DEPARTAMENTO DE ECONOMÍA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DE?L PERÚUNIVERSIDAD CATÓLICA: DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DOCUMENTO DE TRABAJO N° 394 EXTREME VALUE THEORY: AN APPLICATION TO THE PERUVIAN STOCK MARKET RETURNS Alfredo Calderon Vela y Gabriel Rodríguez DOCUMENTO DE TRABAJO N° 394 EXTREME VALUE THEORY: AN APPLICATION TO THE PERUVIAN STOCK MARKET RETURNS? Alfredo Calderón Vela y Gabriel Rodríguez Diciembre, 2014 DEPARTAMENTO DE ECONOMÍA DOCUMENTO DE TRABAJO 394 http://files.pucp.edu.pe/departamento/economia/DDD394pdf © Departamento de Economía – Pontificia Universidad Católica del Perú, © Alfredo Calderon Vela y Gabriel Rodríguez Av. Universitaria 1801, Lima 32 – Perú. Teléfono: (51-1) 626-2000 anexos 4950 - 4951 Fax: (51-1) 626-2874 econo@pucp.edu.pe www.pucp.edu.pe/departamento/economia/ Encargado de la Serie: Jorge Rojas Rojas Departamento de Economía – Pontificia Universidad Católica del Perú, jorge.rojas@pucp.edu.pe Alfredo Calderon Vela y Gabriel Rodríguez Extreme Value Theory: An Application to Peruvian Stock Market Returns? Lima, Departamento de Economía, 2014 (Documento de Trabajo 394) PALABRAS CLAVE: Teoría de Valores Extremos, Valor en Riesgo (VaR), Expected Short-Fall (ES), Distribución de Pareto Generalizada (GPD), Distribuciones Gumbel, Exponencial, Fréchet, Pérdidas Extremas, Mercado Bursátil Peruano. Las opiniones y recomendaciones vertidas en estos documentos son responsabilidad de sus autores y no representan necesariamente los puntos de vista del Departamento Economía. Hecho el Depósito Legal en la Biblioteca Nacional del Perú Nº 2015-02554. ISSN 2079-8466 (Impresa) ISSN 2079-8474 (En línea) Impreso en Kolores Industria Gráfica E.I.R.L. Jr. La Chasca 119, Int. 264, Lima 36, Perú. Tiraje: 100 ejemplares Extreme Value Theory: An Application to the Peruvian Stock Market Returns Alfredo Calderón Vela Gabriel Rodríguez Ponti…cia Universidad Católica del Perú Ponti…cia Universidad Católica del Perú Abstract Using daily observations of the index and stock market returns for the Peruvian case from January 3, 1990 to May 31, 2013, this paper models the distribution of daily loss probability, estimates maximum quantiles and tail probabilities of this distribution, and models the extremes through a maximum threshold. This is used to obtain the better measurements of the Value at Risk (VaR) and the Expected Short-Fall (ES) at 95% and 99%. One of the results on calculating the maximum annual block of the negative stock market returns is the observation that the largest negative stock market return (daily) is 12.44% in 2011. The shape parameter is equal to -0.020 and 0.268 for the annual and quarterly block, respectively. Then, in the …rst case we have that the non-degenerate distribution function is Gumbel-type. In the other case, we have a thick-tailed distribution (Fréchet). Estimated values of the VaR and the ES are higher using the Generalized Pareto Distribution (GPD) in comparison with the Normal distribution and the di¤erences at 99.0% are notable. Finally, the non-parametric estimation of the Hill tail-index and the quantile for negative stock market returns shows quite instability. JEL Classi…cation: C22, C58, G32. Keywords: Extreme Value Theory, Value-at-Risk (VaR), Expected Short-Fall (ES), Generalized Pareto Distribution (GPD), Distributions Gumbel, Exponential, Fréchet, Extreme Loss, Peruvian Stock Market. Resumen Utilizando observaciones diarias de los índices y retornos del mercado de valores Peruano desde Enero 3, 1990 hasta Mayo 31, 2013, se modela la distribución de probabilidad de pérdida diaria, se estiman los cuantiles máximos y las probabilidades de la cola de esta distribución así como también se modelan los extremos a través de un umbral máximo. Esto se utiliza para obtener mejores mediciones del Valor en Riesgo (VaR) y el Expected Short-Fall (ES) a 95% y 99%. Uno de los resultados en el cálculo del bloque máximo anual de los retornos negativos del mercado de valores es la observación de que el mayor retorno negativo del mercado de valores (diario) es 12.44% en 2011. El parámetro de forma es igual a -0.020 y 0.268 para los bloques anual y trimestral, respectivamente. En el primer caso tenemos que la función de distribución no degenerada es de tipo Gumbel. En el otro caso, tenemos una distribución de espesor de cola de tipo Fréchet. Las estimaciones del VaR y el ES son más altos mediante la distribución generalizada de Pareto (GPD) en comparación con la distribución Normal y las diferencias al 99.0% son notables. Por último, la estimación no paramétrica de la cola usando el índice de Hill y el cuantil de rentabilidad negativa del mercado de valores muestra gran inestabilidad. Classi…cación JEL: C22, C58, G32. Palabras Claves: Teoría de Valores Extremos, Valor en Riesgo (VaR), Expected Short-Fall (ES), Distribución de Pareto Generalizada (GPD), Distribuciones Gumbel, Exponencial, Fréchet, Pérdi- das Extremas, Mercado Bursátil Peruano. Extreme Value Theory: An Application to the Peruvian Stock Market Returns1 Alfredo Calderón Vela Gabriel Rodríguez2 Ponti…cia Universidad Católica del Perú Ponti…cia Universidad Católica del Perú 1 Introduction As part of the Peruvian economy’s good performance in recent years, the …nancial sector has played a signi…cant role in terms of the objective of economic growth and capital accumulation. Nonetheless, the world …nancial crisis that began in the …nal quarter 2007 a¤ected the Peruvian capitals market and brought about a sharp fall in the General Index of the Lima Stock Exchange (IGBVL) of 59.78%, and in the Selective Index (ISBVL) of 59.73%. This event illustrates that big losses occur as a result of extreme movements in the markets, and hence that …nancial risk is related to the possible losses that investors can su¤er in these markets; see Jorian (2001). In general, the series of stock market returns have heavy-tailed distribution, due to which, unlike traditional distributions, the distribution of stock market returns possess greater probabilistic density on the tails. The above has, as a consequence, greater probability of extreme losses and it is necessary to analyze the tails of the distribution through the use of methodologies in the context of the Extreme Values Theory (EVT). We seek to capture in the best way possible the sudden movements of the performances of …nancial assets associated with the tails of the distribution, and thus allow better measurement of the behavior of …nancial asset performance3. The recent …nancial crisis put in evidence the existence of multiple faults in the form of risk modeling, and this in turn prompted notable criticism of the di¤erent mathematical models and traditional statistics employed by companies in attempts to predict the risk. In 1993, the members of the Bank for International Settlements (BIS) gathered in Basel and amended the Basel Accords to require that banks and other …nancial institutions keep su¢ cient capital in reserve to cover ten days of potential losses based on the 10-day Value at Risk (VaR)4. The estimation of the VaR by way of traditional models is not entirely adequate, because many of the techniques employed are based on the assumption that the …nancial returns follow a Normal distribution. In this context, the measurement of risk through traditional measures occasions large losses to market participants because of the unexpected falls in …nancial market returns. Another 1This paper is drawn from the Thesis of Alfredo Calderón Vela at the Master Program of Economics, Graduate School, Department of Economics, Ponti…cia Universidad Católica del Perú. We thank useful comments of Paul Castillo (Central Reserve Bank of Peru). 