Effects of Outliers and Parameter Uncertainties in Portfolio Selection

Transcription

1 Effects of Outliers and Parameter Uncertainties in Portfolio Selection Luiz Hotta 1 Carlos Trucíos 2 Esther Ruiz 3 1 Department of Statistics, University of Campinas. 2 EESP-FGV (postdoctoral). 3 Department of Statistics, University Carlos III de Madrid. March, 217 Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 1 / 36 o

2 Outline 1 Introduction 2 Model 3 Point estimation 4 Forecast densities 5 Portfolio selection 6 Empirical Application 7 Concluding Remarks Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 2 / 36 o

3 Introduction We consider the effects of outliers and parameter uncertainties in portfolio selection and propose some alternative robust methods. Basic set up: The portfolio selection is based on the one-step-ahead prediction of the volatility matrix given by the cdcc (corrected DCC) model. In the study we consider: The effect of outliers in the volatility prediction and portfolio selection. The effect of outliers in the estimation of the portfolio variance. The uncertainties in the VAR estimate. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 3 / 36 o

4 Introduction Specifically, we Analyze the effect of outlier on the volatility and correlation estimation when a non-robust estimator, as QML, is used and when a robust estimator is used. Analyze the effect of outliers on the forecast densities returns, volatilities and correlations constructed using the Fresoli and Ruiz algorithm. Assess the effect of outliers on VaR estimation and on the portfolio selection. Propose a robust bootstrap algorithm to obtain forecast densities of returns, volatilities and correlations. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 4 / 36 o

5 Introduction - series used in the application AUD CAD EUR GBP Figure : Daily AUD/USD, CAD/USD, EUR/USD and GBP/USD exchange rates returns observed from January 5, 2 to April 8, 216. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 5 / 36 o

6 Standard forecast density estimator Pascual, Romo and Ruiz (CSDA, 26) (PRR) proposed a method to construct bootstrap forecast densities for returns and volatilities. The method does not require to assume any particular distribution for the innovations and can also handle the effect of uncertainty parameters. Fresoli and Ruiz (IJF, 215) extend the PRR procedure in a multivariate framework to obtain forecast densities for returns, volatilities and also for conditional correlations. Both procedures are based on ML estimators and standard filters. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 6 / 36 o

7 Model The contaminated cdcc Model The DCC model was proposed by Engle (JBES, 22) and corrected by Aielli (JBES, 213). Following Boudt et al. (IJF, 213) the cdcc model contaminated by additive outliers is defined as Y t = Z t + A ti t(t B), (1a) Z t = H 1/2 t ɛ t (1b) H t = D tr td t, (1c) R t = diag(q t) 1/2 Q tdiag(q t) 1/2, Q t = (1 a b)s+a diag(q t 1 ) 1/2 v t 1 v t 1 diag(q t 1) 1/2 + bq t 1, (1d) (1e) where: Y t = (y 1,t,..., y p,t ) is a p-dimensional vector of returns observed at time t A t a p-dimensional vector of contaminations, I t ( ) is the indicator function, B is the set of contaminated observations, Z t is the uncontaminated cdcc process with H t being its conditional covariance matrix that depends on past uncontaminated vector returns, ɛ t is a p-dimensional independent process with identity covariance matrix D t is a diagonal matrix containing the volatilities σ i,t, v t = Dt 1 Z t and S is the unconditional correlation matrix of diag(q t ) 1/2 v t Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 7 / 36 o

8 Second stage (Gaussian) estimator. Not discussed today Aielli Estimator Aielli (JBES, 213) proposes a three-step estimator based on the maximization of the Gaussian log-likelihood function. Suppose that there are no outliers, the Gaussian log-likelihood, conditional to Y 1, D 1 and R 1, can be written as T l (θ, φ, S) = 1 2 t=2 [ ] log (2π) + 2log (det (D t)) + log (det (R t)) + v t Rt 1 v t, (2) where v t = Y tdt 1 are returns which are assumed to be conditionally normally distributed with zero mean and covariance matrix R t. The Aielli estimator use the standard filters for the volatility and for the matrix Q t. Thus, σt 2 = ω + αrt 2 + βσt 1 2 and (3) Q t = (1 a b) S + a diag(q t 1 ) 1/2 v t 1 v t 1 diag(q t 1) 1/2 + b Q t 1. (4) where v t = Dt 1 Y t and Ŝ = T 1 T 1/2 t=1 ˆQ t ˆv t ˆv 1/2 t ˆQ t Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 8 / 36 o

