第五章 結論與建議
第二節 後續建議
本文對於研究仍有許多遺漏之處，例如：礙於資料庫之限制，故研究範圍僅五 年，考量衡量風險應以長期與多元評估，若能加長研究範圍、增加標的資產或是加 進股利之計算，應可助於對應用 t 分配於財金領域有更全面之發展，期望後續研究 者能加以改進，做更具完整且信度更高之研究。
R 語言之功能齊全，與一般市售套裝統計軟體可說是有過之而無不及，其乃集 世界著名學家與權威之大成，若能普及於台灣學界，必定能給予更多研究者幫助，
故期待 R 語言能更加利用於各種學術研究上，望後續研究者亦能多多善用 R 語言。
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附錄
一、R 語言程式
(1) pdf 之繪製
#尺度標準化之 t 分配理論 pdf windows()
x = seq(-5,5,length=10000) #建立一組有 10000 個-5 到 5 之間的樣本空間 plot(x,dt(x,3),type="l",lwd="3",col="1",ylim=c(0,0.45),main="",xlab="", ylab="機率密度" )
lines(x,dt(x,10),lty=2,lwd="3",col="2") lines(x,dt(x,30),lty=3,lwd="3",col="3") lines(x,dnorm(x),lty=4,lwd="3",col="4")
legend("topleft",lty=c(1,2,3,4),lwd="3",col=c("1","2","3","4"), legend=c(expression(paste(nu," = 3")),
expression(paste(nu," = 10")), expression(paste(nu," = 30")),
expression(paste(nu," = +",infinity," = 標準常態分配"))),bty="n")
#尺度標準化之 t 分配實際 pdf windows()
x = seq(-5,5,length=10000) set.seed(12345)
t1 = rt(10000,3)
plot(density(t1),type="l",lwd="3",col="1",xlim=c(-10,10),ylim=c(0,0.40),main="",xlab="", ylab="機率密度" )
t2 = rt(10000,10)
lines(density(t2),lty=2,lwd="3",col="2") t3 = rt(10000,30)
lines(density(t3),lty=3,lwd="3",col="3") lines(x,dnorm(x),lty=4,lwd="3",col="4")
legend("topleft",lty=c(1,2,3,4),lwd="3",col=c("1","2","3","4"),
legend=c(expression(paste(nu," = 3")), expression(paste(nu," = 10")), expression(paste(nu," = 30")),
expression(paste(nu," = +",infinity," = 標準常態分配"))),bty="n")
(2) 機率估計 library(quantmod)
twii = getSymbols("^ TWII",scr="yahoo",from="2010-09-01",to="2015-09-01") #yahoo 資料庫抓取台灣加權股價指數資料
p = coredata(twii$TWII.Open) #去時間化 twiir = 100*diff(log(p)) #轉換成對數報酬率 mu = mean(twiir) #平均數
sigma = sd(twiir) #標準差
# 常態分配
z1a = (-2-mu)/sigma z1b = (2-mu)/sigma
pnorm(z1b)-pnorm(z1a) # 落於-2%到 2%間之機率 z2 = (-5-mu)/sigma
pnorm(z2) # 小於-5%之機率
#t 分配
nu = 4.3841 #自由度
# 使用尺度標準化
# 尺度 = 變異數*(nu-2)/nu lambda^2 = var(twiir)*(nu-2)/nu t1a = (-2-mu)/sqrt(lambda^2) t1b = (0-mu)/sqrt(lambda^2)
# 介於-2%與 2%之間的機率 pt(t1b,nu)-pt(t1a,nu)
t2 = (-5-mu)/sqrt(lambda^2)
# 小於-5%的機率 pt(t2,nu)
(3) 常態分配之對數概似函數求法
#使用 fitdistr 函數
fitnorm = fitdistr(twiir,"normal") summary(fitnorm)
fitnorm$estimate fitnorm$estimate[2]
fitnorm$loglik sqrt(fitnorm$vcov) fitnorm$sd
#使用 nlm 函數
fn <- function (theta,x) #常態分配之 pdf {
m = theta[1]
s = abs(theta[2]) s2 = s^2
0.5*log(2*pi)+0.5*log(s2)+ (0.5/s2)*mean((x-m)^2) }
theta0 = c(0,1) # 設期初值
fitnorm1 = nlm(fn, theta <- theta0, x= twiir) fitnorm1$iterations
-T*fitnorm1$minimum #需取負號即為對數概似值
(4) t 分配之對數概似函數求法
#使用 fitdistr 指令求解
fitt = fitdistr(twiir,"t") #計算台灣加權股價以 t 分配為基礎之概似函數 fitt$estimate #其值分別為平均值、尺度以及自由度
fitt$sd #標準差
fitt$vcov #所有估計值之共變異數 fitt$loglik #對數概似函數
