The method initially known as RiskMetrics corresponds to the estimation of an IGARCH (1,1) type model (Integrated GARCH), which assumes that the returns on an asset or portfolio of assets follow a normal distribution and have a conditional variance described by equation 1 (Morettin, 2011). However, distributions that consider tails heavier than normal and also asymmetry of log-returns may be considered.

σ t 2 = λ σ t − 1 2 + ( 1 − λ ) r t − 1 2 ; t = 1, … , T ;   0 < λ < 1 (1)

In which σ t 2 is the conditional variance of the return on an asset in the period t and T the number of observations. Performing σ 1 2 = V a r ( r t ) , which corresponds to the unconditional variance of returns, 999 processes were simulated in software R for λ=0.001;0.002;…;0.999, in order to obtain the respective mean squared errors (MSE) of each fit, described by the following equation:

M S E = ∑ t = 1 T ( r t 2 − σ t 2 ) 2 T (2)

The parameter λ which minimizes the MSE will be used in equation 1 to make estimates of the conditional variance of returns. The estimation of VaR for k periods ahead is done by means of the following equation:

V a R [ k ] = [ q ( p ) ] k σ ^ t C (3)

Where k is the number of days ahead for the VaR calculation, q(p) are the p-quantiles of the probability distribution used, in which p=1-α and σ ^ t corresponds to the conditional variance estimated at time t. The p-quantiles were obtained for α=1;0.5;0.25;0.1, for the normal and Student’s t distributions. For the latter, the number v of degrees of freedom by maximizing the logarithmic function of the likelihood of the standard Student’s t distribution adjusted to the series of log-returns. This function is given by l(v,μ,σ|r), represented in equation 4, in which T is the number of observations used in the sample, r is the vector of log-returns, μ is the position parameter, σ the scale parameter and logΓ(.) represents the natural logarithm of the Gamma function.

l ( v , μ , σ | r ) = T [ l o g Γ ( v + 1 2 ) − l o g Γ ( v 2 ) − l o g σ − 1 2 l n ( π v ) ] − ( v + 1 ) 2 ∑ t = 1 T l o g [ 1 + ( r t − μ σ v ) 2 ] (4)

According to Berkowitz et al. (2008), the calculation of VaR by Historical Simulation is simply done by obtaining the empirical p-quantile observed from T days passed multiplied by the square root of the number days associated with the projection horizon (k). Thus, the VaR calculated by the Historical Simulation method, with coverage level p and time horizon k, is given by equation 5:

V a R p [ k ] = C q ( p ) k (5)

The estimation of quantiles is a non-parametric alternative for the calculation of VaR (Morettin, 2011), that is, no assumption is made on the probability distribution of the log-returns, only that it will remain the same during the forecast period. The estimator of the p-quantile q(p) is a consistent estimator for the parameter Q(p) and is given by equation 6:

q ( p ) = { r ( j ) , i f p = p j = j − 0,5 n , j = 1, … , n ( 1 − f j ) r ( j ) + f j r ( j + 1 ) , s e p j < p < p j + 1 r ( 1 ) , i f 0 < p < p 1 r ( T ) , i f p T < p < 1 (6)

where f _j =(p-p _j )/(p _j+1 -p _j ).

The Historical Simulation method assumes that the frequency distribution of the log-returns will remain the same in the forecast horizon, because it does not consider the possibility of conditional volatility of log-returns.

Without necessarily imposing the normality hypothesis of the returns of the assets under consideration, a VaR model, estimated by the GARCH method, first proposed by Bollerslev (1986), is a model that estimates the conditional variance of an asset’s returns as a function of past returns and conditional variances.

