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Case Alex Sharpe's Portfolio

In: Business and Management

Submitted By sepi85
Words 1238
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| Case 2: Alex Sharpe’s portfolio | 8.5.2014 |

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Tehtävä 1.
CAPM kaava on:

Osakkeen tuotto-odotus E(Ri) = 10,25 %
Riskittömän sijoituskohteen tuotto Rf = 6 %
Markkinaportfolion tuotto-odotus E(Rm) = 14,5 % (riskitön korko + riskipreemio)

Sijoittamalla nämä yllä olevaan kaavaan saadaan:
10,25 % = 6 % + β(14,5 % - 6 %) josta saadan betan β arvoksi 0,5.
Kun osakkeen tuottojen ja markkinaportfolion välinen kovarianssi kaksinkertaistuu, β-kerroin kaksinkertaistuu, kun markkinaportfolion varianssi ei muutu. Betan kaksinkertaistuessa osakkeen riskipreemio puolittuu.
Kun β kerrotaan kahdella ja luvut sijoitetaan samaan kaavaan, saadaan osakkeen tuotto-odotukseksi E(ri) eli sijoittajien tuottovaatimukseksi 14,5 %.
E(ri) = 6 % +2 *0,5 (14,5 % - 6 %) => E(ri)=14,5 %.

Yrityksen maksaessa 50 € * 0,1025 = 5,125 € osinkoa, osakkeen arvoksi saadaan:
CFvr= 5,1250,145= 35,34 €

Tehtävä 2.
a)
Merkitään raaka-ainerahaston osuutta salkusta muuttujalla x.
12,5 % x + 9 % *(1-x) = 10 % josta saadaan x = 28,6 %

b)
Markkinaportfolion beta β on 1, joten salkun β on oltava 1.
Merkitään raaka-ainerahaston osuutta salkusta muuttujalla x.
1,5x + 0,8*(1-x) = 1 josta saadaan x = 28,6 %, eli raaka-ainerahastoon tulee sijoittaa 28,6 %.

c)
Koska riskittömän sijoituksen volatiliteetti on 0, voidaan suoraan laskea salkun volatiliteetti painotetulla keskiarvolla.
Merkitään nykyisen salkun osakkeita muuttujalla x.
Seuraavien epäyhtälöiden täytyy olla voimassa:
0,8 x < 0,6 ja 35 x < 15
Näistä määräävänä ehtona x:n puitteissa on jälkimmäinen.
Ratkaisuksi saadaan x < 42,9 %, joten riskittömään sijoitukseen täytyy sijoittaa vähintään 1-x = 57,1%

Case 2: Alex Sharpe’s portfolio

Tehtävä 1.
Tuottoihin ja vaihteluun perustuen Reynolds näyttää riskisimmältä osakkeelta. Reynoldsilla on historiallisesti suurin keskimääräinen tuotto (1,42 %) mutta myös suurin volatiliteetti (eli keskihajonta, 9,37 %). Tästä voidaan päätellä, että sen beta – kerroin on luultavimmin korkein vertailujoukosta. Hasbron keskimääräinen tuotto (0,87 %) on Reynoldsin matalampi mutta myös Hasbron volatiliteetti on pienempi (8,12 %).
Molempien osakkeiden keskihajonta on viiden vuoden tarkasteluajanjakson aikana laskenut kohtuullisen suuristakin luvuista lähemmäs S&P 500 indeksin vaihtelua. Indeksiin verrattuna yritysten osakkeiden keskimääräinen tuotto on korkeampi mutta toisaalta osakkeiden volatiliteetti on huomattavasti korkeampi. Tämä selittyy sillä, että indeksiin vaikuttaa suuri joukko eri osakkeita. Eri alojen ominainen vaihtelu tasoittuu, kun indeksillä on suuri hajautuminen.
Keskimääräisen tuoton laskemiseen käytimme geometrista keskiarvoa aritmeettisen keskiarvon sijaan, sillä käytössämme oli toteutuneita tuottoprosentteja aiemmilta vuosilta. Tällöin geometrinen keskiarvo kuvaa menneisyyden keskimääräisiä tuottoja aritmeettista paremmin.

