Normalzed Expoetal Tltg: Prcg ad easurg ultvarate Rsks Shau Wag Departmet o Rsk aagemet ad Isurace Georga State Uversty Abstract: Ths paper dscusses expoetal tltg o the probablty desty ucto o a uderlyg rsk wth respect to some reerece rsk Y By troducg a ormalzato procedure o the reerece rsk Y t shows that ormalzed expoetal tltg s equvalet to applyg Wag trasorm dstorto to the cumulatve dstrbuto o ad s a exteso o CAP to rsks wth geeral-shaped dstrbutos The secod part o the paper deals wth multvarate cases It shows how multvarate ormalzed expoetal tltg s related to multvarate probablty dstortos It provdes ecet routes or computg rsk-adusted multvarate probablty dstrbutos ad gves examples o prcg cotget clams o multple rsks Itroducto: Adustmet o probablty measure s a commo theme prcg ad valuato o rsks The eed or chagg multvarate probablty measures ote arses prcg o cotget clams o multple uderlyg assets or labltes measurg portolo rsks ad whe allocatg total compay rsk captal to varous busess uts Part o the paper revews two methods o adustmet o probablty measure or a gve rsk (radom varable : applyg a expoetal tltg (wth respect to a reerece varable Y to the probablty desty ucto o ; applyg a dstorto drectly to the cumulatve dstrbuto ucto o The paper the troduces a ormalzato procedure va percetle mappg that coverts the reerece Y to a stadard ormal varable It s show that ormalzed expoetal tltg o the probablty desty ucto o s equvalet to applyg the Wag trasorm dstorto to the cumulatve dstrbuto o ad s a exteso o the Captal Asset Prcg odel to rsks wth geeral-shaped dstrbutos Shau Wag 005
Part o the paper exteds both the ormalzed expoetal tltg ad probablty dstorto to multvarate cases ad establshes a mportat lk betwee these two parallel approaches the multvarate case It gves ecet routes or computg the rsk-adusted multvarate probablty dstrbuto ad provdes examples prcg cotget clams o multple rsks ad measurg portolo rsks PART NORAIZED EPONENTIA TITING & DISTORTION --- THE UNIVARIATE CASE Normalzed Expoetal Tltg We shall cosder rsks that are radom varables some probablty space (Ω Ρ For ay radom varable we let We let F represet ts cumulatve dstrbuto ucto (CDF represet the probablty desty ucto (pd o ( the dscrete case the pd s also reerred to as the probablty ucto Cosder two rsks ad Y We say that s absolutely cotuous wrt Y Y > 0 or all pots x wth > 0 Deto Assume that s absolutely cotuously wrt Y We dee the expoetal tltg o wth respect to reerece Y as ollows: E[ exp( λ Y x] (eq- E exp( λy [ ] where ad represet the pd or beore ad ater the expoetal tltg respectvely The λ (eq- s a real-valued parameter cotrollg the magtude o rskadustmet Wth the expoetal tltg (eq- the rato E[ exp( λ Y x] RN( x E exp( λy gves the Rado-Nkodym dervatve o wrt [ ] I the cotext o a ecoomc model or optmal rsk exchage Buhlma (980 derved that the Pareto-optmal equlbrum prce or a rsk ca be represeted as the expected value o the rsk-adusted dstrbuto as (eq- whereas the reerece Y represets portolo aggregate rsk Shau Wag 005