2Address for Correspondence: Gabriel Rodríguez, Department of Economics, Ponti…cia Universidad Católica del Perú, Av. Universitaria 1801, Lima 32, Lima, Perú, Telephone: +511-626-2000 (4998), Fax: +511-626-2874. E-Mail Address: gabriel.rodriguez@pucp.edu.pe. 3 Important texts include Embrechts et al. (1997), and Coles (2001). Other references applied to …nances and the …nancial risk management are Diebold et al. (1998), Danielsson and De Vries (1997), McNeil (1998a, 1998b), and Longin (2000). 4Danielsson et al. (2001) hold that the Committee on Banking Supervision was wrong to consider the risk to be endogenous, and a¢ rm that the VaR can destabilize an economy and generate breaks would not otherwise occur. In this way, the authors leave open the possibility that traditional …nancial models employed to measure and diagnose the risk have a certain degree of inconsistence, primarily because certain assumptions of these models are incapable of capturing the behavior of the indices that are used to measure risk. In particular, it is found that traditional models have a poor performance against sudden movements of these indices in a context of crisis. 1 measure of risk is that proposed by Artzner et al. (1999), called expected shortfall, or expected loss (Expected Shortfall - ES), which is an expectation of loss conditioned to exceeding the indicated VaR level. One of the objectives of …nancial risk management is the exact calculation of the magnitudes and probabilities of big …nancial losses that are produced at times of …nancial crisis. It is thus of relevance to model the probability of loss distribution and estimate the maximum quantiles and tail probabilities associated with this distribution; see Zivot and Wang (2006). The modern EVT started with von Bortkiewicz (1922). Thereafter, Fisher and Tippett (1928) laid the foundations of the asymptotic theory of the distributions of extreme values. Hill (1975) introduces a general approach for inference around the behavior of the tail of a distribution, while Danielsson and De Vries (1997) believe that a speci…c estimation of the form of the tail of foreign currency returns is of vital importance for adequate risk assessment. On the other hand, Embrechts et al. (1997) present the probabilistic models and techniques with the aim of mathematically describing extreme events in the unidimensional case. McNeil (1998a) reduces data from the S&P500 index to 28 annual maximums corresponding to the period 1960-1987, and adjusts them to a Fréchet distribution. In this way, they calculate the estimations of various levels of returns, as well as the con…dence interval at 95% for a 50-year level of return - which on average must be exceeded in just one year- every …fty years. The most probable calculated value is 7.4, but there is a great deal of uncertainty in the analysis as the con…dence interval is approximately [4.9, 24]. Moreover, McNeil (1998b) considers the estimation of quantiles in the marginal distribution tail in the series of …nancial returns, utilizing statistical methods of extreme values based on the distribution limit of maximum blocks of stationary time series. The author proposes a simple methodology for the quanti…cation of the worst possible scenarios, with losses of ten or twenty years. Diebold et al. (1998) hold that the literature on the EVT is more accurate for the exact estimation of the extreme quantiles and tail probabilities of the …nancial assets5. McNeil (1999) shows a general vision of the EVT in the management of risks as a method for modeling and measuring extreme risks, concentrating on the peaks through a threshold. McNeil and Frey (2000) propose a method for estimating the VaR and relate it to the risk measurements that describe the conditional distribution tail of a series of heteroskedastic …nancial yields. Moreover, Longin (2000) present an application of the EVT to calculate the VaR of a position in the market. For Embrechts et al. (2002), the modern risk management requires an understanding of stochastic dependence. The authors conduct a discussion on joint distributions and the use of copulas as descriptions of dependency among random variables. Tsay (2002) applies the EVT to the logarithm of pro…tability of IBM shares for the period from July 3, 1962 to December 31, 1998 and …nds that the range of ‡uctuation of the daily yields, excluding the crisis of 1987, ‡uctuates between 0.5% and 13%. He also estimates the Hill estimator and …nds stable results for a minimum and maximum value of the biggest n-th observation of this estimator. Tsay (2002) performs the estimation for di¤erent sample sizes (monthly, quarterly, weekly, and yearly) and concludes that the estimation of the scale and location parameters increase in modulus when the sample size increases. The shape parameter is stable for extreme negatives values when the sample size is greater than 62 and is approximately equal to a -0.33. The estimator of the shape parameter is small, signi…cantly di¤erent to zero, and less stable for positive extremes. 5Diebold et al. (1998) demonstrate the existence of a trade o¤ between the bias error and the variance when the largest n-th observation increases in Hill’s tail index estimator (1975). 2 The result for the annual sample size has high variability when the number of subperiods is relatively small. According to Del…ner and Gutiérrez (2002), the returns in developing markets are characterized by being more leptokurtic compared to the returns of more developed economies; see also Humala and Rodríguez (2013) for stylized facts in the Peruvian stock market. The authors estimate an autoregressive AR-GARCH model of stochastic volatility, and then apply the EVT to the distri- bution tail of standardized residuals of the model by estimating a generalized Pareto distribution with a view to obtaining a better estimation of the probability when extreme losses are presented. Finally, McNeil et al. (2005) provide two main types of models of extreme values. The most traditional models are maximum block, which are models for the biggest ordered observations of big samples of identically distributed observations. The other group of models are for threshold exceedances and apply to all big observations that exceed a high level. They are generally considered very useful for practical applications, given their more e¢ cient use (often limited) of the data on the extreme results. Using daily observations of the index and stock market returns for the Peruvian case from January 3, 1990 to May 31, 2013, this paper models the distribution of daily loss probability, estimates maximum quantiles and tail probabilities of this distribution, and models the extremes through a maximum threshold. This is used to obtain the best measurements of the Value at Risk (VaR) and the Expected Short-Fall (ES) at 95% and 99%. One of the results on calculating the maximum annual block of the negative stock market returns is the observation that the largest negative stock market return (daily) is 12.44% in 2011. The shape parameter is equal to -0.020 and 0.268 for the annual and quarterly block, respectively. Then, in the …rst case we have that the non-degenerate distribution function is Gumbel-type. In the other case, we have a thick-tailed distribution (Fréchet). Estimations of the VaR and the ES are higher using the Generalized Pareto Distribution (GPD) in comparison with the Normal distribution and the di¤erences at 99.0% are notable. Finally, the non-parametric estimation of the Hill tail-index and the quantile for negative stock market returns shows quite instability. This paper is structured as follows: Section 2 describes the main de…nitions associated with EVT, as well as the method for estimating the main measurements of risk, the VaR and the ES. Section 3 presents the results, utilizing a sample of daily returns of the Peruvian stock market. Section 4 presents the main conclusions. 