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Modeling Multiple Securities

Modeling Multiple Securities. Downloads. Today’s work is in: matlab_lec06.m Functions we need today: crosssection.m. Diversification (no aggregate risk). Lets create a large number of uncorrelated securities, then form portfolios R(t,i)= µ (i)+ σ (i)*x(t,i)

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Modeling Multiple Securities

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  1. Modeling Multiple Securities

  2. Downloads • Today’s work is in: matlab_lec06.m • Functions we need today: crosssection.m

  3. Diversification (no aggregate risk) • Lets create a large number of uncorrelated securities, then form portfolios • R(t,i)=µ(i)+σ(i)*x(t,i) • We will see that by forming a large enough portfolio, we can diversify away all risk >>N=500; T=1000; >>mui=ones(1,N)*1.15; >>sigmai=.1+.55*rand(1,N);

  4. Diversification (no aggregate risk) >>for i=1:N; R(:,i)=mui(i)+sigmai(i)*randn(T,1); end; >>for i=2:N; P=mean(R(:,1:i)')'; portstd(i)=std(P); end; >>plot(portstd,'b'); hold on;

  5. Diversification (with aggregate risk) • Now lets add a systematic element to the risk • R(t,i)=µ(i)+σ(i)*x(t,i)+βM*y(t) • You can diversify away all idiosyncratic risk, but not the aggregate risk >>N=500; T=1000; >>mui=ones(1,N)*1.15; >>sigmai=.1+.25*rand(1,N); >>betaM=ones(1,N); Y=.16*randn(T,1);

  6. Diversification (with aggregate risk) >>for i=1:N; RA(:,i)=mui(i)+sigmai(i)*randn(T,1)+betaM(i)*Y; end; >>for i=2:N; P=mean(RA(:,1:i)')'; portstdA(i)=std(P); end; >>plot(portstdA, 'r'); hold off; • In the second case, individual securities are rigged to be less volatile than in the first case • Nevertheless, portfolio volatilities in the first case are close to zero, in second case, above 16%!

  7. Correlation of Securities • Many securities are correlated • This includes within industry (ie BT and O2), between industry (ie BT and BP), domestic, as well as international (ie BT and Ford) • Oftentimes we can narrow down the correlation to a small number of factors • This correlation represents aggregate risk

  8. Correlation and Diversification • You cannot diversify away systematic correlation! • 1000 uncorrelated securities, each with annual vol 30%; an equal weighted portfolio will have vol 0% • 1000 perfectly correlated securities, each with annual vol 30%; equal weighted portfolio will have vol 30%! • Estimating correlation is extremely important for portfolio management but we are bad at it • This problem is more extreme during extreme times

  9. Factor Models • There is a small number of relevant factors, ie F1 and F2 • Each security loads on the factors in a different way • Rit=μi+B1iF1t+B2iF2t+εit • μi typically depends on the covariance of R with the factors, that is on B • CAPM is a one factor model, where the factor is the aggregate market return • Factors themselves may be correlated

  10. More on Factors • What are factors? Anything associated with aggregate risk: Market, P/E, interest rate, HML, SMB, Momentum, GDP growth, consumption growth, volatility… • Note that they are contemporaneous, not predictive • They may have predictive power, if the factors themselves are predictable • How to model them? AR(1), log-normal…

  11. One Factor Model • Let rM be our only factor, and lets model it as a normal process >>T=1000; µ=1.1; σ=.16; x=randn(T,1); >>rM=µ+σ*x; • Suppose a risk-free bond is available in this economy, and it pays 2% annual interest >>rf=1.02;

  12. The Securities • Lets model returns in the following way: • rti=µi+Bi(rtM-µ)+σiεti • What should E[rti]=µi be? There is theoretical justification to let it be: µi=rf+Bi(µ-rf) • Create a function that will output a cross-section of securities, all of them load on the factor rtM in a unique way • Vector B will define the loadings for each security

  13. crosssection.m function R=crosssection(mu,sigma,rf,B,sigmai,T) [a nsec]=size(B); R=zeros(T,nsec); X=randn(T,1); Xi=randn(T,nsec); rm=mu+sigma*X; mui=rf+B*(mu-rf); for i=1:nsec; R(:,i)=mui(i)+(rm-mu).*B(i)+Xi(:,i)*sigmai(i); end;

  14. SimulateCross-section >>N=10000; >>B=-.5+2.5*rand(1,N); >>sigmai=.05+.55*rand(1,N); >>mui=rf+B*(mu-rf); >>R=crosssection(mu,sigma,rf,B,sigmai,T);

  15. Mean-StDev Frontier • Form portfolios of assets, each portfolio will contain assets with a similar mean return • We will form portfolios of 2 assets with similar mean, 5 assets with similar mean, 10, 20, 50, and 100 • We will then plot the mean of these portfolios against the standard deviation • All else equal, we prefer assets with lower volatility

  16. Create Portfolios A=[2 5 10 20 50 100]; for j=1:6; for i=1:22; x(i)=(.98+.01*i); s=0; in=zeros(1,N); for k=1:N; if s<=A(j) & abs(mean(R(:,k))-x(i))<.005; s=s+1; in(k)=1; end; end; P=mean(R(:,in==1)') '; portmean(i,j)=mean(P); portstd(i,j)=std(P); end; end;

  17. Plot Mean-StDevFrontier >>plot(portstd(:,1),portmean(:,1),'c'); >>hold on; >>plot(portstd(:,2),portmean(:,2),'r--'); >>plot(portstd(:,3),portmean(:,3),'b'); >>plot(portstd(:,4),portmean(:,4),'m--'); >>plot(portstd(:,5),portmean(:,5),'g'); >>plot(portstd(:,6),portmean(:,6),'k--'); >>plot(0,rf,'rd');

  18. Suggested Homework (4) • Model two securities with jumps • Make the securities uncorrelated • Allow for potential correlation of the jumps and have a parameter which can set that correlation to high or low • Calculate the correlation of the two assets and the value at risk for this portfolio • How does correlation change when jump correlation increases? How does value at risk change?

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