I the specal case o Y (eq- dees a expoetal tltg o wrt tsel: exp( λx E exp( λ [ ] The relato s also wdely kow as the Esscher trasorm Gerber ad Shu (994 have successully appled the Esscher trasorm prcg optos Wth the expoetal tltg (eq- we do ot have a cosstet terpretato o the λ parameter except or the specal case whe Y s a ormal (Gaussa varable Ideed we keep the value o λ xed the scale ad shape o the reerece varable Y ca make a huge derece the results o expoetal tltg I order to get a cosstet terpretato o λ here we propose a ormalzato procedure to be perormed o the reerece varable Y We dee the verso o the CDF o Y as F ( p { y F ( y p } Y Y Y F Y ( Φ( Z et Z be a stadard ormal varable such that where Φ s the CDF o ormal(0 We shall reer to Z as a ormalzed varable o Y ad ext we shall use Z to replace Y (eq- Deto et Z be a ormalzed varable o the reerece Y We dee a ormalzed expoetal tltg o wth respect to reerece Y (or say wrt reerece Z as ollows: E[ exp( λ Z x] (eq- E exp( λz [ ] To keep the otatos straght we summarze the above as ollows: Beore: Expoetal tltg o wrt Y Normalzato: Trasorm Y to a stadard ormal varable Z: Ater: Normalzed expoetal tltg o wrt Z Note that here Z essetally replaces Y as a ormalzed reerece varable The ratoales ad beets o troducg the above ormalzato procedure wll be gve secto 3 Probablty Dstortos Now t s tme to troduce probablty dstortos as aother approach that parallels to the expoetal tltg method Shau Wag 005 3
Deto 3 et g:[0 ] [0 ] be a deretable ucto wth g(00 ad g( Gve the CDF F(x or a radom varable the trasormed CDF ( F( x F g dees a rsk-adusted probablty measure The probablty dstorto deto 3 mples the ollowg Rado-Nkodym dervatve: o I the dscrete case where takes o values { x } RN g ( x ( x g ( x ( F ( x g( F ( x F ( x F ( x x : o I the cotuous case where has a postve probablty desty at x: ( F ( RN g g' x As show Wag (000 00 the ollowg specc orm o dstorto s reerred to as the Wag trasorm: [ ( F λ] F g( F Φ Φ (eq-3 Whe s a cotuous varable we have ( F exp( λ Φ ( F exp( λ RN g g' A plot o the Rado-Nkodym dervatve s gve Fgure Shau Wag 005 4
Fgure Rado-Nkodym Dervatve Impled By The Wag Trasorm 45 4 35 Value o RN(x 3 5 5 lambda0 lambda04 05 0 00005 005 05 035 0435 0545 0655 0765 0875 0985 Value o F(x Both ormal ad logormal dstrbutos are preserved uder the Wag trasorm (eq- 3: I F has a Normal(μσ dstrbuto F s also a ormal dstrbuto wth μ μ λσ ad σ σ I F has a log-ormal(μσ dstrbuto such that l( ~ Normal(μσ F s aother log-ormal dstrbuto wth μ μ λσ ad σ σ 3 k Betwee Expoetal Tltg ad Dstorto Uvarate Case Theorem : Assume that ad Y have bvarate ormal copula wth a correlato coecet o Y The ormalzed expoetal tltg (eq- s equvalet to applyg the ollowg Wag trasorm dstorto: [ ( F β ] F g( F Φ Φ wth β λ Y A proo ca be oud Wag (003 Theorem establshes a mportat lk betwee ormalzed expoetal tltg ad the Wag trasorm dstorto Ths result s a geeralzato o the Captal Asset Prcg odelg whch reveals the meag o the λ parameter Shau Wag 005 5
