-
Notifications
You must be signed in to change notification settings - Fork 17
/
lab7.r
551 lines (439 loc) · 15.6 KB
/
lab7.r
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
# lab7.r script file for lab7 calculations
#
# author: Eric Zivot
# created: October 20, 2003
# revised: July 17, 2012
#
# comments:
options(digits=4, width=70)
# make sure packages are installed prior to loading them
library(PerformanceAnalytics)
library(zoo)
library(boot)
library(tseries)
# get monthly adjusted closing price data on VBLTX, FMAGX and SBUX from Yahoo
# using the tseries function get.hist.quote(). Set sample to Sept 2005 through
# Sep 2010. Note: if you are not careful with the start and end dates
# or if you set the retclass to "ts" then results might look weird
# get the last five years of monthly adjusted closing prices from Yahoo!
VBLTX.prices = get.hist.quote(instrument="vbltx", start="2005-09-01",
end="2010-09-30", quote="AdjClose",
provider="yahoo", origin="1970-01-01",
compression="m", retclass="zoo")
# change class of time index to yearmon which is appropriate for monthly data
# index() and as.yearmon() are functions in the zoo package
#
index(VBLTX.prices) = as.yearmon(index(VBLTX.prices))
class(VBLTX.prices)
colnames(VBLTX.prices)
start(VBLTX.prices)
end(VBLTX.prices)
FMAGX.prices = get.hist.quote(instrument="fmagx", start="2005-09-01",
end="2010-09-30", quote="AdjClose",
provider="yahoo", origin="1970-01-01",
compression="m", retclass="zoo")
index(FMAGX.prices) = as.yearmon(index(FMAGX.prices))
SBUX.prices = get.hist.quote(instrument="sbux", start="2005-09-01",
end="2010-09-30", quote="AdjClose",
provider="yahoo", origin="1970-01-01",
compression="m", retclass="zoo")
index(SBUX.prices) = as.yearmon(index(SBUX.prices))
# trial Google data (GOOG), not used in assignment
GOOG.prices = get.hist.quote(instrument="goog", start="2004-09-01",
end="2013-01-31", quote="AdjClose",
provider="yahoo", origin="1970-01-01",
compression="m", retclass="zoo")
index(GOOG.prices) = as.yearmon(index(GOOG.prices))
# create merged price data
lab4Prices.z = merge(VBLTX.prices, FMAGX.prices, SBUX.prices)
# rename columns
colnames(lab4Prices.z) = c("VBLTX", "FMAGX", "SBUX")
colnames(GOOG.prices) = "GOOG"
# calculate cc returns as difference in log prices
lab4Returns.z = diff(log(lab4Prices.z))
GOOGReturns.z = diff(log(GOOG.prices))
#
# 3. Create timePlots of data
#
plot(lab4Returns.z, plot.type="single", lty=1:3, col=1:3, lwd=2)
legend(x="bottomleft", legend=colnames(lab4Returns.z), lty=1:3, col=1:3, lwd=2)
abline(h=0)
title("Monthly cc returns")
plot(GOOGReturns.z, lwd=2)
abline(h=mean(GOOGReturns.z))
title("Monthly cc returns for GOOG")
#
# 4. Create matrix of return data and compute pairwise scatterplots
#
ret.mat = coredata(lab4Returns.z)
colnames(ret.mat)
head(ret.mat)
VBLTX = ret.mat[,"VBLTX"]
FMAGX = ret.mat[,"FMAGX"]
SBUX = ret.mat[,"SBUX"]
pairs(ret.mat, col="blue")
#
# 5. Compute estimates of CER model parameters
#
muhat.vals = apply(ret.mat, 2, mean)
muhat.vals
sigma2hat.vals = apply(ret.mat, 2, var)
sigma2hat.vals
sigmahat.vals = apply(ret.mat, 2, sd)
sigmahat.vals
cov.mat = var(ret.mat)
cov.mat
cor.mat = cor(ret.mat)
cor.mat
covhat.vals = cov.mat[lower.tri(cov.mat)]