2 Methodology In this Section, we closely follow and employ the notation of Zivot and Wang (2006). The EVT provides the statistical tools to model the unknown accumulated distribution function of the random variables that represent the risk or losses, especially in those situations where large losses are produced. Let fX1; X2; :::; Xng be random variables i:i:d: that symbolize the risk or expected losses, which have an unknown accumulated distribution function F (x) = Pr[Xi  x]. Mn = max[X1; X2; :::; Xn] is speci…ed as the worst loss in a sample of losses of n in size. Of the assumption i:i:d:, the cumulative distribution function of Mn is: Pr[Mn  x] = Pr[X1  x;X2  x; :::;Xn  x] = nY i=1 F (x) = Fn(x). It is assumed that the function Fn is unknown, and, moreover, it is known that the function of empirical distribution is not a good approximation of Fn(x). According to 3 the Fisher-Tippett Theorem (1928)6, an asymptotic approximation is obtained for Fn based on the standardization of the maximum value; that is, Zn = Mnn n where it is found that n > 0 and n as measurements of scale and position, respectively. In this way, for Fisher and Tippett (1928) the maximum standardized value converges to a distribution function of generalized extreme value (GEV), de…ned as: H(z) = 8<: exp[(1 + z)1=], para  6= 0; 1 + z > 0exp[ exp(z)], para  = 0; 1  z  1 9=; where  is denominated the shape parameter and determines the behavior of the tail of H(:)7. This distribution is not degenerated and is generalized in the sense that the parametric shape summarizes three types of known distributions. Moreover, if  = 0, H is a Gumbel distribution; if  > 0, H is a Fréchet distribution; if  < 0, H is a Weibull distribution. The parameter shape  is associated with the behavior of the tail of the distribution F and decays exponentially for a function of power 1 F (x) = x1=L(x) where L(x) changes slowly. The GEV distribution is not changed for the transformations of location and scale: H(z) = H( x  ) = H;;(x). For a large size of n, the Fisher-Tippett Theorem (1928) can be interpreted as follows: Pr[Zn < z] = Pr[ Mnn n < z]  H(z). Assuming x = nz + n, then: Pr[Mn < x]  H;;(xnn ) = H;n;n(x). This expression is useful for performing inference related to the maximum loss Mn. The expression depends on the parameter of  in form and the standardized constants n and n, which are estimated for maximum likelihood. To perform the estimation of maximum likelihood, there is supposed to be a set of identically distributed losses from a sample of size T represented for fX1; X2; :::; XT g that have an accumu- lated density function F . A sub-sample method is utilized to form the likelihood function for the parameters ; n and n from the GEV distribution for Mn. In this way, the sample is divided into m non-overlapping blocks of equal size n = T=m, with which we have [X1; :::; XnjXn+1; :::; X2nj:::j X(m1)n+1; :::; Xmn] and where M (j) n is de…ned as the maximum value of Xi in the block j = 1;    ;m. The likelihood function for the parameters ; n and n of the GEV distribution is constructed from the maximum block sample of fM (1)n ; :::;M (m)n g. The likelihood log function as- suming observations i:i:d: of the GEV distribution when  6= 0 is log(; ; ) = m log() (1 + 1  ) Pm i=1 log[1 + ( M (i) n   )] Pm i=1[1 + ( M (i) n   )] 1= with the restriction that 1 + (M (i) n   ) > 0. When  = 0, we have a Weibull distribution8. It is important to discuss the limit distribution of extremes on high thresholds and the gener- alized Pareto distribution (GPD). When there is a succession of random variables fX1; X2; :::; Xng i:i:d: associated with an unknown function of distribution F (x) = Pr[Xi  x], the extreme values are de…ned as the Xi values that exceed the high threshold , so the variable X  represents the excesses on this threshold. The distribution of excesses on the threshold  are de…ned as conditional probability: F(y) = Pr[X   yjX > ] = F (y+)F ()1F () for y > 0. This is interpreted as the probability that a loss exceeds the threshold  for a value that is equal to or less than y, given that the threshold of  has been exceeded. For Mn = maxfX1; X2; :::; Xng, de…ned as the worst loss in a sample of n-sized losses, the distribution function F satis…es the Fisher-Tippett Theorem 6This Theorem is anolagous to the Central Limit Theorem for extreme values. 7The previous expression is continuous in the parameter of form . 8The distribution in the domain of attaction of the Gumbel-type distribution are thin-tailed distributions where practically all moments exist. If they are Fréchet-type, they include fat-tailed distributions such as Pareto, Cauchy, t-Student, among others. Not all moments exist for these distributions. 4 (1928), and for a su¢ ciently large  there is a positive function (); thus, the surplus distribution is approximated through the GPD: G; ()(y) = 8<: 1 [1 + y= ()]1= for  6= 0, () > 01 exp[y= ()] for  = 0, () > 0 9=; de…ned for y  0 when   0 and 0  y  () when  < 0. For a su¢ ciently high threshold  it is found that F(y)  G; ()(y) for a wide range of loss functions F (:). To apply this result, the value of the threshold must be speci…ed and the estimates of  and () can be estimated. There is a close connection between the GEV limit distribution for maximum blocks and the GPD for excesses with respect to the threshold. For a given value of , the parameters ,  and  of the GEV distribution determine the parameters  and (). It is clear that the shape parameter  of the GEV distribution is the same parameter  in the GPD and is independent of the threshold value . In consequence, if  > 0, the function F is Fréchet-type and the expression G; ()(y) is denominated classic Pareto distribution; when  = 0, the function F is Gumbel-type and G; ()(y) is of exponential distribution. Finally, it is found that 0  y  ()= when  < 0, so the function F is Weibull-type and G; ()(y) is the type II Pareto distribution. The parameter  is the shape or tail-index parameter and is associated with the rate of decay of the tail of the distribution, and the decreasing parameter is the shape parameter and is associated with the position of the threshold .9 Now, assuming that the parameter of form is  < 1, then the mean excess function above the threshold 0 will be E[X 0jX > 0] = (0)1 for any  > 0, and it is found that the excess function of the mean e() = E[X jX > ] = (0)+(0)1 . Analogously, for any value of y > 0: e(0+ y) = E[X (0+ y)jX > 0+ y] = (0)+y1 . Therefore, to graphically deduce the threshold value for the GPD, we get the excess function of the empirical mean: en() = 1n Pn i=1[x(i) ], where x(i) (i = 1; 2; :::; n) are the values of xi such that xi > . With the previous expression, a Figure of en() is constructed with the mean excess on the vertical axis. This Figure can be interpreted as follows: if the slope is rising, it indicates thick tail behavior, but if there is a downward