Ottimizzazione di portafoglio, Capital Asset Pricing Model e Arbitrage Pricing Theory: un’analisi empirica sull’indice S&P 500

EOG EOGC to AAPL 0,321093335 JPM to 0,301802 BAC to 0,272666AAPL AAPLC to AVGO 0,40635589 JPM to 0,342615 BAC to 0,361022AVGO AVGOC to CSCO 0,387815718 JPM to 0,393603 BAC to 0,337701CSCO CSCOC to ORCL 0,396124951 JPM to 0,36916 BAC to 0,330237ORCL ORCL 46C to ABT 0,390727464 JPM to 0,35719 BAC to 0,335631ABT ABTC to JNJ 0,231116308 JPM to 0,241767 BAC to 0,164123JNJ JNJC to BIIB 0,361606045 JPM to 0,339699 BAC to 0,327776BIIB BIIBC to SYK 0,19009215 JPM to 0,180348 BAC to 0,129114SYK SYKC to BA 0,468367808 JPM to 0,446019 BAC to 0,424969BA BAC to UNP 0,488597791 JPM to 0,512875 BAC to 0,506612UNP UNPC to RTN 0,211646915 JPM to 0,258293 BAC to 0,22423RTN RTNC to PH 0,547108034 JPM to 0,562756 BAC to 0,538693PH PHMS to C 0,869595707 CVX to C 0,433838 VLO to C 0,434287MS to JPM 0,863586298 CVX to 0,452752 VLO to 0,427058JPM JPMMS to BAC 0,867929037 CVX to 0,384591 VLO to 0,423811BAC BACMS to MS 1 CVX to 0,366758 VLO to 0,397207MS MSMS to CVX 0,366758422 CVX to 1 VLO to 0,301152CVX CVXMS

to VLO 0,397206681 CVX to 0,301152 VLO to 1VLO VLOMS to OKE 0,30888536 CVX to 0,579821 VLO to 0,190995OKE OKEMS to EOG 0,331129978 CVX to 0,671998 VLO to 0,217252EOG EOGMS to 0,311625506 CVX to 0,203519 VLO to 0,222565AAPL AAPL AAPLMS to 0,39552089 CVX to 0,166669 VLO to 0,293383AVGO AVGO AVGOMS to 0,395255667 CVX to 0,337191 VLO to 0,32613CSCO CSCO CSCOMS to 0,38792198 CVX to 0,286285 VLO to 0,293266ORCL ORCL ORCLMS to ABT 0,39936621 CVX to 0,185322 VLO to 0,295878ABT ABTMS to JNJ 0,230539813 CVX to 0,268015 VLO to 0,20949JNJ JNJMS to BIIB 0,370064892 CVX to 0,129257 VLO to 0,234235BIIB BIIBMS to SYK 0,178233787 CVX to 0,16245 VLO to 0,153215SYK SYK 47MS to BA 0,461458381 CVX to 0,359575 VLO to 0,253061BA BAMS to UNP 0,488019461 CVX to 0,382402 VLO to 0,294028UNP UNPMS to RTN 0,206969205 CVX to 0,14173 VLO to 0,241695RTN RTNMS to PH 0,56070534 CVX to 0,481982 VLO to 0,312353PH PHOKE to C 0,350016671 EOG to C 0,388085 AAPL to 0,321093COKE to 0,310246137 EOG to 0,375153 AAPL to

0,301802JPM JPM JPMOKE to 0,314619723 EOG to 0,373742 AAPL to 0,272666BAC BAC BACOKE to MS 0,30888536 EOG to 0,33113 AAPL to 0,311626MS MSOKE to 0,579820663 EOG to 0,671998 AAPL to 0,203519CVX CVX CVXOKE to 0,190995007 EOG to 0,217252 AAPL to 0,222565VLO VLO VLOOKE to 1 EOG to 0,558527 AAPL to 0,177317OKE OKE OKEOKE to 0,558526744 EOG to 1 AAPL to 0,192036EOG EOG EOGOKE to 0,177316938 EOG to 0,192036 AAPL to 1AAPL AAPL AAPLOKE to 0,192618165 EOG to 0,200472 AAPL to 0,554352AVGO AVGO AVGOOKE to 0,183896539 EOG to 0,25522 AAPL to 0,329829CSCO CSCO CSCOOKE to 0,26979204 EOG to 0,199076 AAPL to 0,305053ORCL ORCL ORCLOKE to 0,101572685 EOG to 0,175022 AAPL to 0,299297ABT ABT ABTOKE to JNJ 0,097736794 EOG to 0,165742 AAPL to 0,180454JNJ JNJOKE to 0,130536149 EOG to 0,086403 AAPL to 0,220906BIIB BIIB BIIBOKE to 0,178648227 EOG to 0,143497 AAPL to 0,233471SYK SYK SYKOKE to BA 0,315035114 EOG to 0,26754 AAPL to 0,274801BA BAOKE to 0,230595952 EOG to 0,311553 AAPL to 0,241174UNP UNP UNPOKE to

