6 themes on risk management

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6 lectures on Risk Management

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  • 1

    . 1. ..........................................2 2. .................................................................................15 3. ................................................................................31 4. ...........................37 5. ..............................................................................42 6. ..........49 ........................................................................................................................................54

    2012

  • 2

    1. , - :

    1. , ;

    2. , ;

    3. , .

    , . - .

    - :

    , ;

    , .., .

    -, (). . (, ) . - () . . , . , , , , . .

    , . , , . .

    . - .

    : , (, )

    , -, - ( );

    , (, ) , - ( ).

  • 3

    , . - maxmin minimax. , . , .

    maxmin (minimax) , - (, ). . , .

    - , , 0 1, maxmin (minimax).

    () , , .. . , , , , .

    . , . . ( ). , () /.

    - , , () (), ( ) .

    () , .. ( ) . (), . . .

    , .

    , RETURN / RISK SIGMA. ( , ) , .

    RETURN / RISK , , , , .. ,

  • 4

    .

    , , , , , , , , () .. () , , .

    / ( RETURN / RISK) , . (-) .

    , : (, , , -

    ); S ; ija e i- Ai j- -

    njmiS j ,1;,1; ;

    qj. j- , )0( jqj .

    n

    jjq

    11 .

    ,

    nq L 1 (

    ). Ai (

    , ) :

    n

    jijii aqa

    1,

    n

    j

    ijLi n

    aa

    1, , .

    ija m n , , .

    RETURN / RISK i- j- .

    , : , , (, , ), - (, ) () ..

    - RETURN / RISK -:

  • 5

    .maxminmin

    ,max

    ,minmaxmax

    ,min

    ijij

    ij

    iji

    i

    ijji

    ii

    ijj

    i

    a

    a

    a

    a

    , , , .

    rij Ai Sj :

    ijijiijjijaaar max

    , rij 0. ijr

    . Sj

    j; i aij. (regret).

    , , : , , , , , , , , .., .

    . : ; ; ; ; ; ; ..

    - ( , )

    n

    jijii rqr

    1,

    n

    j

    ijLi n

    rr

    1, ,

    . (

    ) : 1. :

    iiaaE max .

    2. iir

    rE min . : 3.

    Liia

    aL max . 4.

    Liir

    rL min . 5.

  • 6

    ijji aW minmax . ,

    , . 6.

    iiijjirS minmaxmin , iij

    jr max .

    , , - - ( ).

    7. ( )

    ii

    iii

    ijj

    ijji

    haaH max})1({max}max)1(min{max , 0 1 (0 1)

    , . , 7 a. = 1, ijji aH minmax)1( , , 7 b. = 0, ijji aH maxmax

    )0( , . , 0 1,

    . , , . , .

    , = 0 = 1 .

    8. ( ) :

    ii

    jii

    ijj

    ijji

    frrF min})1({min}inm)1(max{min 0 < < 1

    , . , 8 a. = 1, ijji rF maxmin

    )1( , , 8 b. = 0, ijji rF minmin

    )0( , . , 0 < < 1,

    . , F = 0 = 1 .

    RETURN / RISK () 12 . , , . , , , .

    (SIGMA) - .

  • 7

    ( , , ..) - () .

    SIGMA , :

    (, , - );

    S ; ija i- j-

    .; qj j- ,

    )0( jqj ;

    n

    jjq

    11 .

    ,

    nq L 1 (

    ). Ai :

    n

    jijji aqa

    1,

    n

    j

    ijLi n

    aa

    1, -

    n

    jjiiji qaa

    1

    2 .)( , -

    n

    j

    Liij

    Li n

    aa1

    2 1.)( .

    , (), : 1. ,

    iiaK max .

    2. , )max( ii

    iP aK .

    3. , )max( ii

    iaK .

    4. , )2max( ii

    io aK .

    5. , L

    ii

    L aK max . 6. ,

    )max( LiL

    ii

    LP aK .

    7. , )max( Li

    Li

    i

    L aK . 8. ,

    )2max( LiL

    ii

    Lo aK .

  • 8

    . , .

    :

    9. iii

    aK 10max*0 . 10.

    LiLi

    iL aK 10max*0 .

    , , ija - , .. , : . .

    (, , , ) Ai ( ), . A, )( AAi i . ( ) , , , , , , - AAi . - ijx , i ( ), j ( ), a xij i- j. , j Kj.

    , : ( ) ( ), . , (), . : , , - . - . - (, ). . , , , , .

    (.. ijx ) , . , - . . .

