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1、外文资料（原文）及译文AN ITERATIVE ALGORITHEM FOR THE RISK MINIMIZATION OF PORTFOLIO INVESTMENTSAbstract An iterative algorithem is proposed for the risk minimization of portfolios, and it is proved to converge to the optimal portfolio. The algorithem is easily operated and no calculation of matrix inverse is
2、needed in the algorithem. Furthermore, calculation difficulty will not increase by using this algorithm when short selling is not allowed. A linear programming model for the risk minimization of portfolios in which short selling is not allowed is also put forward.Key words portfolio; risk; convergen
3、ce; iterative algorithmPortfolio investment is the valid path to disperse and decrease the investment risk.paper2，3 has given the compute method to portfolio investment risk minimization, its essential expression is as following：Supposed n sorts bond compose portfolio，each bonds corresponding invest
4、ment proportion（that is combination weighting coefficient） 、profit proportion and expectation profit proportion is ， and ，the variance and the covariance of iwriR)1;2,1(niwLprofit proportion between each bond and bond is 和 ，so, 2iij );2,1(2ijinLthe profit proportion、expectation profit proportion of
5、portfolio is（1）niiprw1（2）niipR1（3）nijijipw12If use profit proportion variance to measure the risk，then the problem of risk portfolio risk minimization is（4）1.min2pRwtsand ,the best result of the nnijnR12 ),(,)(,),( LLequation（4） is 3,(5.1)nn11*/(5.2)p2)(外文资料（原文）及译文The key to calculate equation(5.1)
6、and equation(5.2) is the problem of inverse of the covariance matrix . Actually, because portfolio usually compose kinds of bond，and the high dimensions matrix inverses are very discomposed. Another problem to calculate equation(5.1) is maybe negative 的 investment proportion，that is short selling. T
7、his means the bond of short selling negative weighting coefficient 的,use the money to positive weighting coefficient bonds investment. While short selling is not allowed, model（4）will have good method to deal with. If introduce not weighting coefficient control to model （4） ，we can get the portfolio
8、 investment risk minimization model while short selling not allowed.(6.1)npwW2mins. t. (6.2、3)01nRThe result of model （6）should accord with Kuhn-Tucker，supposed the first k kinds of bond investment proportion of the best result of equation（6）is positive，the other investment proportion of bond is zer
9、o （ that is not the short selling not allowed minimize risk） ，so Kuhn-Tucker qualification is (7.1)njpijw12)2,1;0(kiiL(7.2)njpij12 ),;(nii(7.3)11kjjnjwUse Kuhn-Tucker qualification，recommend capacity ,will change the model （6）to 1the linear programming model of portfolio investment risk minimization
10、 in which short selling is not allowed.(8)2minpzs. t. (8.1、2、3)0;2,1(0,12iiinij ipijwwLTo result the linear programming model（8）,we can consult paper5. But as the times of the increasing of the capacity in model （8）,disappeared lots of difficulties.Base on the above analysis，this text put forward a
11、kind of interactive algorithm of risk minimization，and proof which at allow the short selling with disallow the short selling circumstance bottom，this 外文资料（原文）及译文algorithem all refrain from rash action in the best result.1 interactive algorithem of risk minimization and its convergence1.1 interactiv
12、e algorithmFor the circumstance of two kinds of portfolios，the calculation of equation（5.1）and （5.2）is very simple，take the 1, the combination of 2 kinds of bonds as an example，its best investment proportion and minimize risk is:（9.1）)2/()(12121 k（9.2）22（9.3）)/()( 12211Because the risk of the superi
13、or portfolio not higher than( general circumstance bottom not lower than) attend risk that each bond that combine（see to lead the reason below） ，therefore we can proceed the combination again with the 3 bond the superior combination of 1 and 2 kinds of bonds，make risk further lowered，proceed like th
14、is and one by one in order，we must to as follows interactive algorithm.1） First compute the best portfolio of and ；let as a new 1r21nr1nbond，compute the best portfolio of it and ；such iterative proceed until 3and s best combination . that is the result of the first round 2nrn 12nriterative, that is
15、，we know11k（10）nnnnrkrrrr121232M2） is similar to the best result，if need to be lowered further risk，can 1nrbe the second round iterative, that is ：let and compose the best combination 12nr；compute again and s best combination ；such iterative proceed nr2nr2 12nruntil and s best combination . More rou
16、nd iterative can also proceed the 3n 13nr外文资料（原文）及译文same.3）Every time the result of iterative all can become the n kinds of bond of combination，its variance can be computed by the equation（9.3） ，among them each bond weighting coefficient is launched by the result of every iterative this time and before，record th
