Quantitative Portfolio Theory & Performance Analysis

Transcription

1 Quattatve Portfolo heory & Performace Aalyss Week February 11, 2013 Cocepts (fsh-up) Basc Elemets of Moder Portfolo heory Assgmet For Feb 11 (hs Week) ead: A&L, Chapter 2 ( Cocepts) ead: A&L, Chapter 3 (Basc Elemets of Moder PF heory) For Feb 18 (Next Week) ead: A&L, Chapter 3 (Basc Elemets of Moder PF heory) ead: A&L, Chapter 4 (Captal Asset Prcg Model ad ts Applcato to Performace Measuremet) ead: E&G, BA Cocepts Performace Measuremet Frst Stage Performace Aalyss ad Maagemet Quatfy etur o a Asset/Portfolo AIM Stadard equremets/ecommedatos Frst Elemets of Performace Evaluato efereces to be used to Compare eturs: Bechmarks ad Peer Groups sk 1.3 etur Calculato Portfolo etur Basc Formula w/captal Flow hree methods Iteral ate of etur (I) Captal-Weghted ate of etur (CW) me-weghted ate of etur (W) 1.4 1

2 etur Calculato Portfolo etur Basc Formula w/captal Flow Captal-Weghted ate of etur (CW), CW V V0 Ct 1 CW where t V0 Ct 1 V s the value of the PF at the ed of the perod V0 s the value of the PF at the begg of the perod Ct s the th captal/cash flow for the PF at t (pos f a cotrbuto/flow & eg f a wthdrawal/outflow) 1.5 etur Calculato Portfolo etur Basc Formula w/captal Flow atoale t t CW V0 Ct CWV0 CW Ct V V0 Ct Ad t 1CW V 0 1CW Ct V etur Calculato Portfolo etur Basc Formula w/captal Flow Iteral ate of etur (I), I 1 Ct V V0 t 1 1I 1I atoale 1 1 I 1I V0 C t V t 1 1 I Features Iteratve (a challege to determe o paper) Iflows vested the PF gog forward 1.7 etur Calculato Portfolo etur Basc Formula w/captal Flow me Weghted ate of etur (W), W Idea: Break dow perod to elemetary subperods (cash flows); usg the otato as before For each sub-perod V V C t V C t t1 t1 t1 t1 he retur for the whole perod s geometrc mea 1 1 V t W 1t Vt C 1 t

3 etur Calculato Portfolo etur Basc Formula w/captal Flow me Weghted ate of etur (W), W atoale 1W 1t 1 Features o mplemet, eed to kow amout ad tmg of cash as well as value of PF at each date I practce, cash s assumed to occur at moth ed stead of o exact dates; cotuous verso helps 1.9 r W etur Calculato Portfolo etur Basc Formula w/captal Flow r Cotuous W, r W e W 1 W 1 V t l 1 Vt C 1 t1 1 l( Vt1 ) l( V0 ) l( Vt2 ) l( Vt1Ct1 ) l( V ) l( Vt Ct ) 1 1 V V t l l V 0 1 Vt C t 1.10 etur Calculato Portfolo etur Basc Formula w/captal Flow atoale e V V V 0 1 Vt C 1 t V t C 1 t1 1 rw t exp l l V 0 1 Vt C t 1 V V t V t 1.11 etur Calculato Portfolo etur Basc Formula w/captal Flow hree methods: I, CW, W Comparso W allows maager to be evaluated separately from movemet of captal best for evaluatg maager CW allows for total performace of fud to be measured I more precse tha CW whe there s a sgfcat umber of captal flows of dfferet szes Frequet evaluatos of PF reduce mpact of captal flow Daly s best

4 Choce of etur Measure CW vs. W Cosder the Fud wth NAV as follows Choce of etur Measure CW vs. W here are two vestors Each start the year purchasg 100 uts at the ed of Dec Ivestor 1 makes 2 subsequet purchases of 100 uts each at the ed of May (NAV = 14) ad ed of August (NAV = 15) Ivestor 2 makes 2 subsequet purchases of 100 uts each at the ed of Aprl (NAV = 8) ad ed of September (NAV = 9) A summary of ther actvty s show as follows Choce of etur Measure CW vs. W A summary of ther actvty s show as follows Choce of etur Measure CW vs. W I each case accordg to the W they obta the same retur as the fud 1.15 Whch makes o sese to ether vestor, eve f t makes perfect sese to the maager I #1 = % & I #2 = 35.16% he vestors do t see credt or blame for ther choces 1.16 captal tmg 4

