financial management - topic 6 - capm

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Topic Topic 66

The The CapitalCapital Asset Asset PricingPricingModelModel (CAPM)(CAPM)

Copyright of Spanish version from David MorenoTranslation into English by Francisco Romero

Universidad Carlos III Financial Economics

Teacher: Beatriz Mariano


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Topic 6- The Capital Asset Pricing Model (CAPM)

OutlineOutline

1.1. THE CAPITAL ASSET PRICING MODEL THE CAPITAL ASSET PRICING MODEL

1.1. Assumptions Assumptions andand foundationsfoundations of of thethe modelmodel2.2. The The CML (CapitalCML (Capital MarketMarket LineLine ) )

3.3. The The SML ( SML ( Security Security MarketMarket LineLine ) )

4.4. The The Beta ( Beta ( ) )

5.5. The The Beta ( Beta ( ) ) of of aa portfolioportfolio


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1- Assumptions and foundations of the model

The CAPM (Capital Asset Pricing Model) is fundamentalfor modern finance, even though it was developed 50 years

ago.

The CAPM was developed by William Sharpe (1962). He

received the Nobel Prize..

It is a model that works when financial markets are

equilibrium.Therefore, Supply=Demand

The CAPM builds on the Markowitz mean-variance model .


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1- Assumptions and foundations of the model

The CAPM is a theoretical model and therefore it is based onassumptions:

a) It is a static model; economic agents only focus on what happens oneperiod ahead (1 quarter ahead, 1 year ahead etc.)b) Perfect competition in financial markets. There are many investors (each

with a different utility function and initial wealth). Furthermore, investorsare price takers.

c) Risky financial assets are perfectly divisible and supply is exogenous.d) The same interest rate applies to lending and borrowing.e) There are no transaction costs and taxes.

f) Investors optimize according to Markowitz theory (i.e. care about mean- variance trade-off).

g) Investors share the same information, therefore, their risk and returnexpectations for each asset are identical.


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22-- The The CML (CapitalCML (Capital MarketMarket LineLine ))

Given that investors share the same information and follow themean-variance model, all choose the tangency portfolio T as te

optimal portfolio of risky assets.But they differ in the percentage of his/her wealth allocated tothe risk-free asset and to portfolio T.

E[Rp]

Riesgo

Rf

Cartera ptima

Cartera ptimadel inversor

Cartera tangente

CML

Tangent portfolio

Investors optimal portfolio

Risk


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THE THE MARKET PORTFOLIOMARKET PORTFOLIO COINCIDES WITH THECOINCIDES WITH THE TANGENCY PORTFOLIO: TANGENCY PORTFOLIO:

This is easily shown given our previous assumptions.WhatWhat isis thethe marketmarket portfolio portfolio ??

The portfolio that includes all risky assets available.In equilibrium when we add together everyone s holdings of portfolio T, we must have every share in every company in themarket. So T is also called the market portfolio , M.The weight of asset j in the market portfolio is equivalent to

the total value of asset j in the economy, divided by the total value of all risky assets in the economy.

*

1

*

portfoliomarkettheof Value j""assetof Value

i

N

ii

j j j

P n

P nW

=

==

P* represents equilibriumprices, as S =D.


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If every investor holds the tangency portfolio as his risky asset(obtained from the mean-variance model) and the CAPM is an

equilibrium model (i.e. Excess Supply =0), then the weight of an asset in the tangent portfolio is equal to the weight of thisasset in the market portfolio.

EXAMPLE:EXAMPLE: suppose that the weight of TEF shares in thetangent portfolio is 23%. This means that all agents have 23%of their wealth invested in TEF shares. Therefore, if themarket portfolio is the sum of all portfolios from all economicagents, TEF shares will also represent a 23% of the market.

The same applies for the remaining assets.


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This is why we can replace the tangency portfolio with themarket portfolio in the mean-variance graph:

E[Rp]

Riesgo

Frontera Eficiente

CAL

Cartera ptima

Rf

E[Rc]

MarketPortfolio M

CML

Efficient Frontier

Risk


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33-- The The SML (SML ( SecuritySecurity MarketMarket LineLine ))

Under the CAPM agents hold diversified portfolios, hencethey demand a risk premium which depends on the systematic

risk of each asset (and not on specific risk).

The systematic risk wdepends on , therefore, investors

demand a return that will be a function of .

