"Capital Markets & Risk Management" (8611)

Transkript

1 Wirtschaftswissenschaftlies Zentrum der Universität Basel "Capital Markets & Risk Management" (8611) Dozenten: Prof. Dr. Heinz Zimmermann Dr. Patrick Wegmann Wintersemester 00/003

2 Inhalt! "#$% &#$'() *!+, )-./!! (0 (((((1 0 (!& )(1 (3,(!! ('( $ 4..!! 5('"!, )6!,,*( '((%#$ 4(0! )((!7 )#$8#$((!7 )(6! " #,!% ''),&9#!:(!' * (%%7/,&9( 0,);80,,<!! 3,/*( '(( $ %& '() #&* + 0* '(# =9(> 0* #=9>, 0<((/ 0).$( /

3 1. Einführung und Value-at-Risk 1.1 Grundlagen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

4 o #$8 ' o K#$D!(#$A$ +@ %=@3,HA33HA333H> #$!%!& 8 8 A'! +"% #$ 8!=L&9#!-$( 'L> -!!%8#$ =+(> o 7 #$ (A!$ (! C%8#$ ( & 8 8 8!#$8#$ o %7 #$=<! A > o + =@! -!!( >%$ +=@A/,I+3,H> ( '"!&9#!-$( ' F ( x x q) q < = q x f f ( x) ( x) dx dx E & 1I+G! (!<! + ( ( '"!-!!%8#$ F σ = ( x i x) 1 N N 1 i= 1 1. Wünschbare Eigenschaften eines Risikomasses + GG!! &9#! -$( ' > (((F &(((((A!$ F 9 *( '((!9 9 A('('" ((((A!=9 > =9 > 5( FD K! *((A!($K$!( &#$'8!(+ '" >. =9M>I=9>N 6 $=!!$%#$ (%$'>A9 (%$' '((!(@ *( '((9 $!! #$!(! &#$'8!(+ '" > 6((F =% 9>I% =9>!*( '((9D!&@(( G%G'#$=@!(>A( G %G'#$#$ #$!( &#$'8!(+ '" )>- %!!F =9 M9 > RM(x 1 ) + RM(x ) '('' (!'" ( A!! ( ( *( '((!-! &@ &#$'8!(+#$ '"O + & 8!! )

5 F<! ( & %$ D 6 P! *(( $ C '8$ #$#$(H +'" 6 = 3,H! $ #$! "#A( 30H! > +'" %!*((F o C '! ( (!F*I30 30I3)3Q+I o C '! ( (!F*I30 I,R,H+ I o C '! ( (!F*I I3S,H+ I o.(f+ M+ M+ I &(%+K#$A!! +*( '(( K!-! &@G+ #$! D!(#$ A!(#$!*$( %(< =$ <!*( '((> '<!%! (#$%!8!(! % #$ 1.3 Parameter des VaR + GG 8! $ #$#$ (@8 ( %' F! #$ (#$! K!*( '(( (A!$D!(#$!&' <('!@ F!@#$A!$K$ #$ #=@3,HA33H>$K$ ($P$K$ D!(#$!#A!(8 (%#$ %8! #$8 A!8#$% "! (!$@ ''%@8 L #$L (! C!!8!A8! +#$ =+ #$>@8!A $ '# 8$ ' + #$@@8#$@8 <!( '(( KA! <(@! 8 (! (! #$ 8#$ +% #$@8!A!%K<@ @A8T% #$! +! $ &9@%! ($ 8!!( $ %@8 4%( C '8$ G < #$ + #$#$ =#> C H P$ 333H 0H P$ 33H H P$ 30/H K +(#$A8#$ C( '(( ( A! 8!.(! P$ $ 8! 4''#$#$ 8!!*( '((P$ $ #$8.A #$!! +=I# > $K$,

