Differences between mean-variance and mean-cvar portfolio optimization models

D I FF E R E NC ES B E TW EE N M E AN- V AR I ANC E AND M E AN-C V AR P O RTF O L I O O P T I M I ZAT I O N M O D E L S P a nn a M isk o lc z i U n i e si y o D eb e c en, F a c u l y o E c ono mics and B u si ne ss , In s i u e o A cc oun i ng and Fi nan c e, D eb e c en, H unga y misk o lczi panna @ g m a il . c o m Ab s ac : E e y bod y hea d a l ead y ha one s hou l d no e x pe c h i gh e u n s wi hou h i gh isk , o one s hou l d no e x pe c s a e y wi hou l o w e u n s . T he goa l o po o li o heo y is o i nd he ba l an c e be w een m a ximizi ng he e u n and mi n imizi ng he isk . T o do s o w e ha e o i s unde s and and m ea s u e he isk . N a u a lly a good isk m ea s u e ha s o s a is y s e e a l p ope i e s - i n heo y and i n p a c is e. M a k o wi z s ugge s ed o u s e he a i an c e a s a isk m ea s u e i n po o li o heo y . T h is l ed o he s o c a ll ed m ean - a i an c e m ode l - o w h ic h M a k o wi z e c e i ed he N obe l P iz e i n 1990. T he m ode l ha s been c i iciz ed be c au s e i is w e ll s u i ed o e lli p ic a l d is i bu i on s bu i m a y l ead o i n c o e c c on cl u si on s i n he c a s e o non - e lli p ic a l d is i bu i on s . S i n c e hen m an y isk m ea s u e s ha e been i n odu c ed, o w h ic h he V a l ue a Risk ( V a R) is he m o s wi de ly u s ed i n he e c en y ea s . D e s p i e o he wi de s p ead u s e o he V a l ue a Risk he e a e s o m e unda m en a l p ob l e ms wi h i . I doe s no s a is y he s ubadd i i i y p ope y and i i gno e s he s e e i y o l o ss e s i n he a a il o he p o i - and -l o ss ( P& L ) d is i bu i on. M o eo e , i s non -c on e xi y m a k e s V a R im po ssi b l e o u s e i n op imiz a i on p ob l e ms . T o c o m e o e he s e iss ue s he E x pe c ed S ho a ll ( ES ) a s a c ohe en isk m ea s u e w a s de e l oped. E x pe c ed S ho a ll is a ls o c a ll ed C ond i i ona l V a l ue a Risk (C V a R) . C o m pa ed o V a l ue a Risk , ES is m o e s en si i e o he a il beha i ou o he P& L d is i bu i on un c i on. In he i s pa o he pape I s a e he de i n i i on o he s e h ee isk m ea s u e s . In he s e c ond pa I dea l wi h my m a i n que s i on: W ha is happen i ng i w e ep l a c e he a i an c e wi h he E x pe c ed S ho a ll i n he po o li o op imiz a i on p o c e ss . D o w e ha e d i e en op im a l po o li o s a s a s o l u i on? A nd hu s , doe s he s o l u i on s ugge s s o de ci de d i e en ly i n he w o c a s e s ? T o an sw e o he s e que s i on s I ana lys e s e en H unga i an s o ck e xc hange c o m pan i e s . Fi s I u s e he m ean - a i an c e po o li o op imiz a i on m ode l , and hen he m ean -C V a R m ode l . T he e s u l s a e s ho w n i n s e e a l c ha s and ab l e s . K ey w o d s : isk ; V a l ue a Risk ; E x pe c ed S ho a ll ; M ean - V a i an c e P o o li o O p imiz a i on; M ean -C V a R P o o li o O p imiz a i on J E L classi ica i on: G 11 1. Ho w o m eas u e isk ? In li e a u e isk and isk m ea s u e s ha e no un i que de i n i i on and u s age. H o w e e i is o en u s e u l o e x p e ss isk wi h one nu m be ( E mm e , K a z and T a sc he, 2013 ) . T he h is o ic a lly m o s im po an isk m ea s u e s i n i nan c e a e he a i an c e and he s anda d de i a i on and i goe s ba ck o M a k o wi z and he m ode n po o