or synthetic materials [START_REF] Rajak | Fiber-reinforced polymer composites: Manufacturing, properties, and applications[END_REF]. Coconut fiber ropes and cordage have been used since prehistoric times, as coconut was also referred to as "Indian Nut" by explorer Marco Polo (Deccan Herald, 2020). Polynesian coir, also known as sennit, was once again utilized as a building material to weave together boats, houses, swords, and tools. Coir was used as a ship's cable by Indian navigators who crossed the seas to Malaya, Java, China, and the Gulf of Arabia centuries ago. Coir as a reinforcing fiber is gaining prominence in the composite research and industry due to its versatile character, as well as its high biodegradability and availability. They have a number of potential advantages, particularly in terms of environmental performance; this is because when natural fiber composite trash is burned, no net carbon dioxide emissions are released into the environment.

The history of natural fiber reinforced composites began in the last 20 years, when there was a lot of interest in using natural fibers to make polymer bound composites. They are environmentally friendly materials that can compete with glass/polyester in terms of strength, performance, and cost.

The combination of natural fiber reinforced polyester composites has been shown to be an effective approach for designing materials that meet a variety of criteria [START_REF] Idicula | Thermophysical properties of natural fibre reinforced polyester composites[END_REF]. [START_REF] Brahmakumar | Coconut fibre reinforced polyethylene composites: effect of natural waxy surface layer of the fibre on fibre/matrix interfacial bonding and strength of composites[END_REF], shown that coir fibers may be effectively reinforced and bonded in a polyester matrix.

Coconut is widely cultivated in several tropical Asian nations. Indonesia, the Philippines, and India account for 73 percent of global coconut output, with Indonesia being the greatest producer, the Philippines second, and India third. Brazil is the fourth biggest producer of coconut, which is used to make a broad range of floor furnishing materials including mats, yarn, and ropes. Despite all of these applications for coir fiber, just a small portion of the fiber produced is used. As a result, numerous studies have been conducted to investigate the use of coir as reinforcement in composites, [START_REF] Geethamma | Dynamic mechanical behavior of short coir fiber reinforced natural rubber composites[END_REF]; S. [START_REF] Mahzan | Mechanical Properties of Medium Density Fibreboard Composites Material Using Recycled Rubber and Coconut Coir[END_REF].

III. ATURE OF COIR FIBER

Coir, often known as coconut fiber, is a hard structural commercial product derived from the coconut husk. Coir is an excellent replacement for cypress mulch or peat moss, since unlike peat and cypress, it is a renewable resource. Its harvest does not harm the environment in the same way as peat mining does, and it does not contain disease organisms that may be transmitted to plants.

Individual coir fiber measures 0.3-1.0 mm in length, 0.01-0.0.2 mm in diameter, and has an aspect ratio of 35. It features a medium -to large-sized lumen that is polytonally rounded or elliptically formed. The vascular bundle is collateral and is encased in a thick sclerenchymatous sheath. With increasing fiber age, lignin and hemicelluloses, which comprise the cementing elements of fiber cells, increase, whereas pectin declines.

The husk includes coir fiber as well as a corky tissue known as pith, according to [START_REF] Shandilya | Banana fiber reinforcement and application in composites: A review[END_REF]. Due to its high lignin concentration, the husk is composed of water, fibers, and small quantities of soluble solids, and it has a higher biodegradability than most natural fibers. The lignin makes coir fiber excellent for applications in cars, biomedical, railway coaches, maritime, and other long-lasting uses [START_REF] Kakou | Hybrid composites based on polyethylene and coir/oil palm fibers: Http://Dx[END_REF]. Coir is relatively resistant to salinity and microbial degradation, and its current study as a polymer matrix reinforcement has shown promising results [START_REF] Mohanty | Biocomposites : An Introduction. Natural Fibers[END_REF]. Coir fiber polymer composite material is a viable alternative to other natural fiber polymer composites due to their exceptional performance in a wide variety of applications. Differences in coir fiber characteristics might be related to the source of the coconut plant from which the fibers were extracted or to the technique of extraction utilized.

Processing

The term coir is derived from the Malayalam word kayar, which means cord or rope (traditionally a kind of rope is made from the coconut fiber). Coir has been hand-processed throughout its history, initially by climbing the palms to get the coconuts and then by gathering them from the ground. Palm climbers discovered that utilizing a knife tied to a bamboo pole was the most effective method, allowing them to collect fruit from up to 250 trees each day.

