Report on lab session, Monday 27 February, 5-6pm

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1 Report on la session, Monday 7 Feruary, 5-6pm Read the data as follows. The variale of interest is the adjusted closing price of Coca Cola in the 7th column of the data file KO.csv. dat=read.csv("ko.csv",header=true) View(dat) price=dat[,7] Compute the log returns. logreturns=diff(log(price)) Compute the ES y the historical method n=length(logreturns) pp=((seq(1,n))-0.5)/n EShist=cumsum(-sort(logreturns))/(floor(n*pp)+0.5) Fitting of the distriutions (Cauchy, Laplace and Student s t) requires downloading of the following packages. install.packages("vgam") lirary(vgam) install.packages("metrology") lirary(metrology) install.packages("adequacymodel") lirary(adequacymodel) Fit the two-parameter Cauchy distriution to the log returns and compute the ES Note that the two-parameter Cauchy distriution is given y the pdf f(x) = σ π [(x µ) + σ ] for < x < +, < µ < + and σ > 0. den=function(par,x){dcauchy(x,location=par[1],scale=par[])} cum=function(par,x){pcauchy(x,location=par[1],scale=par[])} med1=suppresswarnings(goodness.fit(pdf=den, cdf=cum, starts=c(0,1), data=logreturns, method="bfgs")) p=seq(0.01,0.99,0.01) EScauchy=p for (i in 1:length(p)) {EScauchy[i]=-0.01*sum(qcauchy(p[i]*seq(0.01,1,0.01), location=med1$mle[1],scale=med1$mle[]))} 1

2 Now fit the two-parameter Laplace distriution and compute the ES. Note that the two-parameter Laplace distriution is given y the pdf f(x) = 1 ( σ exp x µ ) σ for < x < +, < µ < + and σ > 0. den=function(par,x){dlaplace(x,location=par[1],scale=par[])} cum=function(par,x){plaplace(x,location=par[1],scale=par[])} med=suppresswarnings(goodness.fit(pdf=den, cdf=cum, starts=c(0,1), data=logreturns, method="bfgs")) ESlap=p for (i in 1:length(p)) {ESlap[i]=-0.01*sum(qlaplace(p[i]*seq(0.01,1,0.01), location=med$mle[1],scale=med$mle[]))} Now fit the three-parameter Student s t distriution and compute the ES. Note that the three-parameter Student s t distriution is given y the pdf ( ) f(x) = Γ ν+1 [ ] ν+1 σ νπγ ( (x µ) ) ν 1 + σ ν for < x < +, < µ < +, ν > 0 and σ > 0. den=function(par,x){dt.scaled(x,df=par[1],mean=par[],sd=par[3])} cum=function(par,x){pt.scaled(x,df=par[1],mean=par[],sd=par[3])} med3=suppresswarnings(goodness.fit(pdf=den, cdf=cum, starts=c(1,0,1), data=logreturns, method="bfgs")) ESt=p for (i in 1:length(p)) {ESt[i]=-0.01*sum(qt.scaled(p[i]*seq(0.01,1,0.01), df=med3$mle[1],mean=med3$mle[],sd=med3$mle[3]))} Now plot the estimates of ES otained y the historical method, fit of the Cauchy distriution, fit of the Laplace distriution and the fit of the Student s t distriution. yrange=range(eshist,escauchy,eslap,est) yrange[]=0.4 xrange=range(p,pp) plot(pp,eshist,xla="p",yla="estimates of ES",type="l",xlim=xrange,ylim=yrange) par(new=true) plot(p,escauchy,xla="",yla="",type="l",xlim=xrange,ylim=yrange,col="red") par(new=true) plot(p,eslap,xla="",yla="",type="l",xlim=xrange,ylim=yrange,col="lue") par(new=true) plot(p,est,xla="",yla="",type="l",xlim=xrange,ylim=yrange,col="rown") legend(0.,0.4,legend=c("historical estimator","cauchy fit","laplace fit","t fit"), col=c("lack","red","lue","rown"),lty=1)

3 You will see the following plot Estimates of ES Historical estimator Cauchy fit Laplace fit t fit p Figure 1: Non-parametric and parametric estimates of ES for the log returns of Coca Cola stock prices. The Student s t distriution gives the est fit, followed y Laplace and then the Cauchy. The fits of the three distriutions can also e assessed using the outputs of med1, med and med3. med1 med med3 gives the following output. The output for the fit of the Cauchy distriution: $W [1].5956 $A [1] $KS One-sample Kolmogorov-Smirnov test 3

4 data: data D = , p-value = 8.11e-06 alternative hypothesis: two-sided $mle [1] $AIC [1] $ CAIC [1] $BIC [1] $HQIC [1] $Erro [1] $Value [1] $Convergence [1] 0 The output for the fit of the Laplace distriution: $W [1] $A [1] $KS One-sample Kolmogorov-Smirnov test data: data D = , p-value = alternative hypothesis: two-sided $mle 4

