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QRM 8 - Laptop session
Creation date：2024-04-27 16:23:07
Belonging category：计算机computer science
Tag QRM 8
Laptop session

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QRM 8 - Laptop session

Introduction
The objective of the session is
(a). To understand how to use the ghyp package to undertake parametric estimation
for normally distributed returns, and to obtain risk measures, in both the case
of a single asset portfolio, and a multiple asset portfolio.
The ghyp package
This package has a number of very useful features which allow the user to go from
fitting parametric model to data (single or multiple assets), to take those fitted mod-
els and obtain the PDF of the loss distribution, and thence to calculate some risk
measures. Furthermore, it allows us to perform these tasks for a range of quite flexi-
ble probability distribution, which is very important as it means we are not restricted
to using the Normal distribution.
To load the package, first make sure you have installed it (remember you need do
this only once), then type
library(ghyp)
and if all goes well it will load. There may be some benign messages; that is OK.
Before turning to estimation it would be helpful to gain some basic insights about
how to work with this package. The fundamental idea is that we work with ghyp
objects. These can be created either directly, as we see how to do in a moment, or
as a result of fitting, which we look at later.
As a simple example, we create a ghyp object which represents the PDF of the exam-
ple we looked at in lecture 5 of a Single Asset. The returns were assumed Normal,
with mean 0.01 and standard deviation 0.02.
To create this object and store it, we can do
g <- gauss(mu=0.01, sigma=0.02)
g
Typing g on its own shows that it has the Gaussian (ie Normal) distribution, and what
the parameters are, in terms of the mean and standard deviation.
Once we have created an object, we can get information about it using functions
which the package supplies. Some particularly useful ones are exhibited here.
Try these out and then refer to the table below for more details.
mean(g)
vcov(g)
plot(g,type='l')
dghyp(0.03,g)
pghyp(0.03,g)
qghyp(0.95,g)
rghyp(10,g)
In the following, the object’s name is represented by the term obj.
Function Description
mean(obj) Returns the expected value
vcov(obj) Returns the Variance in the case of a Univariate
object, and calculates the Variance-Covariance
matrix in the case of a multivariate object
dghyp(x,obj) Returns the PDF evaluated at the points con-
tained in the vector x
pghyp(x,obj) Returns the CDF evaluated at the points con-
tained in the vector x
qghyp(p,obj) Returns the quantiles evaluated at the probabil-
ities contained in the vector p
rghyp(n,obj) Returns a sample of n values drawn from the
density of obj (useful for Simulation)
Perhaps the most useful additional function is the transform function. Recall how
we wanted to find the loss distribution, given that the returns were distributed like
g, and given the current portfolio valuation was £1000. To transform returns into
losses, the rule was to multiply returns by −Pt, so in this case we should multiply by
-1000. The transform function performs this sort of task for us. We give the object
to be transformed, and the multiplier which is to be used in the transformation (here
-1000). Hence to find the loss distribution
pv <- 1000
g.l <- transform(g,multiplier = -pv)
g.l
plot(g.l, type = 'l')
which tells us the loss is Normal with mean -10 and standard deviation 20. We knew
this already, of course, but this saves a lot of time. It gets even better, as ghyp has
the function ES.ghyp to calculate the Expected shortfall from the parametric loss
distribution, and of course we can use qghyp to get the quantiles, and hence VaR.
ESghyp(0.95,g.l,distr="loss")
qghyp(0.95,g.l)
Note we give ESghyp the option ‘distr’ to tell is we are using the loss distribution. Of
course, these calculations are done without historical data, and simply rely on the
properties of the parametric probability distributions.
This makes the VarCov method very easy to apply. We don’t need to recall the rules
for relating the returns to the losses, and we don’t need to recall the formulae for
working out ES and VaR.
