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资产收益的时间序列分析及量化风险建模

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risk-management

资产收益的时间序列分析及量化风险建模

Individual Index VaR (Group A)

  1. Describe your Index.
  1. Show the first four moments mean / variance / skewness / kurtosis for the index.
  2. What kind of distributions do they have?
  3. Are the returns auto-correlated? Are the returns –squared auto-correlated? Do you need any volatility modeling?

Answer: Individual Index: MSFT. The following graph depicts the stock price trend and stock return trend, respectively, in the past five years from 1/1/2008-12/31/2012. The stock price mainly fluctuated between the range in $15 and $30. The stock return display a normal distribution like shape, which we will conduct further test later.

  1. First four moments mean / variance / skewness / kurtosis

  2. With a mean of -0.0001 and variance of 0.0004, the return has a higher kurtosis than the normal distribution, though the skewness is close to zero. So normal distribution may not exactly capture the feature of return of Microsoft.

  3. From the autocorrelation test we can conclude that there is little autocorrelation in the return. However, there is large autocorrelation in the square-return, which indicates GARCH effect, that is, we need to conduct volatility modeling.

  1. Try different volatility models and find the best one for your Index.
  1. State what different volatility modeling you tried. Show all results.
  2. State how do you check which volatility is a better one? Show the test results.

Answer:

  1. I chose three volatility models EWMA

sigma(t)^2=0.06R(t-1)^2+0.94sigma(t-1)^2

GARCH

sigma(t)^2= 3.0072e-006 + 0.0499 *R(t-1)^2+ 0.9412 *sigma(t-1)^2

NGARCH

sigma(t)^2= 2.6431e-006 + 0.0399 (R(t-1)-0.6521sigma(t-1))^2+ 0.9365*sigma(t-1)^2

  1. R(t+1)^2=B0+B1*sigma(t+1)^2+e(t+1)

From the regression test, we conclude that NGARCH is the best model of the three, with the intercept B0 equal to 0 and B1 close to 1.

  1. Calculate 1% VaR for the security, assuming normal distribution, t-distribution, Cornish-Fisher and EVT and estimating Hill parameter. Show graphically the predicted VaR with different distributions for the testing period.
  1. Normal Distribution We chose the NGARCH to be our volatility model, since from the regression test, it shows the best result. “lsigma” stands for leveraged sigma VaRnormal=-eps.*lsigma; eps ~ N(0,1)

From the graph we can see that the quantile does not fit very well to the normal distribution. The return’s quantile is fatter than normal distrubution

  1. T-Distribution From the QMLE method, we get the d estimate to be: d= 6.0519 VaRtd=-lsigma*tinv(0.01,d)*sqrt((d-2)/d);

From the graph below, we can see that t-distribution actually fits very well, which capture the feature of return standardized by the NGARCH model.

  1. Cornish-Fisher CF= -3.4221; VaRCF=-lsigma*CF;

  2. EVT Hill parameter = 0.3325; C = 0.1898

If we compare the VaR from the distributions analyzed above, we get the following picture. From the graph we can see that the VaR got from Cornish-Fisher is the highest one, while VaR got from normal distribution is the lowest one. Testing Period Historical Period

  1. Calculate 1%, 1 day VaR using:
  1. Monte Carlo Simulation (t-distribution / GARCH)
  2. FHS (GARCH) for the testing periods.
  3. Show graphically how VaR changes for testing period given below.

The design period is 1 Jan, 2008 – 31 December 2010. Testing period is 1 Jan., 2011 – 31 December 2012.

Answer:

  1. Monte Carlo Simulation ~t(d)/GARCH Number of design period: 671; Number of testing period: 463; GARCH Model for the design period:

The last sigma in the design period, which we use to set the first number in the testing period: sigmatest =0.0118. Generating random number from the MATLAB built-in function, we obtain the following graph which depicts the VaR in the testing period.

  1. Filtered Historical Simulation

In the filtered historical method, we draw the return from the design period and use it as a sample pool, and then we get the random number z to calculate the volatility evolution.