rhohat.vals = cor.mat[lower.tri(cor.mat)]
names(covhat.vals) <- names(rhohat.vals) <-
c("VBLTX,FMAGX","VBLTX,SBUX","FMAGX,SBUX")
covhat.vals
rhohat.vals
muhat.GOOG = mean(GOOGReturns.z)
sigma2hat.GOOG = apply(GOOGReturns.z, 2, var)
sigmahat.GOOG = apply(GOOGReturns.z, 2, sd)
# summarize the CER model estimates
cbind(muhat.vals,sigma2hat.vals,sigmahat.vals)
cbind(covhat.vals,rhohat.vals)
cbind(muhat.GOOG, sigma2hat.GOOG, sigmahat.GOOG)
# plot mean vs. sd values
plot(sigmahat.vals, muhat.vals, pch=1:3, cex=2, col=1:3,
ylab = "mean", xlab="sd (risk)")
abline(h=0)
legend(x="topright", legend=names(muhat.vals), pch=1:3, col=1:3, cex=1.5)
#
# 6. Compute stndard errors for estimated parameters
#
# compute estimated standard error for mean
nobs = nrow(ret.mat)
nobs
se.muhat = sigmahat.vals/sqrt(nobs)
se.muhat
# show estimates with SE values underneath
rbind(muhat.vals,se.muhat)
nobs.GOOG = nrow(GOOGReturns.z)
se.muhat.GOOG = sigmahat.GOOG / sqrt(nobs.GOOG)
rbind(muhat.GOOG, se.muhat.GOOG)
# compute approx 95% confidence intervals
mu.lower = muhat.vals - 2*se.muhat
mu.upper = muhat.vals + 2*se.muhat
cbind(mu.lower,mu.upper)
mu.GOOG.lower = muhat.GOOG - 2*se.muhat.GOOG
mu.GOOG.upper = muhat.GOOG + 2*se.muhat.GOOG
cbind(mu.GOOG.lower, mu.GOOG.upper)
# compute estimated standard errors for variance and sd
se.sigma2hat = sigma2hat.vals/sqrt(nobs/2)
se.sigma2hat
se.sigmahat = sigmahat.vals/sqrt(2*nobs)
se.sigmahat
rbind(sigma2hat.vals,se.sigma2hat)
rbind(sigmahat.vals,se.sigmahat)
se.sigma2hat.GOOG = sigma2hat.GOOG / sqrt(nobs.GOOG/2)
se.sigmahat.GOOG = sigmahat.GOOG / sqrt(2*nobs.GOOG)
rbind(sigma2hat.GOOG, se.sigma2hat.GOOG)
rbind(sigmahat.GOOG, se.sigmahat.GOOG)
# compute approx 95% confidence intervals
sigma2.lower = sigma2hat.vals - 2*se.sigma2hat
sigma2.upper = sigma2hat.vals + 2*se.sigma2hat
cbind(sigma2.lower,sigma2.upper)
sigma.lower = sigmahat.vals - 2*se.sigmahat
sigma.upper = sigmahat.vals + 2*se.sigmahat
cbind(sigma.lower,sigma.upper)
sigma2.GOOG.lower = sigma2hat.GOOG - 2*se.sigma2hat.GOOG
sigma2.GOOG.upper = sigma2hat.GOOG + 2*se.sigma2hat.GOOG
cbind(sigma2.GOOG.lower, sigma2.GOOG.upper)
sigma.GOOG.lower = sigmahat.GOOG - 2*se.sigmahat.GOOG
sigma.GOOG.upper = sigmahat.GOOG + 2*se.sigmahat.GOOG
cbind(sigma.GOOG.lower, sigma.GOOG.upper)
# compute estimated standard errors for correlation
se.rhohat = (1-rhohat.vals^2)/sqrt(nobs)
se.rhohat
rbind(rhohat.vals,se.rhohat)
# compute approx 95% confidence intervals
rho.lower = rhohat.vals - 2*se.rhohat
rho.upper = rhohat.vals + 2*se.rhohat
cbind(rho.lower,rho.upper)
#
# 7. Compute 5% and 1% Value at Risk
#
# function to compute Value-at-Risk
# note: default values are selected for
# the probability level (p) and the initial
# wealth (w). These values can be changed
# when calling the function. Highlight the entire
# function, right click and select run line or selection
Value.at.Risk = function(x,p=0.05,w=100000) {
x = as.matrix(x)
q = apply(x, 2, mean) + apply(x, 2, sd)*qnorm(p)
VaR = (exp(q) - 1)*w
VaR
}
# 5% and 1% VaR estimates based on W0 = 100000
Value.at.Risk(ret.mat,p=0.05,w=100000)
Value.at.Risk(ret.mat,p=0.01,w=100000)
Value.at.Risk(GOOGReturns.z,p=0.05,w=100000)
Value.at.Risk(GOOGReturns.z,p=0.01,w=100000)
