Homework 4 Problem 3
Eva Nurwita
March 23, 2016
Problem 3
First we need to import EUR/USD exchange rate into R and do a log transformation:
exrate1 <- Quandl("ECB/EURUSD", type="zoo")
exrate <- exrate1[,4]
lexrate <- diff(log(exrate))EWMA and GARCH(1,1) Volatility Model
For the Exponentially Weighted Moving Average or EWMA, we use the following code to calculate and plot EWMA:
lexrate.ewma <- EWMA((lexrate-mean(lexrate))^2,lambda=.05)
plot(lexrate.ewma, type="l", main="Exponentially Weighted Moving Average")
GARCH(1,1) Estimation
For GARCH(1,1) estimation with normal innovation, we use the following code:
garch11 <- garchFit( ~ garch(1,1), data=coredata(lexrate), trace=FALSE)
summary(garch11)
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = coredata(lexrate), trace = FALSE)
Mean and Variance Equation:
data ~ garch(1, 1)
<environment: 0x0000000012944e60>
[data = coredata(lexrate)]
Conditional Distribution:
norm
Coefficient(s):
mu omega alpha1 beta1
4.8764e-05 1.2316e-07 2.8535e-02 9.6896e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 4.876e-05 8.531e-05 0.572 0.56760
omega 1.232e-07 4.199e-08 2.933 0.00336 **
alpha1 2.854e-02 3.064e-03 9.313 < 2e-16 ***
beta1 9.690e-01 3.145e-03 308.091 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log Likelihood:
16250.51 normalized: 3.684087
Description:
Fri Mar 25 23:09:23 2016 by user: evanu
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 467.2629 0
Shapiro-Wilk Test R W 0.9897908 0
Ljung-Box Test R Q(10) 6.083835 0.8081713
Ljung-Box Test R Q(15) 15.8161 0.3943783
Ljung-Box Test R Q(20) 18.80033 0.5348369
Ljung-Box Test R^2 Q(10) 6.912306 0.733697
Ljung-Box Test R^2 Q(15) 7.365904 0.9466897
Ljung-Box Test R^2 Q(20) 12.06334 0.9138786
LM Arch Test R TR^2 7.00247 0.8574503
Information Criterion Statistics:
AIC BIC SIC HQIC
-7.366361 -7.360565 -7.366363 -7.364317 5% Var and 1% Var
Calculating 1% VaR and 5% VaR for EWMA and GARCH(1,1):
var5ewma <- mean(lexrate) - 1.645 * sqrt(lexrate.ewma)
var5garch <- mean(lexrate) - 1.645 * garch11@sigma.t
var1ewma <- mean(lexrate) - 2.326 * sqrt(lexrate.ewma)
var1garch <- mean(lexrate) - 2.326 * garch11@sigma.tPlotting our previous calculation:
timeindex <- index(lexrate)
par(mfrow=c(2,2), cex=0.6, mar=c(2,2,3,1))
plot(timeindex, lexrate,type="l", main ="5% VaR EWMA")
lines(timeindex, var5ewma, col = "blue")
plot(timeindex, lexrate,type="l", main ="1% VaR EWMA")
lines(timeindex, var1ewma, col = "blue")
plot(timeindex, lexrate,type="l", main ="5% VaR GARCH")
lines(timeindex, var5garch, col ="blue")
plot(timeindex, lexrate, type="l", main ="1% VaR GARCH")
lines(timeindex, var1garch, col ="blue")
Calculating fraction of times in the sample where the log change fall below the 5% VaR and 1% VaR.
sum(lexrate < var5ewma)/length(lexrate) # fraction of sample where loss exceeds EWMA 5% VaR[1] 0.04624802sum(lexrate < var1ewma)/length(lexrate) # fraction of sample where loss exceeds EWMA 1% VaR[1] 0.007481297sum(lexrate < var5garch)/length(lexrate) # fraction of sample where loss exceeds GARCH 5% VaR[1] 0.04670143sum(lexrate < var1garch)/length(lexrate) # fraction of sample where loss exceeds GARCH 1% VaR[1] 0.01314895Largest Depreciation
Code for calculation and time identification:
min(lexrate) # onservation with the largest depreciation of EUR to USD[1] -0.04735441index(lexrate[lexrate==min(lexrate)]) # day of the largest depreciation[1] "2008-12-19"min(var5ewma) # largest depreciation by EWMA 5% VaR[1] -0.02911564min(var1ewma) # largest depreciation by EWMA 1% VaR[1] -0.04116379min(var5garch) # largest depreciation by GARCH 5% VaR[1] -0.02439871min(var1garch) # largest depreciation by GARCH 1% VaR[1] -0.03449413The largest depreciation for EUR compared to USD happened on 12/19/2008.