li o heo y (M a k o wi z , 1952; 1968 ) . In he l a s y ea s and a ls o he s e da ys he m o s popu l a isk m ea s u e is he V a l ue a Risk ( V a R) . V a l ue a Risk an sw e s he o ll o wi ng que s i on: W ha is he mi n im u m po en i a l l o ss ha a po o li o c an s u e i n he % w o s c a s e s (i n a g i en im e ho iz on ) ( A c e b i and T a sc he, 2002 ) : , w he e is he p o i d is i bu i on un c i on o and ( ) is he c on i den c e l e e l . In o he w o d s , V a R is he % quan il e o . V a R is c i icis ed be c au s e o s e e a l ea s on s . T he m o s im po an a e ha V a R is no s ubadd i i e, w h ic h m ean s ha po o li o d i e si ic a i on m a y l ead o an i n c ea s e o isk , hen ha i doe s no m ea s u e l o ss e s e xc eed i ng V a R and i s non -c on e xi y m a k e s V a R im po ssi b l e o u s e i n op imiz a i on p ob l e ms ( S z egö, 2002 ) . T o s o l e he s e p ob l e ms a c ohe en isk m ea s u e c a ll ed E x pe c ed S ho a ll ( ES ) w a s i n odu c ed: . I is c on i nuou s , he de i n i i on o e x pe c ed s ho a ll is equ i a l en o , hen c e a l e na i e na m e s lik e c ond i i ona l V a l ue a Risk (C V a R) and c ond i i ona l a il e x pe c a i on a e i n u s e i n i nan c e ( E m b e c h s and H o e , 2014 ) . In h is pape I u s e C V a R and ES a s sy non yms . 2. M ea n- V a ia n ce s. M ea n-C V a R P o o li o O p i m i z a i on 2.1. Th e d a a I w o k wi h H unga i an da ily s o ck da a be w een 01.07.2005 and 29.06.2015. T he da a w a s do w n l oaded o m B udape s S o ck E xc hange ho m epage (www .be .hu ) . I ana lys e he da ily l oga i h mic e u n s o s e en s o cks , na m e ly FH B , M O L, MT E L EK O M , O T P , P annene g y , R aba and Ric h e , i n h is en y ea s . T o ana lys e he da a and o an sw e o my que s i on s I u s e he Rm e ics m a he m a ic a l s o w a e (W ü z e a l ,, 2009 ) . N o e ha I w o k wi h da ily da a and hu s he e s u l s ( e u n s , a i an c e, V a R , ES , e c . ) a e s a ed on a da ily ba sis . 2.2. M ea n- V a ia n ce P o o li o O p i m i z a i on In he m ean - a i an c e (M V ) m ode l , e u n and isk a e e s im a ed b y he s a m p l e m ean and he s a m p l e a i an c e o he a ss e e u n s . T he goa l is o mi n imiz e he isk o a g i en a ge e u n. I w o k wi h l ong - on ly po o li o s on ly (s ho s e lli ng is no a ll o w ed ) and he e o e he w e i gh s a e be w een z e o and one a e an app op i a e no m a liz a i on. W e c an s ee he e s u l s o he m ean - a i an c e po o li o op imiz a i on i n Fi gu e 1 and i n T ab l e 1. In Fi gu e 1 he h y pe bo l a is he s o l u i on o he M a k o wi z m ode l . T he s e i n si de he h y pe bo l a c a ll ed he ea si b l e s e , he uppe bo de ( b l a ck po i n s) is he e ici en on i e and he l o w e bo de is he mi n im u m a i an c e l o c u s ( g e y po i n s) . T he e ici en on i e p l o i n cl ude s a ll si ng l e a ss e s isk s . e u n po i n s (c o l ou ed ci cl e s) , he angen cy li ne ( b l ue s o li d li ne ) , he angen cy po i n o he isk- ee a e ( b l ue po i n ) , he equa l w e i gh po o li o ( E W P ) ( g e y s qua e ) and he g l oba l mi n im u m isk po o li o ( ed po i n ) . T he y e ll o w s o li d li ne i nd ic a e s he S ha pe a i o and he ange o he S ha pe a i o is p i n ed on he i gh hand si de a xis . W e c an s ee, ha i n he c a s e w e de ci de o i n