From the graph we can see there is a lifting trend of VaR in the filtered historical simulation.

  1. Back-testing for the unconditional 1% VaR as well as for conditional tests using historical period and testing period as shown above.

5a) Compare VaR with different measure in unconditional test (LRuc), independent test (LRind) and conditional coverage test (LRcc)

5b) Find the best method of calculating VaR for all your indices, and show enough evidence to support your conclusion.

5a) Answer:

Re-estimate the coefficients for the GARCH and NGARCH.

This time, estimate coefficients in the design period only:

  1. LRuc: Unconditional Coverage Testing

From the testing results showing above, we can see that at the 5% significance level, the NGARCH model and GARCH model is in the acceptable range. These two volatility models perform very well, while the EWMA model does not fit very well according to the back testing.

  1. LRind: Independent Testing

From the test results showing above, we can see that the LRind, which fits the chi-square distribution, has a high p-value. That is, we should reject the null hypothesis that the volatility (All of the three, EWMA, GARCH, NGARCH) have no clustered effect. Actually, the results show that volatilities are not independent. They are clustered in time.

  1. LRcc : Conditional Coverage Testing

Finally, we consider the joint testing for independence and correct coverage. From the testing results we can see that in the EWMA method, we should reject the null hypothesis. However, for the other two, the conclusion depends on what level of significance we choose. If we choose the significant level to be less than 12%, then we should reject the two methods. If we choose the significant level to be greater than 12%, we may accept the null hypothesis.

5b) Answer: Since my role is a capital constraint bank, I will choose a volatility model which gives me a small VaR. Since some of the capital requirements are set as “k times of VaR”, given a small VaR, there’s less capital requirement for me to put on the risk capital, which means I have a lot more to put on other kind of investment. From the results above, I will choose a model which gets the smaller VaR. GARCH or NGARCH may be a better choice for me than EWMA.

EXTRA QUESTION FOR ONE INDEX GROUP A Calculate 1%, 1 day VaR in HS, WHS, Monte Carlo Simulation (t-distribution / GARCH) and FHS (GARCH) for the testing periods. Show graphically how VaR changes for testing period given above.

Answer: For the Monte Carlo Simulation and FHS, we have calculated before. For the HS and WHS, we used the design period as the observation period and calculate the percentile as time moves by. Finally, I get the following graph, which depicts the evolution of VaR. The VaR got from models have more fluctuation compared with the historical simulation.

Appendix: MATLAB Code

Main Code % 1 load stockindex % load data

price=stockindex; logreturn=price2ret(price);

%plot price and return subplot(211);plot(price);title('Stock Price'),grid on; subplot(212);plot(logreturn);title('Stock Return'),grid on;

%distribution: Pricelognormal; returnnormal histfit(logreturn);title('Stock Return Distribution'),grid on;

% mean, standard deviation, skewness, kurtsis meanreturn=mean(logreturn) stdreturn=std(logreturn);varreturn=stdreturn^2 skewreturn=skewness(logreturn) kurtreturn=kurtosis(logreturn)

% conducted autocorrelation analysis towards the return % there is little autocorrelation in the return itself [ACF1, Lags1, Bounds1] = autocorr(logreturn, 100); subplot(211),autocorr(logreturn, 100),title('return Auto-correlation');

%The return is not independent, therefore, we need volatility modeling squarereturn=logreturn.^2; [ACF2, Lags2, Bounds2] = autocorr(squarereturn, 100); subplot(212),autocorr(squarereturn, 100);title('return square Auto-correlation');

%********************************************************************** %2 % Several volatility models % 1) sigma(:,1)~EWMA % 2) sigma(:,2)~GARCH % 3) sigma(:,3)~NGARCH spec=garchset('display','off','P',1,'Q',1); coef=garchfit(spec,logreturn);