9 Second stage robust estimator Boudt et al. Estimator Boudt et al. (IJF, 213) proposes a robust M-estimator for the cdcc model that can be sum up as: Estimate the GARCH parameters as ( ( ˆθ = (ˆα i, ˆβ i ) = argmax 1 T ρ log T t=1 The M-estimator for conditional correlation models is ( ))) y 2 i,t σi,t 2 ( ) ˆφ = argmax 1 T [log (det (R t)) + σ p,4 ρ (d t)], (6) T t=1 p where σ p,4 is a correction given by σ p,4 = ] and the loss function ρ( ) is given by E [ρ tp,ν (u)u ρ(x) = x + σ p,4 ρ tp,4 (exp(x)) with ρ tp,4 (u) = (p + 4)log ( 1 + u 2 ). (5) Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217Department 9 / 36 o

10 Second stage robust estimator Boudt et al. Estimator (continuation) Additionally, they propose use a bounded innovation propagation (BIP) filter to estimate volatilities and correlations. Thus, instead use σt 2 = ω + αr t 2 + βσ2 t 1 and (7) Q t = (1 a b) S + a diag(q t 1 ) 1/2 v t 1 v t 1 diag(q t 1) 1/2 + b Q t 1. (8) They propose to use ( ) y 2 σi,t 2 = ˆγ i (1 α i,1 β i ) + α i,1 yi,t 1 2 c i,t 1 δ,p r c σi,t β i σi,t 1 2 and (9) Q t = (1 a b)s + a c δ,p r c(d t 1 )diag(q t 1 ) 1/2 v t 1 v t 1 diag(q t 1) 1/2 + bq t 1, (1) where r c(x) = { 1, if x c c/x, if x > c, (11) Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 1 / 36 o

11 Second stage robust estimator Boudt et al. Estimator (continuation) In this work, following Carnero et al. (Econ. Lett, 212) and Trucíos et al. (UC3M-WP, 215), we use r c(x) = { 1, if x c E(x)/x, if x > c. Finally, Boudt et. al (IJF, 213) estimate the matrix S as (12) Ŝ = diag(rc 1/2 11,..., RC 1/2 pp ) RC diag(rc 1/2 11,..., RC 1/2 pp ). (13) where RC = c.95,p T L t t=1 T v tv t L t, (14) t=1 with L t = I (v t 1 SCt v t < χ 2 p(.95)) and SC t = 2 sin ( ) π 6 Spt and Spt is the Spearman correlation matrix. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 11 / 36 o

12 Monte Carlo experiments Influence of outliers on the volatility estimates. DGP: cdcc process using Gaussian error distribution with parameters a =.1, b =.8, ω 1 =.5, ω 2 =.1, α 1 =.1, α 2 =.15, β 1 =.85, β 2 =.75 and S = ( ) Sample size: 1 and number of replications: 5. Isolated and consecutive outliers of size ω = 5 and 1 standard deviation of the uncontaminated process for one and both series are considered. Table : Pattern of contamination. Case Series 1: outlier position Series 2: outlier position (1) 5 - (2) (3) 5 and 51 - (4) 998 and (5) 5 5 (6) (7) 5 and 51 5 and 51 (8) 998 and and 999 Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 12 / 36 o

13 Monte Carlo experiment 1. No outliers Series 1: 5 Series 1: Series 1: 999 Series 1: Series 1: 5, Series 2: Series 1: 5-51, Series 2: 5-51 Series 1: 999, Series 2: 999 Series 1: , Series 2: BVT BVT2 QML Figure : Estimated conditional correlation at time t = T for uncontaminated and contaminated series using QML (blue squares), BVT1 (red circles) and BVT2 (green triangles) estimators against true conditional correlations at time t = T Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 13 / 36 o

14 Multivariate forecast densities Standard procedure: Fresoli and Ruiz (IJF, 215), extending the univariate procedure of Pascual et al. (CSDA, 26), propose a bootstrap procedure to obtain forecast densities for returns, volatilities and correlations in cdcc model incorporating the parameter uncertainty without assuming any particular assumption about the error distribution. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 14 / 36 o