The optimal period to gather coir was believed to be 10 months, during which harvesters husked the coconuts and soaked them for a year to yield the coir, because extracting the coir from the shell needs soaking (known as retting).

Depending on the availability of water source, harvesters soaked coconut husks in either salt or fresh water. Soaking the coir in salt water resulted in stronger coir. The results varied, but once the coir was further processed by rolling the fibers from three to ten nuts, depending on their size, the material was claimed to create roughly one pound of yarn. Mechanical processing was introduced by industrialization in 1950, it has led to modern-day dehusking machines that process around 200 coconuts per hour.

Extraction

Coir fiber is divided into two types: brown fiber from mature coconuts and finer white fiber from immature coconuts. Green coconuts, which are collected after six to twelve months from the palm, have flexible white fibers, while brown fiber is acquired by collecting completely ripe coconuts at the point when the nutritive layer surrounding the seed is ready to be processed into copra and dried coconut. During the coir fiber extraction process, a large amount of coir residue is generated.

To extract white fiber, retted husks are taken from the retting enclosure, washed to remove any clinging slime, dirt, or sand, and the exocarp is easily peeled off. The husks are pounded with wooden mallets on tree trunks or stones to separate the fibers. The fibers are washed, dried, and stacked together. The wet husks are put through spiked breaker drums to remove the fragile fiber, while the drums are pushed at high speeds in opposing directions to separate the long and short fibers. The brittle fibers that pass between the drums are gathered and cleaned by running them through another pair of drums with closer nails, followed by washing and drying. Cellulose has a strong crystalline structure and is resistant to hydrolysis, nevertheless, acid hydrolyzes it rapidly. Hemicellulose has a random, amorphous, and hydrophilic structure that serves as a cellulose support matrix. It has a low specific gravity, is equally soluble in alkali, and may be hydrolyzed in acid [START_REF] John | Biofibres and biocomposites[END_REF]. Lignin is an amorphous hydrophobic complex polymer (a thermoplastic) that gives plants stiffness and is not hydrolysable in acid, but is soluble in hot alkali, whereas pectin is a structured heteropolysaccharide that offers plants flexibility [START_REF] John | Biofibres and biocomposites[END_REF].

Physical and Mechanical Composition of Coir Fiber

The diameter, density and weight gain by water absorption of a material are referred to as its physical characteristics, whereas it's mechanical properties include its tensile, flexural, and impact strengths. It also relies on the cellulose type and the shape of the basic cell. The cellulosic chains are arranged parallel to one another, forming bundles that each contain cellulosic macromolecules linked by hydrogen bonds, and the cellulosic chains confer stiffness to fibers called micro fibrils via links with amorphous hemicelluloses and lignin [START_REF] Omrani | State of the art on tribological behavior of polymer matrix composites reinforced with natural fibers in the green materials world[END_REF].

Table 2 summarizes the physical and mechanical characteristics of coir fiber. There is a great possibility to fabricate coir-based composites for a wide range of applications in building and construction, such as boards and blocks.

Other benefits of coir fiber include:

• Tough and durable • Provides excellent insulation against sound • Easy to clean • Not easily combustible • Flame retardant • Totally static free

The impact of lignin as a compatibilizer on the physical characteristics of coconut fiber-polypropylene composites was investigated by [START_REF] Bledzki | Barley husk and coconut shell reinforced polypropylene composites: The effect of fibre physical, chemical and surface properties[END_REF]. The coconut fiber polypropylene composites using lignin as a compatibilizer were shown to have greater flexural characteristics than the control composites. It is used to make a broad range of floor furnishing materials, yarn, rope, and other products due to its hard-wearing qualities, durability, and other benefits. [START_REF] Satyanarayana | Structure property studies of fibres from various parts of the coconut tree[END_REF].

The damping characteristics of randomly oriented coir fiberreinforced polypropylene composites are superior to those of synthetic fiber-reinforced composites. This is because a greater resin content has better damping characteristics, therefore, a lower fiber loading results in more energy absorption. The highest damping ratio of 0.4736 was found in coir composite at 10% fiber content, while increasing fiber content to 30% increased natural frequency of material to 20.92 Hz [START_REF] Munde | Investigation to appraise the vibration and damping characteristics of coir fibre reinforced polypropylene composites[END_REF].