################################################################################
# Hypothesis Testing
################################################################################
#
# 8. Test H0: mu = 0 vs. H1: mu != 0
#
?t.test
t.test(lab4Returns.z[,"VBLTX"])
t.test(lab4Returns.z[,"FMAGX"])
t.test(lab4Returns.z[,"SBUX"])
GOOGReturns.mat = coredata(GOOGReturns.z)
head(GOOGReturns.mat)
t.test(GOOGReturns.mat)
#
# 9. Test H0: rho_ij = 0 vs. H1: rho_ij != 0
#
?cor.test
# VBLTX,FMAGX
cor.test(x=lab4Returns.z[,"VBLTX"], y=lab4Returns.z[,"FMAGX"])
# VBLTX,SBUX
cor.test(x=lab4Returns.z[,"VBLTX"], y=lab4Returns.z[,"SBUX"])
# FMAGX,SBUX
cor.test(x=lab4Returns.z[,"FMAGX"], y=lab4Returns.z[,"SBUX"])
#
# 10. Test H0: returns are normal vs. H1: returns are not normal
#
library(tseries)
?jarque.bera.test
jarque.bera.test(lab4Returns.z[,"VBLTX"])
jarque.bera.test(lab4Returns.z[,"FMAGX"])
jarque.bera.test(lab4Returns.z[,"SBUX"])
#
# 11. 24 month rolling estimates of mu and sd
#
# rolling analysis for VBLTX
roll.mu.VBLTX = rollapply(lab4Returns.z[,"VBLTX"],
FUN=mean, width = 24, align="right")
roll.sd.VBLTX = rollapply(lab4Returns.z[,"VBLTX"],
FUN=sd, width = 24, align="right")
plot(merge(roll.mu.VBLTX,roll.sd.VBLTX,lab4Returns.z[,"VBLTX"]), plot.type="single",
main="24-month rolling means and sds for VBLTX", ylab="Percent per month",
col=c("blue","red","black"), lwd=2)
abline(h=0)
legend(x="topleft", legend=c("Rolling means","Rolling sds", "VBLTX returns"),
col=c("blue","red","black"), lwd=2)
# rolling analysis for FMAGX
roll.mu.FMAGX = rollmeanr(lab4Returns.z[,"FMAGX"], 24)
roll.sd.FMAGX = rollapply(lab4Returns.z[,"FMAGX"],
FUN=sd, width = 24, align="right")
plot(merge(roll.mu.FMAGX,roll.sd.FMAGX,lab4Returns.z[,"FMAGX"]), plot.type="single",
main="24-month rolling means and sds for FMAGX", ylab="Percent per month",
col=c("blue","red","black"), lwd=2)
abline(h=0)
legend(x="bottomleft", legend=c("Rolling means","Rolling sds", "FMAGX returns"),
col=c("blue","red","black"), lwd=2)
# rolling analysis for SBUX
roll.mu.SBUX = rollmeanr(lab4Returns.z[,"SBUX"], 24)
roll.sd.SBUX = rollapply(lab4Returns.z[,"SBUX"],
FUN=sd, width = 24, align="right")
plot(merge(roll.mu.SBUX,roll.sd.SBUX,lab4Returns.z[,"SBUX"]), plot.type="single",
main="24-month rolling means and sds for SBUX", ylab="Percent per month",
col=c("blue","red","black"), lwd=2)
abline(h=0)
legend(x="bottomleft", legend=c("Rolling means","Rolling sds", "SBUX returns"),
col=c("blue","red","black"), lwd=2)
# rolling analysis for GOOG
roll.mu.GOOG = rollmeanr(GOOGReturns.z, 24)
roll.sd.GOOG = rollapply(GOOGReturns.z, width=24, FUN=sd,
align="right")
plot(merge(roll.mu.GOOG, roll.sd.GOOG, GOOGReturns.z), screens=1,
main="24-month rolling means and sds for GOOG", ylab="Percent per month",
col=c("blue", "red", "black"), lwd=2)
abline(h=0)
legend(x="topright", legend=c("Rolling means", "Rolling sds", "GOOG returns"),
col=c("blue", "red", "black"), lwd=2)
# rolling correlation estimates
rhohat = function(x) {
cor(x)[1,2]
}
# compute rolling estimates b/w VBLTX and FMAGX
roll.rhohat.VBLTX.FMAGX = rollapply(lab4Returns.z[,c("VBLTX","FMAGX")],
width=24,FUN=rhohat, by.column=FALSE,
align="right")
class(roll.rhohat.VBLTX.FMAGX)
roll.rhohat.VBLTX.FMAGX[1:5]
my.panel <- function(...){
lines(...)