15 Fresoli and Ruiz algorithm Step 1: Estimate the model parameters by the procedure of Aielli (JBES, 213) and denote by ˆψ = (ˆω 1, ˆα 1, ˆβ 1,..., ˆω p, ˆα p, ˆβ p, â, ˆb) the estimated vector parameters. Obtain ˆɛ t = Ĥ 1/2 t Y t and denote its corresponding empirical distribution function by ˆFˆɛ. Step 2: Using ˆψ and ɛ ˆFˆɛ, generate multivariate bootstrap series Y t Step 3: Fit the cdcc model on Y t and obtain ˆψ. Step 4: Compute h-steps-ahead bootstrap forecast for returns, volatilities and correlations by recursion using the bootstrap estimates parameters ˆψ and the original multivariate series Y t, as follows: Step 5: Repeat steps 2 to 4, B times, and compute (ŶT 1 +h T,..., Ŷ B T +h T ), ( ˆD T 1 +h T,..., ˆD T B +h T ) and (ˆR T 1 +h T,..., ˆR T B +h T ) where h = 1,..., H. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 15 / 36 o

16 Monte Carlo experiment 1 Returns series 1 1 Returns series 2 Coverage (%) Volatilities series 1 1 Volatilities series 2 Coverage (%) Coverage (%) Correlations Above Below Coverage Figure : Estimated coverage of the 95% Bootstrap forecast intervals for returns, volatilities and correlations for uncontaminated series. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 16 / 36 o

17 Monte Carlo experiment Returns Volatilities 25 Figure : Estimated coverage and coverage above and below of the 95% Bootstrap forecast intervals for returns and volatilities for contaminated series. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 17 / 36 o

18 Monte Carlo experiment % One contaminated series Both contaminated series Figure : Estimated coverage and coverage above and below of the 95% Bootstrap forecast intervals correlation for contaminated series. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 18 / 36 o

19 Monte Carlo experiment The Fresoli and Ruiz algorithm has good performance in uncontaminated process. In presence of outliers the performance of the algorithm change. When we have outliers close to the end of the sample period the algorithm has a very poor performance. What happens with the Fresoli and Ruiz algorithm? QML estimator is badly affected by outliers, consequently, volatilities and correlations are also affected, Volatility and correlation equations in the bootstrap algorithm do not avoid the propagation of the influence of outliers. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 19 / 36 o

20 Robust bootstrap algorithm. Not presented the steps We propose some modifications to robustify the Fresoli and Ruiz (IJF, 215) algorithm, obtaining forecast densities for returns, volatilities and correlations that are not badly affected by outliers. These modifications can be summarized as: 1 use a robust estimator and 2 use alternative filters on the volatility and correlation equations. Robust modifications Use the robust estimator and filters of Boudt et al. (IJF, 213) instead the estimator of Aielli (JBES, 213) Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 2 / 36 o

21 Robust bootstrap algorithm. Not presented Another important modification is given in the volatility equations used in steps 2, 3 and 4. Specifically, change σi,t 2 = ˆω i + ˆα i yi,t ˆβ i σi,t 1 2, by σ 2 i,t = ˆω i + ˆα i y 2 ( ) y 2 i,t 1 cγrc i,t 1 σi,t ˆβ i σ 2 i,t 1, (15) where c γ is a correction factor and r c( ) is the filter proposed by Trucíos et al. (215) and defined as { 1, if x c r c(x) = ε 2 i,t /x, if x > c, (16) with c = χ 2 1 (δ) and where ε i,t is the ith element of ε t = 1 R 2 t ɛ t, with 1 R 2 t bootstrap version of R 1 2 t and ɛ t being a random draw of ˆFˆɛ. being a Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 21 / 36 o

22 Robust bootstrap algorithm. Not presented Finally, we also use an alternative filter for the matrix Q. If d t 1 χ 2 p(δ) then ˆQ t = (1 â ˆb [ )Ŝ + â diag( ˆQ t 1 ) 1 2 ˆD t 1 Y t 1 Y t 1 ˆD t 1 diag( ˆQ ] t 1 ) ˆb Qt 1. else ˆQ t = (1 â ˆb [ )Ŝ + â diag( ˆQ t 1 ) 2 1 ε t 1 ε t 1 diag( ˆQ ] t 1 ) ˆb Qt 1, (17) for t = 1,..., T. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 22 / 36 o

23 Monte Carlo Experiments Returns Volatilities 25 Figure : Estimated coverage and coverage above and below of the 95% Bootstrap forecast intervals for returns and volatilities for contaminated series Luizusing Hotta, Carlos our Trucíos, robust Esther procedure Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 23 / 36 o

24 Monte Carlo experiments % One contaminated series Both contaminated series Figure : Estimated coverage and coverage above and below of the 95% Bootstrap forecast intervals correlation for contaminated series using our robust Luizprocedure Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 24 / 36 o