By utilizing 70% fiber content and 28% epoxy in the casting [START_REF] Natsa | Development of a military helmet using coconut fiber[END_REF], created military helmets out of coir fiber reinforced polymer composite. The study used several fiber compositions in epoxy resin to generate a variety of specimen. The manufactured helmet's physical characteristics were measured and compared to those of other helmets. The manufactured helmet weighs less than both the Chinese and British helmets. However, it weighs only a little more than a US Ballistic helmet.

V. LIMITATIONS OF COIR FIBER AS FILLER MATERIAL IN COMPOSITES

Despite its favorable characteristics, untreated coir fiber composites have certain unwanted features such as dimensional instability, flammability that makes it unsuitable for high temperature applications, UV radiation, acids, and bases (Izzuddin [START_REF] Zaman | Influence of Fiber Volume Fraction on the Tensile Properties and Dynamic Characteristics of Coconut Fiber Reinforced Composite[END_REF]. The majority of research on coir fiber composites show that a variety of factors such as fiber volume fraction, fiber length, fiber aspect ratio, fibermatrix adhesion, fiber orientation, and stress transmission at the interface have a substantial impact on their mechanical properties. A variety of studies on coir fibers have been conducted to investigate the influence of various fiber characteristics on the mechanical properties of composite materials.

The effect of fiber content on the mechanical characteristics of coir fiber reinforced composites was studied by (Izzuddin [START_REF] Zaman | Influence of Fiber Volume Fraction on the Tensile Properties and Dynamic Characteristics of Coconut Fiber Reinforced Composite[END_REF]. The coir fiber volume in the composite was varied from 5% to 15% in the research. The results demonstrate that when the fiber volume percentage increases, the tensile strength and Young's Modulus drop. The decrease is caused by inadequate interfacial adhesion between the fibers and the matrix. The decrease is caused by inadequate interfacial adhesion between the fibers and the matrix. Brittleness of the fibers also led to low mechanical strength since stronger fibers have more potential to withstand larger loads. When the fiber volume fraction is raised, the failure strain increases. (Izzuddin [START_REF] Zaman | Influence of Fiber Volume Fraction on the Tensile Properties and Dynamic Characteristics of Coconut Fiber Reinforced Composite[END_REF].

When submerged in water, coir fiber reinforced composites and polymer-based composites absorb moisture in humid environments. In general, moisture diffusion in composites is affected by variables such as fiber volume fraction, void volume, additives, humidity, and temperature. [START_REF] Errajhi | Water absorption characteristics of aluminised E-glass fibre reinforced unsaturated polyester composites[END_REF][START_REF] Weitsman | Anomalous fluid sorption in polymeric composites and its relation to fluid-induced damage[END_REF]. Moisture diffusion in polymer composites has been demonstrated to be controlled by three distinct processes. The first involves water molecules diffusing into the tiny gaps between the polymer strands.