Cosder a stock dex R (represet the market portolo whose prospectve ed-operod retur has a orma l ( μ σ dstrbuto wth mea μ ad stadard devato σ The dscouted ed-o-perod stock prce S ( R exp( ( S (0 exp r σ has a log-ormal μ r σ dstrbuto ad r s the rsk-ree rate o retur I we apply ormalzed expoetal tltg o R wth the reerece Y beg the stock retur R the rsk-adusted dstrbuto or the stock dex retur has a ormal dstrbuto wth E [ ] μ λ σ R Alteratvely we apply ormalzed expoetal tltg o the stock prce S ( wth the reerece Y S ( beg the stock prce the rsk-adusted dstrbuto or the σ stock prce s log-ormal μ λσ r σ To orce the rsk-adusted expected retur equal the rsk ree rate r we get E[ R ] r μ r λ σ σ To orce the rsk-adusted expected value o the dscouted stock prce the curret stock prce S (0 we get the same result λ E[ R ] r σ : μ r σ S ( to equal For the stock dex the rsk adustmet parameter λ s exactly the market prce o rsk (or the Sharpe rato Ths specal case helps us to assg a dete meag to the parameter λ as a exteso o the market prce o rsk (or Sharpe rato to rsks wth geeralshaped dstrbutos Now we cosder a asset o a oe-perod tme horzo et R be the retur or asset et be the correlato coecet betwee R ad R Applyg ormalzed However or the expoetal tltg (eq- wthout the ormalzato procedure usg the stock prce as reerece Y would ot yeld aother logormal rsk-adusted stock prce dstrbuto Shau Wag 005 6
expoetal tltg o R wth the reerece Y beg ether the retur R or the stock dex prce S ( Theorem states that λ λ or equvaletly E[ R ] r E[ R r ] σ σ Ths s exactly the CAP result or the expected retur o stock relato to the market portolo Thus Theorem exteds CAP to the case that {R R } ollow a ormal copula a more geeral case tha the multvarate ormal ramework 4 Valuato o Cotget Clams I h(y be a (mootoe ucto o the varable Y we say that s a (mootoe cotget clam o the uderlyg rsk Y The market prce o rsk or a mootoe cotget clam h(y s the same as that or the uderlyg rsk Y Theorem Whe valug a mootoe cotget clam h(y the ollowg are equvalet: Apply ormalzed expoetal tltg o Y wrt Y ad calculate the expected value o h(y uder the rsk-adusted dstrbuto o the uderlyg rsk Y Apply ormalzed expoetal tltg o wrt Y ad calculate the expected value o uder the rsk-adusted dstrbuto o 3 Apply Wag trasorm to the CDF o Y ad calculate the expected value o h(y uder the rsk-adusted dstrbuto or the uderlyg rsk Y 4 Applyg Wag trasorm to the CDF o ad calculate the expected value o uder the rsk-adusted dstrbuto o Cosder the specal case that max{ 0 Y K} exp( r where Y represets the edo-perod stock prce varable whch has a logormal dstrbuto s the payo o a Europea call opto o Y wth a strke prce K Whe valug ths cotget clam usg the ormalzed expoetal tltg wth λ beg the stock s market prce o rsk we recover the Black-Scholes ormula or Europea call optos (also see Wag 000 I the remag o ths paper we shall exted the ormalzed expoetal tltg to multvarate cases Shau Wag 005 7
PART NORAIZED EPONENTIA TITING & DISTORTION --- THE UTIVARIATE CASE Normalzed ultvarate Expoetal Tltg Frst we exted the expoetal tltg cocept to the multvarate case Deto 4 Cosder varables { } ad k reereces {Y Y Y k } The expoetal tltg o { } wth respect to reereces {Y Y Y k } s deed by the ollowg pd: k [ exp( Y x x x ] ( x x x c ( x x x E λ (eq-4 where {λ λ λ k } are real-valued parameters that cotrol the magtude o rskadustmet ad c s a ormalzg coecet I ths deto we leave much lexblty the choce o the reereces {Y Y Y k } For stace oe ca choose the reereces {Y Y Y k } to be the rsks { } themselves the compay aggregate or the dustry aggregate Sometmes we ca choose {Y Y Y k } as the uderlyg rsks or the cotget clams { } I order to get a meagul terpretato (as well as cross-cotract comparso o the parameters {λ λ λ k } we eed to apply the ormalzato procedure to all reereces {Y Y Y k } Deto 5 Assume that there exst stadard ormal varables {Z Z Z k } such that ( Φ( Z Y F ( Φ( Z Y F ( ( Z Y F Φ k Y Y k Y k We dee the ormalzed expoetal tltg o { } wth respect to reereces {Y Y Y k } by the ollowg: k [ exp( Z x x x ] ( x x x c ( x x x E λ (eq-5 Shau Wag 005 8
