* * PORTFOLIO.RPF * Portfolio optimization calculations. * * RATS User's Guide, from Section 13.2 * open data berndt.xls calendar(m) 1978 data(format=xls,org=columns) 1978:1 1987:12 mobil ibm weyer citcrp market rkfree * * Compute the covariance matrix of returns into omega. Save the means as * mu. * vcv(center,matrix=omega) # mobil ibm weyer citcrp * compute n=%nvar compute mu=%meanv compute [rect] ones=%fill(1,n,1.0) * * Compute the minimum variance portfolio, and its expected return. All * the analysis is constraining portfolios to non-negative weights. If * you want to allow short positions, add the NNEG=0 option to LQPROG. * lqprog(q=omega,a=ones,b=1.0,equalities=1) x compute minr=%dot(x,mu) * * Choose as upper limit the highest expected return. * compute maxr=%maxvalue(mu) * * Create a grid series for expected values running from minr to maxr * set erets 1 101 = minr+(maxr-minr)*.01*(t-1) set srets 1 101 = 0.0 * * Find the minimum variance portfolio with the expected return erets(t). * Note - the problem being solved is to min .5 x'Qx subject to the * constraints. To get the variance (rather than .5 variance), we need to * multiply the optimized value by 2. * do t=1,101 lqprog(noprint,q=omega,a=ones~~tr(mu),b=1.0~~erets(t),equal=2) compute srets(t)=sqrt(2.0*%funcval) end do t scatter(style=lines,\$ header="Efficient Frontier w/o Risk-Free Asset",\$ vlabel="Expected Return",\$ hlabel="Sigma") # srets erets * * With riskfree asset. There's no longer a constraint that the weights * on the portfolio sum to one. Instead, the expected return constraint * is altered to weights * (mu-Rf) = target return - Rf. * stats(noprint) rkfree compute rf=%mean set sretsrf 1 101 = 0.0 compute [rect] arf=tr(mu)-rf do t=1,101 lqprog(noprint,q=omega,a=arf,b=erets(t)-rf,equalities=1) x compute sretsrf(t)=sqrt(2.0*%funcval) end do t * scatter(style=lines,\$ header="Efficient Frontier with Risk-Free Asset",\$ vlabel="Expected Return",\$ hlabel="Sigma") 2 # srets erets # sretsrf erets