Modeling Multiple Securities

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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) 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);

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;

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);

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%!

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

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

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+B1iF1

t+B2iF2

t+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

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…

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;

The Securities Lets model returns in the following

way: rt

i=µi+Bi(rtM-µ)+σiεt

i

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 rt

M in a unique way Vector B will define the loadings for

each security

crosssection.mfunction 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;

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);

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

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;

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');

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?