### Citations

7427 | Convex optimization
- Boyd, Vandenberghe
- 2004
(Show Context)
Citation Context ...tions in (8), and the restriction on the number of VaR violations in (9) are both discontinuous and non-convex due to the presence of the max-operator and the indicator function, respectively. Note that the 3DeMiguel et al. (2009a) recently proposed a unifying approach based on constraints of the portfolio norms that nests several commonly applied restrictions as special cases, including the no-shortselling and diversification constraints. 12 non-convexity imposes important difficulties in terms of computational effort and a potential problem of local minima; see Nocedal and Wright (1999) and Boyd and Vandenbergue (2004). For this reason we next formulate a convex and continuous approximation to the original problem for which a highly accurate solution can be obtained with low computational effort. 2.3. A convex and continuous reformulation Reformulating the objective function In order to obtain a continuous and smooth objective function, (8) can be reformulated by introducing artificial variables to get rid of the max-operator. Specifically, the objective function in (8) can be equivalently expressed as the following linear optimization problem: minimize w,v1,v2 v1 + v2 (12) subject to: v1 ≥ −(w′µt+1 + (w′Ht... |

359 |
The stationary bootstrap
- Politis, Romano
- 1994
(Show Context)
Citation Context ...ction costs equal to zero. Note that, when comparing two alternative portfolio strategies, the one with a higher breakeven cost is to be preferred. To test the hypothesis that the capital requirement levels, the number of VaR exceedances, and the Sharpe ratios obtained with the MCR portfolios and with the benchmark portfolios are equal, we follow DeMiguel et al. (2009a) and use the stationary bootstrap of 12Note that, in the case of an equally weighted (or 1/N) portfolio composition, we have wj,t = wj,t+1 = 1/N , but wj,t+ may be different due to changes in asset prices between t and t+ 1. 25 Politis and Romano (1994) with B=1,000 bootstrap resamples and expected block length b=5.13 The resulting bootstrap p-values are obtained using the methodology suggested in Ledoit and Wolf (2008, Remark 3.2). 3.7. Results Baseline empirical analysis We first consider a baseline empirical analysis based on a version of the MCR portfolios in (14) using as objective function the capital requirement formula in (1). Table 2 reports the daily capital requirements, the number of VaR violations, and the performance of each portfolio strategy in terms of average gross returns, standard deviation of returns, Sharpe ratio, turno... |

285 | Multivariate GARCH models: A survey
- Bauwens, Laurent, et al.
- 2006
(Show Context)
Citation Context ...ns. For each of these three inputs, both nonparametric and parametric specifications may be adopted. Given that in this paper we focus on highdimensional portfolios consisting of a large number of assets N , parametric specifications may be more appropriate. In this case, the expected returns may be obtained from linear (vector) autoregressive [(V)AR] models as well as nonlinear models (Carriero et al., 2009; Pesaran et al., 2009; DeMiguel et al., 2010). Similarly, alternative specifications for the conditional covariance matrix Ht+1 can be considered, including multivariate GARCH models (see Bauwens et al. (2006) and Silvennoinen and Terasvirta (2009) for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assuming a particular distribution for εp,t+1, such as the normal or the Student’s t distribution. This distribution may also be considered in order to obtain the quantile q in (6). For instance, when assuming normali... |

258 |
A simpli model for portfolio analysis
- Sharpe
- 1963
(Show Context)
Citation Context ...ead of all distinct pairs, in this work, we implement the CL estimator using all contiguous pairs of data, which implies O(K) calculations; the extensive Monte Carlo results reported in Engle et al. (2008) show that the choice of contiguous pairs can be successfully applied in problems involving hundreds of assets. 18 used vis-a-vis traditional estimators such as the sample covariance matrix. In this paper we consider the shrinkage estimator proposed by Ledoit and Wolf (2003), which is defined as an optimally weighted average of the sample covariance matrix and the covariance matrix based on Sharpe (1963) single-index model. The intuition behind this shrinkage estimator is to come up with an optimal convex combination between an unbiased covariance matrix estimator that may be subject to substantial estimation error (i.e. the sample covariance matrix) and another estimator that possibly is biased but has considerably less estimation error (i.e. the covariance matrix from the single factor model). In this model the returns of asset i are described by: rit = ai + birmt + υi,t, (20) where rmt is the market portfolio return. 7 The residuals υi are assumed to be uncorrelated with market returns and... |