0,13438725 EOG to 0,058132 AAPL to 0,202149RTN RTN RTN 48OKE to PH 0,369149613 EOG to 0,454408 AAPL to 0,258189PH PHAVGO to C 0,40635589 CSCO to 0,387816 ORCL to 0,396125C CAVGO to 0,342614747 CSCO to 0,393603 ORCL to 0,36916JPM JPM JPMAVGO to 0,361022448 CSCO to 0,337701 ORCL to 0,330237BAC BAC BACAVGO to 0,39552089 CSCO to 0,395256 ORCL to 0,387922MS MS MSAVGO to 0,166668593 CSCO to 0,337191 ORCL to 0,286285CVX CVX CVXAVGO to 0,293383009 CSCO to 0,32613 ORCL to 0,293266VLO VLO VLOAVGO to 0,192618165 CSCO to 0,183897 ORCL to 0,269792OKE OKE OKEAVGO to 0,200471845 CSCO to 0,25522 ORCL to 0,199076EOG EOG EOGAVGO to 0,554351759 CSCO to 0,329829 ORCL to 0,305053AAPL AAPL AAPLAVGO to 1 CSCO to 0,368186 ORCL to 0,372445AVGO AVGO AVGOAVGO to 0,36818623 CSCO to 1 ORCL to 0,404165CSCO CSCO CSCOAVGO to 0,372445375 CSCO to 0,404165 ORCL to 1ORCL ORCL ORCLAVGO to 0,329254568 CSCO to 0,377339 ORCL to 0,347785ABT ABT ABTAVGO to 0,171306631 CSCO to 0,334757 ORCL to 0,272404JNJ JNJ JNJAVGO to

0,275183126 CSCO to 0,265447 ORCL to 0,248269BIIB BIIB BIIBAVGO to 0,323480787 CSCO to 0,240033 ORCL to 0,288327SYK SYK SYKAVGO to 0,352186547 CSCO to 0,271427 ORCL to 0,302251BA BA BAAVGO to 0,340996569 CSCO to 0,322196 ORCL to 0,278705UNP UNP UNPAVGO to 0,189131955 CSCO to 0,17835 ORCL to 0,179066RTN RTN RTNAVGO to 0,325261296 CSCO to 0,473867 ORCL to 0,346094PH PH PHABT to C 0,390727464 JNJ to C 0,231116 BIIB to C 0,361606ABT to 0,357190323 JNJ to 0,241767 BIIB to 0,339699JPM JPM JPM 49ABT to 0,335631476 JNJ to 0,164123 BIIB to 0,327776BAC BAC BACABT to MS 0,39936621 JNJ to MS 0,23054 BIIB to 0,370065MSABT to 0,185322076 JNJ to 0,268015 BIIB to 0,129257CVX CVX CVXABT to 0,295878171 JNJ to 0,20949 BIIB to 0,234235VLO VLO VLOABT to 0,101572685 JNJ to 0,097737 BIIB to 0,130536OKE OKE OKEABT to 0,175022191 JNJ to 0,165742 BIIB to 0,086403EOG EOG EOGABT to 0,299296766 JNJ to 0,180454 BIIB to 0,220906AAPL AAPL AAPLABT to 0,329254568 JNJ to 0,171307 BIIB to 0,275183AVGO AVGO AVGOABT to

0,377338982 JNJ to 0,334757 BIIB to 0,265447CSCO CSCO CSCOABT to 0,347784852 JNJ to 0,272404 BIIB to 0,248269ORCL ORCL ORCLABT to 1 JNJ to 0,372281 BIIB to 0,377711ABT ABT ABTABT to JNJ 0,372281209 JNJ to 1 BIIB to 0,278447JNJ JNJABT to 0,377710505 JNJ to 0,278447 BIIB to 1BIIB BIIB BIIBABT to 0,414534363 JNJ to 0,398838 BIIB to 0,185025SYK SYK SYKABT to BA 0,265281125 JNJ to BA 0,202082 BIIB to 0,261292BAABT to 0,235196711 JNJ to 0,176248 BIIB to 0,194136UNP UNP UNPABT to 0,156589109 JNJ to 0,272339 BIIB to 0,209574RTN RTN RTNABT to PH 0,328693148 JNJ to PH 0,262905 BIIB to 0,243772PHSYK to C 0,19009215 BA to C 0,468368 UNP to C 0,488598SYK to 0,180347664 BA to 0,446019 UNP to 0,512875JPM JPM JPMSYK to 0,129114364 BA to 0,424969 UNP to 0,506612BAC BAC BACSYK to MS 0,178233787 BA to MS 0,461458 UNP to 0,488019MSSYK to 0,16244959 BA to 0,359575 UNP to 0,382402CVX CVX CVXSYK to 0,153215323 BA to 0,253061 UNP to 0,294028VLO VLO VLO 50SYK to 0,178648227 BA to 0,315035 UNP to 0,230596OKE