ultvarate Dstortos Now we exted the dstorto method to multvarate cases Cosder multvarate rsks { } that have margal CDFs { F x F ( x F ( x } ( respectvely Assume that { } have a ot CDF speced by F ( F ( x F ( x F ( x ( x x x C where C( s a multvarate uorm dstrbuto (or a copula ucto eg Embrechts et al 00 Deto 6 We dee separate dstortos { g g } dstrbuto has the ollowg margal dstrbutos: F ( x g g such that the resultg multvarate [ F ( x ad the same correlato structure term o copula: F ] ( F ( x F ( x F ( x ( x x x C Deto 7 We dee ot dstortos { g g g } terms o the ot pd: ( x x x RN g g g ( x x x ( x x x where the Rado-Nkodym dervatve s gve by: o I the dscrete case or the pot k ( x x x we have RN ( x x g g g ; ; x ; ; ; x ; c g ( F ( x g ( F ( x F ; ( x F ; ; o I the cotuous case or the pot ( x x x we have RN x ( F ( x g g g ( x x x c g ' Theorem 3 Whe { } have ucorrelated margal dstrbutos both the separate dstortos ad the ot dstortos yeld the same adusted multvarate probablty dstrbuto wth ucorrelated margal dstrbutos ( x ; Shau Wag 005 9
( x x x ( x F ( x g [ F ( x ] wth Remark: Ths result may have mplcatos the aggregato o rsks For stace surace rsks ad market rsks are assumed to be ucorrelated the we ca apply separate adustmets Whe { } are correlated the separate dstortos ad the ot dstortos ca yeld deret results Jot dstortos relect the ter-relato betwee ad the probablty adustmet whle separate dstortos do ot Cosder the specal case that ad g g beg the Wag trasorm wth parameter λ The ot dstortos {g g } s equvalet to applyg a sgle Wag trasorm to wth parameter λ whle the separate dstortos {g g } s equvalet to applyg Wag trasorm to wth parameter λ Whe g u Φ[ Φ u + λ ] ( ( or we shall reer to the separate dstortos deto 6 as the separate Wag trasorms wth parameters {λ λ λ }; ad we shall reer to the ot dstortos deto 7 as the ot Wag trasorms wth parameters {λ λ λ } 3 k betwee Expoetal Tltg ad Dstorto ultvarate Case Theorem 4 Assume that the varables ad the k reereces { ; Y Y } Y k ollow a ormal copula The multvarate ormalzed expoetal tltg (eq-5 s equvalet to applyg separate dstortos to wth: g ( u Φ Φ [ ( u + β ] k ad β Y λ (or The correlato matrx betwee { } expoetal tltg: Σ Σ s uchaged ater the ormalzed Shau Wag 005 0
Example Assume that the rsks { } have a bvarate ormal(0 wth correlato coecets: Σ 06 06 Accordg to Theorem 4 uder bvarate ormalzed expoetal tltg (eq-5 wth reereces ad Y Y the adusted ot dstrbuto or { } s also bvarate ormal wth correlato coecets: Σ Σ 06 06 For llustrato we choose 3 0 λ ad 0 λ The adusted margal dstrbutos are equvalet to applyg separate Wag trasorms [ ] ( ( ( x F x F β Φ Φ or wth + + 038 04 λ λ λ λ λ λ β β Shau Wag 005