We estimate the parameters of a GARCH model(m,n) for the returns on an asset or portfolio of assets by means of the following system of equations:

r t = ε t h t ; ε t ~   R B ( 0, σ 2 ) (7)

h t = ω + ∑ i = 1 m α i r t − i 2 + ∑ j = 1 n β j h t − j (8)

Equation 8 is subject to the following restrictions:

ω > 0, α _i ≥ 0, i = 1,…, m - 1, α _m ≠ 0, β _j ≥ 0, j = 1,…, n-1, β _n ≠ 0

Where ꞷ, α _i , β _j are the model parameters to be estimated, h _t is the conditional variance of returns in the period t and ε _t is a white noise (WN), with mean 0 and variance 1. In addition, is a condition for stationarity of log-returns that ∑ i = 1 q ( α i + β i ) < 1 in which q=max(m,n). Based on this model, the conditional variances for the k horizons is given by:

h t ^ [ k ] = E [ h t + k | F t   (9)

In which F _t is the information filtration available in period t. In turn, by assuming r t = ε t h t , the conditional default forecast errors, et[k], are calculated as follows:

e t [ k ] = h t ^ [ k ] (10)

The cumulative forecast variance k, further ahead, which is given by the following:

V t [ k ] = h t ^ [ k ] + h t ^ [ k − 1 ] + h t ^ [ k − 2 ] + … + h t ^ [ 1 ] (11)

The standard errors of the cumulative returns forecasts are obtained as follows:

e t [ k ] * = V t [ k ] (12)

In this way, we are able to calculate the conditional confidence intervals for forecasts. Considering an interval with probability p, for the VaR calculation of a long position, we calculated the lower limit of the confidence interval, P(rt+k<LIt+k)=p. Thus, the VaR[k] is calculated as follows:

V a R [ k ] = C ( q ( p ) e t [ k ] * ) (13)

We estimated 25 combinations of the GARCH(m,n) Family, with the following series of combinations {(m,n)}={(1,1),...,(5,5)}, assuming that the white noise term εt follows asymmetric Student’s t-distribution. The criteria for the model selection were the joint analysis of the Bayesian Information Criterion (BIC), mean VaR and the adherence backtesting and independence of violations.

It is assumed that log-returns rt are independently and identically distributed, with cumulative distribution function F(x). In the EVT-based model, we will be interested in studying the behavior of the probability distribution tails of log-returns. For a more detailed review on EVT, see Tsay (2010).

In order to model the tails of the distribution of log-returns, we will use the Generalized Extreme Value Distribution (GEVD), whose distribution function is given by:

F ( r n , i ) = { e − [ 1 + ξ ( r n , i − μ σ ) ] − 1 ξ , i f     ξ ≠ 0 e − e − ( r n , i − μ σ ) ,     i f     ξ = 0 , (14)

Defined in { r n , i : 1 + ξ ( r n , i − μ σ ) > 0 } , i f ξ ≠ 0 , for − ∞ < μ < + ∞ , − ∞ < ξ 〈 + ∞ , σ 〉 0. .

The distribution family is determined by the parameter ξ, so that if ξ=0 we get the Gumbel Type I family, if ξ>0 we obtain Fréchet Type II family, and if ξ<0 the inverse Weibull Type III family.

Assuming that we have T log-returns available { r j } j = 1 T , we divide the data into g sub-samples of identical size n, that is, T=gn, such that:

{ r t } t = 1 T = { r 1 , … , r n ∨ r n + 1 , … , r 2 n ∨ r 2 n + 1 , … , r 3 n ∨ … ∨ r ( g − 1 ) n + 1 , … , r g n } (15)

Given the relationship g = T n , depending on the choices of T and n, g may result as not integer. The solution to this is the exclusion of a minimum of the first observations of the series. The minimum number of observations excluded (NE) to make g integer is given by equation 16:

N E = T − | g | . n (16)

In which |g| is the lowest integer of g.

With r _n,i being the minimum log-return observed in sub-sample i multiplied by -1, where the subscript n denotes the size of the subsample, the series of positive values of the minima will be given by:

{ r n , i } = { − m i n { r ( i − 1 ) n + j } } ,   i = 1, … , g , j = 1, … , n (17)

The estimation of the scale parameters σ, position μ and form ξ can be obtained by the maximum likelihood method. In the event that ξ≠0, the log likelihood function is given by equation 18:

l ( σ n , μ n , ξ n | r n ,1 , … , r n , g ) = − g l n σ n − ( 1 + 1 ξ n ) ∑ i = 1 g l n [ 1 + ξ n ( r n , i − μ n σ n ) ] − ∑ i = 1 g [ 1 + ξ n ( r n , i − μ n σ n ) ] − 1 ξ n (18)

For ξ=0, we have:

l ( σ n , μ n | r n ,1 , … , r n , g ) = − g ln σ n − ∑ i = 1 g r n , i   −   μ n σ n − ∑ i = 1 g e −   r n , i   −   μ n σ n (19)

Nonlinear optimization computational procedures should be used to find the estimators ( σ ^ n , μ ^ n , ξ ^ n ) that maximize the value of the respective likelihood functions above.

