Value-at-Risk Computed Using Various Methods (Historical Simulation, Delta-normal method, Delta-Normal method using the exponentially weighted covariance estimator, Historical simulation using rescaled returns)

By Dongliang “Larry” Yi

Problem:

1.     Historical Simulation

#thousand.000

VaR <- vector("numeric", length=246)

for (i in 1:246)

{

PLGBP = (1000/(data$GBPUSD[i:(i+1999)]*data$UKX[i:(i+1999)]))*data$GBPUSD[(i+1):(i+2000)]*data$UKX[(i+1):(i+2000)]-1000

PLEUR = (1000/(data$EURUSD[i:(i+1999)]*data$DAX[i:(i+1999)]))*data$EURUSD[(i+1):(i+2000)]*data$DAX[(i+1):(i+2000)]-1000

PLUSD = (1000/data$SPX[i:(i+1999)])*data$SPX[(i+1):(i+2000)]-1000

Total = PLGBP + PLEUR + PLUSD

VaR[i]<-abs(unname(quantile(Total,0.01,na.rm=T)))

}

VaR

#VaR stores the historical VaR

2.     Delta-normal method

VaRdelta <- vector("numeric", length=246)

for (i in 1:246)

{

  RFTGBP = (1/(data$GBPUSD[i:(i+1999)]*data$UKX[i:(i+1999)]))*data$GBPUSD[(i+1):(i+2000)]*data$UKX[(i+1):(i+2000)]-1

  RDAXEUR = (1/(data$EURUSD[i:(i+1999)]*data$DAX[i:(i+1999)]))*data$EURUSD[(i+1):(i+2000)]*data$DAX[(i+1):(i+2000)]-1

  RSP500 = (1/data$SPX[i:(i+1999)])*data$SPX[(i+1):(i+2000)]-1

  stdRSP500 = sqrt(var(RSP500))

  stdRDAXEUR = sqrt(var(RDAXEUR))

  stdRFTGBP = sqrt(var(RFTGBP))

  covSPFT = cov(RSP500,RFTGBP)

  covSPDA = cov(RSP500,RDAXEUR)

  covFTDA = cov(RFTGBP,RDAXEUR)

 

 stdPortfolio=sqrt(var(RSP500)+var(RDAXEUR)+var(RFTGBP)+2*cov(RSP500,RFTGBP)+2*cov(RSP500,RDAXEUR)+2*cov(RFTGBP,RDAXEUR))

 

  VaRdelta[i]=2.33*stdPortfolio*1000

 

}

VaRdelta

#below assume the average daily return is zero.

VaRdeltabook <- vector("numeric", length=246)

for (i in 1:246)

{

  RFTGBP = (1/(data$GBPUSD[i:(i+1999)]*data$UKX[i:(i+1999)]))*data$GBPUSD[(i+1):(i+2000)]*data$UKX[(i+1):(i+2000)]-1

  RDAXEUR = (1/(data$EURUSD[i:(i+1999)]*data$DAX[i:(i+1999)]))*data$EURUSD[(i+1):(i+2000)]*data$DAX[(i+1):(i+2000)]-1

  RSP500 = (1/data$SPX[i:(i+1999)])*data$SPX[(i+1):(i+2000)]-1

  stdRSP500 = sqrt((sum(RSP500^2))/2000)

  stdRDAXEUR = sqrt((sum(RDAXEUR^2))/2000)

  stdRFTGBP = sqrt((sum(RFTGBP^2))/2000)

  covSPFT = (sum(RSP500*RFTGBP))/2000

  covSPDA = (sum(RSP500*RDAXEUR))/2000

  covFTDA = (sum(RFTGBP*RDAXEUR))/2000

 

  stdPortfolio=sqrt((sum(RSP500^2))/2000+(sum(RDAXEUR^2))/2000+(sum(RFTGBP^2))/2000+2*covSPFT+2*covSPDA+2*covFTDA)

 

  VaRdeltabook[i]=2.33*stdPortfolio*1000

 