6 &*( '((8!"% * (!.I.$CD!6!%8 #$! %$(+(! #$+! +%@8!-!!%8#$!*( '(('" I. + =.>I+ =>M+ =>M+ =>MUM+ => + =.>I. + => σ ( T ) = σ ( t) T! 8! &@ (! #$ %$ ( *( '((! % "##$ (! A M! <( (F σ R R = σ R + σ R + Cov R, R t + t + 1 t t + 1 t t + 1 ( ) ( ) ( ) ( ) 8= >I= M >I=>A(8 F σ R + R + = σ R + σ R + ρ σ R ( t t 1) ( ) ( ) ( ) σ ( R + R + ) = σ ( R) ( + ρ) t t 1 R!( T A8S#$8#$ T ; %#$A!*( '((! $%! : '@=@! #$-(:( C ' >!- %' 1.4 Stress Testing + GG#$ "%!(!@8! &9#!-$( '% #$ D!(#$ =<! (!(!%(!! +! >( CC! +! -. '(!-@ (7A% #$(! C! -. #$ $! =+ #$ A *( '((! >% -@ ( 'F!+! (-#$" %=5 > o #$8 A #$-@ (@!' 6( #$- ( o 8 $ #$ *( '((%%& =30G $A V A -% #> $ * (-@ o 8 8W o &! ;(-.#$A(!+ #$ 8#A (!! -(@ $ #$ -@ /

7 . Marktrisiko.1 Grundlagen des Marktrisikos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ekomposition von Risikoquellen *' -! 7 $ + '! <(%(($ &@'( C#$ '&9( "% (!! 0

8 ( % &9( %@#$!-! ; K "%! 7 F(! ( (!'! ((!!' F P DM y P = ( ( F P = ( DM P) y = DD y =( > (( ; $@8#$X!!!*! = G>!(!( '(($(!*!(! %+ '! *(($8!! '! +! &9( (!! '! +!!X! &$K$ &9( =$ $K$ (>$#$(#$&''A $K$ %!C 8 +!!! 7=$!X!>! &9( (!( *(( " C#$!(! C! (!(((@8#$ (!!&9(! C"%! F R i = a i + i R M + i i R M + (#$!&9(! +!! + %-( $@'#$(=! #$ (> '!.F Pi Ri = A!F Pi = βi Pi RM + εi Pi P i ('(F Var(P i ) = i P i Var(R M ) + P i Var( i ) + Cov(...) $ ('( =@ #$!@!9>(! '( 8!A('"$!@ Y &8!!( A!!Y (!!!Y D!A(!! #$! =* >$ '! & 8 8 &=Y >IA! '@ #$!Y 8" (*( '((!#$! (! '(''#$$.3 Diskontinuitäten und Event Risks $ 8!@ + '!! (A8( =@!%#$8>(((( '(5! % A ( C%! 8! ( 'F =< G> - " ; (! (#$ ( $ $% A#$ 7#$& =&> A8!&%@#$ &'" (#$&! K # $(330A!C!@ $%6!!NH ( %$ % #$C@! ( =&9(!+>#$#$'"! (& &9( %! C (9( #$ JA! '" < %8 +'" (

9 < #$ #$ A8!+! #$ #$!%! ; ((#$&"-.! #$'"$ 8! (! % #$#$ *$(%&C #$ $% ( '*7(''A 8 -!! 7%&9 7 '(! "% - ( - "!( (#$ 5!! +GC@ (!! &9( (''#$#$.4 Risikoquellen im Marktrisiko =( &9#$AB>F '9%#$ F#$! '9#$ FC 'G!C%8 =I&> 8#$'9'9%.7#$+(F/HNH -@%BG* (! F ( %F- =-*> - =-&> #$F- =- - >I=&*> #$F I M N ; (! =5 A5>F (!! X!! J( J(! o y ~ N o (y) = const (!! X!! (( (! y o ~ N y y o σ = const y <( F y y o σ ( y) = σ y = y σ y y C (! =&1 A&?>F ( 8!! F o F t e r = S t e y o F I A I;@A- IC A7I!X!A I '@ ((!F ( 8!! F o F t e r = S t e y o $ F7I(#X! o (#X!F+(! : $ AD#$+ '"% $ ( 3

10 .5 Modellierung der Risikofaktoren "! $(! $#$ (!! ('( >(8!'#$(#$(!!E( A! %(! +( $ -% F : (%( ( E(!8! ( FE(! * (!!8!% #$#$! (!( F P1 P0 = R1 ~ N P 0 ( ) '(F ~ P N P µ ( Pσ ) P , D!(#$!K#$ * * C!!8!! (( + 8!'"!F P 1 r = ln ~ N P * Z(E 0 (( + '" (*!'!* @( $#$& %8!!( *! #$ #$A( 'A!#$!('#$ (! #$!E( %%! 5%(! %(%#$!!*$(! L.LA! ( #$ < ( $%!"%!! E( + +=33HG? > I G I G/3/, I) G0)0 I/ G) I[ G/ E G/ $! + $! C 8 '!+5!.%! + '" #$! + ''"$ P$ $!G+ $A!(!! 33H? P 8 $!A!(!!.=!($K$!< (>G+ ( & #$(C!"%.8!%! #$G+ I) $!%#$ % & 8 #$ % ( =6 (!!!> C! 8*$ C$ E(!!(8'( F E(! #$+ & #$ + =I)> ; %( +(*$ $$ E(!

