VaR
2024-03-20
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# BIBLIOTECAS
#----------------
library(PerformanceAnalytics)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numeric##
## Attaching package: 'PerformanceAnalytics'## The following object is masked from 'package:graphics':
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## legendlibrary(quantmod)## Loading required package: TTR## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(rugarch)## Warning: package 'rugarch' was built under R version 4.3.3## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
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## sigmalibrary(car)## Loading required package: carDatalibrary(FinTS)## Warning: package 'FinTS' was built under R version 4.3.3library(scales)
library(data.table)##
## Attaching package: 'data.table'## The following objects are masked from 'package:xts':
##
## first, last#Datos
#---------------
symbols <- c("^GSPC")
getSymbols(symbols, from = "2000-01-03", to = "2021-12-31")## [1] "GSPC"GSPC.ret <- CalculateReturns(Cl(GSPC), method = "log") # rentabilidad Log
GSPC.ret <- GSPC.ret[-1,] #eliminamos la primera fila
colnames(GSPC.ret) <- "GSPC"# Análisis EDA
#----------------------------
table.Stats(GSPC.ret)## GSPC
## Observations 5534.0000
## NAs 0.0000
## Minimum -0.1277
## Quartile 1 -0.0047
## Median 0.0006
## Arithmetic Mean 0.0002
## Geometric Mean 0.0001
## Quartile 3 0.0058
## Maximum 0.1096
## SE Mean 0.0002
## LCL Mean (0.95) -0.0001
## UCL Mean (0.95) 0.0005
## Variance 0.0002
## Stdev 0.0124
## Skewness -0.4004
## Kurtosis 11.0561# Plot series
#-----------------------------
dataToPlot <- cbind(GSPC.ret, GSPC.ret^2,abs(GSPC.ret))
colnames(dataToPlot) <- c("Retornos", "Retornos^2", "abs(Retornos)")
plot.zoo(dataToPlot, main = paste(symbols, "Rentabilidades diarias"), col = "blue")
# Computar el rezago del retorno de ayer
GSPC.ret$yesterday.ret <- data.table::shift(GSPC.ret$GSPC, n = 1, type = "lag")
GSPC.ret <- na.omit(GSPC.ret)
# Test de autocorrelación
lm.autocorr <- lm(coredata(GSPC.ret$GSPC) ~ coredata(GSPC.ret$yesterday.ret))
# visualizar
# crear gráficos 2x2
par(mfrow = c(2,2))
# plot 1: Autocorrelación
plot(coredata(GSPC.ret$yesterday.ret), coredata(GSPC.ret$GSPC), col=scales::alpha("black",0.6), pch=16, cex=0.8, main="Autocorrelación de retornos", xlab="retorno t-1", ylab="retorno t")
abline(lm.autocorr, col="red")
legend("topleft", legend=c(paste0("Pendiente: ",round(summary(lm.autocorr)$coefficients[2,1],3)," ( t-value ",round(summary(lm.autocorr)$coefficients[2,3],3),")"),
paste0("R^2: ",round(summary(lm.autocorr)$r.squared,3))), bty="n")
box(col="grey")
# plot 2: serie histórica residuales
plot(lm.autocorr$residuals, type="l", main="Residuales")
# plot 3: distribución residuales
hist(lm.autocorr$residuals, breaks=64, main="PDF", col="grey", freq = FALSE, xlab="residuales")
rug(lm.autocorr$residuals)
box(col="grey")
lines(density(lm.autocorr$residuals, bw=0.004), col="red")
# plot 4: QQ-plot: revisión normalidad
car::qqPlot(lm.autocorr$residuals, main="QQ-Plot")
## [1] 5080 2206# Test de autocorrelación
# ----------------------------
par(mfrow=c(1,1))
summary(lm.autocorr)##
## Call:
## lm(formula = coredata(GSPC.ret$GSPC) ~ coredata(GSPC.ret$yesterday.ret))
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.117897 -0.004927 0.000563 0.005477 0.107994
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0002462 0.0001654 1.488 0.137
## coredata(GSPC.ret$yesterday.ret) -0.1126219 0.0133484 -8.437 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0123 on 5531 degrees of freedom
## Multiple R-squared: 0.01271, Adjusted R-squared: 0.01253
## F-statistic: 71.19 on 1 and 5531 DF, p-value: < 2.2e-16P valor = 0.000 < 0.05. Rechazamos H0 y tenemos un problema de autocorrelación.