5 Choce of etur Measure CW vs. W Cosder a secod stuato A maager makes allocato betwee three asset classes ad geerates the followg retur Choce of etur Measure CW vs. W Cosder a secod stuato Utl we cosder captal flows hs seems to be coutertutve; how ca a portfolo exceed the retur of ts compoets Utl we cosder captal flows Choce of etur Measure CW vs. W Cosder a secod stuato Utl we cosder captal flows By aptly allocatg amog the classes the maager captures the retur If we were to use CW (I) the last row would appear as Choce of etur Measure CW vs. W he key to choosg CW or W s to detfy who cotrols the flow of captal he retur should be calculated assocated wth cotrol of the flow of captal I the frst example, the maager dd ot cotrol the captal flow o descrbe hs performace use W I the frst example, the vestor cotrolled the captal flow o descrbe the performace of hs accout use CW I the secod example, the maagers retur the asset classes eeded to reflect hs decso o Where rather tha geometrcally combg perod returs, we take accout of captal flows he maager gets credt for hs dscreto the asset allocato decso for teral captal movemet allocato thus asset class returs are hgher

6 etur Calculato HW A Example (pg 32) CW V V C CW V 0 1 t t C t =45.28% Covertg to a stadard referece perod rate CW 1CW CW 13.26% I = 13.17% 1 V t W W % 1 Vt C 1 t1 W cotuous tme = 13.33% 1.21 etur Calculato Iteratoal Ivestmet Up to ow, assumpto of sgle currecy Exchage rate deftos Forward premum (or dscout), f : fwd premum ( dscout) ( fwd x - rate - spot x - rate) / spot x -rate fwd x - rate (1 fwd premum) spot x -rate emember: rr - f F0 S0e S0 (1 + fwd premum) where S : spot prce of currecy referece ($), also d F : fwd prce of currecy referece ($), also F Q / 0 0 Q/ 0 0 rr, f : ref ($) rsk free rate, currecy rsk free rate 1.22 etur Calculato Iteratoal Ivestmet Evoluto of the spot exchage rate percetage terms s called the currecy retur he varable C s used to refer to ths quatty I a teratoal portfolo, the currecy retur eeds to be bfurcated from the asset retur (whe the asset s prced currecy) etur s expressed as the sum of the retur o the asset ad the currecy retur 1.23 etur Calculato Iteratoal Ivestmet Exteso of retur models (arthmetc & logarthmc) provdes (asset currecy) bfurcato of retur Startg wth the prce represetato the eferece currecy of asset at tme t terms of the Quotato currecy: Q Q/ Q / P & t Pt dt d t : Prce of quote currecy the referece currecy Allows the developmet of the arthmetc retur: Q Q/ Q Q/ Pt d t P t d t Q Q/ 1t 1 1 Q Q/ Q Q/ t Ct Pt 1dt1 Pt1dt1 Q Q/ Q Q/ Q Q/ 1t Ct t Ct 1 t Ct ad Q Q/ C 1.24 t t t 6

7 etur Calculato Iteratoal Ivestmet Smlarly for the Logarthmc etur we have the exact equalty Q Q/ Q Q/ Pt d t P t d t Q Q/ t l l l Q Q/ Q Q/ t Ct Pt1dt1 Pt1 dt1 S F0 For the portfolo, we also have the expected ref result FC S Q Q/ Q/ 0 P t t t t t & f quote = ref currecy, t 1 Q / where (1+ ) s the forward premum o 0, Q Q/ Q Q/ Q / P t t t t t t t t t : arthmetc & C f hs retur s sometmes referred to as the forward surprse Q Q/ P t t t t t t : logarthmc V P P d d x x x C x C x x x C etur Calculato Iteratoal Ivestmet For bfurcato whe hedge strumets are used Note the retur, FC, for a forward exchage cotract 1 dff betwee currecy retur & fwd premum retur C f f etur Calculato Iteratoal Ivestmet So retur o a asset hedged w/fwd for currecy rsk s Q Q/ Q/ 1t 1t 1 Ct hct ft where the hedge rato h 1,0 Expadg (settg asde the cross term ad add/sub f t ) Q Q/ Q Q/ Q/ 1 1 C C h C f so t t t t t t t Q Q/ Q Q/ Q/ t t Ct t Ct hct ft Q Q/ t ft 1hCt ft he sum of the completely hedged posto plus the uhedged proporto of the exchage exposure 1.27 etur Calculato Iteratoal Ivestmet Smlarly for the hedged portfolo of assets m Q Q/ Q/ x x x C h C f P t t t t t t jt jt jt j1 where h s the hedge rato for each of m curreces j