11 r N(, σ ) µ t ~ t C! %( 8! C6* (@F GARCH(1,1): h t = r t 1 + h t 1 &$($8 N @ ;% '#$$ (A! TG#$$ 8!8! (! C6* (@! &C =9(7 8$! ( >F $ I $ G M=G> G & #$ %#$ '" ( I3)'" #$! I30'" (! &! C@ 8$! $(!! +(A!! +(! ; K ( 5 +( ' %A! +( % # %$ ( $(!! %$ (! (! ( A 8 #$ '!=(G>+!( C@! ( % (! $(!! ((! 5%(! #$!$!((%8 (! (8 E( $(!! (8 8!'"! &9( C (9( A 8$!'"! ( E( + ( : (%( (&9(! ( '"$ E( C$! (! #A%! ; ;! (! 8!= (> &9(!(! "% ;! #$ %8!F P P = y = DP y y &9( #$ + %=I'"!&9( > #$!$%% C@'"!&9( ( E( $(! E( $(!#$ #$'#$% #$A% '" (\!! ( A 8! C (9( '" (!! E( $(! #$ #$ &'' &9( %"##$ -$*!+A-)

12 && E( $(!! $(!! C C (9(!&9( (9( $(A#$F P 1 P P = y + ( y) y y P = DP y + 1 C P ( y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

13 3. Risikomanagement mit Derivaten 3.1 Charakterisierung '( ( #$ '#$ L& 5 A! #$ (! 5 % & 4( ' KL! % '(1 D!@ G. #$' G ( 8! G !! G G <'((!+ '#$ #$!! *!'( ( 8! =!A 8 #$ % >A ( #$! #$@8 < '@ #$! 8! 4.G =4.$ ( > (.!!G =%K $! > #$! 5 A! #$ =% + '#$ > %"!A! 5 A!%!+ '#$ % "! &%! + '#$ $ & 8 8! E A8! ( 8! #$% #$ 8! ( 8! % #$ #$ '! (!! + '#$ $% 7 #$ (%@8 7 #$8+ * (' 4($%@ -!! %@$* ( 4(7 #$$!- * # % A$D!(#$&' '!=(>! 4(!! '! 4( #$! 7 #$ 8 % #$ J 8K$#$J ( A 8$!%%! + '#$! + #$'! :(( 8! -7$#:( -7$# :( 5 # * 3. OTC vs. Traded Derivatives G G (( G -$ G 4( G.!!4( :( ( 8!! *7('':(( 8!:( :(( 8! - * :(!! 7 8$! $ %% &*( '((A! * (( < %#$ =*( '(( 5 #A-7$#:( >A $ + A!#$7 #$ + K! *! * ((!#$K#$ 4.$L4.$( L4.* (!!#$%K $!A(! #$! <!8"#$ ( 8! 8! ( B #$' $ ' 8!=. #$'>(#$( 8! '#$#$!* C $ A! * -#$@ (-@ ' 5

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

15 5! '".!! ( ' =@ (! K >A!#$ 3.3 Implikationen für das Risk Management 4.* (!! L7(!LA '!!" '! D8 <! %& #$!!4. '@'#$&@ ($% %!!C%#$ #$$ '"!4.* (! (E 1!+! #$!8#$ "!!C%#$ %!" '!( #$8 A!$ ( +!! *(( (8! 4. * (! :1! (! (!@ (=<! (>A! ( K AD$ 8 %( C%#$ (' A 8 8 +! #$.!!!"%!$(#$1!A#$% '"!C%#$ =@ -#$8! C (! $%! 6 8$ >$($-!! #$ C% '! &9( K#$ C ( G!:1! ( %!.!!! (A!( #$ &8#! (((!! 7! $ "$ 3.4 Hedging mit Futures *((@ #$& #$@8 F (! C #$W %@$#$ '!( C! C #$W %@$#$ '! #$! + ( $!*( F! + (!! 7!! = '>+ (!6!=!! ( >&F ]+I]*M ] &%! ]+!+!!*( '((8 =+ >A]*!+!!! 7=*((>A]!! <(!!C@$ <( R #$( (*((A8S( $( *((;F + =]*M ]> (F ^+ =]*>M + =]>M(=]*A]>_ 4 '! %@( 'A!!C@$<(!@ %' % K Var( ) = 0 = n Var( F ) + Cov( P, F ) n Cov( P, F ) (!F n* = Var( F ),