# Correlogramas
#-------------------------
par(mfro = c(3,2))## Warning in par(mfro = c(3, 2)): "mfro" is not a graphical parameteracf(GSPC.ret$GSPC, main = " FAS ^GSPC Retornos")
pacf(GSPC.ret$GSPC, main = " PACF ^GSPC Retornos")
#-------------------------------------------------
acf(GSPC.ret$GSPC^2, main = " FAS ^GSPC Retornos^2")
pacf(GSPC.ret$GSPC^2, main = " FAS ^GSPC Retornos^2")
#----------------------------------------------------
acf(abs(GSPC.ret$GSPC), main = "^GSPC Abs Retornos")
pacf(abs(GSPC.ret$GSPC), main = "^GSPC Abs Retornos")
par(mfrow=c(1,1))
# Testeando por efectos ARCH/GARCH en retornos
#----------------------------------------------------------------
Box.test(coredata(GSPC.ret$GSPC^2), type="Ljung-Box", lag =12)##
## Box-Ljung test
##
## data: coredata(GSPC.ret$GSPC^2)
## X-squared = 6403.1, df = 12, p-value < 2.2e-16Los errores al cuadrado tienen autocorrelación significativa –> Proceso ARCH/GARCH
GARCH(1,1)
# Construir el modelo GARCH
#---------------------------------
spec = ugarchspec() # Default: GARCH(1,1)
fit = ugarchfit(data = GSPC.ret$GSPC, spec = spec)
fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000637 0.000087 7.3005 0.000
## ar1 0.742638 0.117394 6.3260 0.000
## ma1 -0.792624 0.106892 -7.4152 0.000
## omega 0.000002 0.000001 3.2905 0.001
## alpha1 0.124890 0.009814 12.7264 0.000
## beta1 0.857993 0.010230 83.8696 0.000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000637 0.000086 7.43654 0.000000
## ar1 0.742638 0.167454 4.43488 0.000009
## ma1 -0.792624 0.154175 -5.14107 0.000000
## omega 0.000002 0.000004 0.56761 0.570297
## alpha1 0.124890 0.028058 4.45108 0.000009
## beta1 0.857993 0.041024 20.91423 0.000000
##
## LogLikelihood : 17915.46
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.4737
## Bayes -6.4665
## Shibata -6.4737
## Hannan-Quinn -6.4712
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.004055 0.94923
## Lag[2*(p+q)+(p+q)-1][5] 3.967018 0.07233
## Lag[4*(p+q)+(p+q)-1][9] 6.410172 0.19555
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.8043 0.3698
## Lag[2*(p+q)+(p+q)-1][5] 5.5473 0.1147
## Lag[4*(p+q)+(p+q)-1][9] 7.1614 0.1860
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.04944 0.500 2.000 0.8240
## ARCH Lag[5] 2.11410 1.440 1.667 0.4466
## ARCH Lag[7] 2.70822 2.315 1.543 0.5703
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 23.5854
## Individual Statistics:
## mu 0.57419
## ar1 0.04229
## ma1 0.03849
## omega 2.13973
## alpha1 0.37355
## beta1 0.90989
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 3.6308 2.851e-04 ***
## Negative Sign Bias 0.8713 3.836e-01
## Positive Sign Bias 2.6190 8.843e-03 ***
## Joint Effect 42.4167 3.273e-09 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 205.4 2.927e-33
## 2 30 234.1 6.035e-34
## 3 40 281.1 2.057e-38
## 4 50 293.9 1.230e-36
##
##
## Elapsed time : 0.5137441# Plot
# -----------------------------------
plot(fit, which = "all")##
## please wait...calculating quantiles...
# media incondicional
uncmean(fit)## [1] 0.0006372291# varianza incondicional: omega/(alpha1 + beta1)
uncvariance(fit)## [1] 0.0001386205# persistencia: alpha1 + beta1
persistence(fit)## [1] 0.9828833# half-life: tiempo (días) que toma recuperar la mitad del incremento en la varianza
halflife(fit)## [1] 40.14784# Plot
#----------------------
dataToPlot <- cbind(residuals(fit), sigma(fit))
colnames(dataToPlot) <- c("Residual", "Volatilidad_condicional")
plot.zoo(dataToPlot, main = paste(symbols, "Retornos Diarios"), col = "blue")
SIMULACIÓN
garch11.sim = ugarchsim(fit, n.sim=nrow(GSPC.ret), rseed=12345)
# plotear retornos reales y simulados
par(mfrow=c(2,1))
plot(GSPC.ret$GSPC, main=paste(symbols,"Retornos realizados"), grid.col=NA, lwd=1)
plot(as.xts(garch11.sim@simulation$seriesSim,
order.by=index(GSPC.ret)),
main="GARCH(1,1) Retornos Simulados", grid.col=NA, lwd=1)
par(mfrow=c(1,1))
precio.actual = as.numeric(head(GSPC$GSPC.Adjusted[1]))
precio.futuro.sim1 <- precio.actual*(cumprod(1+garch11.sim@simulation$seriesSim))
GSPC$precio.futuro.sim1[3:nrow(GSPC)] <- precio.futuro.sim1
GSPC <-na.omit(GSPC)
plot.sim <- plot(GSPC$precio.futuro.sim1["2000"], type="l", ylim=c(min(GSPC$precio.futuro.sim1["2000"],GSPC$GSPC.Close["2000"]),
max(GSPC$precio.futuro.sim1["2000"],GSPC$GSPC.Close["2000"])),main="Futuro alterno simulado - data sintética", grid.col=NA)
plot.sim <- lines(GSPC$GSPC.Close, col="red", on=1)
plot.sim <- addLegend("topleft", legend.names=c("Realizado","Simulado"), col=c("red","black"), lwd=2)
plot.sim
CONCLUSIONES
En la práctica, el modelo GARCH(1,1) con innovaciones Normales no se adapta adecuadamente.
Importantes limitaciones:
La especificación de la función generadora modelada ¿es la más aproximada a la volatilidad empírica? Sobre la distribución de los errores, ¿es la mas adecuada? No permite la asimetría. El mercado no reacciona de la misma manera a las noticias positivas y negativas. El modelo debería acomodar la posibilidad de saltos. Conclusión: Es necesario realizar un fit iterativo que pruebe distintos modelos, en distintas configuraciones, incluyendo otras distribuciones de probabilidad. Entonces, elegir el modelo que minimiza el error.