trend, this shows thin tail behavior in the distribution; …nally, if the slope of the line is equal to zero, the behavior of the tail is exponential. If the line is straight and has a positive slope located above the threshold, it is a Pareto-type sample of behavior in the tail. For the values of the maximum losses that exceed the threshold; that is, when xi > , the threshold excess is de…ned as yi = x(i)  for i = 1; :::; k, in which the values of the x1; ::::; xn have been denoted as x(i); ::::; x(k). When the threshold value is su¢ ciently large, then the sample fy1; :::; ykg can be expressed within a likelihood that is based on the unknown parameters  and (); that is, a random sample of a GPD. When  6= 0, the log likelihood function of G; ()(y) has the following form: log[ ()] = k log[ ()] [1+ 1 ] Pk i=1 log[1+ yi () ] where yi  0 when  > 0 and 0  yi  ()= when  < 0. If the parameter of form is  = 0, then the log likelihood function is: log[ ()] = k log[ ()] ()1 Pk i=1 yi. To estimate the tails of the loss distribution for F (x), and where x > , is utilized F (x) = [1 F ()]G; ()(y) + F (). The previous expression is ful…lled for a su¢ ciently large threshold and in which F(y)  G; ()(y).10 9For  > 0 (the most relevant case for risk administration purposes) it can be shown that E[Xk] = 1 for k  = 1=. If  = 0:5, E[X2] =1 and the distribution of losses X does not have …nite variance. Equally, if  = 1, then E[X] =1. 10 It is assumed that x = + y. 5 There are two common risk measurements: Value at Risk (VaR) and Expected Shortfall (ES). The VaR is the largest quantile of the distribution of loss; that is, V aRq = F1(q).11 For a given probability q > F (), it is found that dV aRq = + b ()b [nk (1 q)b1]. The ES is the expected size loss, given that the V aRq is exceeded: ESq = E[XjX > V aRq]. This equation is related to the V aRq in accordance with ESq = V aRq + E[X V aRqjX > V aRq], where the second term is the mean of excess of the distribution FV aRq(y) on a threshold V aRq. The approximation of the GPD (due to the translation property) to FV aRq(y) has the shape parameter  and the scale parameter () + [V aRq ] : E[X V aRqjX > V aRq] = ()+(V aRq)1 provided that  > 1. Moreover, it is found that the GPD approximates dESq = dV aRq 1b + b ()b1b .12 It is also possible to perform the non-parametric estimation of the shape  or tail-index parameter a = 1= of the distributions H(z)and G; ()(y) utilizing Hill’s method (1975), in which  > 0 ( > 0), is generated by the same thick-tailed distributions in the domain of attractions of a Fréchet GEV. Considering a sample of losses fX1; X2; :::; XT g, the statistical order is de…ned as X(1)  X(2)  :::  X(T ) for a positive whole k, and the Hill estimator of  is de…ned as bHill(k) = 1kPkj=1[logX(j) logX(k)]. The Hill estimator of is b Hill(k) = 1bHill(k) .13 3 Empirical Evidence Figure 1 shows the series for the closing prices of the General Index of the Lima Stock Exchange (IGBVL)14. The series is of daily frequency and covers the period January 3, 1990 to May 30, 2013. The returns are de…ned as rt = log[ PtPt1 ], which are shown in Figure 2. Empirically, the returns display certain properties as marginal thick-tailed distributions, nonexistence of correlation, and dependency across these; though they are highly correlated if it concerns the squared results or their absolute value; see Humala and Rodríguez (2013) for a more detailed description about the stylized facts. By way of motivation, the Figure 3 shows the GEV accumulative distribution function for the distribution function H(:), which adopts the Fréchet, Weibull and Gumbel forms of distribution when the shape parameter  = 0:5,  = 0:5 and  = 0, respectively, and for general values of z, the parameter of position of  and the parameter of scale . In this particular case, the Fréchet distribution is de…ned for z > 2 and the Weibull distribution is only de…ned for z < 2. The Figure 4 shows the GEV probability density function H(:) for the non-degenerate Fréchet, Weibull and Gumbell distribution functions when the shape parameter has values of  = 0:5,  = 0:5 and  = 0 respectively. The Fréchet and Weibull distributions are de…ned for z > 2 and z < 2, respectively. 11 If it is assumed that X  N(; 2), then V aR0:99 = + q0:99. 12 If X  N(; 2), it is found that ES0:99 = +  (z)1(z) . 13 It can be seen that if F is located in the domain of attraction of a GEV distribution, then bHill(k) converges in probability to  when k )1 and k n ) 0, and that bHill(k) is Normally asymptotically distributed with asymptotic variance: avar[bHill(k)] = 2 k . Via the delta method, b Hill(k) is Normally asymptotically distributed with asymptotic variance avar[b Hill(k)] = a2 k . 14The closing prices of the General Index of the Lima Stock Exchange are taken into account from Monday until the closure price on Friday. Moreover, it should be recalled that holiday days are not considered, and more generally the days on which the market was closed. 6 The Figure 5 shows the GEV density function for negative stock market returns. The horizontal axis represents the standardized value Zn of the maximum value of the block Mn with respect to the measurements of scale and position. The vertical axis shows the probability associated with the GEV density function. It is observed that this distribution does not have the form of a known distribution and the maximum probability shows positive asymmetry. The Quantile-p of a distribution functionG is denominated to the valueXp such thatG(p) = Xp; that is, to the value of Xp that leaves the percentile p of probability to its left. If a distribution function G is continuous and thus strictly growing, the quantile function is the inverse of the distribution function G and is usually denoted as G1. The Figure 6 shows the qq-plot, taking the Normal distribution as theoretical distribution to contrast it with the distribution of stock market returns. It is observed that a straight line close to zero is not observed (approximately), and so it is concluded that the distribution of the variable is not the same as the comparison distribution, showing evidence that the distribution of negative stock market returns is unknown. Subsequently, the annual maximum block of the negative stock market returns is calculated. The Figure 7 shows four representations for this annual maximum block. In the upper left corner is the largest negative return of the period analyzed, which reaches 12.44% in 2011. The upper right extreme of Figure 7 shows the histogram where the horizontal axis represents the annual maximum blocks. In the lower left extreme, the qq-plot is shown, contrasting again the distri- bution of stock market returns for the period of analysis. In the vertical axis, the quantiles of the referential theoretical distribution are represented (Gumbel distribution, H0), which satis…es H10 (p) = log[ log(p)] and the horizontal axis represents the empirical quantiles for the annual maximum blocks of the distribution of stock market returns. It is observed that a straight line close to the centre of this Figure is approximately obtained, which suggests that the distribution of the variable of real data (empirical distribution) is the same as the distribution of comparison (Gumbel distribution). The lower right extreme shows the development of the records (new maximum) for the negative stock market returns, together with the expected number of returns for the data i:i:d:. In this Figure, it is observed that the data was not within the interval