PSM I_Week 6_Contextual
Kanaya Azlia
2023-09-29
Financial Returns: Normal Distribution
library(readxl)
week6 = read_excel("TLKM.JK.xlsx")
stock.return = week6$Return*100Finding Mean and Variance
stock.mean = mean(stock.return)
stock.mean## [1] 0.001097486stock.var = var(stock.return)
stock.var## [1] 3.3989875% Quantile
The Lowest 5% in Return Percentage
lower.05 = qnorm(0.05, stock.mean, sqrt(stock.var))
lower.05## [1] -3.031411The Highest 5% in Return Percentage
high.05 = qnorm(0.95, stock.mean, sqrt(stock.var))
high.05## [1] 3.033606The Probability of Having No Return (return = 0)
return.0 = dnorm(0, stock.mean, sqrt(stock.var))
return.0## [1] 0.216389The Probability of Loss (return < 0)
loss = pnorm(0, stock.mean, sqrt(stock.var))
loss## [1] 0.4997625The Probability of Getting Profit (return > 0)
profit = 1 - pnorm(0, stock.mean, sqrt(stock.var))
profit## [1] 0.5002375Loss Distribution: Pareto Distribution
library(CASdatasets)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: sp## The legacy packages maptools, rgdal, and rgeos, underpinning the sp package,
## which was just loaded, will retire in October 2023.
## Please refer to R-spatial evolution reports for details, especially
## https://r-spatial.org/r/2023/05/15/evolution4.html.
## It may be desirable to make the sf package available;
## package maintainers should consider adding sf to Suggests:.
## The sp package is now running under evolution status 2
## (status 2 uses the sf package in place of rgdal)library (fitdistrplus)## Loading required package: MASS## Loading required package: survivallibrary(MASS)data(danishuni)
head(danishuni)## Date Loss
## 1 1980-01-03 1.683748
## 2 1980-01-04 2.093704
## 3 1980-01-05 1.732581
## 4 1980-01-07 1.779754
## 5 1980-01-07 4.612006
## 6 1980-01-10 8.725274Data used in this distribution fitting is from a data set called “danishuni”, containing of Danish reinsurance claim data. The smallest value of this data is is 1.00, while the largest value is 263.250.
x = danishuni$Loss
head(x, n = 96)## [1] 1.683748 2.093704 1.732581 1.779754 4.612006 8.725274
## [7] 7.898975 2.208045 1.486091 2.796171 7.320644 3.367496
## [13] 1.464129 1.722223 11.374817 2.482739 26.214641 2.002430
## [19] 4.530015 1.841753 3.806735 14.122076 5.424253 11.713031
## [25] 1.515373 2.538589 2.049780 12.465593 1.735445 1.683748
## [31] 3.323852 1.821025 2.415813 1.464129 5.875902 1.792439
## [37] 3.338214 1.584488 1.931321 2.049780 1.573758 1.756955
## [43] 4.931186 5.417277 1.537335 17.569546 1.691123 1.581612
## [49] 1.467116 2.049780 1.928712 3.263154 1.704246 2.074608
## [55] 3.294290 1.839975 7.320644 7.320644 1.576215 4.856098
## [61] 1.542275 13.620791 1.756955 1.639824 2.036378 21.961933
## [67] 1.799026 2.452416 2.669107 2.024146 1.724744 1.643864
## [73] 1.961933 4.626647 1.464129 3.963250 5.563852 4.392387
## [79] 3.640313 1.677892 1.449488 263.250366 2.500732 1.825769
## [85] 9.882870 2.146618 2.870259 6.319914 2.123461 2.923865
## [91] 2.036376 1.736457 2.049780 1.903367 1.610542 1.717423summary(x)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 1.321 1.778 3.385 2.967 263.250Histogram
From histogram below, we know that this data distribution mostly pooled in the smaller value below 50 rater that above.
hist(x, breaks = 100,main = "Histogram of Loss", xlab = "Loss")
lines(density(x), lty = 5, col = "red")
We will first fit this data set with gamma distribution using multiple methods, such as:
Maximum Likelihood Estimation (MLE)
Moment Matching Estimation (MME)
Quantile Matching Estimation (QME)
Maximum Goodnes-of-Fit Estimation (MGE)
Maximum Likelihood Estimation
fgamMLE <- fitdist (x, "gamma", method="mle")
summary(fgamMLE)## Fitting of the distribution ' gamma ' by maximum likelihood
## Parameters :
## estimate Std. Error
## shape 1.2976762 0.03548677
## rate 0.3833939 0.01273545
## Loglikelihood: -4767.096 AIC: 9538.191 BIC: 9549.554
## Correlation matrix:
## shape rate
## shape 1.0000000 0.8232389
## rate 0.8232389 1.0000000Moment Matching Estimation
fgamMME <- fitdist (x, "gamma", method="mme")
summary(fgamMME)## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.15839499
## rate 0.04679198
## Loglikelihood: -6665.992 AIC: 13335.98 BIC: 13347.35Quantile Matching Estimation
fgamQME <- fitdist (x, "gamma", method="qme", probs=c (0.95, 0.05))
summary(fgamQME)## Fitting of the distribution ' gamma ' by matching quantiles
## Parameters :
## estimate
## shape 0.13187546
## rate 0.07444676
## Loglikelihood: -7023.213 AIC: 14050.43 BIC: 14061.79Maximum Goodness-of-Fit Estimation
fgamMGE <- fitdist (x, "gamma", method="mge", gof="CvM")
summary(fgamMGE)## Fitting of the distribution ' gamma ' by maximum goodness-of-fit
## Parameters :
## estimate
## shape 3.750012
## rate 1.772281
## Loglikelihood: -6882.448 AIC: 13768.9 BIC: 13780.26Model Fitting Comparison
We try to compare, which method is more suitable by comparing each of the AIC and BIC values. The smaller the value, the more suitable the method.
data.frame("Method" = c("MLE", "MME", "QME", "MGE"),
"AIC" = c(fgamMLE$aic, fgamMME$aic, fgamQME$aic, fgamMGE$aic),
"BIC" = c(fgamMLE$bic, fgamMME$bic, fgamQME$bic, fgamMGE$bic),
"Log-Likelihood" = c(fgamMLE$loglik, fgamMME$loglik,
fgamQME$loglik, fgamMGE$loglik),
check.names=FALSE)## Method AIC BIC Log-Likelihood
## 1 MLE 9538.191 9549.554 -4767.096
## 2 MME 13335.984 13347.346 -6665.992
## 3 QME 14050.426 14061.788 -7023.213
## 4 MGE 13768.895 13780.257 -6882.448In comparing both AIC and BIC value, MLE seems like the most suitable model. From this statement, we will use Gamma Distributed Loss with MLE method with the following parameters:
fgamMLE$estimate## shape rate
## 1.2976762 0.38339395% Quantile
The Lowest 5% in Loss
lower.05 = qgamma(0.05, fgamMLE$estimate[1], fgamMLE$estimate[2])
lower.05## [1] 0.306883The Highest 5% in Loss
high.05 = qgamma(0.95, fgamMLE$estimate[1], fgamMLE$estimate[2])
high.05## [1] 9.260058