e s j u s i n one s o ck , po si i e e u n is ea c hab l e j u s wi h he R aba s o ck . T he MT E L EK O M ha s he l o w e s w h il e he Ric h e s o ck ha s he h i ghe s isk . Fi s I c a lc u l a ed he E W P , w h ic h is a ea si b l e po o li o wi h g i en equa l w e i gh s . T he E W P ha s qu i e a sm a ll isk ( a i an c e = 0.0161 ) bu i n h is c a s e he ea c hab l e e u n is nega i e (- 0.03 % ) . M o eo e , one c ou l d ea c h a h i ghe e u n wi h he s a m e isk o he s a m e e u n wi h a sm a ll e isk . T he g l oba l mi n im u m isk po o li o is he e ici en po o li o wi h he l o w e s po ssi b l e isk and he eb y he g l oba l mi n im u m isk po i n s epa a e s he e ici en on i e o m he mi n im u m a i an c e l o c u s . In my c a s e he g l oba l mi n im u m a i an c e po o li o ha s a nega i e e u n (- 0.02 % ) . T he angen cy po o li o is c a lc u l a ed b y m a ximizi ng he S ha pe a i o o a g i en isk- ee a e. T he S ha pe a i o is he sl ope o he c ap i a l a ll o c a i on li ne g i en b y he a i o o he a ge e u n l o w e ed b y he isk- ee a e and he po o li o isk . In he c a s e o he H unga i an da ily da a he isk- ee a e is 0.008 % . T he angen cy po i n c o i n ci de s wi h he R aba si ng l e a ss e , s o one c an ea c h he h i ghe s e u n ( 0.02 % ) wi h he R aba si ng l e a ss e po o li o. B u on h is e u n l e e l he isk is qu i e h i gh ( a i an c e = 0.0213 ) . T he S ha pe a i o is be w een - 0.00026 and 0.000206. F i gu e 1 : E ici en on i e : s o l u i on o he m ean - a i an c e po o li o op imiz a i on ( a l pha = 0.05 ) S ou c e: o w n c a lc u l a i on, Rm e ics T a b le 1 : R e s u l o he m ean - a i an c e po o li o op imiz a i on ( a l pha = 0.05 ) P o o li o R e u n V a ia n ce V a R ES E qua l W e i gh - 0.0003 0.0161 0.0211 0.0374 Mi n im a l Risk ( a ge e u n = 0 ) 0.0000 0.0156 0.0231 0.0372 Mi n im a l Risk ( a ge e u n = 0.0001 ) 0.0001 0.0177 0.0257 0.0413 T angen cy 0.0002 0.0213 0.0282 0.0482 G l oba l Mi n im u m Risk - 0.0002 0.0139 0.0208 0.0343 S ou c e: o w n c a lc u l a i on, Rm e ics W e c an s ee he w e i gh s o ou d i e en po o li o s i n Fi gu e 2. A s I m en i oned ea li e , i n he c a s e o he angen cy po o li o he op im a l po o li o c on sis s on ly o he R aba s o ck . I w e i n e s ed on ly i n he R aba s o ck w e c ou l d ea c h a 0.02 % e u n, ha i ng he o ll o wi ng isks : a i an c e = 0.0213, V a R = 0.0282 and ES = 0.0482 (s ee T ab l e 1 ) . T he mi n im u m isk po o li o is an e ici en po o li o wi h he l o w e s isk o a g i en a ge e u n. Wi h my da a a h i ghe s ea c hab l e e u n is 0.02 % ha is w h y a s a i s e x a m p l e I c a lc u l a ed he mi n im u m isk po o li o wi h 0 % a ge e u n. T he op im a l po o li o c on sis s o ou s o cks ( o m he s e en ) : R aba ( 51.8 % ) , MT E L EK O M ( 25 % ) , M O L ( 16.7 % ) and FH B ( 6.5 % ) . A s a s e c ond e x a m p l e I c a lc u l a ed he mi n im u m isk po o li o wi h 0.01 % a ge e u n. T he c o m po si i on o he op im a l po o li o is he s a m e a s i n he p e i ou s c a s e, bu he w e i gh s a e d i e en : R aba ( 69.6 % ) , M O L ( 21.3 % ) , MT E L EK O M ( 7.7 % ) , and FH B ( 1.4 % ) . T he l a s p l o s ho ws he w e i gh s i n he g l oba l mi n im u m op im a l po o li o. A ll he s o cks e xc ep i ng O T P a e p e s en : MT E L EK O M ( 45.1 % ) , R aba ( 