The second method includes capillary transport through the gaps and faults at the fiber-matrix contacts. The third includes the movement of tiny fractures in the matrix caused by fiber swelling. [START_REF] Dhakal | Effect of water absorption on the mechanical properties of hemp fibre reinforced unsaturated polyester composites[END_REF][START_REF] Thwe | Effects of environmental aging on the mechanical properties of bamboo-glass fiber reinforced polymer matrix hybrid composites[END_REF], investigated the mechanical performance of coir fiber/polyester composites, the mechanical behavior of coir fiber/polyester composites with ineffective coir fiber reinforcement is related to their low modulus of elasticity when compared to bare polyester resin. (Harish et al). Because of their ability to absorb moisture from the environment and their poor interfacial interaction with the hydrophobic polymer matrix, untreated coir fibers are limited in their usage in industrial applications. [START_REF] Sudhakara | Fabrication of Borassus fruit lignocellulose fiber/PP composites and comparison with jute, sisal and coir fibers[END_REF]. [START_REF] Asasutjarit | Development of coconut coirbased lightweight cement board[END_REF], investigated the optimal treatment conditions and fiber length for producing coconut coir fiberbased cement board from portland cement and coir fiber. He observed that a composite made using 1-13cm lengths of raw coir fiber did not conform appropriately due to the spring back effect and the difficulty of performing measurements. As a result, the composite failed to meet the standard specification required for testing samples. Moisture can cause the leaching of water-soluble components and/or the breakdown of fibers or particles into low molecular weight degradation products in natural fiber reinforced cement board samples. As a result, the bonding of fibers and cement in composites may be disrupted. [START_REF] Balea | Nanocelluloses: Natural-Based Materials for Fiber-Reinforced Cement Composites[END_REF][START_REF] N E Naveen | Experimental Analysis of Coir-Fiber Reinforced Polymer Composite Materials[END_REF], studied the effect of fiber length on coir fiber reinforced composites, he discovered that the mechanical properties of the composites such as tensile strength, flexural strength, impact strength, and so on are also greatly influenced by the fiber lengths, and the micro-hardness decreases with the increase in fiber length up to 20 mm. Small changes in the physical nature of fibers for a given volume content of fibers can cause significant variations in the overall mechanical characteristics of composites. These composite materials sometimes depart from their designed condition as some defects, such as manufacturing defects, cause them to digress from the expected improvement in mechanical properties. These manufacturing flaws include fiber displacement, waviness, and occasionally breakage, fiber/matrix de-bonding, decortication, and the creation of voids in a composite material's matrix. 1% increase in void content in composites results in a drop in tensile strength (10-20%), flexural strength (10%), and inter-laminar shear strength (5-10%), correspondingly [START_REF] Mehdikhani | Voids in fiber-reinforced polymer composites: A review on their formation, characteristics, and effects on mechanical performance[END_REF]. [START_REF] Bhaskar | Water Absorption and Compressive Properties of Coconut Shell Particle Reinforced-Epoxy Composite[END_REF], investigated the water absorption and compressive characteristics of a coconut shell particle reinforced epoxy composite. It was determined that the water absorption capacity of coconut shell particles was limited to a maximum of 30%. Coir fiber is made from lingo cellulose and has highly polarized hydroxyl groups. As a result, they tend to be hydrophilic [START_REF] Harish | Mechanical property evaluation of natural fiber coir composite[END_REF]. The extremely hydrophilic character of coir, along with the hydrophobic nature of the polymers employed for matrix production, makes the manufacture of coir polymer composites difficult, resulting in mechanical property loss following moisture uptake. Because of the poor compatibility, the coir fiber surface must be treated chemically to make it less hydrophilic and enhance the interface contact between the fiber and the matrix. [START_REF] Khan | Swelling and Thermal Conductivity of Wood and Wood-Plastic Composite[END_REF][START_REF] Tajvidi | Water uptake and mechanical characteristics of natural filler-polypropylene composites[END_REF].

VI. CONCLUSION

This paper demonstrates that coir fiber can be utilized as reinforcement in composites, as it has the benefits of being lightweight, having a high strength-to-weight ratio, inexpensive, and being widely available. However, the primary drawbacks are its relative moisture and fiber volume content, as well as other limits due to length and fiber orientation, which are common to natural fibers. To minimize or eliminate the rate of moisture absorption, the coir fiber surface must be chemically modified.

More study is required for the modification of the chemical properties of coir which will thus eliminate or decrease the limitations of coir fiber. This is important for broadening the application of coir fiber composites.

A

  www.ijltemas.in Page 14 II. HISTORY OF COIR FIBER According to Coir Team (2021), coconuts are thought to have originally appeared in the Ramayana era about the third century B.C. they explained that a Greek sailor wrote in the first century A.D. about observing coco fibers from an East African hamlet being utilized in the construction of a boat, which was done by stitching together planks with twine spun from coir fibers. Mentions of coir characterization have also been discovered dating back to the 11th and 13th centuries A.D. In Sri Lanka, India, and the Persian Gulf, these fibers were utilized as building materials in place of nails. The coir strands, sometimes referred to as lashings, were used by early Hawaiian settlers to bind their boats together. Maritime explorers have traditionally used coir rope as cables and rigging on their ships. Coir was also commercially sold in the United Kingdom in the mid-1800s, notably for use in carpets and fabrics used in flooring.