16 C! # #$#$ %8!F Cov( RP, RF ) P n* = Var R ( ) F F #$ ( "% #$ 8!! *( #$!< '=%@8+ '>! C@$ ( #$ + # *( #$!$% ( + # 6! 6 ' % #$! +*( '(( (%$' %$ (! <( (@8#$*( '((!! 7! =@ & F 8! ' 8A $(#$!!! 7 %(> 4%! 8#$ 6K$ (%$A#$% #$#$'(! ( F σ = σ ρ H * P 1 PF 5( F8!*( '(((!(!A!((! %$!8! #$!8!! C! # 1 ρpf #$? 6!%@#$ 5 C A! *( '(( (! %!#& 5( $!$A #$!#$.*( '(( #$! %@ #$ 8$A!!( '((8 *( ' #$!#$ %$!A ( $K$ #$! <( (@8#$ 6!!*( '((A! ( '" % ' :1!!6!< 'A8@'#$ 1! $!8! " 8K$#$ ((! *( '((%#$ 8!! C (( #$!F. # A! (. # %@#$!(A8#$ #$ + #$% ( *( '!! 7! $A%@#$!( #$ 8 (!! 7A!! (%@#$!(A8#$ #$ (( C%#$!*( '((! #$! 6! 5!.$( ( #$( 8!<( #$!5!9G! 78#A 8'" #$ %F *( '. # F arb ( (( = I e ( 5!9 ( r δ )( T t ) t. T t ( + #6!6` ( 6` * /

17 -$ 8!!(!'"!; '#$!A!! #$!$!5!9!'" (' #$C % (!(%#$ A(! 5 I 5! #$(%#$ % 5 SA8 ;! (!!! #$ $$;!!! ''@! 5!9A!$* '(#$!#$#$ C % #$ ''@A8* <( %!!! :1!@8! 3.5 Portfolioabsicherung mit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

20 ~ ~ r* E( r r < r *) = r f ( r) dr &9#!:( '"8#$&#$'!C'(! (A!(($ & #$! 8#$(! (7 > :*F:(+(7 :(+(7%#$ %A8 -$( '=!$A! + K!.$ $(!>!&9#!:( <(@ "!8!D!(#$!-!!%8#$ 8!F Loss Volatility LPM = Var( ~ r ~ r < r *) = C(@ -$ (8!! (-$( '=.->!' F R R RTS1 P RP R = RTS = LPM1 LPM 4. Commodities 4.1 Commodity Futures ((!$($8!($(''A% #$ %('';! ((!@$! F&! =-% A (!A*> A 5! =< ' A-$A C >A:!8 <''>! & &#$! A8#$* (! (#$!! A8*! #$ ((!7 %#$ 8! K % 8! C$ (''F% (AD! "'* %@ 8!& %? (! #$#$ 9 (A!$ * 8#! ( F ;!% '!@ "'*! * (! '9 % 8&!! * (!% #$#$#$! ( F ~π = ~ p q c( q) + n ( ~ p f ) (%F ~ p F *! * (!=#$ $%$'A!$%!Z> 1F A!8$!! * (! '8! #=1>F* (! (( F C@$. ( 'F. =$ * '" &!! * (!' > * 1!8!(* K! 9(%(! %#$ $%$'A8!!*(%! + #+ '$ K& %#$F

21 1 max E π η Var π q, n max E p q c q + n, n C% F ( ) ( ) %@8F ( ) ( ) [ E( p) f ] η Var( π ) q 0 q = 0 E p f η q + n Var p n = E( p) c' ( q) η ( q + n) Var( p) ( ) ( ) ( ) '"$ (''#$#$@ F 'I#J=1> #$! "#!(* (! *(! ( #$!. #$F!. $! * G * (! (A#$ %D* 8 $A8!! C% #$ 'K & %#$F ( p) f Var( p) E n* = q + η (! #$@8= %$>.&.A! #$ '!(* (! (#$! %@$A!.A! #$!& 8!5!! "%! F n = q E( p) f - F n = η Var( p) <(( (''#$#$! E A 8&=> I' A 8 #$ (!!! & 8 D! =(!. >!#!5!! (& 8 $A!( '%@88 '! #$! * (! 8! C!! ( #$!- ($( %F L* (! (G!- (#$! K =! "> (''8!L 4. Gleichgewicht im Commodity Futures Markt 5 #$8#$ " #$ C%(! E#$' #$ C%#$! $A(F 1