%Estimate coefficient of NGARCH L1=@(mu)sum(rmgQ(mu,3,logreturn,[])); A = [0 1 0 1]; %alpha + beta<1 B = 1; LB = [0 0 0 0]; initials= [0.0001 0.05 0.6 0.94]; options = optimset('Display','off','MaxIter',999999999,'MaxFunEvals',999999999,'Algorithm','sqp'); [muvalue,fval1] = fmincon(@(mu)(-L1(mu)),initials,A,B,[],[],LB,[],[],options); omega=muvalue(1),alpha=muvalue(2),theta=muvalue(3),beta=muvalue(4)

n=length(logreturn); sigma=zeros(n,3); sigma(1,:)=stdreturn; for i=2:n; %EWMA sigma(i,1)=sqrt(0.06*(logreturn(i-1)^2)+0.94*(sigma(i-1,1).^2)); %GARCH sigma(i,2)=sqrt(coef.K+coef.GARCH*(sigma(i-1,2).^2)+coef.ARCH*(logreturn(i-1).^2)); %NGARCH sigma(i,3)=sqrt(omega+alpha*(logreturn(i-1)-thetasigma(i-1,3)).^2+betasigma(i-1,3).^2); end

plot(sigma),grid on; title('Three Volatility Models'); legend('EWMA','GARCH','NGARCH');

%Test y=logreturn.^2;% R^2 and sigma^2 x1=zeros(n,2); x1(:,1)=sigma(:,1).^2; x1(:,2)=1; %EWMA x2=zeros(n,2); x2(:,1)=sigma(:,2).^2; x2(:,2)=1; %GARCH x3=zeros(n,2); x3(:,1)=sigma(:,3).^2; x3(:,2)=1; %NGARCH [B1,BINT1,R1,RINT1,STATS1] = regress(y,x1); [B2,BINT2,R2,RINT2,STATS2] = regress(y,x2); [B3,BINT3,R3,RINT3,STATS3] = regress(y,x3);

%After testing, we found that the NGARCH is the best model of all. lsigma=sigma(:,3); %leveraged garch std standret=logreturn./lsigma; %leveraged garch standard return

%********************************************************************** %3 eps=norminv(0.01,0,1);

%Normal VaRnormal=-eps.*lsigma; %sigma is calculated using the three methods above stand=repmat(logreturn,1,3)./sigma; figure,qqplot(stand);grid on title('QQplot');legend('EWMA','GARCH','NGARCH');

%t(d) Distribution L=@(d)n*(gammaln((d+1)/2)-gammaln(d/2)-log(pi)/2-log(d-2)/2) ... -1/2*(1+d)*sum(log(1+standret.^2/(d-2))); [dvalue,fval] = fminsearch(@(d)(-L(d)),10);

for i = 1: n tp=(i-0.5)/n; if tp<=0.5 t(i)= -abs(tinv(2*tp,dvalue))sqrt((dvalue-2)/dvalue); else t(i)= abs(tinv(2(1-tp),dvalue))sqrt((dvalue-2)/dvalue); end end figure,qqplot(standret,t);grid on title('Standard T distribution'); VaRtd=-lsigmatinv(0.01,dvalue)*sqrt((dvalue-2)/dvalue);

%Cornish-Fisher skew=skewness(standret);kurt=kurtosis(standret);F=norminv(0.01); CF=F+skew/6*(F^2-1)+kurt/24*(F^3-3F)-skew^2/36(2F^3-5F); VaRCF=-lsigma*CF;

%EVT u=0.05; Tu = fix(nu); evtret=sort(standret);evtret=evtret(1:Tu); uret= evtret(Tu); tailpara= sum(log(abs(evtret./uret)))/Tu; C= Tu/nabs(uret)^(1/tailpara); VaRevt=lsigmaabs(uret)(0.01/(Tu/n))^(-tailpara); for i = 1: Tu evt(i)= uret*(((i-0.5)/n)/(Tu/n))^(-tailpara); end figure,qqplot(evt,evtret),grid on title('qqplot ~ EVT') % Compare the quatiles only

%plot VaR=[VaRnormal,VaRtd,VaRCF, VaRevt]; plot(VaR) title('VaR'),legend('VaRnomal','VaRtd','VaRCF','VaRevt'),grid on