}

VaRdeltabook

3.     Delta-Normal method using the exponentially weighted covariance estimator

VaRdeltaexpo <- vector("numeric", length=246)

weight <- vector("numeric", length=100)

weight[100]=0.06

for (i in 1:99)

{

  weight[i]=(0.94^(100-i))*0.06

 

}

for (i in 1:246)

{

  RFTGBP = (1/(data$GBPUSD[(i+1900):(i+1999)]*data$UKX[(i+1900):(i+1999)]))*data$GBPUSD[(i+1901):(i+2000)]*data$UKX[(i+1901):(i+2000)]-1

  RDAXEUR = (1/(data$EURUSD[(i+1900):(i+1999)]*data$DAX[(i+1900):(i+1999)]))*data$EURUSD[(i+1901):(i+2000)]*data$DAX[(i+1901):(i+2000)]-1

  RSP500 = (1/data$SPX[(i+1900):(i+1999)])*data$SPX[(i+1901):(i+2000)]-1

  stdRSP500 = sqrt(sum((RSP500^2)*weight))

  stdRDAXEUR = sqrt(sum((RDAXEUR^2)*weight))

  stdRFTGBP = sqrt(sum((RFTGBP^2)*weight))

  covSPFT = sum((RSP500*RFTGBP)*weight)

  covSPDA = sum((RSP500*RDAXEUR)*weight)

  covFTDA = sum((RDAXEUR*RFTGBP)*weight)

  stdPortfolio=sqrt(sum((RSP500^2)*weight)+sum((RDAXEUR^2)*weight)+sum((RFTGBP^2)*weight)+2*sum((RSP500*RFTGBP)*weight)+2*sum((RSP500*RDAXEUR)*weight)+2*sum((RDAXEUR*RFTGBP)*weight))

 

  VaRdeltaexpo[i]=2.33*stdPortfolio*1000

 

}

VaRdeltaexpo

4.     Historical simulation using rescaled returns

VaRexpohist <- vector("numeric", length=246)

sigmaFT <- vector("numeric", length=2246)

sigmaDAX<- vector("numeric", length=2246)

sigmaSP <- vector("numeric", length=2246)

for (i in 1:2247)

{

  RFTGBP = (1/(data2$GBPUSD[i:(i+99)]*data2$UKX[(i):(i+99)]))*data2$GBPUSD[(i+1):(i+100)]*data2$UKX[(i+1):(i+100)]-1

  RDAXEUR = (1/(data2$EURUSD[(i):(i+99)]*data2$DAX[(i):(i+99)]))*data2$EURUSD[(i+1):(i+100)]*data2$DAX[(i+1):(i+100)]-1

  RSP500 = (1/data2$SPX[(i):(i+99)])*data2$SPX[(i+1):(i+100)]-1

  sigmaSP[i] = sqrt(sum((RSP500^2)*weight))

  sigmaDAX[i] = sqrt(sum((RDAXEUR^2)*weight))

  sigmaFT[i] = sqrt(sum((RFTGBP^2)*weight))

 

}

for (i in 1:246)

{

 

  PLGBP = ((1000/(data$GBPUSD[i:(i+1999)]*data$UKX[i:(i+1999)]))*data$GBPUSD[(i+1):(i+2000)]*data$UKX[(i+1):(i+2000)]-1000)*sigmaFT[i+2000]/sigmaFT[(i+1):(i+1999)]

  PLEUR = ((1000/(data$EURUSD[i:(i+1999)]*data$DAX[i:(i+1999)]))*data$EURUSD[(i+1):(i+2000)]*data$DAX[(i+1):(i+2000)]-1000)*sigmaDAX[i+2000]/sigmaDAX[(i+1):(i+1999)]

  PLUSD = ((1000/data$SPX[i:(i+1999)])*data$SPX[(i+1):(i+2000)]-1000)*sigmaSP[i+2000]/sigmaSP[(i+1):(i+1999)]

  Total = PLGBP + PLEUR + PLUSD

  VaRexpohist[i]<-abs(unname(quantile(Total,0.01,na.rm=T)))

}

VaRexpohist

5.     The result