102 | Bayesian dynamic factor models and portfolio selection
- Aguilar, West
- 2000
(Show Context)
Citation Context ...hdimensional portfolios consisting of a large number of assets N , parametric specifications may be more appropriate. In this case, the expected returns may be obtained from linear (vector) autoregressive [(V)AR] models as well as nonlinear models (Carriero et al., 2009; Pesaran et al., 2009; DeMiguel et al., 2010). Similarly, alternative specifications for the conditional covariance matrix Ht+1 can be considered, including multivariate GARCH models (see Bauwens et al. (2006) and Silvennoinen and Terasvirta (2009) for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assuming a particular distribution for εp,t+1, such as the normal or the Student’s t distribution. This distribution may also be considered in order to obtain the quantile q in (6). For instance, when assuming normality of εp,t+1, q = −2.33 for α = 1%. See Santos et al. (2009) for an empirical comparison among alternative procedures for computing the invers... |

91 | Multivariate GARCH models. - Silvennoinen, Terasvirta - 2009 |

84 | Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading’,
- Barndorff-Nielsen, Hansen, et al.
- 2011
(Show Context)
Citation Context ...turns may be obtained from linear (vector) autoregressive [(V)AR] models as well as nonlinear models (Carriero et al., 2009; Pesaran et al., 2009; DeMiguel et al., 2010). Similarly, alternative specifications for the conditional covariance matrix Ht+1 can be considered, including multivariate GARCH models (see Bauwens et al. (2006) and Silvennoinen and Terasvirta (2009) for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assuming a particular distribution for εp,t+1, such as the normal or the Student’s t distribution. This distribution may also be considered in order to obtain the quantile q in (6). For instance, when assuming normality of εp,t+1, q = −2.33 for α = 1%. See Santos et al. (2009) for an empirical comparison among alternative procedures for computing the inverse of the cumulative distribution function of the portfolio returns. Finally, it is worth noting that the MCR portfolio construction methodology developed in... |

84 | How accurate are value-at-risk models at commercial banks? - Berkowitz, O'Brien - 2002 |

55 |
Economic implications of using a mean-var model for portfolio selection: A comparison with mean-variance analysis.
- Gordon, Alexander
- 2002
(Show Context)
Citation Context ...uction problem in which the optimal portfolio composition is found by minimizing the level of CR subject also to a given (i.e. user specified) target performance and to direct constraints on the portfolio weights. We apply the proposed methodology to two different asset portfolios: (i) a mixed portfolio composed of 30 futures on a variety of assets including equities, bonds, commodities and currencies, and (ii) an equity portfolio comprising 48 US industry indices. The minimum capital requirement (MCR) portfolio is compared to various benchmark portfolios, including the minimum-VaR portfolio (Alexander and Baptista, 2002), the minimum-sVaR portfolio, 5 and the equally weighted (1/N) portfolio. In our empirical analysis we pay particular attention to the consequences of the introduction of the sVaR-based CR. For this purpose, in addition to ‘normal’ market conditions we consider several alternative, realistic scenarios in which expected returns, volatilities and cross-correlations are modified to reflect a stressed environment. Furthermore, we consider different models for obtaining forecasts of expected returns, volatilities and correlations, which are crucial inputs for the asset allocation decisions. We also... |

23 |
Forecasting exchange rates with a large Bayesian VAR.”
- Carriero, Kapetanios, et al.
- 2009
(Show Context)
Citation Context ...definition in (6) requires estimates of the conditional expected returns µt+1, the conditional covariance matrix Ht+1 and the quantile q or, more generally, the distribution of the standardized unexpected returns. For each of these three inputs, both nonparametric and parametric specifications may be adopted. Given that in this paper we focus on highdimensional portfolios consisting of a large number of assets N , parametric specifications may be more appropriate. In this case, the expected returns may be obtained from linear (vector) autoregressive [(V)AR] models as well as nonlinear models (Carriero et al., 2009; Pesaran et al., 2009; DeMiguel et al., 2010). Similarly, alternative specifications for the conditional covariance matrix Ht+1 can be considered, including multivariate GARCH models (see Bauwens et al. (2006) and Silvennoinen and Terasvirta (2009) for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assumin... |