  • 9

    , - : () MAXIMIN, () () LINCOM.

    ij - , . - . , . , (-) . , .. , - - .

    , , ,

    ijx . ij j . Ai

    ),(min ijjji xt . Ai ti. ,

    ( ) .

    , , .

    , xij ( Ai Kj j . - (, , ) - j . )0( jj

    jj 1 . Ai

    j

    ijji xs . si.

    MAXMIN LINCOM ,

    nL 1 , , (

    ) . MAXMIN LINCOM

    . () : , :

    1 ; 2 ; 3 ; 4 , .

  • 10

    : S1 . ; S2 ; S3 ()

    . aij (43), .3,1;4,1 ji

    SjAi

    S1 S2 S3

    A1 85 5 45 A2 20 30 15 A3 75 20 35 A4 25 80 25

    :

    .4.0;5.0;1.0 321 qqq , , 2.0 . : , 13Lq . : 8 . TBP1, TBP2 TBP3.

    TBP1 S1 S2 S3 i i ia ir Lia Lir

    A1 85

    0 5

    75 45

    0 5 75 29 37.5 45* 25*

    A2 20

    65 30

    50 15

    30 12 65 23 43.5 21.7 48.3

    A3 75

    10 20

    60 35

    10 20 60* 31.5 35 43.3 26.7

    A4 25

    60 80

    0 25

    20 25* 60* 52.5* 14* 43.3 26.7

    qj 0.1 0.5 0.4

    qL 3-1 3-1 3-1

    j 85 80 45 .1;1

    1

    Ln

    j qq

    TBP1 : Ea, Er, La, Lr, W S. aij , rij.

  • 11

    :

    ;5.5225)4.0(80)5.0(25)1.0(

    ;5.3135)4.0(20)5.0(75)1.0(

    ;2315)4.0(30)5.0(20)1.0(

    ;2945)4.0(5)5.0(85)1.0(

    4

    3

    2

    1

    a

    a

    a

    a

    ;1420)4.0(0)5.0(60)1.0(

    ;3510)4.0(60)5.0(10)1.0(

    ;5.4330)4.0(50)5.0(65)1.0(

    ;5.370)4.0(75)5.0(0)1.0(

    4

    3

    2

    1

    r

    r

    r

    r

    ;3.43)258025(3

    ;3.43)352075(3

    ;7.21)153020(3

    ;45)45585(3

    14

    13

    12

    11

    L

    L

    L

    L

    a

    a

    a

    a

    .7.26)20060(3

    ;7.26)106010(3

    ;3.48)305065(3

    ;25)0750(3

    14

    13

    12

    11

    L

    L

    L

    L

    r

    r

    r

    r

    TBP2

    i i i i 2.0ih 2.0if 0ih 0if 1ih 1if

    A1 5 85 0 75 69* 15 85* 0* 5 75

    A2 12 30 30 65 27 37 30 30 15 65

    A3 20 75 10 60 64 20 75 10 20 60*

    A4 25 80 0 60 69* 12* 80 0* 25* 60* TBP2 : H0.2, F0.2, H0, F0, H1 F1, }1;2.0;0{ . :

    ;6980)8.0(25)2.0(

    ;6475)8.0(20)2.0(

    ;2730)8.0(15)2.0(

    ;6985)8.0(5)2.0(

    2.04

    2.03

    2.02

    2.01

    h

    h

    h

    h

    .120)8.0(60)2.0(

    ;2010)8.0(60)2.0(

    ;3730)8.0(65)2.0(

    ;150)8.0(75)2.0(

    2.04

    2.03

    2.02

    2.01

    f

    f

    f

    f

    TBP1 TBP2 * , .

    TBP3 Ea Er La Lr W S H0.2 F0.2 H0 F0 H1 F1

    A4 A4 A1 A1 A4 A3 A4

    A1 A4

    A4 A1 A1 A4

    A4 A3 A4

    TBP3 , ,

    6 4, 3 A1, 1 A3, A2

  • 12

    . , , .

    - . SIGMA : 1, 2 3 S1, S2 S3

    .3.0;5.0;2.0 321 qqq aij () (43), 3,1i ; 3,1j .

    Sj

    Ai S1 S2 S3

    A1 -5 +10 +20 A2 -1 +6 +14 A3 +4 +7 +9 qj 0.2 0.5 0.3 qL 31 31 31

    : , qL = 3-1. : 8 . TBP4, TBP5 TBP6.

    TBP4

    ia i Lia Li iia iia iia 2 LiLia LiLia LiLia 2A

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