22 &F (!F j ( p) E f n j* = 0 = q j + η Var j j η M j j j 1 1 =!q j = Q η ( p) f 1 ( p) ηm E 0 = Q + Var 1 η 'KF E( p) f = Q Var( p) M j ( p) & 8! #$ 5!9 $A 8 #$ #$8#$ 8 ' (? 8!=?R>A(! #$-! #$ 'D! K!(&=>R' #$!(J%J - (@ $A"-! ( A'"! ( #$!8!8(( %#$! '! 8 &=> &=>R'A(#$!!(!-!'" A! '. '&%$(-$( 6!* =!* $>&%$(C%#$ "% $! * $ &=> S S A %$ ( C%#$ "% $ '<( #$(:(6!* 5(&=>I'?IA! '"$ #$@ - ( #$(E 5((! K(( ' A8+ C 8K$#$%! CG! ((!7.7#$ 8! (! ((!7 (! #$ :1!@ 8#$& % G- 8! <!G%@8 (!! 6! %'! #$#$! <(8G6(8! %(! * (! #$' 4.3 Cross Hedging $ 8! A! '8! 6 '%$% '"!@'#$%@ #$! (! #$#$. = 1!A ( #> & ( = > 6! '! 8!A!! %@ #$! ( $ #$! #$% % K#$ #$! *! (6!!AG%@8.! ( &K#$(!A! (%@ %!F ~ ~ ~ π = q p C q + n f f ( ) [ ] 1 &%! F1F* ' F. % 0

23 F* % ' F. $ F8 =1>F* (! (( &8!8%(!($ (8!!+ K9 F 1 max V = E( π ) η Var( π ) q, n 8(%F E( π ) = q E( ~ p) C( q) + n [ E( ~ f1) f 0] ~ ~!F Var( π ) = q var( ~ p) + n var( f1) + qn cov( ~ p, f1) &#$8(!- ($( A(! V = 0 q * =* (! (> q V! = 0 n * =C%#$ > A!#J=1`>I' - #$F!!! * 8! * (! (#$!8!8 #$ ($ ( (''!!'" A!' 5'( (@ %; =''@ ; >&(!&(! (@!#$ K#$A%#$%#$ % =8#$G!. # > ( 6! $! ( *(! (#$! ( '( #$ (!!! & 8 F c' q * η var p 1 ρ = 1 β E p + β f E B + pf pf pf 0 ( ) ( ) ( )( ) ( ) ( ) ( ) F!F β pf cov = var (6!( ( p, f ) ρ pf 1 = ( f1) σ f σ B = f 1 p E B = E f p = E f (!F ( ) ( ) ( 1) 0 1 f p * (! ( ( aa( #$!! * 8 ('"! * (! (#$! " * 8 '"!* (! (#$! P8 #$6!!* (! $=!$D '!<( (@8#$!'>A!(#$#$! 6!!!($ IP$ IA!(8#$ 8!%! %D* 8 %$!(A(% %#$ 8!(! #$ =!- ($( O> * (! ( 8#$ #$#$ ( (%D 8! %D* 8 - %DF=N ' > &=> F = ' > ' * (! ('"!* (! (#$! #$ ( 6! ( ( $!! #$! %@ #$! (!6!A(8! (< #$%(8!

24 ( 6! (%$! ( '"!E#$GC%#$ %!* ( 6<%! <(((! * (! ( C%#$ %! =(!<( ( ' >$(&'' '!( * (! (A 8! %$! $! & ' <( ( #$(%! #$8!! #$A!( * (! A!% (!6!A8( (6!(9 A! C%#$ #$! <(9 8 #$'#$ $ 5. Kreidtrisiko 5.1 Grundbegriffe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`A, 1`A3 =b ' I,> =b ' I3> 1 )