8 etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe Smply, the cotrbuto of the dervatve s the performace of the uderlyg aloe dffered from the performace cludg the dervatve; we ca say more about whece the dervatve performace comes he prce performace of the call ca be wrtte Ct Now suppose the theoretcal value of the call at 0 s V 0 he t s possble to break the performace to 2 terms C C V C C V t t 0 C etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe C C V C C V t t 0 he frst term measures the theoretcal vs. quoted prce based o the spot prce of the uderlyg o date of purchase 2 d term measures the dfferetal betwee the curret quote ad the tal equlbrum prce he frst ca measure the ablty of the maager to pck udervalued optos he secod measures the ablty to select optos wth udervalued uderlyg assets (as we shall see) Are the outcomes from luck or skll? 1.31 etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe C C V C C V t t 0 Each term ca be broke dow aga he frst ca be broke dow to a volatlty proft ad a formula proft () () V C C s C V C s Note 3 formulas: the true, the market, ad the bechmark We use the bechmark (market) wth the mpled vol. at 0 vs. the real vol. (s) to determe the vol. proft he formula proft s foud usg the bechmark (true) wth 1.32 the real vol. etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe C C V C C V t t 0 Each term ca be broke dow aga he secod term ca be broke dow to the proft from asset udervaluato ad oe due oly to the opto overlay he asset proft s measured relatve to a bechmark strategy a forward cotract o the uderlyg r dt St S0e Assumg the market s effcet ad rsk eutral, ths proft s zero

9 etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe Ct C0 V0 C0 Ct V0 Each term ca be broke dow aga he secod term ca be broke dow to the proft from asset udervaluato ad oe due oly to the opto overlay Fally, the proft from usg a opto vs. a forward r d t Ct V0 St S0e So we have Ct C0 C0() s C0 V0 C0() s rdt St S0e Ct V0 St S0e rdt 1.34 etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe C C C () s C V C () s t rdt rd t St S0e Ct V St S0e Vol. term + formula term + asset term + opto aloe If optos are valued effcetly, the sum of the frst two s zero (o average), rrespectve of the bechmark formula beg close to the oe used by the market If optos are valued effcetly ad the bechmark cocdes w/ market formula, both terms are zero1.35 etur Calculato Usg Dervatves Cosder the smple case of a call opto to apprecate what ca be doe C C C () s C V C () s t rdt rd t St S0e Ct V St S0e If the assets are valued effcetly, term 3 s zero If ot, the market s ot rsk eutral ad ts value measures the compesato correspodg to the rsk take If assets are ot valued effcetly, the term 3 s ot zero ad ts value measures the maagers ablty to select optos that have a correctly valued uderlyg 1.36 I a effcet market cotext, term 4 s also zero etur Calculato he GIPS Performace Must be o a total retur bass usg tme weghted rate of retur Best s to value o a daly bass ad combe results geometrcally Calculated at least quarterly, but recommeds mothly (et of trasacto costs ad cludg retur o cash)

10 elatve etur Calculato Better tha absolute retur, t s more relevat to measure performace relato to a referece Ca thereby hghlght the addtoal share of retur that comes from the vestmet strategy used ad the maagers skll he referece ca be A Bechmark, or A group of portfolos wth the same characterstcs, called a Peer Group elatve etur Calculato Bechmarks A bechmark s smply a referece portfolo It must be chose to reflect Dversty of Assets elgble for the measured portfolo he Ivestmet Strategy he same Calculato ules as the measured portfolo (the portfolo beg evaluated) Especally, w/r to dvded/cash flow ad revestmet ypes of Bechmarks elatve etur Calculato ypes of Bechmarks Market Idces Quoted o Exchages smple ad readly accessble May ot suffcetly represet objectve or strategy of measured PF Broad (cludes less lqud, e.g. Wlshre) vs. More Lqud Idces (more lqud, more restrcted, e.g. Dow) vs. Md-Sze Geerc Ivestmet Style Idces Developed by Specalzed Frms & Allow Dfferet Styles Growth, Value, Small-Cap, etc. Approprate for maagers wth a well-defed, stadard 1.40 vestmet style elatve etur Calculato ypes of Bechmarks Geerc Ivestmet Style Idces (cotued) For o-stadard styles there are bechmarks that better descrbe a maager s style Dffcult to select compoets for ths type of dex Is a low P/E stock a value or a udervalued growth stock No Stadard Lots of Competto Low correlato Frak ussell, S&P-Barra, & others

