CAL CML SML - The difference among CML CAL SML. PDF

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The difference among CML CAL SML....

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CALvs.CMLvs.SML

PostedbyBillCampbellIII,CFAonSeptember29,2013

Postedin:LevelI.

TheCapitalAllocationLine(CAL),CapitalMarketLine(CML),andSecurityMarketLine(SML)can beconfusedeasily,andforgoodreason:the graphs look virtually identical, the assumptions under which they are constructed are essentially the same, and their implications are similar. We’ll characterizeeachoneandtrytoeliminatetheconfusion.

Theassumptionscommontothesethreelinesarethat:

There is a riskfree asset (whose standard deviation of returns is zero, and whose expected returninknown) There are risky assets (whose expected returns, standard deviations of returns, and correlationsofreturnsareknown) Aninvestorcanlendorborrowanyamountattheriskfreerate Aninvestorcanbuyanyamountofariskyasset

CapitalAllocationLine(CAL)

Technically, there isn’t a CAL; there are infinitely many CALs: one for each risky asset (or portfolio of risky assets).  Given a risky asset (portfolio)A andtheriskfreeasset, allcombinationsof portfolio Aandthe riskfreeassetlie alongtheCALfor portfolioA. If,forexample, the riskfree assethasanexpectedreturnof3%,andportfolioAhasanexpectedreturnof8%withastandard deviationofreturns of7%,thenthe graphoftheCALforportfolioAis:

Fiveexampleportfoliosareshownontheline,movingfromthelowerlefttotheupperright:

100%riskfreeasset 80%riskfreeasset,20%portfolioA 30%riskfreeasset,70%portfolioA 100%portfolioA 20%riskfreeasset(i.e.,borrowing20%),120%portfolioA

There is a CAL for every risky asset/portfolio.  Here is a graph showing CALs for three risky portfolios:

Recallthattheslopeofalineisthechangeinyvaluesbetweentwopointsdividedbythechangein xvaluesbetweenthesametwopoints:

ForportfolioA,thetwopointsare

,and

,(where

),so,

whichis the Sharpe ratio for portfolio A. Similarly, the slopeof the CAL for portfolioB isportfolio B’sSharperatio,andtheslopeoftheCAL forportfolioCisitsSharperatio. CapitalMarketLine

Oneof theassumptions ofModern PortfolioTheory (MPT)isthat allinvestors havethe same marketexpectations;given that,we cantalk about theexpectedreturnonaportfolio,andthe Sharperatio,andsoon.Amongstallofthepossibleriskyportfolios,thereis(atleast)onethathas the highest Sharpe ratio. (Most people saythat there is onesuch portfolio, but itis theoretically possible to have morethan one; it’s nota big deal eitherway.)Thatportfoliowilllieontheefficient frontier(seethegraph below),andiscalledtheMarket Portfolio.(Somepeoplethinkthatthisis the portfolio with all investible assets at their total market weights.  As far as I can tell, there’s no good reason to believe that that’s true.  It’s probablynotimportant.)TheCALfortheMarket PortfolioiscalledtheCapitalMarketLine(CML),illustratedhere,alongwithafewotherCALs(so youcanseethattheCMLhasthehighestSharperatio(slope)),theefficientfrontier,andthe optimalfrontier:

NotethattheMarketRiskPremium(MRP)isdefinedas:

sotheslopeoftheCML–theSharperatiofortheMarketPortfolio–is:

SecurityMarketLine

The security marketline (SML) is derived from theCML; the only difference isthat the horizontal axis for theSML is systematic risk – beta(β) – whereasthehorizontalaxisfortheCMListotal risk–standarddeviation(σ)ofreturns:

TheslopeoftheSMLis:

ImplicationsoftheCMLandtheSML

OneoftheassumptionsofMPTisthatsecuritiesarefairlypriced.Ifasecurityisfairlypricedthenit isgeneratingreturnsthatareappropriatefor the risk that it poses: it will plot on the CML (expected return vs. σ of returns (total risk)), and it will plot on the SML (expected return vs. β (systematic risk)).

If a security plots above the CML or above the SML, then it is generating returns that are too high for the risk it poses: the security is underpriced.(Forexample,supposethatasecurityispricedat$10andisearninga10%return($1.00)when,accordingtoitsriskitshouldbe earninga5%return.Areturnof$1.00is5%of$20,sothesecurityshouldbepricedat$20:itisunderpriced.)

IfasecurityplotsbelowtheCMLor belowtheSML,thenitisgeneratingreturnsthataretoolowfor theriskitposes:thesecurityis overpriced....