abline(h=0)
}
plot(merge(roll.rhohat.VBLTX.FMAGX, lab4Returns.z[,c("VBLTX","FMAGX")]),
main="Rolling Correlation b/w VBLTX and FMAGX", screens=c(1, 2, 2),
lwd=2, col=c("blue", "darkgreen", "black"), panel=my.panel)
# compute rolling estimates b/w VBLTX and SBUX
roll.rhohat.VBLTX.SBUX = rollapply(lab4Returns.z[,c("VBLTX","SBUX")],
width=24,FUN=rhohat, by.column=FALSE,
align="right")
class(roll.rhohat.VBLTX.SBUX)
roll.rhohat.VBLTX.SBUX[1:5]
plot(roll.rhohat.VBLTX.SBUX, main="Rolling Correlation b/w VBLTX and SBUX",
lwd=2, col="blue", ylab="rho.hat")
abline(h=0)
# compute rolling estimates b/w FMAGX and SBUX
roll.rhohat.FMAGX.SBUX = rollapply(lab4Returns.z[,c("FMAGX","SBUX")],
width=24,FUN=rhohat, by.column=FALSE,
align="right")
class(roll.rhohat.FMAGX.SBUX)
roll.rhohat.FMAGX.SBUX[1:5]
plot(roll.rhohat.FMAGX.SBUX, main="Rolling Correlation b/w FMAGX and SBUX",
lwd=2, col="blue", ylab="rho.hat")
abline(h=0)
#
# 12. Evaluate bias and SE formulas using Monte Carlo
#
# generate 1000 samples from CER and compute sample statistics
mu = muhat.vals["FMAGX"]
sd = sigmahat.vals["FMAGX"]
n.obs = 60
set.seed(123)
n.sim = 1000
sim.means = rep(0,n.sim)
sim.vars = rep(0,n.sim)
sim.sds = rep(0,n.sim)
for (sim in 1:n.sim) {
sim.ret = rnorm(n.obs,mean=mu,sd=sd)
sim.means[sim] = mean(sim.ret)
sim.vars[sim] = var(sim.ret)
sim.sds[sim] = sqrt(sim.vars[sim])
}
par(mfrow=c(2,2))
hist(sim.means,xlab="mu hat", col="slateblue1")
abline(v=mu, col="white", lwd=2)
hist(sim.vars,xlab="sigma2 hat", col="slateblue1")
abline(v=sd^2, col="white", lwd=2)
hist(sim.sds,xlab="sigma hat", col="slateblue1")
abline(v=sd, col="white", lwd=2)
par(mfrow=c(1,1))
# generate 1000 samples from CER and compute sample statistics for GOOG
mu = muhat.GOOG
sd = sigmahat.GOOG
n.obs = nobs.GOOG
set.seed(123)
s.sim = 1000
sim.means = rep(0, n.sim)
sim.vars = rep(0, n.sim)
sim.sds = rep(0, n.sim)
for (sim in 1:n.sim) {
sim.ret = rnorm(n.obs, mean=mu, sd=sd)
sim.means[sim] = mean(sim.ret)
sim.vars[sim] = var(sim.ret)
sim.sds[sim] = sd(sim.ret)
}
par(mfrow=c(2,2))
hist(sim.means,xlab="mu hat", col="slateblue1")
abline(v=mu, col="white", lwd=2)
hist(sim.vars,xlab="sigma2 hat", col="slateblue1")
abline(v=sd^2, col="white", lwd=2)
hist(sim.sds,xlab="sigma hat", col="slateblue1")
abline(v=sd, col="white", lwd=2)
par(mfrow=c(1,1))
#
# 13. compute MC estimates of bias and SE
#
c(mu, mean(sim.means))
mean(sim.means) - mu
c(sd^2, mean(sim.vars))
mean(sim.vars) - sd^2
c(sd, mean(sim.sds))
mean(sim.sds) - sd
# compute MC SE value and compare to SE calculated from simulated data
c(se.muhat.GOOG, sd(sim.means))
c(se.sigma2hat.GOOG, sd(sim.vars))
c(se.sigmahat.GOOG, sd(sim.sds))
#
# 14. bootstrapping SE for mean, variance, sd and correlation
#
?boot