Fgure Scatter plot bvarate varables { } RN( x x ad Rado-Nkodym dervatves < > Scatter Plot RN(x x 5 4 3 0-4 -3 - - 0 3 4 - - -3-4 -5 Fgure shows a scatter plot o { } ad ther correspodg Rado- Nkodym dervatves The rght-most blue damod s a scatter plot o (x 395 x 505 The rght-most red square gves ts value o Rado-Nkodym dervatve RN(395 5053884 Oe ca see that the Rado-Nkodym dervatves crease expoetally whe the pot (x x moves rom the lower let quadrat to the upper rght quadrat 4 Valug Cotget Clams o ultple Uderlyg Rsks Cosder cotget clams o multple uderlyg varables: h Y Y Y ( k Whe valug the cotget clam h Y Y Y the market prce o rsk should ( k be speced through the uderlyg rsks { Y Y } Y k Theoretcally we should rst adust the multvarate probablty measure or the uderlyg rsks ad the valug cotget clams as expected payo uder the rskadusted probablty measure Accordgly we should rst apply ormalzed expoetal tltg o { Y Y Y k } wrt themselves ad calculate the expected value o h Y Y Y uder the rsk-adusted dstrbuto o the uderlyg rsks ( k Y Y k { Y } Shau Wag 005
Theorem 5 I we let Y be the uderlyg rsks themselves or k The multvarate ormalzed expoetal tltg (eq-5 o { Y Y } equvalet to ot Wag trasorms wth parameters {λ λ λ k } Y k wrt themselves s Ths result has mplcatos prcg cotget clams o multple uderlyg rsks { Y Y } Y k Example Applcatos Prcg Cotget Clams Suppose that the uderlyg rsks (Y Y have the ollowg bvarate emprcal dstrbuto Note that (Y Y are correlated wth a lear correlato coecet o 038 However ther correlato structure does ot ollow a ormal copula Shau Wag 005 3
Scearo Y Y - 95408-39 3-974 4-37538 5-33593 6-65696 7-95463 8-399995 9-47963 0-4593 - 6799-9344 3-350798 4-575943 5-696738 6-86387 7-4890905 8 63880 3333 9 533 0474 0 5036 5907 638795 574 638795 894058 3 89433 494935 4 4983-5 53309-6 779-7 357754 563479 8 9364 4573 9 0075 68706 30 963860 3409674 3 460043 36459 3 50366 3964 33 50366 503030 34 6665050 734035 o Cotract # has a cotget payo the amout o Y excess o 00000 That s the payo { Y 00000 0} max o Cotract # has a cotget payo o 50% o the amout o Y That s 05Y o Cotract #3 has a cotget payo the amout o Y excess o 00000 plus 50% o the amout o Y Techcally Cotract 3 s smply the combato o Cotract # ad Cotract #: 3 + Shau Wag 005 4
Wthout rsk-adustmet the expected payos or Cotract # # #3 are $3357 $7399 $50656 respectvely Suppose that the market prce o rsk or the uderlyg rsks Y ad Y are λ 03 ad λ 0 respectvely We derve a rsk-adusted dstrbuto by applyg ormalzed expoetal tltg o Y ad Y wth respect to themselves usg λ 03 ad λ 0 Theorem 5 acltates a umercal method or calculatg the rsk adusted probabltes or each o the 34 scearos Based o the adusted probabltes or each scearo we calculated the prces or Cotract # # #3 beg $6840 $4847 ad $93087 respectvely Expected Payo o Cotract # Expected Payo o Cotract # Expected Payo o Cotract #3 No Rsk Adustmet $ 3357 $ 7399 $ 50656 Wth Rsk Adustmet $ 6840 $ 4847 $ 93087 oadg 05% 43% 84% Note that the obtaed prces are addtve Ideed the oly way to esure prce addtvty s by a chage o bvarate probablty measure Example 3 Numercal Techques Ivolvg Dscrete Dstrbutos Cosder the ollowg bvarate dstrbuto (that does ot ollow a ormal copula 3 4 5 00 007 006 005 004 006 005 004 003 003 3 005 004 003 003 00 4 003 003 00 00 00 5 003 00 00 00 00 Shau Wag 005 5