25 Portfolio selection We consider that one-step-ahead prediction of the volatility matrix is used to choose the minimum variance portfolio (MVP). The portfolios are selected using the standard and the robust procedures for point and density estimators. We compare the variance of the selected portfolios with the variance of the minimum variance portfolio when the volatility is known. Suppose we selected a portfolio with weights ˆω T +1, chosen using the robust predictor. We analyze how far are the estimated variances of the selected portfolio (ˆω T +1ĤT +1 T ˆω T +1 ) with respect to the true variances of the selected portfolio (ˆω T +1 H T +1 T ˆω T +1 ) when ĤT +1 T is estimated by the standard and the robust methods. We obtain forecast intervals of the variance of the selected portfolio using both bootstrap procedures presented previously. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 25 / 36 o

26 Portfolio selection: portfolio variance No outliers Series 1: 5 Series 1: 5-51 QML MSE = 1e-4 QML MSE = 1e-4 QML MSE = 2e-4 BVT MSE =.215 BVT MSE =.25 BVT MSE = Series 1: 999 Series 1: Series 1: 5, Series 2: 5 QML MSE =.1253 QML MSE =.2426 QML MSE = 2e-4 BVT MSE =.213 BVT MSE =.214 BVT MSE = Series 1: 5-51, Series 2: 5-51 Series 1: 999, Series 2: 999 Series 1: , Series 2: QML MSE = 2e-4 QML MSE =.1337 QML MSE =.1939 BVT MSE =.163 BVT MSE =.227 BVT MSE = BVT QML Figure : Variances of MVP against variances of selected portfolio at time t = T + 1 using QML (green triangles) and BVT (red circles) estimators. 5 replications. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 26 / 36 o

27 Portfolio selection: estimated variance No outliers Series 1: 5 Series 1: QML MSE =.47 BVT MSE =.36 QML MSE =.58 BVT MSE =.341 QML MSE =.64 BVT MSE = Series 1: 999 Series 1: Series 1: 5, Series 2: 5 15 QML MSE =.414 BVT MSE =.427 QML MSE =.569 BVT MSE =.457 QML MSE =.74 BVT MSE = Series 1: 5-51, Series 2: 5-51 Series 1: 999, Series 2: 999 Series 1: , Series 2: QML MSE =.82 BVT MSE =.281 QML MSE = BVT MSE =.558 QML MSE = BVT MSE = BVT QML Figure : Estimated variances of selected portfolio (ˆω T +1ĤT +1 T ˆω T +1 ) against the true variance (ˆω T +1 H T +1 T ˆω T +1 ) at time t = T + 1 using QML (green triangles) and BVT (red circles) estimators. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 27 / 36 o

28 Portfolio selection: CI for the estimated portfolio variance Table : Average coverage of 95% and 99% bootstrap forecast intervals for the variance of the selected portfolio using the Fresoli and Ruiz algorithm and our robust propose algorithm. Simulation based on 5 replications for time series of size 1 with and without outliers. Fresoli and Ruiz Algorithm Robust Algorithm Coverage 95% Coverage 99% Coverage 95% Coverage 99% No outlier One series Both series One series Both series One series Both series One series Both series Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 28 / 36 o

29 Porfolio analysis: CI for the VaR. Not discussed the steps. To construct forecast intervals for VaR h-step-ahead we need to obtain the empirical density distribution of the VaR. This is obtained for each h = 1,..., H. in the following four step algorithm. Step 1: For each b compute the bootstrap covariance matrix H b T +h T. Step 2: For each HT b obtain M bootstrap vector returns r m for m = 1,..., M, +h T T +h T where for each m we have rt m +h T = H b T +h T ɛ m T +h with ɛ T +h being a random draw with replacement of Fˆɛt. Step 3: For a given α obtain the VaRT +h as a α1% quantile of the empirical distribution of the bootstrap portfolio returns rp,t 1,..., r M where the jth +h T p,t +h T bootstrap portfolio return r j p,t +h T is obtained, as usual, as K k=1 ω kr j k,t +h T with ω = (ω 1,..., ω K ) being a vector of weights and r j being the kth element of the k,t +h T bootstrap vector return r j T +h T. Step 4: Repeat steps 1-3 for each b = 1,...B and obtain VaR 1 T +h T,..., VaR B T +h T. Finally, using the quantile bootstrap intervals we obtain the forecast intervals for VaR T +h T. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 29 / 36 o