of trust (dotted lines), due to which it can be concluded that the data is not consistent with the behavior i:i:d. Similarly to the Figure in the lower left extreme of Figure 7, Figure 8 shows the qq-plot, using as referential distribution the Gumbel distribution H0, and unlike Figure 7, the horizontal axis repre- sents the standardization of maximum value Zn. As shown previously for the Gumbel distribution, the quantiles satisfy H10 (p) = log[ log(p)] and the points of the quantiles correspond to the standardization of the maximum value Zn and indicate a GEV distribution with  = 0. Then, the entire annual value of the number of observations in each maximum block is deter- mined for M (i)n i = 1; :::;m for the stock market returns, with m = 24. The shape parameter  is statistically insigni…cant (b = 0:020, tb = 0:126) and so the value of this parameter is equal to zero ( = 0). Moreover, the asymptotic interval at 95% of con…dence for  is [0:337; 0:2968] and indicates the considerable uncertainty related to the value of . This result determines the tail behavior of the GEV distribution function of stock market returns, and it is concluded that the non-degenerate distribution function is Gumbel-type. The shape and scale parameters (standard- ized constants) are statistically signi…cant: bn = 4:232, tbn = 8:713 and bn = 2:098, tbn = 5:954, respectively. Utilizing the estimation by maximum likelihood of the adjusted GEV distribution for the max- imum annual block of negative stock market returns, the following question can be answered: how probable is it that the maximum annual negative pro…tability for the following year exceeds the 7 above negative returns? This probability is calculated utilizing the expression H;n;n(x) where the maximum block is equal to 1.68%, and so there is a 1.68% possibility that a new maximum record of negative performance will be established during the following year. A similar analysis is possible by considering the GEV distribution adjusted for the quarterly maximum block for the data from the series of stock market returns. The maximum block for the return of this series is m = 94. It is observed that estimated standard asymptotic errors are much lower when quarterly blocks are employed. The shape estimator is b = 0:268 (tb = 2:031) and in this case, the asymptotic interval at 95% of con…dence for  is [0:004; 0:532] and contains only positive values for the shape parameter, indicating a thick-tailed distribution, with the estimated probability equal to 0.0172. Finally, the estimations of the shape parameter and the standardized constants are signi…cant: bn = 2:419, tbn = 9:186 and bn = 2:419, tbn = 13:705, respectively. In Figure 9, the asymmetric form of the asymptotic con…dence interval can be observed. This Figure allows for a response to the question, what is the level of stock market return for the last forty years? The estimated point of the level of return (11.67%) is at the point where the vertical line cuts at the maximum point of the asymmetric curve. The upper extreme point of the con…dence interval of 95% is approximately 22%; this point is located where the asymmetrical curve cuts at the straight horizontal line. In addition, the Figure 10 shows the estimation of the expected yield level of the negative stock market returns for the forty years with a con…dence level band of 95% based on the model of GEV for an annual maximum block. The Figure 10 has the horizontal line situated at the mean corresponding to the expected level of return (11.67%), the pointed horizontal line that is located below the expected level of return corresponding to the lowest level of return (9.33%), and the pointed line above the expected level of return corresponding to the highest level of return (22.21%). In this Figure, the 24 annual maximum blocks (m = 24) obtained from the real data of the stock market returns (point cloud) can also be seen, in which only two points exceed the lower extreme. Following Zivot and Wang (2006), the 40-year level of return can also be estimated based on the GEV …tted to quarterly periods as a maximum, where forty years correspond to 160 quarters, obtaining the lowest and highest level of return; see Figures 11 and 12. In Figure 12, the horizontal line located on the mean corresponds to the expected level of return (17.18%); the dotted horizontal line is located below the level of expected return corresponds to the lowest level of return (10.88%) and the dotted line above the expected level of return corresponds to the highest level of return, being equal to 40.68%. This Figure also shows the 94 (m = 94) quarterly maximum blocks obtained from the data on stock market returns (point cloud) below the lower band of con…dence of the con…dence interval, except for two points, which means that the return for these 160 quarters must be above these values. According to Zivot and Wang (2006), modeling only the maximum block of data is ine¢ cient if there is other data available on the extreme values. A more e¢ cient, alternative approach that utilizes more observations is to model the behavior of extreme values above a given high threshold. This method is called peaks over thresholds (POT). Another advantage of the POT method is that the common risk measurements such as the, VaR and ES, can be calculated easily15. To motivate the importance of the foregoing in Figure 13, the calculation of the accumulated 15For risk administration purposes, insurance companies may be interested in the frequency of occurrence of a large demand above a certain threshold, as well as the average value of the demand that exceeds the threshold. In addition, they may be interested in the daily VaR and ES. The statistical models for extreme values above a threshold can be used to tackle these questions. 8 distribution and probability functions are shown with () = 1 for a Pareto ( = 0:5), exponential ( = 0), and Pareto type II ( = 0:5) distributions. The Pareto type II distribution is de…ned only for y < 2. According to Zivot and Wang (2006), to infer the tail behavior of the observed losses, a qq-plot is created using the exponential distribution as reference distribution. If the excess on the threshold is a thin-tailed distribution, then the generalized Pareto distribution is exponential with  = 0 and the qq-plot should be linear. Deviations from the linearity in the qq-plot indicate thick-tailed behavior ( > 0) or bounded tails ( < 0). In Figure 14, the qq-plot is observed for the distribution of negative stock market returns through the threshold when this is equal to one ( = 1). The selection of the threshold under this methodol- ogy is complicated, so for the identi…cation of the threshold, there are a number of methodologies, such as parametric and graphic methods16. The Figure 14 shows a slight deviation from the linear- ity for negative stock market returns, due to which it is concluded that the distribution of negative stock market returns is a thick-tailed distribution. The main distributional model for excess through the threshold is the GPD, so on de…ning the excess function of the empirical sample mean, a graph can be prepared in which the expectation of the values above the threshold  is represented, once the threshold has been exceeded on the vertical axis associated with each of the thresholds. This is useful for discerning tails of a distribution against the di¤erent possible levels of threshold  on the horizontal axis. This Figure must be approximately linear at the level of