5 [1] $AIC [1] $ CAIC [1] $BIC [1] $HQIC [1] $Erro [1] $Value [1] $Convergence [1] 0 The output for the fit of the Student s t distriution: $W [1] $A [1] $KS One-sample Kolmogorov-Smirnov test data: data D = , p-value = alternative hypothesis: two-sided $mle [1] $AIC [1] $ CAIC [1]

6 $BIC [1] $HQIC [1] $Erro [1] $Value [1] $Convergence [1] 0 The output includes the following: maximum likelihood estimates of the parameters (mle), standard errors (Erro), 95 percent confidence intervals, value of Cramer-von Misses statistic (W), value of Anderson Darling statistic (A), value of Kolmogorov Smirnov test statistic (KS) and its p-value, value of Akaike Information Criterion (AIC), value of Consistent Akaike Information Criterion (CAIC), value of Bayesian Information Criterion (BIC), value of Hannan-Quinn information criterion (HQIC), minimum value of the negative loglikelihood (Value) function and convergence status. Smaller the values of Cramer-von Misses statistic, Anderson Darling statistic, Kolmogorov Smirnov test statistic, Akaike Information Criterion, Consistent Akaike Information Criterion, Bayesian Information Criterion, Hannan-Quinn information criterion and minimum of the negative log-likelihood the etter the fit. The Student s t distriution gives the smallest values for these and hence provides the est fit. Can you find other distriutions giving even etter fits than the Student s t distriution? Try fitting the following distriutions: Gumel distriution, g = gumel, starts = c(loc = a, scale = ) and g(x) = 1 ( exp x a ) [ ( exp exp x a )] for < x < +, < a < + and > 0. The contriuted R package evd is used to compute this pdf g. Skew normal distriution, g = sn, starts = c(xi = a, omega =, alpha = c) and ( ) ( x a f(x) = φ Φ c x a ) for < x < +, < a < +, > 0 and < c < +, where φ( ) and Φ( ) denote, respectively, the pdf and cdf of the standard normal distriution. The contriuted R package sn is used to compute this pdf g. Skew t distriution, g = st, starts = c(xi = a, omega =, alpha = c, df = n) and ( ) Γ n+ [ f(x) = π n(n + 1)Γ ( n ) 1 + ] n+1 [ ] (x a) x n+ n 1 + y dy n + 1 6

7 for < x < +, < a < +, > 0, < c < + and n > 0, where x = c x a. The contriuted R package sn is used to compute this pdf g. n+1 (x a) +n Asymmetric Laplace distriution, g = ALD, starts = c(mu = a, sigma =, p = p) and [ ( )] p(1 p) x a f(x) = exp ρ p for < x < +, < a < +, > 0 and 0 < p < 1, where ρ p (u) = u [p I {u < 0}]. The contriuted R package ald is used to compute this pdf g. Normal Laplace distriution, g = nl, starts = c(mu = a, sigma =, alpha = c, eta = d) and f(x) = cd ( ) [ ( x a c + d φ R c x a ) ( + R d + x a )] for < x < +, < a < +, > 0, c > 0 and d > 0, where R(u) = [1 Φ(u)] /φ(u). The contriuted R package NormalLaplace is used to compute this pdf g. Generalized logistic distriution, g = glogis, starts = c(location = a, scale =, shape = c) and f(x) = c ( exp x a ) [ ( 1 + exp x a )] c 1 for < x < +, < a < +, > 0, and c > 0. The contriuted R package glogis is used to compute this pdf g. Exponential power distriution, g = normp, starts = c(mu = a, sigmap =, p = c) and [ 1 f(x) = ( ) exp 1 ( ) x a c ] c 1/c Γ c c for < x < +, < a < +, > 0, and c > 0. The contriuted R package normalp is used to compute this pdf g. Ex Gaussian distriution, g = exgaus, starts = c(mu = a, sigma =, nu = c) and f(x) = 1 ( c exp a x ) ( + x a c c Φ exp ) c for < x < +, < a < +, > 0 and c > 0. The contriuted R package gamlss is used to compute this pdf g. Sinh arcsinh distriution, g = SHASH, starts = c(mu = a, sigma =, nu = c, tau = d) and f(x) = β exp ( α / ) π + (x a) 7

8 for < x < +, < a < +, > 0, c > 0 and d > 0, where α = 1 { [ ( )] [ ( )]} x a x a exp d arcsinh exp c arcsinh and β = 1 { [ d exp d arcsinh ( x a )] [ + c exp c arcsinh ( x a )]}. The contriuted R package gamlss is used to compute this pdf g. 8