# note: boot requires user-supplied functions that take
# two arguments: data and an index. The index is created
# by the boot function and represents random resampling with
# replacement
# function for bootstrapping sample mean
mean.boot = function(x, idx) {
# arguments:
# x data to be resampled
# idx vector of scrambled indices created by boot() function
# value:
# ans mean value computed using resampled data
ans = mean(x[idx])
ans
}
VBLTX.mean.boot = boot(VBLTX, statistic = mean.boot, R=999)
class(VBLTX.mean.boot)
names(VBLTX.mean.boot)
# print, plot and qqnorm methods
VBLTX.mean.boot
se.muhat["VBLTX"]
# plot bootstrap distribution and qq-plot against normal
plot(VBLTX.mean.boot)
# compute bootstrap confidence intervals from normal approximation
# basic bootstrap method and percentile intervals
boot.ci(VBLTX.mean.boot, conf = 0.95, type = c("norm","perc"))
#
# boostrap SD estimate
#
# function for bootstrapping sample standard deviation
sd.boot = function(x, idx) {
# arguments:
# x data to be resampled
# idx vector of scrambled indices created by boot() function
# value:
# ans sd value computed using resampled data
ans = sd(x[idx])
ans
}
VBLTX.sd.boot = boot(VBLTX, statistic = sd.boot, R=999)
VBLTX.sd.boot
se.sigmahat["VBLTX"]
# plot bootstrap distribution
plot(VBLTX.sd.boot)
# compute confidence intervals
boot.ci(VBLTX.sd.boot, conf=0.95, type=c("norm", "basic", "perc"))
# bootstrap correlation
# function to compute correlation between 1st 2 variables in matrix
rho.boot = function(x.mat, idx) {
# x.mat n x 2 data matrix to be resampled
# idx vector of scrambled indices created by boot() function
# value:
# ans correlation value computed using resampled data
ans = cor(x.mat[idx,])[1,2]
ans
}
VBLTX.FMAGX.cor.boot = boot(ret.mat[,c("VBLTX","FMAGX")],
statistic=rho.boot, R = 999)
VBLTX.FMAGX.cor.boot
se.rhohat[1]
# plot bootstrap distribution
plot(VBLTX.FMAGX.cor.boot)
# bootstrap confidence intervals
boot.ci(VBLTX.FMAGX.cor.boot, conf=0.95, type=c("norm", "perc"))
#
# 15. Bootstrap VaR
#
# 5% Value-at-Risk
ValueAtRisk.boot = function(x, idx, p=0.05, w=100000) {
# x.mat data to be resampled
# idx vector of scrambled indices created by boot() function
# p probability value for VaR calculation
# w value of initial investment
# value:
# ans Value-at-Risk computed using resampled data
q = mean(x[idx]) + sd(x[idx])*qnorm(p)
VaR = (exp(q) - 1)*w
VaR
}
VBLTX.VaR.boot = boot(VBLTX, statistic = ValueAtRisk.boot, R=999)
VBLTX.VaR.boot
boot.ci(VBLTX.VaR.boot, conf=0.95, type=c("norm", "perc"))
plot(VBLTX.VaR.boot)
GOOG.VaR.boot = boot(GOOGReturns.mat, statistic=ValueAtRisk.boot, R=999)
GOOG.VaR.boot
boot.ci(GOOG.VaR.boot, conf=0.95, type=c("norm", "perc"))
plot(GOOG.VaR.boot)