11 elatve etur Calculato ypes of Bechmarks Sharpe Bechmarks Explas a maagers style from performg a multple regresso o several specalzed dces (asset classes ad markets that the maager s ) Normal Portfolos (the tred) alor-made for each maager Uses the prcple that the PF maager s returs should be compared wth the returs of a referece PF whose structure ad composto are as smlar as possble to those of the PF beg evaluated 1.42 elatve etur Calculato ypes of Bechmarks Normal Portfolos (the tred) Portfolo s made up of a set of securtes that cotas all the securtes from whch the maager wll choose; weghted the same way as would be expected the maagers PF Objectve: a portfolo that obtas a average characterzato of the PF to be evaluated hree techques for choosg securtes (to reduce the umber from the uverse (2); or express as a style) Based o hstorcal composto Based o hstorcal rsk/factor exposure Usg Style Idces (Sharpe) 1.43 elatve etur Calculato ypes of Bechmarks Normal Portfolos (the tred) wo techques for choosg weghtgs (more later) eturs-based: Aalyss of the maagers PF returs Portfolo-Based: Aalyss of style characterstcs of the securtes that makeup the maagers PF Hstorcal Composto (most commo for choosg securtes) Defe a tal uverse Choose securtes cluded the ormal PF (maagers rules compared to actual hstory) elatve etur Calculato ypes of Bechmarks Practcal Use of the Bechmark Maagers Value-Add s calculated as the dfferece betwee the retur o the PF beg evaluated ad the bechmark We wll look at precse measures as we go; later a larger portfolo theory cotext (what the course s all about) Actve etur the percetage of retur that s due to the maager s decsos Maager s Skll over a gve perod of tme t s defed as At Pt Bt Choose Weghtgs (market cap, equal, market cap wth breakpots o captalzato)

12 elatve etur Calculato ypes of Bechmarks Practcal Use of the Bechmark At Pt Bt he calculato may be made perodcally (daly, mothly) ad the compouded for the whole perod If (sub-perod) returs are calculated arthmetcally ad the compouded geometrcally, the the cumulatve actve retur for the whole perod s ot equvalet to calculatg the dfferece betwee the cumulatve PF retur ad the cumulatve BM retur If we have worked wth logarthmc returs, the two calculatos are equvalet logarthmc returs allow 1.46 smpler formulas elatve etur Calculato ypes of Bechmarks Practcal Use of the Bechmark At Pt Bt Cosder sub-perods, the the logarthmc returs for the PF ad the BM are wrtte as log P log B t1 t1 log Pt log Bt ad gves log log log log log log P B Pt Bt At A t1 t elatve etur Calculato ypes of Bechmarks Practcal Use of the Bechmark Allows us to compare a maager to hs bechmark, but does ot allow us to compare fuds that use dfferet bechmarks Peer Groups Dstgush betwee Uverses ad Peer Groups Uverse s a group of PFs vested the same market sector Peer Group s a group of maagers who vest the same class of assets or have the same vestmet style Peer Groups are smaller ad more precse 1.48 elatve etur Calculato Peer Groups Peer Groups are put together by choosg cohort maagers e.g., same style: market cap, growth, value, emergg markets, Calculate returs ad establsh a rakg wth the group Comparsos are made betwee real PFs wth trasacto costs, taxes, etc. ad ot dces as theoretcal, cost-free PFs echcal rakg cosderatos

13 elatve etur Calculato Portfolo Opportuty Dstrbutos (New) Seeks to combe advatages of bechmarks ad peer groups whle elmatg ther dsadvatages Prcple: Compare maagers result wth that acheved by chace Start wth the securtes the maager s lable to clude Geerate a large umber of PFs represetatve of the maagers style he maager s performace s compared to the result uverse Overcomes problem of establshg cohorts; survvorshp bas; statstcally sgfcat umbers; ad luck vs. skll Quattatve Portfolo heory & Performace Aalyss Week February 11, 2013 Cocepts (fsh-up) Basc Elemets of Moder Portfolo heory 1.51 sk etur s (of course) ot suffcet to aalyze the results of a portfolo t s ecessary to add a measuremet of the rsk take the PF Some smple statstcal rsk measures to get started, we expad greatly later 2 Varace, (ad stadard devato, ) t where t1 t s retur o asset for subperod t s mea retur o asset for whole perod s the umber of subperods 1.52 sk 2 Varace, (ad stadard devato, ) A good estmate of rsk s obtaed by usg mothly returs over a 3 year perod hs s the most commo rsk measure used theoretcal dscussos Has the dsadvatage of symmetry Above average returs ad below average returs are equally weghted vestors do t see t that way, though

