: a.chinchuluun@imperial.ac.uk

.

.

(1952).

X ?

X ?

2 ( ( ).

A = 20% , 20%

B = 10% , 20%

X 2 20% - ?

: = , =

-

2.0 0.2,Ar A ==

2.0 0.1,Br B ==

01.0=-- 50%-50% :

:

01.0AB =

0.5== BA xx

15.0)1.0(5.0)2.0(5.0r =+=16.0)2.0()5.0(01.0)5.0(2)2.0()5.0( 22222 =++=

1952 - .

.

: . .

(Efficient or Pareto optimal) .

?

5 ()

: ?

?

5 ()

: ?

X: .

?

5 ()

P1 (A-60%,C-40%),

P2(A-50%,D-50%),

P3(A-100%),

P4(A-30%,D-40%,E-30%),

P5(E-100%)

?

5 ()

P1 (A-60%,C-40%),

P2(A-50%,D-50%),

P3(A-100%),

P4(A-30%,D-40%,E-30%),

P5(E-100%)

?

e ?

e ?

.

e ?

.

.

:

: 10%, 20%,30% .

3.0,2.0,1.0 321 === rrr .

: x1, x2, x3

()

(20%)

233213311221

23

23

22

22

21

21

222

xxxxxx

xxx

+++

+++

2.03.02.01.0 321 =++ xxx

1321 =++ xxx

Min ()

:

(1) =

233213311221

23

23

22

22

21

21

222min

xxxxxx

xxx

+++

+++

=

(2) (20%)

(3)

2.03.02.01.0 321 =++ xxx

1.. 321 =++ xxxts

0,, 321 xxx

?

?

. .

.

x x .

x? .

.

.

( ).

:

Mean Standard Deviation ()

VaR (Value-at-Risk)

CVaR (Conditional Value at Risk)

Mean-Absolut Deviation

.

.

,

X

.

.

.

( )

X ( 20, S&P 500, Nikkei 225, x FTSE 100).

1992-1995, 475 v vv 96.1 - vv.vv 96.1 - vv.

1995 vv.

M- x: 2005 - 1 834 174 222.27 T, 2006 - 1 870 346 551.47T, 2007 - 2 458 153 074.76T

: 20

1. A. Chinchuluun, P.M. Pardalos, A. Migdalas, and L.Pitsoulis, editors. Pareto Optimality, Game Theory andEquilibria, Springer (2008)

2. F. Fabozzi, P. Kolm, D. Pachamanova, and S. Focardi.Robust Portfolio Optimization and Management, WileyRobust Portfolio Optimization and Management, Wiley(2007)

3. R. Fernholz. Stochastic Portfolio Theory, Springer-Verlag(2002)

4. D. Luenberger. Investment Science, Oxford UniversityPress (1998)

5. H. Markowitz. Portfolio Selection, Journal of Finance,7(1):77-91 (1952)

6. W. Sharpe. Portfolio Theory and Capital Markets.McGraw-Hill (2000)

!

, ?