It is even more convenient when we consider Bivariate returns. We look at the second
example from the lecture using ghyp. The steps are similar to the above. With
multivariate densities it is usual to write the means as a vector and the variances
and covariances as a matrix. Referring to the example from Lecture 8, we had two
assets, A and B,
ˆ μA = 0.09, μB = 0.07
ˆ V(RA) = 0.004, V(RB) = 0.003
ˆ Cov(RA, RB) = 0.002
This information is succinctly represented by a vector of means (pronounced “mew”)
μ =
•
0.09
0.07
˜
and by the Variance-Covariance Matrix (pronounced sigma)
 =
•
0.004 0.002
0.002 0.003
˜
In this matrix, top left is the variance of A, bottom right is the variance of B, and the
off-diagonal elements record the Covariance between A and B.
To create the object we first set up the mean vector and the Variance-Covariance
matrix
m <- c(0.09,0.07)
sig <- cbind(c(0.004,0.002),c(0.002,0.003))
m
sig
and next we can define the object:
gm <- gauss(mu=m,sigma=sig)
gm
Of the functions we used above, possibly the most useful in the Bivariate case is the
rghyp function, which now gives us a sample of pairs from the Bivariate normal:
rghyp(10,gm)
plot(rghyp(10000,gm),pch = '.')
For risk measurement, we need to convert the asset returns to losses. Again, we will
use the transform function. We need to identify what each asset gets multiplied by
to convert the returns to losses. In the bivariate case we saw in Lecture 5 that this is
achieved by the following transformation
Loss = −Pt × (A × RA + B × RB)
ie return A gets multiplied by the negative of the portfolio value times the weight of
A, and B gets multiplied by the negative of the portfolio value times the weight of B.
So the multiplier of A is −PtA and the multiplier of B is −PtB. We need to give the
transform function a vector containing each multiplier, in order first A then B:
pv <- 1000
wa <- 0.25
wb <- 0.75
mult <- -pv*c(wa,wb)
mult
gm.l <- transform(gm,multiplier = mult)
gm.l
As we can see, the loss distribution is univariate (as of course it must be). Given the
loss distribution, we find our risk measures as before:
ESghyp(0.95,gm.l,distr="loss")
qghyp(0.95,gm.l)
The above is mainly for illustration, since it supposes we have already obtained the
parameters of the returns’ distribution. However, you may appreciate how useful
this approach is when used in conjunction with parameter estimation, as we now
investigate.
Estimation
We now explore how to fit the parameters of a model using returns data, for the case
the reutrns are assumed Normal. We see there is very little extra work involved.
Essentially all we have to do is get estimated for the mean and standard deviation of
the returns from out data. ghyp comes with some data on Swiss stock returns, which
we now use for illustration. They are stored in the data set called smi.stocks, which
we first have to load into the workspace with the data command, see below. Then I
pick out columns 4 and 5 (Nestle and Swisscom) and store in d. Then I pick out the
first column of the result and store in d1 for Univariate estimation. are already
data(smi.stocks)
d <- smi.stocks[,c(4,5)]
d1 <- d[,1]
First we estimate the mean and standard deviation:
md1 <- mean(d1)
md1
vard1 <- var(d1)
vard1
Now we can use these to construct the gauss object:
g <- gauss(mu=md1, sigma=sqrt(vard1))
g
Then we can calculate VaR and ES as before:
pv <- 1000
g.l <- transform(g,multiplier = -pv)
g.l
plot(g.l, type = 'l')
ESghyp(0.95,g.l,distr="loss")
# qghyp(0.95,g.l)
For multivariate estimation using both stocks, the idea is similar, we estimate the
means and the covariance using the sample data, and use these values to greate
the ghyp object.
md <- colMeans(d)
md
covd <- cov(d)
covd
g <- gauss(mu = md, sigma = covd )
g
Now transform to loss and calculate risk measures:
pv <- 1000
wa <- 0.25
wb <- 0.75
mult <- -pv*c(wa,wb)
mult
g.l <- transform(g,multiplier = mult)
g.l
ESghyp(0.95,g.l,distr="loss")
qghyp(0.95,g.l)
Thanks for taking this tutorial.
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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