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aR - ' +,

26 5. Expected Loss und Default Probability &9#!:(A(! 8 + A@#$! K@ F - C '8$ #$#$F8!%! #$!(! - & (F8#$ #( 7G:(G@ #$W - & 8 &9( C '8$ #$#$=!' $ #$#$ % * (!> % C '8$ #$#$ P$ '! %A " C '8$ #$#$% #$ 8!! #$#$#$ ' G $ #$#$ #$ '" C '% #$8!A (F PD = 1 [( Pnd [ t = 1] )( Pnd [ t = ] )] F P = 1 [( 0.8) ( 0.9) ] = = 0. 6 D D$ C '8$ #$#$ % #$ #$ ( ( #$ #$#$A 8! ' G$ #$#$ ( (#$ C '$ #$#$% #$#$@ F d = 1 1 d m 1 y + $! C '8$ #$#$"%!;$%(C " $($ =CCC% >! C ''$ #$#$"%!;@ A 8 #$8 A($($-#$! $A8$!%'C =%>!C '8$ #$#$"%!;%A8L!- "% %L< $%8#$D$!! (A(! 8! $(#$(! $!"<( 5! #$ (#$' $ #$#$@ $(#$ 5!; $ ( T% 8$ #$#$ (F $!(% 8!!$ #$#$%A 9! 8!F M A B C D #$! C '8$ #$#$ P$ P$ (!' I!' I (!'!' C P$ I3 I C / C '8$ #$ P$ A B C D %@#$!C '8$ #$#$ 9$ #$ '" *((=*((A!A $#$#$A(%GCA(! 8 > /

27 5.3 Exposure &9( K8 <! #$!!(A%!&9( %$(=;G>! *((A( &9( I9=+ Q> + F ; A'( F' 8 *(&9( %@( '@ "'; P#$* (!! *(&9( % ( =@%-8A ( 8!A:( 4(>(! =% -$( 4(> & 8 8!*(&9( #$! 8! %#$'' 8! A 8! &9( %;. &9#!!&9( &&F ECE max( x, 0) f ( x) = dx +! (! &9( &!' &%! &9( A! % $ #$#$#$"% #$ 8! WCE α = f ( x) 1 dx " $%E( =&9( '" ( 9(!' > 8 % #$#$!&&F 1 6%E( ECE = E[ x x > 0] 1 ECE = σ π σ ECE = π &! #$!F WCE = = 1.65 =8cI> 5.4 Zinsswaps & =!> & &9( P$ (;$ (* (! 8#A! #$ <!G!; '%%!!!!" '#$* (!! ;8A%! %;'9;@$ #$8!A($!C #$! E( '! *7 '#$ #$A '9; #!'" ;$ (:54M9%=@:54M%> 0

28 + K!# % #$#$#$! ( F V t,rec = B var (F, t, T, c, r t ) B fix (F) 8(%F F#=E(> #F( ( F$ ; F;! * (!.F+' -8%@$#$ '!: '@!(!A(!! '9;( #%$( (!(8$A!-#$! * H % 5! 8# ( '' (''=$ ; #$ $ ' C( ('' =! '! ;$ 8!A! 8 ;$ (#$ $!A8(! #$! &9( %@8! (%>& %#$(@8 : '@!-8!'' ("% 7! C( (''!6 ( %!&'' -8(( '' ( <('!@G C( (= (> ( <(%( "!!&9( A!! -!G (! + '! :%!! -8 'A ( $! #$! ( '9! + &9#!!&9(! KA 8! = 8 '9! &!! : '@8! %A8 (#$ 8;$ $!!! C$ %$! ( %@"#$! '@! (( ( ;( #$! 9! *(! &9( %F D = k( T t) mod ( ) = σ t σ r t r0 &'(F σ ( V ) k( T t) σ t = -89( && -8+.

29 (!'' #$ F dσ ( V ) 1! = k( 1) σ t + k( T t) σ = 0 dt t 1 t = ( T t) t T t = 3 &9( (9#$C% '(! : '@ &%@$#$ '!9&9(!! ; E( A( %#$'"!& 3,H F WCE = 1.65 σ T WCE = 1.65 kt σ 3 T 3 T T WCE = 1.65 k σ 3 3 " &;% 'F 1 3 WCE95% = 1.65 k σ T 3 3 &8#$(. 5.5 Kreditderivate E%!-8#$5 A ; (@ 'A% L#$ L <!! A! #$! <! (! * % +( A!$($-( G(! ((@ ($ <!( '( <!! #$<'@!! 7= (! $ >A! C "% = %' (! #$(%(8!>!#$!*7(''='9(! + %> '#$(!<!!!!' -8 * (#( 7 (!#$;$!* (#(-! $!'"! &%!;$ (* (#(- @8#$!&9(!! #( 7 C * (#( 7!' -8 (!;$ =L* L> %!;$ =L+ #$ L> * (#( - F C $ ( (!A P$ '@! BX;& #$! <! (%!@$!!'" D$ #$* (,% =I,JGG> &!P$ 'BX; =' >A! (! /H! )H! *(( K #( 7 8! 8( 7!E(A(/( 3