( σ ^ n , μ ^ n , ξ ^ n ) = a r g m a x σ n , μ n , ξ n l ( σ n , μ n , ξ n | r n ,1 , … , r n , g ) (20)

Estimates of the parameters of this distribution were performed using the evd library of R.

Value at Risk, with time horizon k and coverage level p, will be given by:

V a R p [ k ] = { C   k ξ ^ n   { μ ^ n − σ ^ n ξ ^ n [ 1 − [ − n ln ( p ) ] − ξ ^ n ] } ,     i f   ξ ^ n ≠ 0   C   k ξ ^ n   { μ ^ n − σ ^ n  ln [ − n ln ( p ) ] } ,     i f     ξ ^ n = 0 (21)

In the present article, VaR models with moving windows with variations in n were estimated. Considering the size of the series of 3845 daily log-returns, and given the graphical behavior of the volatility of the log-returns series, the sub-prime crisis began to take effect in the IBOVESPA at the end of November 2007. Thus, for the first estimation to include the period of crisis, we defined T=2100, associated with the date of 06/23/2010. Thus, 1745 estimates were made for VaR of 1 day with moving windows of size T=2100. For the VaR of 10 days, 1736 estimates were conducted with the same size for the windows. The models were estimated with n=5, 10 and 21. The choice of n = 5 is associated with the number of working days in a week, whereas n=21, the number of working days in a month. The value of n 10 was an intermediate choice between both.

Kupiec’s (1995) test, also known as the proportion of failures (POF) test, has the objective of verifying if the proportion of violations in relation to the total observations of a VaR model is adherent to the level of significance chosen for the calculation of this risk measure. Formally, we are interested in testing the hypothesis of unconditional coverage (adherence). A violation I _t (α) occurs when the Value at Risk ex-ante is lower/higher than the return ex-post in a given time t, considering a long/short position. Assuming that I _t (α)Bernoulli(α), for a long position, we will have:

I t ( α ) = { 1, i f r t < V a R t | t − k ( α ) 0, o t h e r w i s e (22)

Under the null hypothesis, the number of violations V in a given time interval [1,T] follows binomial distribution with parameters (T,α), such that:

V = ∑ t = 1 T I t ( α )   ~ B i n ( T , α ) (23)

The null and alternative hypothesis of the Kupiec’s test are { H 0 : α = α ^ = V T H 1 : α ≠ α ^

The test statistic is obtained by the likelihood ratio test between the null hypothesis and the alternative, described by:

L R K = − 2 l n ( α V ( 1 − α ) ( T − V ) ( V T ) V [ 1 − ( V T ) ] ( T − V ) ) (24)

By the properties of logarithms, we can rewrite it as:

L R K = − 2 ( l n [ α V ( 1 − α ) ( T − V ) ] − l n { ( V T ) V [ 1 − ( V T ) ] ( T − V ) } ) (25)

In which LRK follows asymptotic distribution χ2 with 1 degree of freedom.

Christoffersen (1998) assumed that under the alternative VaR inefficiency hypothesis, the process of violations can be modeled by a Markov chain, with transition matrix defined by:

H 1 : Π ^ = [ π 00 π 01 π 10 π 11 ] (26)

In which:

π i j = P [ I t ( α ) = j | I t − 1 ( α ) = i ] (27)

A Markov chain postulates the existence of a process AR(1) for the process of violations. The null hypothesis of Christoffersen’s test is defined by:

H 0 : Π α = [ 1 − α α 1 − α α ] (28)