We wat to compute the adusted ot dstrbuto or the multvarate ormalzed expoetal tltg o ( wth reerece to themselves ad wth λ 03 ad λ 0 We rst apply the Wag trasorm to wth λ 03 x (x F(x F(x (x 04 04 030787 030787 0 063 057 00483 3 07 080 070596 0935 4 0 09 0850 04505 5 009 00 00000 04899 We the apply the Wag trasorm to wth λ 0 x (x F(x F(x (x 037 037 0974 0974 0 058 050076 00334 3 06 074 0674 07049 4 05 089 084768 07644 5 0 00 00000 053 Accordg to Theorem 4 the bvarate Rado-Nkodym dervatves are: RN g ( x x ( x x ( x ( x c ( x x ( x ( x The al rsk-adusted ot probablty (desty ucto s: 3 4 5 078 00497 00469 0043 00406 00470 0047 0046 00344 00405 3 00457 00440 00363 0040 0035 4 0038 00383 008 0030 0083 5 00399 003 0076 00389 009 Shau Wag 005 6
Fgure 3 The Bvarate Rado-Nkodym Dervatves Rado-Nkodym Dervatves 5 0 5 0 05 00 3 4 5 5 4 3 As show Fgure 3 the Rado-Nkodym dervatve creases to ts hghest value at { 5 5} dcatg the largest relatve rsk adustmet at the ot tal o the bvarate varables 5 Portolo Rsk easures Here we meto brely that multvarate dstortos ca be employed to dee portolo rsk measures Cosder a portolo o rsks { } wth CDFs { F x F ( x F ( x } ( respectvely Assume that { } have a ot CDF speced by F ( F ( x F ( x F ( x ( x x x C where C( s a multvarate uorm dstrbuto (or a copula ucto et { } have a multvarate dstrbuto produced by separate dstortos g g g o { } as deto 6 That s { } F ( g [ F ( x ] g [ F ( x ] g [ F ( x ] ( x x x C wth margal dstrbutos: F ( x g [ F ( x ] Shau Wag 005 7
et W + + + represet the sum o radom varables{ } Ay rsk measure o the aggregate varable W dees a portolo rsk measure or { } We ca eve apply aother dstorto ucto h o the CDF o W ad dee the portolo rsk measure o rsk-adusted dstrbuto: PR 0 { } as the expected value o W uder the ( h F dx + W h( FW 0 ( dx Such a portolo rsk measure has applcatos measurg portolo rsks portolo optmzato ad allocatg rsk captals That s a mportat subect ad deserves a separate dscusso Cocluso: We have troduced the cocept o ormalzed expoetal tltg ad establshed a mportat lk wth probablty dstortos rst or the uvarate case the or the multvarate case Normalzed expoetal ttlg provdes a geeral ramework or prcg rsks wth respect to some reerece rsks or or valug cotget clams o some uderlyg rsks The paper also provdes ecet umercal routes or adustg multvarate probablty dstrbutos I a sequel paper we shall explore urther ormalzed expoetal tltg o multvarate rsks wth practcal cosderatos such as adustmet or parameter ucertaty ad teractos amog reerece rsks Shau Wag 005 8
Reereces: Buhlma H (980 A ecoomc premum prcple ASTIN Bullet 5-60 Embrechts P cnel A ad Strauma D (00 Correlato ad Depedece Rsk aagemet: Propertes ad Ptalls Rsk aagemet: Value at Rsk ad Beyod Dempster (Ed Cambrdge Uversty Press 00 pp 76-3 Gerber HU ad Shu ESW (994 Opto prcg by Esscher trasorms Trasactos o the Socety o Actuares vol VI 99-40 Wag S (000 A Class o Dstorto Operators or Prcg Facal ad Isurace Rsks Joural o Rsk ad Isurace 67 (000 arch: 5-36 Wag S (00 A Uversal Framework or Prcg Facal ad Isurace Rsks ASTIN Bullet 3 (00 November: 3-34 Wag S (003 Equlbrum Prcg Trasorms: New Results Usg Buhlma s 980 Ecoomc odel ASTIN Bullet 33 (003 ay: 57-73 Shau Wag 005 9