23 | Do banks overstate their Value-at-Risk? - Perignon, Deng, et al. - 2008 |

21 |
Multivariate stochastic volatility, in:
- Chib, Omori, et al.
- 2009
(Show Context)
Citation Context ...consisting of a large number of assets N , parametric specifications may be more appropriate. In this case, the expected returns may be obtained from linear (vector) autoregressive [(V)AR] models as well as nonlinear models (Carriero et al., 2009; Pesaran et al., 2009; DeMiguel et al., 2010). Similarly, alternative specifications for the conditional covariance matrix Ht+1 can be considered, including multivariate GARCH models (see Bauwens et al. (2006) and Silvennoinen and Terasvirta (2009) for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assuming a particular distribution for εp,t+1, such as the normal or the Student’s t distribution. This distribution may also be considered in order to obtain the quantile q in (6). For instance, when assuming normality of εp,t+1, q = −2.33 for α = 1%. See Santos et al. (2009) for an empirical comparison among alternative procedures for computing the inverse of the cumulative ... |

21 | Forecasting economic and financial variables with global VARs.”
- Pesaran, Schuermann, et al.
- 2009
(Show Context)
Citation Context ...res estimates of the conditional expected returns µt+1, the conditional covariance matrix Ht+1 and the quantile q or, more generally, the distribution of the standardized unexpected returns. For each of these three inputs, both nonparametric and parametric specifications may be adopted. Given that in this paper we focus on highdimensional portfolios consisting of a large number of assets N , parametric specifications may be more appropriate. In this case, the expected returns may be obtained from linear (vector) autoregressive [(V)AR] models as well as nonlinear models (Carriero et al., 2009; Pesaran et al., 2009; DeMiguel et al., 2010). Similarly, alternative specifications for the conditional covariance matrix Ht+1 can be considered, including multivariate GARCH models (see Bauwens et al. (2006) and Silvennoinen and Terasvirta (2009) for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assuming a particular distrib... |

20 | Correlation stress testing for value-at-risk: an unconstrained convex optimization approach.
- Qi, Sun
- 2010
(Show Context)
Citation Context ...n we apply a haircut to the conditional correlations, 10 In unreported results, we considered a fourth stress scenario in which only correlations are stressed. The results are similar to those obtained under the stress scenarios reported here and are available upon request. 21 i.e. Rt = Rt + ∆R, where ∆R is the haircut level for the correlations. In this case, sVaRt = w ′µt + (w ′DtRtDtw) 1/2q. An important issue that arises when “stressing” the correlation matrix is that the resulting matrix Rt may not be positive definite. To circumvent this problem, we employ the approach proposed by Qi and Sun (2010), which is designed to obtain the nearest positive definite correlation matrix in the context of the sVaR. Definition of the haircut parameters In order to obtain the stressed conditional moments for each of the stress scenarios defined above, it is necessary to define specific values for the haircut levels to be applied to expected returns, volatilities, and correlations. Our choices for the haircut levels are similar to those considered in the stress testing exercise conducted by the European regulatory authorities, see Committee of European Banking Supervisors (2010). Specifically, we set t... |

9 |
Comparing Univariate and Multivariate Models to Forecast Portfolio Value–at–Risk.
- Santos, Nogales, et al.
- 2013
(Show Context)
Citation Context ... for comprehensive reviews), stochastic volatility models (Harvey et al., 1994; Aguilar and West, 2000; Chib et al., 2009), as well as realized covariance matrices based on high-frequency intraday data (De Pooter et al., 2008; Barndorff-Nielsen et al., 2008). Finally, the models for the conditional mean and variance are usually estimated by assuming a particular distribution for εp,t+1, such as the normal or the Student’s t distribution. This distribution may also be considered in order to obtain the quantile q in (6). For instance, when assuming normality of εp,t+1, q = −2.33 for α = 1%. See Santos et al. (2009) for an empirical comparison among alternative procedures for computing the inverse of the cumulative distribution function of the portfolio returns. Finally, it is worth noting that the MCR portfolio construction methodology developed in the remainder of this section is independent of the method used to obtain the VaR and sVaR measures. However, and perhaps obviously, we may expect that more accurate modeling of the expected returns and conditional covariance matrix leads to improved portfolio characteristics. 2.2. MCR Portfolios The problem of constructing an MCR portfolio consists of findin... |