23.3 % ) , FH B ( 11.4 % ) , P annene g y ( 8.9 % ) , M O L ( 8 % ) and Ric h e ( 3.4 % ) . F i gu e 2 : M V po o li o w e i gh s : mi n im a l isk po o li o wi h 0 % a ge e u n (l e up ) , mi n im a l isk po o li o wi h 0.01 % a ge e u n ( i gh up ) , g l oba l mi n im u m po o li o (l e do w n ) , angen cy po o li o ( i gh do w n ) , a l pha = 0.05 S ou c e: o w n c a lc u l a i on, Rm e ics 2.3. M ea n-C V a R P o o li o O p i m i z a i on In he m ean -C V a R po o li o m ode l he a i an c e is ep l a c ed b y he C V a R . In c on a s o he m ean - a i an c e po o li o op imiz a i on, he s e o a ss e s a e no l onge e s ic ed o ha e an e lli p ic a l d is i bu i on. T he e s u l s a e s ho w n i n Fi gu e 3 and T ab l e 2. I u s e he s a m e no a i on a s i n he c a s e o he m ean - a i an c e po o li o op imiz a i on. S imil a ly o he e s u l o he m ean - a i an c e op imiz a i on a ll si ng l e a ss e s ha e nega i e e u n s e xc ep he R aba s o ck . M o eo e R aba is he s o l u i on o he angen cy po o li o op imiz a i on hu s h is si ng l e a ss e po o li o ha s he h i ghe s S ha pe a i o. Wi h h is s o ck one c ou l d ea c h he m a xim u m 0.02 % a ge e u n, ha i ng a i an c e = 0.0213, V a R = 0.0282 and ES = 0.0482 (s ee T ab l e 2 ) . T he S ha pe a i o is be w een 0.000532 and 0.000206. F i gu e 3 : E ici en on i e : he s o l u i on o he m ean -C V a R po o li o op imiz a i on ( a l pha = 0.05 ) S ou c e: o w n c a lc u l a i on, Rm e ics T a b le 2 : R e s u l o he m ean -C V a R po o li o op imiz a i on ( a l pha = 0.05 ) P o o li o R e u n V a ia n ce V a R ES E qua l W e i gh - 0.0003 0.0161 0.0211 0.0374 Mi n im a l Risk ( a ge e u n = 0 ) 0.0000 0.0156 0.0231 0.0372 Mi n im a l Risk ( a ge e u n = 0.0001 ) 0.0001 0.0177 0.0256 0.0413 T angen cy 0.0002 0.0213 0.0282 0.0482 G l oba l Mi n im u m Risk - 0.0002 0.0142 0.0203 0.0340 S ou c e: o w n c a lc u l a i on, Rm e ics I a ls o c a lc u l a ed he m ean - a i an c e po o li o w e i gh s (s ee Fi gu e 4 ) . T he op im a l angen cy po o li o is c on sis s o on ly he R aba s o ck . T he mi n im u m isk po o li o wi h 0 % a ge e u n c on sis s o ou s o cks : R aba ( 51.8 % ) , MT E L EK O M ( 25.2 % ) , M O L ( 16.8 % ) and FH B ( 6.2 % ) . A s a s e c ond e x a m p l e I c a lc u l a ed aga i n he mi n im u m isk po o li o wi h 0.01 % a ge e u n. T he c o m po si i on o he op im a l po o li o c on sis j u s h ee s o cks : R aba ( 69.1 % ) , M O L ( 22.7 % ) and MT E L EK O M ( 8.2 % ) . S o wi h a li l e b i o h i ghe e x pe c ed e u n he FH B is no an y m o e i n he op im a l po o li o. T he l a s p l o s ho ws he w e i gh s i n he g l oba l mi n im u m op im a l po o li o. A ll he s o cks a e i n h is po o li o e xc ep O T P : MT E L EK O M ( 43.7 % ) , R aba ( 21.1 % ) , P annene g y ( 15.8 % ) , FH B ( 12.1 % ) , M O L ( 6.8 % ) and Ric h e ( 0.5 % ) . F i gu e 4 : M ean -C V a R po o li o w e i gh s : mi n im a l isk po o li o wi h 0 % a ge e u n (l e up ) , mi n im a l isk po o li o wi h 0.01 % a ge e u n ( i gh up ) , g l oba l mi n im u m po o li o (l e do w n ) , angen cy po o li o ( i gh do w n ) , a l pha = 0.05 S ou c e: o w n c a lc u l a i on, Rm e ics 3. Co m p a is on a nd s u mm a y I w ou l d lik e o an sw e o he que s i on w he he w e w ou l d de ci de d i e en ly u si ng a i an c e o ES a s a