Figure 1 :

 1 Figure 1: Coconut tree and fruits

Figure 2 :

 2 Figure 2: Extracted and Un-Extracted Coir Fiber. Source: Okpala, Onukwuli, and Ezeanyim (2021)Chemical Properties of Coir FiberMany studies have investigated the chemical composition of coir fiber; Table1summarizes their findings. According to (Carlos -Brazil -et al., 2000), water is the most important chemical component of a living tree, but on a dry weight basis, all plant cell walls are primarily composed of sugarbased polymers (carbohydrates) combined with lignin, with minor amounts of extractives, protein, starch, and inorganics.

Table 1 :

 1 Chemical Composition of Coir

	Cellulose (%)	Hemicellulose (%)	Lignin (%)	Pectin (%)	Source
	36-43	0.2	41-45	1.8	(Yan et al., 2016)
	45-50	-	30	-	(Roy et al., 2012)
	37	-	42	-	(Verma et al., 2013)
	43.4	0.25	45.8	3	(Verma and Gope, 2015)
	36-43	0.15-0.25	41-45	3-4	(Zhang and Hu, 2014)
	42.14	15-17	35.25	-	(Siakeng et al., 2018)
	32-43	0.15-0.25	40-45	-	(Yusoff et al., 2016)
	38-46	10-15	37-41	-	(Pérez-Fonseca et al., 2016)
	42.44	0.25	45.4	3	(A. Khan and Mehra, 2014)
	32-43	0.15-0.25	40-45	-	(Zainudin et al., 2014)
	45.67	0.12-0.25	41-45	-	(Sudhakara et al., 2013)
	39.3	-	29.8	-	(Chollakup et al., 2013)
	43.44	0.25	45.84	3	(Haque et al., 2009)

Table 2 :

 2 Physical and Mechanical Composition of Coir Fiber

	Average diameter (mm)	Water Absorption (%)	Density (g/cm3)	Young (GPa) Modulus	Tensile (MPa) Strength	Elongation (%) at break	Source
	0.025	-	1.2	2.74	286	20.8	(Yan et al., 2015)
	0.4	130-180	1.2	4-6	175	30	(Anupama et al., 2014)
	0.01-0.46	-	1.15-1.46	2.2-6	95-230	15-51.4	(Yan et al., 2016)
	0.25	-	1.2	2.74	286	20.8	(Yan et al., 2015)
	0.1-0.45	10	1.3-1.5	4-6	105-175	17-47	(Sanjay et al., 2018)
	0.38		1.2	2	144	4.5	(Yusoff et al., 2016)
	-	-	-	3.23	165.2	39.45	(Andiç-Çakir et al., 2014)
	-	-	-	4-6	144	15-40	(Zainudin et al., 2014)
	0.1-0.4	-	1.15	4-6	108-252	15-40	(Sanjay et al., 2018)
	-	-	1.37	3.19-3.23	158-165	39-41	(Yan et al., 2015)
	-	-	-	4-5	250	20-40	(Tran et al., 2013)
	0.1-0.45	-	-	3-6	106-175	47 (max)	(Yan et al., 2015)
	-	-	-	4-6	131-175	47.2	(Sudhakara et al., 2013)
	0.2	-	1.3	3.11	144.6	32.3	(Saw et al., 2012)
	-	93	1.17	8	95-118	-	(Sen and Reddy, 2011)
	-	-	1.2	4-6	593	30	(Ku et al., 2011)
	IV. BENEFITS OF COIR FIBER AS FILLER MATERIAL IN	With a growing emphasis on fuel efficiency, natural fibers
		COMPOSITES			such as coir-based composites are finding more usage in
	Many researchers have reported the benefits of coir fibers, including the fact that they are abundant in nature, non-toxic, renewable, and cost-effective, as well as providing the	vehicles, railway coaches, and buses for public transportation. Coir fiber reinforced composites are also used to create seat cushions for Mercedes vehicles (Naveen and Yasaswi, 2013).
	necessary bonding with the cement-based matrix for
	significant improvements in material properties such as
	ductility, toughness, flexural capacity, and impact resistance
	(Ardanuy et al., 2011).
	Coir fiber is more durable than other natural fibers because of
	its lignin content, it also has the lowest thermal conductivity
	and bulk density. The inclusion of coconut coir decreased the
	heat conductivity of the composite specimens resulting in a
	lighter product. The development of composite materials for
	buildings based on natural fibers such as coconut coir with
	low heat conductivity is an intriguing option that would
	address environmental and energy concerns. (Asasutjarit et
	al., 2005; Khedari et al., 2001)