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

31 4(F PStrike PMarkt Nominal = 100'000'000 = 1'80' C! *7('' #$ '!! (% #$8!F *7(''I-! ( E( F *7(''I=N/> / JJIJ/J #$!C (9(! (% #$ <(9!!:!E(:EF * (! =* (#( 7 > :54 M9 %! ;$ *!:!E( :E %! '-*+ CCCC :54 M7 * :54 M9M7 %! ;$ 5( "!8 (<!! %#$ #$!#$C@F - + #$ $#$ C@ o $( #$C '8$ #$#$ o &9#!:( o + #$ o %$( - L #$ LC@ %(%#$* C o %!% %8 *!''@ o $( "!8 '!(C %#$! ((!A8(% (%$' (! (( (!! $( - ' 8!&!F (F +N-I< N ((! N- 8(%F +F 8 F (( ; -F& : : '@ + <FE(! 8 #$%A8!!! K $ & #$ ( '' 8! 4( '! 8 - #$!!(! &< < (

32 5.6 Portfoliomanagement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perationelle Risiken ein Abriss '(F L ( + '(!!1 (! '$#$ -7!* (@! A#$!9 & L & + #$A ( ( =%@8 %8 A %!! &!! C-&: 5 -!! $ D #$ K&9( Expected Loss + $ ' K((#$ < CVaR (95% Quantil)

33 # (( ( o <%! 'I (8 - -!!@ o <%! 'I= 5!( > - C!#! C (#$ o 5 C (#$ o :( % (C (#$ o -#( #!C (#$=@ C@A!!-71 % "##$> - <7(( o - o + 8! (7J#$E@8 o (1 '@ #$ $#$ $(! + ( ($K!! + (! - :(!"! #$+ #$ ''8!! + $! ( ( (! & '(!! #$'#$%#$8!8 #$'#$(=''@>+( #$!( %A8(! <(! -#$ $! -#$!!# 7. Kapitalallokation und Performancemessung 7.1 Perfomancemessung (ex post) " #$! #$! * '( # % #$ A8 (9(!!8 A(! 8$(#$8!(A!!!@! %! #$!! =4&A 45 #>!(#$ A8!! (!D!C4= G!D! (= G!D!>#>!' FBG!5G6! $%P$ 8(D8( 8 #$' $!% * '( ( + #$8!@!+A! (G!D * '( #@ 8F 8 E( +( + (* B /H 5 ) 3,)H 8(%F +IE( +( #$! E+=33H> =%8B> +I I00d!F C*I8+ =%8B>C*II,0d/H ( =$! +> ( #$ %$A ( GA <!G!( ( A:1!! (!!@! <((!( '((% "##$-!G( #$ 8 =8( <( (@8#$! 6! %$> K!&#$!

34 '#$#$ 8 (% #$ 8! C4G$(!! #$! -#$ F > ( F (&9( A+(!<( ( > <((F $! +A ; #$'" ( #$< > *'( # FC!D!!! 5;!C4GC@$%!C4%! &+C=&#((#+ C!!!>A! #$K' $% * (!"%!(8!!$ F &+CI8N=< > F F<(@%@8 (!D Gewinn ( Kapital k) EVA (!F RAROC = = Kapital Kapital ;A!! C4 K =(! #$> C4(!E * + C@=E*+> G* (!5((! <F Gewinn + Kapital NPV = Kapital 1+ k Gewinn + Kapital Kapital ( 1+ k) NPV = 1+ k 1+ k Gewinn ( Kapital k) EVA NPV = = 1+ k 1+ k E*+ G* (!5( #$(! 8!&+C! * (! &#$! E*+ R! ( (! C4G &#$! " $ (! #$! C8!! C4! (%#$A 8! (!( #$ %#$!!' ( # '! 7#$ 8 8! $( G* (! #$ A 8! ( =@;@$ >D!* (! '%@8#$(* (!@ * (!! ; 5 (! &+CGC4G$(!8!!! %!.! '( '"$ F F$ $!.! H!($ 8 #$'8A('" D!&( ( 7'"$ 8!A! '!&+C% $ C( 8! (!D <(@ ( I,H (!!'! #$ '"!; ( = #$! (>%A! &+C% &$!#$ (F &+C IN,H I0 8(%F F#$#$ 8 F; ( #$! &$!#$ 8 (8! '8(! &+C 8!! #$!% $F )