the selected threshold, and it is possible to determine intervals on whose basis the threshold can be selected. In general, the thick-tailed distributions give way to a mean excess function that tend toward the in…nite for high values of  and display a linear form with a positive slope. The above-mentioned is shown in Figure 15. On the vertical axis, the empirical mean excess is represented for the series of stock market returns, and on the horizontal axis, the threshold  is represented. If the points that are represented have an upward trend (upward slope), this indicates thick-tailed behavior in the sample represented, as well as a GPD with positive shape parameter  > 0. If there is a downward trend (negative slope), this shows the thin-tailed behavior of the GPD with negative parameter  < 0. Finally, if an approximately linear graph is obtained (tends toward the horizontal axis), this indicates a GPD and the tail behavior is exponential (an exponential excess distribution), with the shape parameter approximately equal to zero ( = 0). From the observation of Figure 15 on mean excess, a declining trend for the data up to the value of the threshold  = 1 is detected, which indicates a thin-tailed distribution therein; but from this value for the threshold, there is an upward trend for the data, indicating the thick-tailed behavior in the sample represented17. Once the mean excess function is determined, the tails of the distribution of negative stock market losses are estimated for the period of analysis by way of the maximum likelihood estimation of the parameters () and  of the GPD. To determine this estimation, a threshold , must be speci…ed, which must be big enough for the approximation of the GPD to be valid, but must also be small enough so that a su¢ cient number of observations is available for an exact …t; see Carmona (2004). In the Figure on the excess of the mean (Figure 16) for stock market returns, it is observed that the threshold has a value of one; that is  = 1 and may be appropriate for the GPD to be valid. The estimation of the parameters indicates b = 0:185 (tb = 4:463) and b (1) = 0:941 (tb (1) = 18:801). If the shape parameter estimated for the GPD (b = 0:185) is compared with 16One of these methods is the Figure of the mean of excesses. 17Empirical evidence on di¤erent behavior in the tails of the Peruvian stock market returns is also found in Bedón and Rodríguez (2014) and Language Lafosse et al. (2014). 9 the GEV estimations of the yearly and quarterly maximum blocks, it is seen that this is higher in the case where the analysis is based on quarterly data (b = 0:268), but less if annual data are used (b = 0:020), being close to zero in the latter case. According to Carmona (2004), Figure 16 shows the underlying distribution. On the extreme left the survival function 1F (x) is represented on the vertical axis instead of the cumulative distribution function F (x) and it is seen that the curve moves very close to the horizontal axis, so it is extremely di¢ cult to correctly quantify the quality of the …t. This Figure is not very useful, so on the extreme right a Figure is observed that represents the survival function in logarithmic scale on the vertical axis, which helps ensure that the …t of the distribution is adequate by taking into account the available data. Observing both Figures, it is concluded that the …t is good. Changing the value of the threshold brings about changes in the estimation of , so the stability of the shape parameter must be considered. It is optimal not to depend on a procedure that is too sensitive to small changes in the threshold selection. In e¤ect, given that there is no clear procedure for the selection of the threshold with a high level of accuracy, the estimation of the shape parameter must remain robust in the face of variations in the errors in the selection of this threshold. The best way of verifying the stability of the parameter is through visual inspection. Now, to show how the estimation by maximum likelihood in the shape parameter  varies with the threshold selected, we observe Figure 17 where the lower horizontal axis represents the maximum number of threshold excesses, and is assumed to be equal to six hundred. On the upper horizontal axis, the threshold is represented, and on the vertical axis, the estimation of the shape parameter with 95% con…dence. In the Figure it is seen that  has very stable behavior close to 0.185 for threshold values lower than 1.91. In accordance with the above, Figure 18 shows how the estimator of the GPD shape parameter varies with the threshold, where the start of the percentile has been speci…ed based on the data equal to 0.9, to be used as a threshold that …ts the model. In the upper part of the Figure (horizontal axis), the proportion of the points included in the estimation is represented. This information is useful when it comes to deciding whether or not it is necessary to take seriously some of the estimations of  that appear on the left and right extremes of the Figure. The central part of the Figure should essentially be horizontal, though this does not always result in a straight line, when the empirical distribution of the data can be reasonably explained by a GPD. Finally, on the lower part of the Figure (horizontal axis) the threshold is represented. In Figure 18, the left-most part of the Figure should be ignored, as if the threshold is too small, much of the data (that must be included in the center of the distribution) contributes to the estimation of the tail, skewing the result. Similarly, the right-most part of the Figure must also be ignored, as if the threshold is too big, few points will contribute to the estimation. This is the case in the current situation, and a value of  = 0:185 appears to be a reasonable estimation for the intersection of a horizontal line …tted to the central part of the Figure. On the other hand, according to Zivot and Wang (2006), it is often desirable to estimate the parameters  and () through the maximum likelihood estimation of the GPD seperately for the upper and lower tails of the negative returns (POT analysis). In the analysis of the mean excess through the threshold (Figure 15) the lower threshold is determined, which is equal to 1. Similarly, with the help of this Figure, the upper threshold is selected, which is equal to 1. The estimations for the lower threshold are b = 0:185 and b () = 0:912, while for the upper threshold they are b = 0:217 and b () = 1:087. Note that the estimated value of the parameters  and () are the same as the estimates in the analysis of excess on the previously realized threshold when 10 the threshold equal to minus one was estimated (see Figure 15). The next analysis is very similar to the previous, with the di¤erence being the presence of two tails instead of one. The Figure 19 shows the qq-plots of the excess on the speci…ed threshold versus the quantiles of the GPD by employing the estimated shape parameters of the upper and lower tails. In this case, the lower tail (left) could start at minus one, and the upper tail (right) at one18. In Figure 19 (in both representations) it is seen that the point sets form a straight line up to a certain stage, so it is reasonable to assume that a GPD …ts the data. Moreover, the two estimations for the shape parameter  are not the same based on the particular selections of the upper (0.217) and lower (0.185) thresholds. If the distribution is not symmetrical, there is no special reason for the