Tehtävä 2.
Jos Sharpella olisi ollut sijoitusajanjakson aikana portfoliossaan 99 % S&P 500:aa ja 1 % Reynoldsia, portfolion arvo olisi vuoden 2007 Tammikuussa kasvanut (1+0,510 %)5 eli 2,57 prosenttia. Jos portfoliossa 1 % olisi Hasbroa, olisi portfolion arvo kasvanut (1+0,504 %)5 eli 2,55 prosenttia. Tämä on laskettu käyttäen portfolion tuoton odotusarvoa, joka on laskettu osakkeiden geometrisista keskiarvoista painottaen ne osakkeiden suhteellisella määrällä portfoliossa. Geometrista keskiarvoa käytetään, koska tehtävässä arvioidaan portfolion nykyistä arvoa historiallisen datan perusteella.

Hasbron korrelaatio on korkea (62,79 %) suhteessa S&P 500 indeksiin, josta johtuen portfolion volatiliteetti nousee suuremmaksi, kuin Reynolds & S&P 500 portfoliolla. Vaikka Hasbron oma keskihajonta onkin Reynoldsia pienempi. Hasbron kurssi vaihtelee enemmän samaan suuntaan S&P 500 kanssa, eivätkä ne kumoa toistensa vaikutusta.

Tässä tilanteessa Reynolds olisi parempi valinta, koska portfolion tuoton odotusarvo (0,510 %) olisi suurempi kuin S&P Hasbron, ja volatiliteetti on pienempi. Tämä havainto on ristiriitainen ensimmäisen tehtävän havaintojen kanssa, jossa Reynolds on yksittäisenä osakkeena riskisempi. S&P 500 pienentää Reynoldsin riskisyyttä portfoliossa. Tässä tilanteessa, kun yhdistetään kaksi osaketta, niiden keskeisellä korrelaatiolla on vaikutusta tuoton odotusarvoon ja portfolion riskisyyteen.

Jos Sharpella olisi ollut sijoitusajanjakson aikana portfoliossaan 99 % S&P 500:aa ja 1 % Reynoldsia, portfolion volatiliteetti olisi ollut 3,605. Hasbron tilanteessa olisi portfolion hajonta 3,629. Tästä nähdään, että osakkeiden yhdistäminen portfolioksi voi pienentää tuottojen vaihtelua pienemmäksi, kuin mitä yksittäisten osakkeiden vaihtelu on. Esimerkiksi näin on käynyt S&P 500 ja Reynoldsin portfoliolla. Hasbron ja S&P 500 tilanteessa volatiliteetti laskee, mutta jää lievästi S&P 500 omaa keskihajontaa suuremmaksi.

Tehtävä 3.
Beta saadaan laskettua kaavalla: βi=σimσm2, jossa σim on osakkeen ja S&P 500 tuottojen välinen kovarianssi ja σm2 markkinaportfolion on S&P 500 tuoton varianssi.

Reynolds on S&P 500 markkinaportfoliota vähäriskisempi sijoitus, eli sillä on pienempi systemaattinen riski kuin Hasbrolla (riski, jota ei voi hajauttamalla poistaa). Reynoldsin beta (0,73) on defensiivinen ja Hasbron (1,41) on aggressiivinen, eli Hasbro reagoi yleisindeksiä voimakkaammin sen heilahteluihin.
Tämän mukaan Reynolds on paljon vähemmän riskisempi kuin Hasbro, kun niitä verrataan yleisindeksin kehitykseen. Tämä tukee tehtävässä 2 todettuja havaintoja, että S&P 500 ja Reynolds portfolio on parempi sijoitus, joka johtui siitä, että sen korrelaatiokerroin oli pienempi.

Tehtävä. 4
Capital asset pricing (CAPM) – mallin mukaan sijoittajien tuottovaatimus sijoituskohteelle kasvaa, jos osakkeen epäsystemaattinen riski, jota beta kuvaa, kasvaa. Sijoittajien tuottovaatimus on sitä korkeampi, mitä riskisempi sijoitus on. Tehtävässä kolme ilmeni, että Reynoldsin beta oli pienempi kuin Hasbrolla, joten Hasbro on riskisempi. Tämä näkyi myös hajontakuvion regressiosuoran kulmakertoimessa, jossa Reynoldsin suora on Hasbron suoraa loivempi.
CAPM kaava: Eri=rf+βi[Erm-rf], jossa E(ri) on yksittäisen sijoituskohteen tuotto-odotus, rf on riskittömän sijoituskohteen tuotto, βi yksittäisen sijoituskohteen beta ja E(rm) on markkinaportfolio S&P 500:n tuotto-odotus.
Historiallisesta datasta voidaan arvioida osakkeen tulevaisuuden tuotto-odotus käyttämällä aritmeettista keskiarvoa. Kun laitetaan S&P 500, Reynoldsin ja Hasbron tiedossa olevat luvut (Eri, βi ja Erm) yllä olevaan kaavaan, saadaan riskittömän koron rf arvot. Kun muut arvot ovat selvillä, niin mitä suurempi arvo rf:ksi saadaan, sitä parempi sijoitus on.