35 I0 99I0I,0Hd3H 5 ( 78!(3H!&+C(!.! %@$ E(#$!.%(-F 8 E( +( + (* B /H 5 ) 3,)H! ( G %#$'"!%!.! F BF &+CIN,H I, ( I&+C 3HI/ 5F &+CIN,H 3I0, ( I&+C 3HI0, 7. Pricing (ex ante) &+CC@#$#$ '" 9(* '( # #$ A(! #$ '"!9* #* (! '"! (!D! *(! 7 8!" ;8 '!%8F - ( KF 8!%!-!%@"#$%"$ - <(F 6!#(A 8 ( ( <(A %( =!! >A- A 8 <!(=I9:(> &+C% #$#$F &+CI& KN<(N= >!&#$! &+CR A8!! -8 A(!F & KR<(M= > %@8F! KI<(M= > 5!* #* (@* (! '(8$$ &@(( A!( K $ #$!#$ + 8( $ (. + F * #"% ( 8! &9 #$ $! <( <!! % #$! ((! 8 +! <! ;! <( $! <! ( '"!C! -!.( 9 E( %#$'" &+CI! * =8O> #$!. ' <( <! E( <( (( %( '@ + 8( F (! *( '((!4''# (. 7 C '! (-'' $% P$ (! #$(A()G* (! ='( >@ %A%(! C.G,

36 #$'=!C. ' *('#!<!#$' %! *( '(( %A! *('# %%%! (!G C '%+ #$ '(8#$@. ' 7.3 Kapitalallokation " #$! A 8 (!# % 8! A! (8! <(! ( * #%@ + <( ( * (! %#$A "% #$@!-! * (! G+!(! #$(@ $($ %KK((#$< * #!$%! #$ + (% * (! %#$F ](I $+* (! N $+($* (! #$! (!* (! % #$! <(@K#$<(@ (!<( (!.((#$% "##$ 8!A #$@! C@!(AF S $G ; :K!<(( (%%#$'"'C@F > <((#$* (! G+ + %#$ <( (A 8!@ < $($<( > <((#$ * (! G+ +8!( A!!(@ 8!%8!% <( (%#$ > <((#$ + (8! #$@A8!@ 8<(@ )> <((#$ + + ' ( (+ A% $ % #$ =! +'" D!&@ (! #$!!(#$ '! ( #$%(8! 8!>,> <((#$ $(!! <((A 8! (%! &@ (! %( 8! <( (% "##$! (! <((@ &D!(#$$ % #$ β = cov 7.4 Theorie Expected Loss (8!< #$#$%K G K((#$< '( unternehmensweiter VaR ( RenditeProd ; RenditeBank ) var( Rendite ) Bank Summe der Produkt VaR 4''#$#$!!C4G<(@( #$* (% %! 8!% (#$#$%8( A(%! C4"% $! 8 9 C! *( '(($( =C*> %A! 7#$ % 8 8!A /

37 8$! 7#$ #$%8! % 1! CW 6!C! '(#$ * (%#$G* (!(!(! =@! #$ $8! 9 A8(! 6! $!% '( 8! A! 6!G! ''@> "!! *((#$@A!F `I #(=Y. A>M #(=Y E AY E > 8(%F `F! Y. F!!$!% (! *(( Y E F!!#$$!% (! *(( Y E F!!#$$!%! F'"!< "! ' (L( (L! &%! F #(=Y. A>F ( #(=Y E A Y E >F * '" #$ $!% Q $K$ % #$#$ '(#$' ( '" *((A8 (*((!%' K $ 5(#$! (''$A($!(5( * (! %(! 5(* (! DA8 '(''%$! %8 <('' (*(( =! #$> $%!&#$! "% '' C4!#$! -(!A8F - %8 ($ ( $!#$($! -! (! 5(! ( '((! %!A(!Ie #(=Y E AY E > - e #(=Y E AY E > #$! L( (L "!5(!C4'((F -! ( = '(! 7#$ > % A(! (#$#$$!% ( $!! -!8!%$!! =%@8-#(!F+ )#$ +> -! <(@8!#$!!! =(#$! 0