8 |
Mean-variance portfolio allocation with a value at risk constraint.
- Sentana
- 2003
(Show Context)
Citation Context ...livers lower levels of capital charges, as proposed recently by McAleer et al. (2010), for instance. Using the terminology of Christoffersen (2009), this approach can be considered a risk measurement or passive risk management approach, since it is applied to a given (i.e. predetermined) portfolio composition. Alternatively, in this paper we propose to perform active risk management by deciding on the portfolio allocations themselves to attain lower levels of CR. Second, portfolios with low levels of CR may be obtained by imposing constraints on the amount of CR or on the portfolio VaR, as in Sentana (2003), Cuoco and Liu (2006) and 6 Alexander et al. (2007). In our approach, the level of CR plays a much more central role as it is taken to be the objective function that should be minimized. The remainder of the paper is organized as follows. In Section 2 we describe the procedure to obtain the optimal portfolios with minimum CR subject to restrictions on the number of VaR violations. In Section 3 we present the empirical applications. We conclude in Section 4. 2. Portfolios with Minimum Capital Requirements The main ingredient required to obtain optimal portfolios with minimum capital requiremen... |

7 | Diversification and value-at-risk. - Perignon, Smith - 2010 |

5 |
Market Risk Analysis, Volume IV: Value at Risk Models.
- Alexander
- 2008
(Show Context)
Citation Context ...the conditional covariance matrix. On the other hand, the VaR estimate, the level of capital requirements, and the number of VaR violations are affected by these methods. 3.4. Stress scenarios The sVaR measures the risk of extreme losses if the relevant market factors were experiencing a period of stress. As mentioned before the amendments to the Basel accord do not provide specific implementation details concerning the stress scenarios, except that they should typically involve lower expected returns, higher volatilities and more extreme correlations in agreement with empirical evidence, see Alexander (2009, Chapter IV.7). We consider three alternative stress scenarios that reflect the impact of relevant changes or ‘haircuts’ to the conditional moments of the asset returns on the estimation of the sVaR.10 Stress scenario 1 (Expected returns) In this scenario we apply a haircut to the conditional expected return, i.e. µt = µt − ∆µ, where ∆µ > 0 is the haircut level for the expected return. In this case, we will have sVaRt = w ′µt + (w ′DtRtDtw) 1/2q. Stress scenario 2 (Expected returns and volatilities) In this case we lower the conditional expected returns as in scenario 1 but also apply a hai... |

3 |
Multi-step estimation of multivariate GARCH models, in:
- Sheppard
- 2003
(Show Context)
Citation Context ...ameters satisfying α + β < 1.6 Our third and final approach to model the covariance matrix is the shrinkage estimator of Ledoit and Wolf (2003) (LW). Shrinkage estimators are becoming very popular in the portfolio construction literature due to their ability to reduce the estimation error in large covariance matrices. For instance, Ledoit and Wolf (2003) and Ledoit and Wolf (2004) report improved results in terms of portfolio performance when the shrinkage estimator is 6The parameters of the DCC model are usually estimated using the two-step procedure proposed by Engle and Sheppard (2001) and Sheppard (2003). However, Engle et al. (2008) point out that when the dimension of the portfolio increases, the two-step estimator can be severely downward biased due to an undiagnosed incidental parameter problem. Therefore, in this paper we estimate the parameters of the DCC model by the composite likelihood (CL) estimator proposed by Engle et al. (2008). Essentially, this method is based on the decomposition of the original estimation problem into many small subproblems in such a way the composite likelihood is constructed by summing up the quasi-likelihood of subset of assets. By summing up over many sub... |

1 |
Mean–variance portfolio selection with value-atrisk constraints and discrete distributions.
- Alexander, Baptista, et al.
- 2007
(Show Context)
Citation Context ...proposed recently by McAleer et al. (2010), for instance. Using the terminology of Christoffersen (2009), this approach can be considered a risk measurement or passive risk management approach, since it is applied to a given (i.e. predetermined) portfolio composition. Alternatively, in this paper we propose to perform active risk management by deciding on the portfolio allocations themselves to attain lower levels of CR. Second, portfolios with low levels of CR may be obtained by imposing constraints on the amount of CR or on the portfolio VaR, as in Sentana (2003), Cuoco and Liu (2006) and 6 Alexander et al. (2007). In our approach, the level of CR plays a much more central role as it is taken to be the objective function that should be minimized. The remainder of the paper is organized as follows. In Section 2 we describe the procedure to obtain the optimal portfolios with minimum CR subject to restrictions on the number of VaR violations. In Section 3 we present the empirical applications. We conclude in Section 4. 2. Portfolios with Minimum Capital Requirements The main ingredient required to obtain optimal portfolios with minimum capital requirements (hereafter MCR portfolios) is a measure of the Va... |