14 sk Sem-Varace A atural exteso for varace aalyzg PF rsk s 1 2 t 0 t t I a aalogy to the relatoshp betwee the stadard devato ad varace, we defe a measure called the dowsde rsk as the square root of the sem-varace If the dstrbuto of returs s symmetrcal (e.g., Gaussa) the the sem-varace s equal to half the varace More terestg whe returs are skewed o-lear rsk exposures Dffcult to estmate from hstorcal returs ot stable over 1.54 tme sk Lower Partal Momets (LPM) of degree for asset 1 LPM m max 0, h t where h s the target retur t 1 Whe =2 we fd the sem-varace expresso by takg the mea retur as the target retur; we ca use other targets too allows the vestor s rsk averso to be represeted <1 : vestor s rsk seekg =1 : vestor s rsk eutral >1 : vestor s rsk averse (more so as creases) 1.55 sk Asset rsk lkage wth multple assets Covarace 1 j t jt j t1 Normalzed verso (w/r stadard devatos) s correlato j j Other rsk measures j max m Varato terval (hghest to lowest) t t 1 t 1 t Absolute mea devato 1 t E 1 Probablty of egatve retur the proporto of egatve asset 1.56 returs over a gve perod sk wo examples retur vs. rsk & dversfcato Cosder the case of bods & stocks Let us coclude from ths ad from aalyst forecasts: S& P 12.5% S& P 14.9% S& P, B 0.45 B 6% 4.8% B

15 sk wo examples retur vs. rsk & dversfcato Usg the stadard formulas formg the PF of stocks ad bods, we get sk wo examples retur vs. rsk & dversfcato Graphcally sk wo examples retur vs. rsk & dversfcato Now cosder the case of domestc & foreg stocks We coclude from hstorcal data ad aalysts forecasts that sk wo examples retur vs. rsk & dversfcato Usg the stadard formulas formg the PF of domestc ad teratoal stocks, we get S& P S& P S& P, IN IN 12.5% 14.9% % 14.0% IN

16 sk wo examples retur vs. rsk & dversfcato Graphcally 1.62 sk Fxed Icome t s durato ad covexty: about whch more later Iteratoal Assets ad Currecy sk We have already see how to bfurcate returs o measure rsk, we apply the above Q Q/ var var C t t t Q Q/ Q Q/ t Ct t Ct 2 Q 2 Q/ 2 Q Q/ C 2, C C var var 2 cov, or 1.63 sk Iteratoal Assets ad Currecy sk Hedgg currecy exposure (H = 1 + h, from earler) / 2 / Q H Q 2H Q Q C, C C If H=0, we are hedged; f H=1, we are u-hedged Extedg to a PF of domestc asset ad a Itl asset Cross terms get to be a usace (as above wth values of H) P xdd xii H xic 2xdxIdl 2HxI xdd, C xii, C he varace of the u-hedged PF wll be greater tha that of the hedged PF f 2 2 x 2x x x 0 I C I d d, C I I, C 1.64 sk Have looked at specfc rsk dcators by strumet: used by traders & sector specalsts Varace & Beta for Stocks Durato & Covexty for Bods Delta & Vega for Dervatves o roll up rsk measures & cosder rsk more broadly through a smplfed represetato Use Value-at-sk (Va)

17 sk he Va Measure o provde a sgle umber to characterze the rsk for loss a portfolo Complemets trader rsk measures for assessg rsk over the whole portfolo sk he Va Measure Let the varable V be the Va of the portfolo It s a fucto of two parameters he tme horzo (N days) he cofdece level (X %) he loss level, V, over N days that we are X % certa wll ot be exceeded sk he Va Measure We usually use the Stadardzed Va terms from regulators (for market rsk) Cofdece level of 99% (loss wo t occur more tha 1 tme 100) 10-day perod s the usual term for a market reversal Va s the 1% quatle of the PF retur pdf over 10 days sk he Va Measure If N days s the tme horzo & X% the cofdece, Va s the loss correspodg to the (100-X)th percetle of the dstrbuto of the chage the value of the portfolo over the ext N days