%********************************************************************** %4 %Monte Carlo n=671; %1/1/2008 - 12/31/2010 n2=length(logreturn)-n; %1/1/2011 - 12/31/2012 m=1000; spec=garchset('display','off','P',1,'Q',1); coef2=garchfit(spec,logreturn(1:n));

%Choose the last sigma in the design period to be the first number of sigma %in testing period. sigmatest=stdreturn; for i=2:n; sigmatest=sqrt(coef2.K+coef2.GARCH*(sigmatest.^2)+coef2.ARCH*(logreturn(i-1).^2)); end MonSi=ones(m,n2+1);MonSi(:,1)=sigmatest; MonR=ones(m,n2);

%Generate random number from t(d), and take the inverse after converting %to student t-distribution z=trnd(dvalue,m,n2)*sqrt((dvalue-2)/dvalue); %used dvalue calculated before

for i=1:n2 %R=zsigma, z generate from normal distrubution by built-in function MonR(:,i)=MonSi(:,i).z(:,i);
%GARCH model MonSi(:,i+1)=sqrt(coef2.K+coef2.GARCH(MonSi(:,i).^2)+coef2.ARCH(MonR(:,i).^2)); end VaRmon=-prctile(MonR,0.01*100); plot(VaRmon),grid on title('VaR in testing period through Monte Carlo Simulation');

%FHS %mx1 column index, using the range from 1~671 historical data as data pool index=randi(n,m,n2);
standret2=logreturn./sigma(:,2); %garch standard return z2=standret2(index); % generate return from the historical data

FSi=ones(m,n2+1);FSi(:,1)=sigmatest; FR=ones(m,n2);

for i=1:n2 %R=zsigma, z generate from historical pool FR(:,i)=FSi(:,i).z2(:,i); FSi(:,i+1)=sqrt(coef2.K+coef2.GARCH(FSi(:,i).^2)+coef2.ARCH(FR(:,i).^2));
end VaRfh=-prctile(FR,0.01*100); plot(VaRfh),grid on title('VaR in testing period through Filtered Historical Simulation');

%********************************************************************** %5 p=0.01; % 1%VaR % Three methods of volatility % coefficients ~ get from design period % test ~ using data in testing period %NGARCH coefficients estimated from design period only L2=@(mu)sum(rmgQ(mu,3,logreturn(1:n2),[])); A = [0 1 0 1]; %alpha + beta<1 B = 1; LB = [0 0 0 0]; initials= [0.0001 0.05 0.6 0.94]; options = optimset('Display','off','MaxIter',999999999,'MaxFunEvals',999999999,'Algorithm','sqp'); [muvalue,fval1] = fmincon(@(mu)(-L2(mu)),initials,A,B,[],[],LB,[],[],options); omega2=muvalue(1);alpha2=muvalue(2);theta2=muvalue(3);beta2=muvalue(4); sigma2=zeros(n2,3); sigma2(1,:)=sigmatest; for i=2:n2; %EWMA sigma2(i,1)=sqrt(0.06*(logreturn(i-1)^2)+0.94*(sigma(i-1,1).^2)); %GARCH sigma2(i,2)=sqrt(coef2.K+coef2.GARCH*(sigma(i-1,2).^2)+coef2.ARCH*(logreturn(i-1).^2)); %NGARCH sigma2(i,3)=sqrt(omega2+alpha2*(logreturn(i-1)-theta2sigma(i-1,3)).^2+beta2sigma(i-1,3).^2); end

VaR=-eps*sigma2; vartest=VaR; test1=logreturn(n+1:end); test1=repmat(test1,1,3);

%LRuc: Unconditional Coverage Testing %condition: return<-var ==> return>var

%compare(:,1)~EWMA %compare(:,2)~GARCH %compare(:,3)~NGARCH compare=(test1>vartest); t=length(compare); t1=sum(compare) t0=t-t1

lp=(1-p).^t0p.t1 l1=(1-t1./t).^t0.(t1./t).^t1 lruc=-2log(lp./l1) freedom=1; pvalue=1-chi2cdf(lruc,freedom)