two values of  to be the same; that is, there is no particular reason why, in general, the polynomial decay of the right and left tails must be identical. At the start of this research it is held that for better understanding of the risk, the V aR and ES should be borne in mind to quantify the …nancial risks. The estimation of these risk measurements is performed for negative stock market returns for the quantiles q = 0:95 and q = 0:99, which are based on the GPD19. For the case of the GPD, it is inferred that with 5% probability, thedV aR0:95 = 2:146% and, given that the return is less than -2.146%, thedES0:95 is -3.562%. Similarly, with 1% probability, the dV aR0:99 = 4:309% with adES0:99 = 6:217% given that the return is less than -4.309%. Compared with the results obtained utilizing a Normal distribution, the dV aR0:95 is less than the estimation of the GPD. Nonetheless, the dV aR0:99 is higher in the case of the GPD in comparison with the Normal distribution. In the case of the dES0:95 and dES0:99, both are higher using the GPD approximation. The di¤erence at 99.0% is notable and important (6.217% in the GPD compared with 4.375% for the Normal distribution). Once adjusted to a model of GPD for the excess of stock market returns above a threshold, we proceed to the estimation of valid asymptotic con…dence intervals for the V aRq and the ESq.20 These intervals can be visualized on Figure 20 with the tail estimate bF (x) = 1 kn [1 + b  xb () ]. The con…dence intervals for the V aR are [2:062; 2:240] and [4:048; 4:643] for 95% and 99%, respec- tively. With respect to the ES, the intervals are [3:358; 3:839] and [5:595; 7:156] for 95% and 99%, respectively. Figure 21 allows for an analysis of the sensitivity of V aRq estimated in response to changes in the threshold . It is observed how the estimation by maximum likelihood of the parameter of form  varies with the threshold. In the Figure, it is estimated that the parameter of form  has behavior that is very stable and close to the estimated value of the Value at Risk (4.309) for threshold values of less than four. According to McNeil et al. (2005), the GPD method is not the only way of estimating the tails of a distribution as has been performed above. The other methodology for the selection of the threshold is based on the Hill estimator, estimating, in a non-parametric way, the Hill tail index = 1= and the quantile xq;k for the negative stock market returns. This estimator is often a good estimator of , or its reciprocal . In practice, the general strategy is to graph the Hill estimator for all possible values of k (numbers of excesses through the threshold). Practical experience suggests that the best options of k are relatively small -for example, between 10 and 50 of statistical orders 18 It should be recalled that the upper and lower thresholds do not necessarily have to be equal in absolute value, as they are in this case. 19Under the assumption of Normally distributed returns, it is found that V aR0:99 =  +   q0:99 and ES0:99 = +   (z) 1(z) for the case of the quantile 0.99. 20Which are based on the delta method of the likelihood log function pro…le. 11 in a sample of 1000. In Figure 22, the Hill estimator fk; b (H)k;n : k = 2; :::; ng is estimated for negative stock market returns of the shape parameter . We expect to …nd a stable region for the Hill estimator where estimations are constructed based on the di¤erent numbers of statistical order. In this Figure, the upper horizontal axis represents the threshold associated with the possible values of k; in the lower horizontal axis, the number of observations included in the estimation is represented, and …nally the con…dence interval is observed at 95% (dotted lines). According to the results, it is observed that the estimation of the parameter does not stabilize as the statistical order increases, due to which bHill(k) is quite unstable. It should be borne in mind that in practice, the ideal situation does not usually occur if the data does not come from a distribution with a tail that changes with regularity. If this occurs, the Hill method is not appropriate. The serial dependence on the data can also impair the performance of the estimator, although this can also be said of the estimator of the GPD. 4 Conclusions Using daily observations of the index and stock market returns for the Peruvian case from January 3, 1990 to May 31, 2013, this paper models the distribution of daily loss probability, estimates maximum quantiles and tail probabilities of this distribution, and models the extremes through a maximum threshold. This is used to obtain the best measurements of VaR and ES at 95% and 99%. One of the results on calculating the maximum annual block of the negative stock market returns is the observation that the largest negative stock market return (daily) is 12.44% in 2011. Moreover, if it is estimated that the probability of the maximum negative annual pro…tability for the following year exceeds all previous negative returns, turning out equal to 1.68, there is a 1.68% possibility of a negative maximum record of the negative yield being stabilized during the following year. Then, by way of the estimator of maximum likelihood, the parameter of form and the asymp- totic interval are estimated at 95% con…dence thereof for the annual and quarterly maximum block. The results indicate that the shape parameter is equal to -0.020 and 0.268, as well as the asymptotic interval [-0.337, 0.2968] and [-0.004, 0.532] for the maximum annual and quarterly block, respec- tively. The shape parameter estimation (-0.020) of the calculation of the maximum annual block of negative stock market returns is insigni…cant, due to which the value of this parameter is equal to zero and determines the tail behavior of the GEV distribution, and it is concluded that the non-degenerate distribution function is Gumbel-type. In the case of the estimation by maximum likelihood for the maximum quarterly block, a positive value was obtained for the shape parameter (0.268), with this being signi…cant, indicating a thick-tailed distribution (Fréchet). For the case of the GPD, it is inferred that with 5% probability, the daily return would be as low as -2.146% and, given that the return is less than -2.146%, the average of the value of the return is -3.562%. Similarly, with 1% probability, the daily returns could be as low as -4.309% with an average return of -6.217%, given that the return is less than -4.309%. Compared with the results obtained utilizing a Normal distribution, the dV aR0:95 is smaller with the estimation of the GPD. Nonetheless, the dV aR0:99 is higher in the case of the GPD, in comparison with the Normal distribution. In the case ofdES0:95 anddES0:99, both are higher using the GPD approximation. The di¤erence in 99.0% is notable and important (6.217% in the GPD, compared with 4.375% for the Normal distribution). 12 Finally, the non-parametric estimation is performed of the Hill tail-index and the quantile for negative stock market returns, expecting to …nd a stable region for the Hill estimator. The results related to the estimation of the parameter do not stabilize as the statistical order increases, due to which the estimator of the Hill tail-index is quite instable. 