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CFRM 462: Introduction to Computational Finance and Econometrics Homework 5
1. Monte Carlo Simulation in the CER Model
a)
Table 1: Parameter Estimates
Asset µi 2
i i
VBISX 0.00428 3.99e-05 0.00632
FBGRX 0.00214 3.34e-03 0.05783
GOOGL 0.00846 1.05e-02 0.10250
Table 2: Parameter Estimates Matrix
Parameters VBISX,FBGRX VBISX,GOOGL FBGRX,GOOGL
ij -1.69e-05 -1.98e-04 3.82e-03
⇢ij -0.0464 -0.3065 0.6450
b)
VBISX FBGRX GOOGL
muhat . val s 0.004282 0.00214 0.00846
se . muhat 0.000815 0.00747 0.01323
VBISX FBGRX GOOGL
sigma2hat . vals 3.99e05 0.003345 0.01051
se . sigma2hat 7.28e06 0.000611 0.00192
VBISX FBGRX GOOGL
sigmahat . vals 0.006316 0.05783 0.10250
se . sigmahat 0.000577 0.00528 0.00936
VBISX,FBGRX VBISX,GOOGL FBGRX,GOOGL
rhohat . vals 0.0464 0.306 0.6450
se . rhohat 0.1288 0.117 0.0754
µ is precise for VBISX, but not for FBGRX and GOOGL. 2 and are precise for all the
assets. ⇢ is only precise for FBGRX, GOOGL.
c) The distribution for the mean looks slightly positively skewed, but for the most
part looks normally distributed. The variance looks very positively skewed and the sd
looks normally distributed.
The Monte Carlo simulations do a good job of estimating the parameters and all of the
parameters are close to their true values.
> c (mu, mean ( sim . means ) )
[1] 0.000800 0.000771
> mean ( sim . means ) mu
[1] 2.9e05
> c ( sd ˆ2 , mean ( sim . va r s ) )
[1] 0.00451 0.00453
> mean ( sim . va r s ) sdˆ2
[1] 2.33e05
> c ( sd , mean ( sim . sd s ) )
[1] 0.0672 0.0670
> mean ( sim . sd s ) sd
[1] 0.000117
> c ( s e . muhat [ ”FBGRX” ] , sd ( sim . means ) )
FBGRX
0.00747 0.00831
> c ( s e . sigma2hat [ ”FBGRX” ] , sd ( sim . va r s ) )
FBGRX
2
0.000611 0.000843
> c ( s e . sigmahat [ ”FBGRX” ] , sd ( sim . s d s ) )
FBGRX
0.00528 0.00625
2. Bootstrapping
a) All of the SE below are very close to the analytic solutions.
boot ( data = VBISX, s t a t i s t i c = mean . boot , R = 999 )
Bootstrap Statistics :
original bias std . error
t1 ⇤ 0.00428 1.6e05 0.000807
> s e . muhat [ ”VBISX” ]
VBISX
0.000815
boot ( data = VBISX, s t a t i s t i c = sd . boot , R = 999 )
Bootstrap Statistics :
original bias std . error
t1 ⇤ 0.00632 8.42e05 0.000714
> s e . sigmahat [ ”VBISX” ]
VBISX
0.000577
boot ( data = VBISX, s t a t i s t i c = va r . boot , R = 999 )
Bootstrap Statistics :
original bias std . error
t1 ⇤ 3.99e05 7.31e07 8.63 e06
> s e . sigma2hat [ ”VBISX” ]
VBISX
7.28e06
boot ( data = r e t . mat [ , c (”VBISX” , ”FBGRX” ) ] , s t a t i s t i c = rho . boot ,
R = 999)
Bootstrap Statistics :
original bias std . error
t1 ⇤ 0.0464 0.0036 0.16
> se . rhohat [1]
VBISX,FBGRX
0.129
b) The mean looks relatively normally distributed, except it is slightly negatively
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skewed and the qq plot has non-normal behavior in the lower quantiles. The sd does not
look symmetric about the mean and exhibits non-normal behavior in the lower quantiles
of the qq plot. The var looks almost bimodal and exhibits non-normal behavior in the
lower quantiles of the qq plot. Rho looks the most normal because it is almost symmetric
and the qq plot is very linear.
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c) The bootstrap estimate suggests only a small loss and error of 131, also the confidence interval is fairly small
boot ( data = VBISX, s t a t i s t i c = ValueAtRisk . boot , R = 999 )
Bootstrap Statistics :
original bias std . error
t1 ⇤ 609 9.76 131
boot . c i ( boot . out = VBISX .VaR. boot , co nf = 0. 9 5 , type = c (” norm ” ,
”perc ”))
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Intervals :
Level Normal Pe rcen tile
95% (888, 354) (875, 335)
3. Class Project
a) The parameter estimates for mu are much less accurate than sigma, which is clear
from the wide confidence intervals that include both negative and positive numbers as well
as the magnitude of the SE compared to mu. Sigma and sigma2 are relatively accurate, the
SE is very small. However, sigma2 has a much larger confidence interval, which suggests
the se is not very accurate.
#Compute s e f o r mean
nobs = nrow ( r e t . mat )
se . muhat = sd . val s / s q r t ( nobs )
# compute approx 95% c o nfi d e n c e i n t e r v a l s
mu. low e r = muhat . v a l s 2⇤ se . muhat
mu. upper = muhat . v a l s + 2⇤ se . muhat
#Sigma SE
se . sigma = sd . vals/ sqrt (2⇤ nobs )
sigma . lower = sd . vals 2 ⇤ se . sigma
sigma . upper = sd . vals + 2 ⇤ se . sigma
#Var SE
se . sigma2 = sd . vals/ sqrt (nobs/2)
sigma2 . lower = var . vals 2 ⇤ se . sigma2
sigma2 . upper = var . vals + 2 ⇤ se . sigma2
#Cov SE
#Cor SE
se . rho = (1rhohat . vals )ˆ2/ sqrt (nobs )
rho . lower = rhohat . vals 2 ⇤ se . rho
rho . upper = rhohat . vals + 2 ⇤ se . rho
#Combine the SE
rbind ( se . muhat , se . sigma , se . sigma2 , se . rho )
vfinx presx hlemx vbllx fshbx vpacx
se . muhat 0.00425 0.00611 0.00619 0.00306 0.000290 0.00501
se . sigma 0.00301 0.00432 0.00438 0.00216 0.000205 0.00354
se . sigma2 0.00601 0.00864 0.00875 0.00432 0.000410 0.00708
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r bi n d (mu. lower ,mu. upper , sigma . lower , sigma . upper )
vfinx presx hlemx vbllx fshbx vpacx
mu. low e r 0.00303 0.00457 0.00935 0.000462 0.000726 0.00538
mu. upper 0.02003 0.01987 0.01541 0.012689 0.001885 0.01465
sigma . lower 0.03005 0.04320 0.04377 0.021616 0.002050 0.03540
sigma . upper 0.04207 0.06047 0.06128 0.030262 0.002870 0.04955
sigma2 . lower 0.01072 0.01459 0.01475 0.007973 0.000814 0.01235
sigma2 . upper 0.01332 0.01997 0.02027 0.009319 0.000826 0.01596
http://computational-finance.uw.edu
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