The fundamental CAPM equation relates an assets risk premium to:

The expected market risk premiumThe assets systematic risk (its beta)


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Therefore:

Risk premiums are represented by capital letters:rM-rf =R M Market Risk PremiumE(ri )-rf =R i Asset i Risk Premium

[ ] [ ][ ] )(][

)(

f M ii

f M i f i

r r E R E

r r E r r E

=

+=


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ExampleExample :: Under the CAPM assumptions, determine AMADEUS shares expected return taking into account thatnext year expected market return is 11.5%, one-year t-bills offera return of 3.5%, and AMADEUS is 1.8.

1.1. IsIs thethe expectedexpected returnreturn onon sharesshares of of AMADEUS AMADEUS higherhigherthanthan thethe expectedexpected returnreturn of of thethe marketmarket ??

2.2. DetermineDetermine thethe expectedexpected returnreturn onon sharesshares of of Amadeus. Amadeus.

Answers Answers ::


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CAPM derivation : Assume a portfolio p with proportion a of wealth in asset

i and porportion (1-a) in the market portfolio.

The risk and return of portfolio p is:

Different portfolios are generated by changing the weighta. These portfolios are represented by the curve I-I .

[ ] [ ]i M p r aE r E a=r E + )1(][

[ ]2/1,2222 a)-2a(1a)-(1 M ii M r a p ++=


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For a=0.

The slope of I-I at M is:

Equating the slopes of I-I and of the CML at point M (as donepreviously):

][][][

0 M i

a

p r E r E a

r E =

=

[ ] [ ] M

M M i M i M M

a

p

a

r

2,,

22/12

0

2221)(

=+=

=

M

M M i

M i

a

p

p

r E r E

a

r a

r E

2,

0

][][)(

][

=

=

M

M M i

M i

M

f M r E r E r r E

2,

][][][

=


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Then:

This This isis thethe CAPMCAPM basicbasic expressionexpression ..

We can represent graphically the expected return required (inequilibrium) for holding a asset depending on its systematicrisk SML (SML ( SecuritySecurity MarketMarket LineLine ))

)][(][

)1)(][(][][

)][(][][ 22

,

f M i f i

i f M M i

M

M M i f M M i

r r E r r E

r r E r E r E

r r E r E r E

+=

+=

=

2,

M

M ii

=

Given that:


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The relationship between expected return and systematic risk isrepresented by the SML.SML.

CanCan anan assetasset bebe representedrepresented outsideoutsidethethe SML?SML?

Slope of SML = market risk premium

E[r i]

r f

E[r M]

SML

BiBM=1


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44-- The The BetaBeta

The The BETA BETA :As we have already seen:

The Beta measures an assets contribution to the risk of a well diversified portfolio (or market portfolio).

The Beta shows the sensitivity of an assets excess return tochanges in the market return.

Total RiskSpecific Risk Systematic Risk

= +Variance BETA


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It can be calculated by aregression of the assetsexcess return on themarket excess return.

Then,

R i-R f

R m-R f

ri

rMt it mt i r r ,,, ++=

2),(

mr

mi r r Cov

=


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ExampleExample : Determine the beta and the expected return of shares of TELEFON, given that the covariance between the returns of thisshares and the IGBM is 0.099, the standard deviation of returns onthe shares is 17%, and the standard deviation of returns on themarket index (IGBM) is 24%. In addition, te return on a one-year T-bill is 4.5%, and the expected market risk premium is 8%.

Answer Answer ::


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55-- The The portfolio portfolio BetaBeta

THE PORTFOLIO BETA: THE PORTFOLIO BETA:Given that the Beta measures non-diversifiable risk, the

portfolio beta is the portfolio-weighted average of thebetas of its individual securities.

2211

assets2With

ww p +=


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55-- The The portfolio portfolio BetaBetaExampleExample :: Suppose that an investment fund holds a portfolio with three risky assets, with the following characteristics:

Asset 1: = 0.05, and portfolio weight 20%.Asset 2: = 1.02, and portfolio weight 35%.The returns of asset 3 have a covariance with the returns of the market of 0.0399.

Considering that the standard deviation of returns of the market portfolio is19%, compute the of this portfolio.

Answer Answer ::


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Readings

Brealey, R.A. and Myers, S.C. (2003). Principles of Corporate Finance . McGraw Hill

Chapter 8

Surez Surez, Andrs S. (2005). Decisiones ptimas de inversin y financiacin en la empresa . Ediciones Piramide.

Chapters 32, 33 and 34.

Brigham E.F. and Daves, P. R. (2002). International Financial Mangement . South-Western.

Chapters 2 and 3.

Grinblatt, M. and Titman, S. (2002). Financial Markets and CorporateStrategy . McGraw Hill

Chapter 5.


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