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[23] Zivot, E., andWang, J. (2006),Modelling Financial Times Series with S-PLUS, Second edition. Springer-Verlag, Carey, NC. 14 Figure 1. Daily Closing Prices of the Stock Market of Peru F-1 Figure 2. Daily Percentage Returns of the Stock Market of Peru F-2 -4 -2 0 2 4 z 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 H (z ) W eibull H(-0.5,0,1) Frechet H(0.5,0,1) Gumbel H(0,0,1) Figure 3. Generalized Extreme Value (GEV) CDFs for Fréchet, Weibull and Gumbel F-3 -4 -2 0 2 4 z 0. 0 0. 1 0. 2 0. 3 0. 4 h( z) Weibull H(-0.5,0,1) Frechet H(0.5,0,1) Gumbel H(0,0,1) Figure 4. Generalized Extreme Value (GEV) pdfs for Fréchet, Weibull and Gumbel F-4 Zn P ro ba bi lid ad -1 0 1 2 3 4 0. 1 0. 2 0. 3 Figure 5. GEV pdf for Daily Returns in Peru F-5 Quantiles of standard normal Q ua nt ile s of IG B V L -4 -2 0 2 4 -1 0 -5 0 5 10 15 Figure 6. Normal qq-plot for the Daily Percentage Returns in Peru F-6 1990 1994 1998 2002 2006 2010 2 4 6 8 10 12 0 2 4 6 8 10 12 14 0 2 4 6 8 Annual maximum 2 4 6 8 10 12 Annual maximum -1 0 1 2 3 4 - lo g( - lo g( pp oi nt s( Xn ))) 1 10 100 1000 Trial 5 10 15 R ec or ds Plot of Record Development Figure 7. Annual Block Maxima, Histogram, Gumbel qq-plot and Records Summary for the Daily Stock Returns in Peru F-7 Figure 8. Gumbel qq-plot of Stock Daily Returns in Peru F-8 rl pa rm ax 10 20 30 40 50 -6 1 -6 0 -5 9 -5 8 -5 7 -5 6 Figure 9. Asymptotic 95% Con…dence Interval for the 40 Year Return Level F-9 5 10 15 20 0 5 10 15 20 da ta Figure 10. Estimated 40-Year Return Level with 95% Con…dence Band for the Stock Daily Returns in Peru F-10 rl pa rm ax 10 15 20 25 30 35 -1 98 -1 97 -1 96 -1 95 -1 94 -1 93 -1 92 Figure 11. Asymptotic 95% Con…dence Interval for the 160-Quarterly Return Level F-11 0 20 40 60 80 0 10 20 30 40 da ta 0 10 20 30 40 da ta Figure 12. Estimated 160-Quarters Return Level with 95% Con…dence Band for the Stock Daily Returns in Peru F-12 yG (y ) 0 2 4 6 8 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Pareto G(0.5,1) Exponential G(0,1) Pareto II G(-0.5,1) y g( y) 0 2 4 6 8 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Pareto g(0.5,1) Exponential g(0,1) Pareto II g(-0.5,1) Figure 13. Generalized Pareto CDFs and pdfs for Pareto F-13 IGBVL negative returns 2 4 6 8 10 12 0 2 4 6 Ordered Data Ex po ne nt ia l Q ua nt ile s Figure 14. qq-plot with Exponential Reference distribution for the Stock Daily Negative Returns over the Threshold  = 1. F-14 -15 -10 -5 0 5 10 2 4 6 8 10 12 14 Threshold M ea n E xc es s Figure 15. Mean Excess Plot for the Stock Daily Negative Returns F-15 5 10 15 0. 0 0. 01 0. 02 0. 03 0. 04 0. 05 x 1- F( x) 5 10 15 20 0. 00 00 5 0. 00 05 0 0. 00 50 0 0. 05 00 0 x (on log scale) 1- F( x) (o n lo g sc al e) Figure 16. Diagnostic Plots for GPD Fit to Daily Negative Returns on Stock Index F-16 600 559 519 478 438 398 357 317 277 236 196 156 115 75 35 -0 .6 -0 .4 -0 .2 0. 0 0. 2 1.34 1.47 1.62 1.75 1.91 2.10 2.38 2.70 3.35 4.43 Exceedances S ha pe (x i) (C I, p = 0. 95 ) Threshold Figure 17. Estimates of the Shape Parameter for the Daily Negative Returns as a Function of the Threshold F-17 Threshold Es tim at e of x i 1.5 2.0 2.5 3.0 -0 .4 -0 .2 0. 0 0. 2 0. 4 0. 6 10 9 8 6 4 3 Percent Data Points above Threshold Figure 18. Estimates of the Shape Parameter with Time-Varying Threshold F-18 GPD Quantiles, for xi = 0.216792687971979 Ex ce ss o ve r t hr es ho ld 0 5 10 15 20 0 2 4 6 8 10 12 14 Upper Tail GPD Quantiles, for xi = 0.185278230304054 Ex ce ss o ve r t hr es ho ld 0 5 10 15 0 2 4 6 8 10 Lower Tail Figure 19. Estimated Tails when Distributions does not have Lower or Upper Limit F-19 1 5 10 0. 00 01 0. 00 10 0. 01 00 0. 10 00 x (on log scale) 1- F( x) (o n lo g sc al e) 99 95 Figure 20. Asymptotic Con…dence Intervals for V aR0:99 and ES0:99 based on the GDP Fit F-20 500 466 433 399 366 332 299 265 232 198 165 132 98 65 31 1 2 3 4 5 1.57 1.67 1.80 1.94 2.10 2.30 2.57 2.99 3.59 4.67 Exceedances 0. 99 Q ua nt ile (C I, p = 0. 95 ) Threshold Figure 21. Estimation of the V aR0:99 as a function of the Threshold F-21 50 113 185 257 329 401 473 545 617 689 761 833 905 977 1058 1148 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 6.45 4.10 3.24 2.72 2.39 2.12 1.86 1.70 1.58 1.42 1.30 1.19 1.11 Order Statistics xi (C I, p =0 .9 5) Threshold Figure 22. Estimates of the Hill  for the Daily Negative Returns F-22 ÚLTIMAS PUBLICACIONES DE LOS PROFESORES DEL DEPARTAMENTO DE ECONOMÍA Libros Máximo Vega-Centeno 2014 Del desarrollo esquivo al desarrollo sostenible. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. José Carlos Orihuela y José Ignacio Távara (Edt.) 2014 Pensamiento económico y cambio social: Homenaje Javier Iguíñiz. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Jorge Rojas 2014 El sistema privado de pensiones en el Perú. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Carlos Conteras (Edt.) 2014 El Perú desde las aulas de Ciencias Sociales de la PUCP. Lima, Facultad de Ciencias Sociales, Pontificia Universidad Católica del Perú. Waldo Mendoza 2014 Macroeconomía intermedia para América Latina. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Carlos Conteras (Edt.) 2014 Historia Mínima del Perú. México, El Colegio de México. Ismael Muñoz 2014 Inclusión social: Enfoques, políticas y gestión pública en el Perú. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Cecilia Garavito 2014 Microeconomía: Consumidores, productores y estructuras de mercado. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Alfredo Dammert Lira y Raúl García Carpio 2013 La Economía Mundial ¿Hacia dónde vamos? Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Piero Ghezzi y José Gallardo 2013 Qué se puede hacer con el Perú. Ideas para sostener el crecimiento económico en el largo plazo. Lima, Fondo Editorial de la Pontificia Universidad Católica del Perú y Fondo Editorial de la Universidad del Pacífico. Serie: Documentos de Trabajo No. 393 “Volatility of Stock Market and Exchange Rate Returns in Peru: Long Memory or Short Memory with Level Shifts?” Andrés Herrera y Gabriel Rodríguez. Diciembre, 2014. No. 392 “Stochastic Volatility in Peruvian Stock Market and Exchange Rate Returns: a Bayesian Approximation”. Willy Alanya y Gabriel Rodríguez. Diciembre, 2014. No. 391 “Territorios y gestión por resultados en la Política Social. El caso del P20 MIDIS”. Edgardo Cruzado Silverii. Diciembre, 2014. No. 390 “Convergencia en las Regiones del Perú: ¿Inclusión o exclusión en el crecimiento de la economía peruana”. Augusto Delgado y Gabriel Rodríguez. Diciembre, 2014. No. 389 “Driving Economic Fluctuations in Perú: The Role of the Terms Trade”. Gabriel Rodríguez y Peirina Villanueva. Diciembre, 2014. No. 388 “Aplicación de una metodología para el análisis de las desigualdades socioeconómicas en acceso a servicios de salud y educación en Perú en 2005- 2012”. Edmundo Beteta Obreros y Juan Manuel del Pozo Segura. Diciembre, 2014. No. 387 “Sobre la naturaleza multidimensional de la pobreza humana: propuesta conceptual e implementación empírica para el caso peruano”. Jhonatan A. Clausen Lizarraga y José Luis Flor Toro. Diciembre, 2014. No. 386 “Inflation Targeting in Peru: The Reasons for the Success”. Oscar Dancourt. Diciembre, 2014. No. 385 “An Application of a Short Memory Model With Random Level Shifts to the Volatility of Latin American Stock Market Returns”. Gabriel Rodríguez y Roxana Tramontana. Diciembre, 2014. No. 384 “The New Keynesian Framework for a Small Open Economy with Structural Breaks: Empirical Evidence from Perú”. Walter Bazán-Palomino y Gabriel Rodríguez. Noviembre, 2014. Departamento de Economía - Pontificia Universidad Católica del Perú Av. Universitaria 1801, Lima 32 – Perú. Telf. 626-2000 anexos 4950 - 4951 http://www.pucp.edu.pe/economia