OKE OKESYK to 0,143497058 BA to 0,26754 UNP to 0,311553

EOG EOG EOGSYK to 0,233471166 BA to 0,274801 UNP to 0,241174

AAPL AAPL AAPLSYK to 0,323480787 BA to 0,352187 UNP to 0,340997

AVGO AVGO AVGOSYK to 0,240032634 BA to 0,271427 UNP to 0,322196

CSCO CSCO CSCOSYK to 0,288327049 BA to 0,302251 UNP to 0,278705

ORCL ORCL ORCLSYK to 0,414534363 BA to 0,265281 UNP to 0,235197

ABT ABT ABTSYK to JNJ 0,398837556 BA to JNJ 0,202082 UNP to 0,176248

JNJSYK to 0,185025262 BA to 0,261292 UNP to 0,194136

BIIB BIIB BIIBSYK to 1 BA to 0,120207 UNP to 0,12448

SYK SYK SYKSYK to BA 0,120207307 BA to BA 1 UNP to 0,400227

BASYK to 0,124479672 BA to 0,400227 UNP to 1

UNP UNP UNPSYK to 0,143255342 BA to 0,388286 UNP to 0,25129

RTN RTN RTNSYK to PH 0,196749287 BA to PH 0,48961 UNP to 0,511345

PHRTN to C 0,211646915 PH to C 0,547108

RTN to 0,258293413 PH to 0,562756

JPM JPMRTN to 0,224230078 PH to 0,538693

BAC BACRTN to MS 0,206969205 PH to MS 0,560705

RTN to 0,141729866 PH to 0,481982

CVX CVXRTN to 0,241694788 PH to

0,312353VLO VLORTN to 0,13438725 PH to 0,36915OKE OKERTN to 0,058132349 PH to 0,454408EOG EOGRTN to 0,202149181 PH to 0,258189AAPL AAPLRTN to 0,189131955 PH to 0,325261AVGO AVGO 51RTN to 0,17834959 PH to 0,473867CSCO CSCORTN to 0,179065781 PH to 0,346094ORCL ORCLRTN to 0,156589109 PH to 0,328693ABT ABTRTN to JNJ 0,272338799 PH to JNJ 0,262905RTN to 0,209573951 PH to 0,243772BIIB BIIBRTN to 0,143255342 PH to 0,196749SYK SYKRTN to BA 0,388286255 PH to BA 0,48961RTN to 0,251289937 PH to 0,511345UNP UNPRTN to 1 PH to 0,309946RTN RTNRTN to PH 0,309945705 PH to PH 1

La maggior parte dei titoli presenta un grado di correlazione molto basso, dunque i portafogli ne derivano beneficiano dell'effetto della diversificazione.

3.3 Portafoglio tangente

Si procede adesso con il calcolo del tangency portfolio, che si ricorda essere il portafoglio situato sul punto di tangenza della CAL con la frontiera efficiente, che rappresenta il massimo valore dello Sharpe ratio. R costruisce il tangency

portfolio attribuendo i seguenti pesi ad ogni titolo azionario: pesi ottimali

ORCL 0C 0 ABT 0JPM 0 JNJ 0,175563755BAC 0 BIIB 0MS 0 SYK 0,24995798CVX 0 BA 0,182136102VLO 0 UNP 0,034733638OKE 0,030700747 RTN 0,137244223EOG 0 PH 0,133794937AAPL 0,055868618AVGO 0CSCO 0

I risultati sono esposti in percentuale nei seguenti grafici:

52AAPL ABT AVGO BA BAC BIIB C CSCC CVX EOG JNJ JPM MS OKE ORCL PH RTN SYK UNP VLO

Fonte: elaborazione personale SYKBA UNP

Fonte: elaborazione personale 53

Dall’esito del processo di ottimizzazione si deduce che il software ha utilizzato soltanto 8 dei 20 titoli iniziali per implementare il tangency portfolio, ovvero: OKE, AAPL, JNJ, SYK, BA, UNP, RTN E PH. In particolare, il titolo a cui è stato assegnato il peso maggiore risulta essere SYK, con un 25%. Le principali statistiche descrittive dei rendimenti del portafoglio così realizzato presentano tali valori:

Media Varianza Mediana Asimmetria Kurtosi Min Max

0,0012 0,00005 0,00009 -0,3239 2,3832 -0,0293

0,0266Questi risultati si basano sui rendimenti campionari. È rilevante notare come la varianza,dunque la rischiosità del portafoglio, risulta inferiore rispetto a quella di ogni singolo titolo, adimostrazione che la diversificazione in tal caso ha dato i suoi frutti. Annualizzando i valori delrendimento e della deviazione standard, si ottiene:

Rendimento medio annualizzato 0,3367

Deviazione standard annualizzata 0,1102

Lo Sharpe ratio del portafoglio risulta pari al 16%, che è il massimo valore raggiungib iletrattandosi del punto situato sulla CAL tangente la frontiera efficiente. I risultati ottenuti sulperiodo di test (2/01/2018-28/06/2019) sono invece i seguenti:

Media Varianza Mediana Asimme