Taulukosta nähdään, että markkinoiden riskittömän koron tulisi olla Reynoldsin kohdalla 5,49 % ja Hasbron kohdalla -1,17 %, jotta odotetut arvot vastaisivat sijoittajien tuottovaatimusta.
Tämän laskelman mukaan Hasbro on riskisempi sijoitus, sillä jos historiallinen data on myös ennuste tulevasta, niin sijoittajien tuottovaatimus kohtaisi Hasbron tuloskehityksen markkinakoron ollessa negatiivinen. Kuten edellä on todettu, Hasbron riskipreemio on myös suurempi. Volatiliteetin perusteella Hasbro on parempi sijoitus Reynoldsiin nähden, koska sen kehitys on hieman tasaisempaa. Mutta mielestämme tässä tapauksessa, beta ja CAPM kuvaavat parhaiten riskin ja tuoton suhdetta.
Edellisten laskujen perusteella voi sanoa, että riski ja tuotto kulkevat käsi kädessä - mitä suurempi tuotto-odotus, sitä suurempi riski.

Tehtävä 5.
Kvantitatiivisesti arvioiden Reynoldsin beta on markkinaportfoliota pienempi, joten se reagoi markkinoiden muutoksiin markkinaportfoliota vähemmän. Vaikka Alex Sharpe oli valmis lisäämään riskiä sijoituksissaan, silti Reynoldsiin sijoittamien on järkevämpää. Reynoldsin odotetut tuotot ovat isommat ja se pienentää portfolion riskiä, koska sen beta on alle 1. Hasbron tuotot ovat Reynoldsia enemmän yhteydessä S&P 500 yleisindeksin muutoksiin markkinatilanteen vaihteluissa. Hasbron tilanteessa portfolion volatiliteetti jopa nousee hieman S&P 500 indeksiä suuremmaksi, kuten tehtävässä 2 havaittiin. Hasbroon sijoittaminen lisääkin portfolion epäsystemaattista riskiä, sillä sen beta on suurempi kuin 1.
Kvalitatiivisesti arvioiden voidaan lisäksi sanoa, että tupakoitsijat polttavat tupakkaa olipa markkinatilanne mikä tahansa. Toisaalta lapsille ostetaan enemmän leluja nousukaudella kun siihen on enemmän varaa, kuin taantumassa.
Sijoituskohteiden valinta perustuu kvantitatiiviseen analyysiin, koska voidaan olettaa, että kvalitatiiviset riskit on hinnoiteltu jo osakkeen arvoon ja ne voidaan hajauttaa pois.
Vastaus:
Suosittelemmekin Sharpea sijoittamaan Reynoldsiin, koska se nostaa portfolion tuotto-odotusta ja samalla pienentäen sen riskisyyttä. Osakkeiden ja Vanguard 500 rahaston optimaalinen suhde riippuu Sharpen preferensseistä ja markkinoiden riskittömästä korosta, jonka avulla voitaisiin määrittää tangenttiportfolio. Loimme S&P 500 indeksin ja Reynoldsin suhteesta riippuvan kuvaajan (seuraavalla sivulla), joka kuvaa tuoton ja riskin välistä suhdetta.

Markkinoiden riskitöntä korkoa ei ollut tässä tilanteessa tiedossa, joten kuvio on vain suuntaa antava. Kuitenkin päätös perustuisi pitkälti tämän kuvion malliin, jossa optimaalisen riskipitoinen portfolio saataisiin tangentin ja suoran leikkauspisteestä. Tehokas alue on valkoisella pohjalla alla olevassa taulukossa.…...

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= 0.07047 
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