fitt$n #樣本數
# 使用 stdFit 指令求解 library(fGarch)
model = stdFit(twiir) #計算台灣加權股價其 t 分配之概似函數 model$par #其值分別為平均值、標準誤以及自由度
-model$objective #需取負值即為對數概似值
# 使用 nlm 指令求解 mlogt <- function(b, x) {
sum(-dstd(x,mean=b[1],sd=b[2],nu = b[3],log = TRUE)) #以 dstd 取代 t 分配之 pdf }
b = c(0,1,3)
out2 = nlm(mlogt,b,x=y1,hessian=T)
-out2$minimum #需取負值即為對數概似值
(5) 計算訊息決策準則 AIC、BIC
#AIC、BIC
#常態分配 T = length(twiir)
fitnorm = fitdistr(twi.r,"normal") summary(fitnorm)
fitnorm$loglik
AIC1 = -2*fitnorm$loglik + 2*3 BIC1 = -2*fitnorm$loglik + log(T)*3 AIC1
BIC1
#t 分配
T = length(twi.r)
fitt = fitdistr(twiir,"t") summary(fitt)
fitt$loglik
AIC1 = -2*fitt$loglik + 2*3 BIC1 = -2*fitt$loglik + log(T)*3 AIC1
BIC1
(6) 風險值計算
#風險值
model1 = stdFit(twiir) param1 = c(model1$par) nu=param1[3]
mu = param1[1]
sigma = param1[2]
alpha = 0.05
mean(twiir)+sd(twiir)*qnorm(alpha) #常態法 quantile(twiir,alpha) #歷史法
mu+sigma*qt(alpha,nu)/sqrt(nu/(nu-2)) #t 分配法
alpha = 0.01
mean(twiir)+sd(twiir)*qnorm(alpha) #常態法 quantile(twiir,alpha) #歷史法
mu+sigma*qt(alpha,nu)/sqrt(nu/(nu-2)) #t 分配法
alpha = 0.001
mean(twiir)+sd(twiir)*qnorm(alpha) #常態法 quantile(twiir,alpha) #歷史法
mu+sigma*qt(alpha,nu)/sqrt(nu/(nu-2)) #t 分配法
#預期損失
#常態分配與歷史法之 ES alpha = c(0.05,0.01,0.001) z = (twiir-mean(twiir))/sd(twiir) esnormal(alpha,mu=0,sigma=1) es = mean(twiir[index])
es
index = twiir < quantile(twiir,0.01) es = mean(twiir[index])
es
index = twiir < quantile(twiir,0.001) es = mean(twiir[index])
es
#t 分配之 ES
modelt = stdFit(twiir) param1t = modelt$par
nuhat = as.numeric(param1t)[3]
esT(alpha,n=nuhat)
(f(qt(alpha))/alpha)*((nu+qt(alpha)^2)/(nu-1)) qt05 = qt(0.05,df=nuhat)
qt01 = qt(0.01,df=nuhat) qt001 = qt(0.001,df=nuhat)
-((dt(qt05,df=nuhat)/0.05)*((nuhat+qt05^2)/(nuhat-1))) -((dt(qt01,df=nuhat)/0.01)*((nuhat+qt01^2)/(nuhat-1))) -((dt(qt001,df=nuhat)/0.001)*((nuhat+qt001^2)/(nuhat-1)))
## -(mu + lambda*(f(qt(alpha))/alpha)*((nu+qt(alpha)^2)/(nu-1))) muhat = as.numeric(param1t)[1]
sigmahat = as.numeric(param1t)[2]
-(muhat+sigmahat*(dt(qt05,df=nuhat)/0.05)*((nuhat+qt05^2)/(nuhat-1))) -(muhat+sigmahat*(dt(qt01,df=nuhat)/0.01)*((nuhat+qt01^2)/(nuhat-1))) -(muhat+sigmahat*(dt(qt001,df=nuhat)/0.001)*((nuhat+qt001^2)/(nuhat-1)))
二、一致性風險衡量工具
Artzner (1999) 提出一致性風險衡量工具，並以數理之觀點，提出風險測度的公 式化理論：假設有兩個投資組合 X、Y，並令 𝜌( ) 為風險衡量工具，如果 𝜌( ) 滿 足下列 4 個性質，即可稱 𝜌( ) 為具有一致性的風險衡量工具。
1. 單調性 (Monotonicity)：𝑌 ≥ 𝑋 ≈ 𝜌(𝑌) ≤ 𝜌(𝑋) 表示較低報酬的資產，具有較高風險。
2. 次可加性 (Subadditivity)：𝜌(𝑋 + 𝑌) ≤ 𝜌(𝑋) + 𝜌(𝑌) 表示增加投資組合資產，並不會多增加風險。
3. 正同質性 (Positive Homogeneity)：𝜌(𝑋) = 𝜌(𝑋) 𝑓𝑜𝑟 > 0 表示改變資產 h 倍風險，測度值也同樣改變 h 倍。
4. 轉移不變性 (Translational Invariance)： 𝜌(𝑋 + 𝑛) = 𝜌(𝑋) − 𝑛 表示若增加數量為 n 之現金，則風險測度值會降低 n 。