18 sk he Va Measure he smplest assumpto s that daly gas/losses are ormally dstrbuted ad depedet It s the easy to calculate Va from the stadard devato σ of gas/losses (1-day Va=2.33 x ) Sce N( 2.33)=0.01 or N(2.33)=0.99 he N-day Va equals N tmes the oe-day Va egulators allow baks to calculate the 10 day Va as 10 tmes the oe-day Va sk he Va Measure Hstorcal Smulato Model Buldg (Varace Covarace) Lear Model Quadratc Model whe gamma s sgfcat Iterest ates Mote Carlo Smulato Stress estg & Back estg Prcpal Compoet Aalyss sk he Va Measure: Hstorcal Smulato Create a database of the daly movemets all market varables. he frst smulato tral assumes that the percetage chages all market varables are as o the frst day he secod smulato tral assumes that the percetage chages all market varables are as o the secod day ad so o 1.72 sk he Va Measure: Hstorcal Smulato Suppose we use m days of hstorcal data Let v be the value of a varable o day here are m-1 smulato trals he th tral assumes that the value of the market varable tomorrow (.e., o day m+1) s v v m v

19 If we are terested the 1- percetle pot of the dstrbuto of chages PF value, estmate ths as the 5 th worst umber the last colum of able 9.2 he 10-day Va for a 99% cofdece level s usually calculated as 10 x that 5 th worst umber sk he Va Measure: Model Buldg he ma alteratve to hstorcal smulato s to make assumptos about the probablty dstrbuto of returs o the market varables ad calculate the probablty dstrbuto of the chage the value of the portfolo aalytcally hs s kow as the model buldg approach or the varace-covarace approach sk he Va Measure: Model Buldg A Detal emder about Volatlty I opto prcg we measure volatlty per year I Va calculatos we measure volatlty per day year day year 252 Strctly speakg we should defe day as the stadard devato of the cotuously compouded retur oe day I practce we assume that t s the stadard devato of the percetage chage oe day 1.76 sk he Va Measure: Model Buldg We have a posto worth $10 mllo Mcrosoft shares he volatlty of Mcrosoft s 2% per day (about 32% per year) We use N=10 ad X=99 he stadard devato of the chage the portfolo 1 day s $200,000 2% of $10 mllo he stadard devato of the chage 10 days s 200, $632,

20 sk he Va Measure: Model Buldg We assume that the expected chage the value of the portfolo s zero (hs s OK for short tme perods) We assume that the chage the value of the portfolo s ormally dstrbuted Sce N( 2.33)=0.01, the (10-day, 99%) Va s , 456 $1, 473, 621 sk he Va Measure: Model Buldg Cosder a posto of $5 mllo A& he daly volatlty of A& s 1% (approx 16% per year) 50, $158, 144 he S.D per 10 days s he Va s 158, $368, sk he Va Measure: Model Buldg Now cosder a portfolo cosstg of both Mcrosoft (X) ad A& (Y) Suppose that the correlato betwee the returs s 0.3 So a portfolo of X ad Y has stadard devato 2 2 X Y X Y 2 X Y I ths case X = 200,000 ad Y = 50,000 ad = 0.3. he stadard devato of the chage the portfolo value oe day s therefore 220, sk he Va Measure: Model Buldg he 10-day 99% Va for the portfolo s 220, $1,622,657 he beeft of dversfcato s (1,473, ,405) 1,622,657=$219,369 What s the cremetal effect of the A& holdg o Va? Drve by the correlato At 0 : 1,842,026 (creases by 368,405) At 0.3 : 1,622,657 (creases by 219,405)

21 sk he Va Measure: Model Buldg For the geeral Lear Model for Va, assume: he daly chage the value of a portfolo s lear the daly returs from market varables he returs from the market varables are ormally dstrbuted ( : amout vested asset ) Px where x s % chage value of asset P j j j 2 j j j 1 j1 1 j where s the volatlty of varable ad s the portfolo's stadard devato P 1.82 sk he Va Measure: Mote Carlo Smulato Value portfolo today usg curret values of market Sample oce from the multvarate dstrbutos of the x Use the x to determe mkt varables at ed of tomorrow evalue the portfolo at the ed of day Calculate P epeat may tmes to buld up a (probablty) dstrbuto for P Va s the approprate fractle of the dstrbuto tmes square root of N For example, wth 1,000 tral the 1 percetle s the 10th worst case