%Independent Testing %1) EWMA t00=0;t01=0; t10=0;t11=0; for i=2:t if compare(i,1)==0 & compare(i-1,1)==0 t00=t00+1; elseif compare(i,1)==1 & compare(i-1,1)==0 t01=t01+1; elseif compare(i,1)==0 & compare(i-1,1)==1 t10=t10+1; elseif compare(i,1)==1 & compare(i-1,1)==1 t11= t11+1; end end

EWMA_tn=[t00 t01 t10 t11] pi01=t01./(t01+t00) pi11=t11./(t10+t11) lpi1=(1-pi01).^t00.pi01.^t01.(1-pi11).^t10.pi11.^t11 lrind=-2log(l1(1)/lpi1) pvalue2=1-chi2cdf(lrind,freedom) lpiwhole=zeros(1,3); lpiwhole(1)=lpi1;

%2) GARCH t00=0;t01=0; t10=0;t11=0; for i=2:t if compare(i,2)==0 & compare(i-1,2)==0 t00=t00+1; elseif compare(i,2)==1 & compare(i-1,2)==0 t01=t01+1; elseif compare(i,2)==0 & compare(i-1,2)==1 t10=t10+1; elseif compare(i,2)==1 & compare(i-1,2)==1 t11= t11+1; end end

GARCH_tn=[t00 t01 t10 t11] pi01=t01./(t01+t00) pi11=t11./(t10+t11) lpi1=(1-pi01).^t00.pi01.^t01.(1-pi11).^t10.pi11.^t11 lrind=-2log(l1(2)/lpi1) pvalue2=1-chi2cdf(lrind,freedom) lpiwhole(2)=lpi1;

%3) NGARCH t00=0;t01=0; t10=0;t11=0; for i=2:t if compare(i,3)==0 & compare(i-1,3)==0 t00=t00+1; elseif compare(i,3)==1 & compare(i-1,3)==0 t01=t01+1; elseif compare(i,3)==0 & compare(i-1,3)==1 t10=t10+1; elseif compare(i,3)==1 & compare(i-1,3)==1 t11= t11+1; end end

NGARCH_tn=[t00 t01 t10 t11] pi01=t01./(t01+t00) pi11=t11./(t10+t11) lpi1=(1-pi01).^t00.pi01.^t01.(1-pi11).^t10.pi11.^t11 lrind=-2log(l1(3)/lpi1) pvalue2=1-chi2cdf(lrind,freedom) lpiwhole(3)=lpi1;

%Conditional Coverage Testing lrcc=-2*log(lp./lpiwhole) pvalue3=1-chi2cdf(lrcc,freedom+1)

Built-in Code function f=rmgQ(mu,casechoice,x,y) n=length(x); %First assign the sigma(1) sigma=zeros(n,1); stdreturn=std(x); sigma(1)=stdreturn; f=zeros(n,1);

switch casechoice case 2 %Garch (1-alpha-beta), alpha, beta for i=2:n; sigma(i)=sqrt(stdreturn^2*(1-mu(1)-mu(2))+x(i-1).^2mu(1)+sigma(i-1).^2mu(2)); end f=log(normpdf(x,0,sigma)); case 3 %Leverage
for i=2:n; sigma(i)=sqrt(mu(1)+mu(2)*(x(i-1)-mu(3)*sigma(i-1)).^2+mu(4)sigma(i-1).^2); end f=log(normpdf(x,0,sigma)); case 9 %t(d) for i=2:n; sigma(i)=sqrt(mu(1)+mu(2)(x(i-1)-mu(3)*sigma(i-1)).^2+mu(4)*sigma(i-1).^2+mu(5)y(i-1).^2/252);
end f=gammaln((mu(6)+1)/2)-gammaln(mu(6)/2)-log(pi)/2-log(mu(6)-2)/2-log(sigma) ... -1/2(1+mu(6))*log(1+(x./sigma).^2/(mu(6)-2)); end

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资产收益的时间序列分析及量化风险建模