Under the null hypothesis, whatever the state of the process in t-1, the probability of a violation occurring at time t is equal to α, the level of significance used for the VaR calculation. Therefore, the probability of occurrence or non-occurrence of a violation at time t is independent of the occurrence or not of a violation in time t-1, so that equations 29 to 32 are valid:

P [ I t ( α ) = 1 | I t − 1 ( α ) = 1 ] = P [ I t ( α ) = 1 ] = α (29)

P [ I t ( α ) = 1 | I t − 1 ( α ) = 0 ] = P [ I t ( α ) = 1 ] = α (30)

P [ I t ( α ) = 0 | I t − 1 ( α ) = 1 ] = P [ I t ( α ) = 0 ] = 1 − α (31)

P [ I t ( α ) = 0 | I t − 1 ( α ) = 0 ] = P [ I t ( α ) = 0 ] = 1 − α (32)

A test of the likelihood ratio denoted by LRCC, allows us to jointly test the hypotheses of adherence and independence associated with Christoffersen’s (1998) test:

L R C C = − 2 { l o g L [ Π α , I t ( α ) , … , I T ( α ) ] − l o g L [ Π ^ , I t ( α ) , … , I T ( α ) ] } d → T → ∞ χ 2 ( 2 ) (33)

The statistic LRCC presented in equation 33 converges to a chi-square asymptotic distribution with 2 degrees of freedom. In this equation, Π ^ is the transition matrix of the process of violations under the alternative hypothesis:

Π ^ = [ n 00 n 00 + n 01 n 01 n 00 + n 01 n 10 n 10 + n 11 n 11 n 10 + n 11 ] (34)

n _ij is the number of times we have I _t (α)=j and I _t-1 (α)=i.

The likelihood function associated with the alternative hypothesis Π ^ is:

L [ Π ^ , I t ( α ) , … , I T ( α ) ] = ( 1 − π ^ 01 ) n 00 π ^ 01 n 01 ( 1 − π ^ 11 ) n 10 π ^ 11 n 11 (35)

Similarly, the likelihood function associated with the null hypothesis Π is:

L [ Π ; I t ( α ) , … , I T ( α ) ] = ( 1 − α ) n 0 α n 1 (36)

In which, n ₀ =n ₀₀+n ₁₀ and n ₁=n ₀₁+n₁₁.

Independence tests based on Markov chains have the limitation of only evaluating the presence of first order dependence in the violations. To circumvent this limitation, it is possible to use an LB test for the VaR violations proposed by Berkowitz et al. (2008). These authors define H _it (α) as a variable indicating the ‘i-th” violation occurred in time t, centered around its expected value α In this way, we have H _it (α),=I _t (α)- α, such that:

H i t ( α ) , = { 1 − α ; i f r t < V a R t | t − k ( p ) − α ; o t h e r w i s e (37)

Berkowitz et al. (2008) start from the fact that the hypothesis of conditional coverage (adherence and independence) is satisfied when the process H _it (α), follows a martingale difference. For more details, see Morettin (2011). Thus, a range of tests for the martingale difference hypothesis can be applied in VaR models for a given level of significance α. The null hypothesis of the LB test is that the stochastic process {H _it (α)} follows a martingale difference.

According to the LB test, the statistic associated with the nullity of the first K empirical autocorrelations of the process of centered violations is described by equation 38:

L B ( K ) = T ( T + 2 ) ∑ k = 1 K ρ ^ k 2 T − k d → T → ∞ χ 2 ( K ) (38)

In which ρ ^ k is the empirical autocorrelation of order k of the process. Each k-th empirical autocorrelation r_k of the process {H _it (α)} was calculated as follows:

ρ ^ k = ∑ t = k + 1 T { [ H i t ( α )    − ∑ t = 1 T ( H i t ( α )   / T ) ] [ H t − k ( α )   − ∑ t = 1 T ( H i t ( α ) / T ) ] } ∑ t = 1 T [ H i t ( α )   − ∑ t = 1 T ( H i t ( α ) / T ) ] 2 (39)

In the present study, the LB test was performed for all the models to jointly test different orders of autocorrelations of the violations, having performed ten tests per model, so that K (eq. 38) was set from 1 to 10.