isk m ea s u e. Le u s a k e a cl o s e l oo k o he w e i gh s on he e ici en on i e i n he c a s e o he m ean - a i an c e and he m ean -C V a R op imiz a i on i n o de o an sw e o h is que s i on. In he Fi gu e 5 w e c an s ee he w e i gh s on he h y pe bo l a c u e. T he i s p l o is he s o l u i on o he m ean - a i an c e po o li o op imiz a i on, he s e c ond and he h i d is he s o l u i on o he m ean -C V a R po o li o op imiz a i on a l pha = 0.05 and 0.01 e s pe c i e ly . N o e ha he s ong s epa a i on li ne m a ks he po si i on be w een he mi n im u m a i an c e l o c u s and he e ici en on i e . T a ge e u n s a e i n c ea si ng o m l e o i gh , w he ea s a ge isks a e i n c ea si ng o he l e and o i gh wi h e s pe c o he s epa a i on li ne. W e c an s ee, ha he h ee p l o s i n Fi gu e 5 s ho w d i e en d is i bu i on s . N o e a ls o ha he O T P s o ck is a lm o s ne e i n he op im a l po o li o (i ndependen ly o he a ge e u n and/o he op imiz a i on m e hod ) . F o an i n e s o i is m o e i n e e s i ng o c on si de he w e i gh s w he e he a ge e u n s a e po si i e. F i gu e 5 : W e i gh s i n he s o ck po o li o s S ou c e: o w n c a lc u l a i on, Rm e ics I w e ix he a ge e u n o e x a m p l e a 0.015 % hen he e is no d i e en c e be w een he m e hod s (s ee Fi gu e 6 l a s c o l u m n ) : he op im a l po o li o c on sis o he R aba ( 80 % ) and he M O L ( 20 % ) . S e c ond ly w e c on si de a 0.005 % ix ed a ge e u n (s ee Fi gu e 6 i s c o l u m n ) . In h is c a s e he c o m po si i on o he h ee op im a l po o li o is he s a m e (FH B , M O L, MT E L EK O M , R ABA ) , bu he w e i gh s a e d i e en . T h is m ean s , ha w e w ou l d c on cl ude d i e en ly u si ng one o he o he isk m ea s u e s . T he e is an e en b i gge d i e en c e o e x a m p l e i n he c a s e o he ix ed 0.01 % a ge e u n (Fi gu e 6 mi dd l e c o l u m n ) be c au s e no on ly he w e i gh s bu a ls o he c o m po si i on s o he po o li o s a e d i e en . T he m ean - a i an c e and he m ean - C V a R po o li o op imiz a i on on he l e e l o 1 % s ugge s ou s o cks i n he op im a l po o li o (FH B , M O L, MT A L AK O M , R aba ) bu aga i n i n a c o m p l e e ly d i e en p opo i on. T he s o l u i on o he m ean -C V a R po o li o op imiz a i on on he l e e l o 5 % consis s o ‘only’ h ee s ocks: MOL, MTELEKOM, Raba. This h i d e x a m p l e s ho ws e en cl ea e ha w e w ou l d m a k e a d i e en de cisi on depend i ng on he isk m ea s u e. F i gu e 6 : W e i gh s wi h ix ed a ge e u n s S ou c e: o w n c a lc u l a i on, Rm e ics In h is pape I ana lys ed he d i e en c e be w een he m ean - a i an c e and he m ean - C V a R po o li o op imiz a i on m e hod s b y u si ng da ily H unga i an s o ck da a and I an sw e ed o he que s i on w he he w e w ou l d de ci de d i e en ly c on c e n i ng he c o m po si i on o he po o li o s i n he w o c a s e s . T he ana lysis s ho ws cl ea ly – s ee o e x a m p l e Fi gu e 5 and Fi gu e 6 - ha he an sw e o h is que s i on is y e s . T he de cisi on o an i n e s o depend s on he isk m ea s u e. R e e e n ces A c e b i , C ., & T a sc he, D . ( 2002 ) E x pe c ed s ho a ll : a na u a l c ohe en a l e na i e o a l ue a isk . E c ono mic no e s , 31 ( 2 ) , 379 - 388. E m b e c h s , P ., & H o e , M . ( 2014 ) S a is ics and quan i a i e isk m anage m en o