30 VaR Table : Proportion of portfolio returns less than the point estimation of 1% VaR, coverage, and fail coverage of the 95% and 99% bootstrap confidence interval procedures for the real 1% VaR. Proportion Interval 95% Interval 99% of failures Below Coverage Above Below Coverage Above Fresoli and Ruiz Algorithm No outlier One series Both series One series Both series One series Both series One series Both series Robust procedure No outlier One series Both series One series Both series One series Both series One series Both series Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 3 / 36 o

31 Empirical Application Four daily currencies: Australian dollar (AUD), Canadian dollar (CAD), Euro (EUR) and Pound (GBP), all against US dollar (USD) from January 4, 2 to April 8, 216. Currencies correspond to buying rates at noon time in New York for cable transfers payable and are available at Returns are computed by r i,t = 1 ( P i,t /P i,t 1 1 ), where P i,t denotes the closing price of the i th currency at day t for i = 1,..., AUD CAD EUR GBP Figure : Daily AUD/USD, CAD/USD, EUR/USD and GBP/USD exchange rates returns observed from January 5, 2 to April 8, 216. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 31 / 36 o

32 Empirical Application Table : Coverage, fail coverage below/above h-step-ahead for the forecast bootstrap intervals 95% and 99% for the portfolio returns with equal weights using the classical bootstrap procedure of Fresoli and Ruiz and our proposed robust procedure. Fresoli and Ruiz algorithm Robust algorithm Steps-ahead below coverage above below coverage above 95% h= h= h= h= h= % h= h= h= h= h= Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 32 / 36 o

33 Empirical Application Table : Proportion of portfolio returns lower than the estimate one-period 1% VaR and 5% VaR and proportion of portfolio returns lower than the 95% lower/upper bounds of the forecast intervals for the 1% VaR and 5% VaR. 1% VaR 5% VaR Lower Limit Point estimate Upper Limit Lower Limit Point estimate Upper Limit FR Robust Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 33 / 36 o

34 Concluding Remarks We proposed a robust algorithm to obtain forecast intervals for returns, volatilities and correlations in the cdcc model even in the presence of outliers. We use the bootstrap algorithm to approach two common problems in financial econometrics: portfolio selection and VaR estimation. The robust algorithm seems to be useful to measure the uncertainty of the variance the selected portfolio and also to obtain correctly forecast intervals for the variance of the minimum variance portfolio and for a fixed portfolio. We also proposed additional bootstrap steps that allow the construction of forecast intervals for the VaR of a fixed portfolio. This bootstrap procedure extends and generalize the procedures of Christoffersen and Goncalves (J. Risk, 25) and Nieto and Ruiz (UC3M-WP, 21) in a multivariate framework. An empirical application using daily exchange rates and both bootstrap procedures presented was implemented and we found a similar performance with a slight better performance in favor of our robust bootstrap procedure proposed in this paper. The results are probably due to the fact that no outliers are present close to the end of the sample period where, as showed in the Monte Carlo simulations, our proposed algorithm is superior to the classical one. Finally, although the bootstrap procedure was proposed for the corrected dynamic conditional correlation model, it can be adapted to other multivariate GARCH models. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 34 / 36 o

35 References Aielli, G. P. (213). Dynamic conditional correlation: On properties and estimation. Journal of Business & Economic Statistics, 31(3): Boudt, K., Danielsson, J., and Laurent, S. (213). Robust forecasting of dynamic conditional correlation GARCH models. International Journal of Forecasting, 29(2): Fresoli, D. E. and Ruiz, E. (215). The uncertainty of conditional returns, volatilities and correlations in DCC models. Computational Statistics & Data Analysis. doi:1.116/j.csda Pascual, L., Romo, J., and Ruiz, E. (26). Bootstrap prediction for returns and volatilities in GARCH models. Computational Statistics & Data Analysis, 5(9): Trucíos, C., Hotta, L. K., and Ruiz, E. (215). Robust bootstrap forecast densities for GARCH models: returns, volatilities and Value-at-Risk. UC3M Working Papers. Statistics and Econometrics, 15(23). Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 35 / 36 o

36 Financial Support The first two authors acknowledge financial support from São Paulo Research Foundation (FAPESP) grants 212/9596- (Trúcios), 216/ (Trúcios), 213/56-1 (Hotta) and 213/2293- (Hotta) and Laboratory EPIFISMA. The third author is grateful for financial support from Project ECO by the Spanish Government. Luiz Hotta, Carlos Trucíos, Esther Ruiz ( Department of Statistics, EESP-217 University of Campinas., EESP-FGV (postdoctoral)., March, 217 Department 36 / 36 o