05+-+CAPM

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Capital Asset Pricing Model

Overview:

Centrepiece of modern financial economics

Most used formulation to price assets

Empirically shown, many many many times, that does not hold

Then why has it permeated all of finance?

Easy to use Simplistic to understand

Captures the essential issues in asset pricing in one neat formula

Serves as a first stab at pricing

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We have learnt from the consumption modelling underlying all ofasset pricing that there are a certain issues to consider

How do we capture the riskiness of the asset?

Use this riskiness to estimate an appropriate premium for takingon that risk

Add the risk free rate…

Get a fair price for the asset!

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Not that simple

Incredibly difficult to estimate the risk-adjustment

Not a single definitive answer to the question CAPM is just one answer

Research in asset pricing remains a hot topic in finance

This problem impedes a lot of other research in finance – dualhypothesis problem

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CAPM gives us a precise relationship between therisk of an asset and its expected return

Allows us to calculate a benchmark return forevaluating investments

Compare ‘fair’ return to what the market gives us

Allows us to price assets that are not yet traded Example: IPO

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We assume a very simplistic world Derive a reasonably realistic and comprehensive model

Ensure conformity of investor behaviour Except with regard to risk aversion and initial level of

wealth

It is the equilibrium model that underlies all modernfinancial theory.

Derived using principles of diversification withsimplified assumptions.

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Assumptions

The assumptions are as follows:

1. There are many investors

Each investor has a ‘small’ initial endowment

Investors are price takers

In essence the perfect competition assumptions we’ve

encountered before

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Assumptions

2. Investor behaviour is myopic

Investor only plans for one holding period

Ignore everything after

3. Investors only trade in publically traded financialassets

Rules out non-traded assets such as education

4. Investor may borrow or lend any amount at a fixedrisk free rate

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Assumptions

5. Investors pay no taxes and incur no transactioncosts

Taxes play an essential role in investment decisions

Only care about net returns

How will your income flow be affected?

Trades are costly – most arbitrage opportunities areunprofitable

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Assumptions

6. Investors are mean-variance optimisers

7. Investors have homogenous expectations Analyse securities in the same way Have the same economic view of the world

Identical probability distributions for returns

Same input list for the markowitz model

All investors generate the same efficient frontierand optimal risky portfolio

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Resulting Equilibrium Conditions

The implication of our simplistic world is as follows:

All invesors will choose to hold a portfolio of risky assets

that exactly duplicates the proportions of assets in themarket portfolio m

The portfolio m includes all traded assets

The market portfolio will be on the efficient frontier

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The market portfolio will be the tangency portfoliothat all investors choose as optimal

The CML is the best attainable CAL

All investors hold m as their optimal risky portfolio

This is the fundamental insight provided by theCAPM

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Risk premium on the market depends on the average riskaversion of all market participants.

Risk premium on an individual security is a function of itscovariance with the market.

The risk premium on the market portfolio will beproportional to its risk and the degree of risk aversion ofthe representative investor

m2

σ

0.01*σ A r)E(r m2

f m

= Systematic risk of our universe of assets

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Risk premium on individual stocks isproportional to the risk premium on themarket portfolio and the beta coefficient of

the security

What is beta?

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Why do all investors hold the market portfolio?

If we sum over all portfolios of all investors

Lending and borrowing cancel out

Result is that the aggregate risky portfolio is the market portfolio

m

Will equal the entire wealth of the economy

The CAPM implies that as each individual attempts to optimise

their portfolio selection they each arrive at the same portfolio

The weights in each portfolio equal the market portfolio

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Our assumptions ensure this result

Investors all use markowitz portfolio analysis

Apply it to the same universe of risky securities

For the same time horizon And use the same input list

Must arrive at the same optimal risky portfolio

As a result the optimal risky portfolio for all investorsis to hold a share of the market portfolio

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If a security is not included in the portfolio – disequilibrium

As all investors avoid the stock – price falls - becomes more andmore attractive

Eventually the stock is more attractive than other stocks and isincluded in the portfolio

Such a price adjustment guarantees that all stocks will be

included in the optimal risky portfolio

ALL assets therefore have to be included in the market portfolio

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Capital Market Line

E(r)

E(rM)

rf

M

CML

m

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Capital Market Line

The passive strategy is efficient

Market portfolio that all investors hold is based on a commoninput list

Incorporates all information

An investor can attain an efficient portfolio simply by holding themarket portfolio m

Different optimal risky portfolios formed by investment managersare as a result of a different input list

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Risk premium of the market portfolio

Before we simply stated the following result

The risk premium on the market portfolio is given by:

The equilibrium risk premium on the market portfolio (LHS)is proportional to the average degree of risk aversion of theinvestor population and the risk of the market

0.01*σ A r)E(r m2

f m

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So far we have established that as a result of the assumptionsimposed on the model, all investors will hold the same riskyportfolio

This portfolio will have the same weights as the market portfoliom

We now need to decide the optimal allocation between the riskyportfolio m and the risk free asset

How much to put at risk

Recall that each individual will allocate an optimal proportion y*to the risky asset –in this case portfolio m

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In the simplified CAPM environment, all investmentsin risk free asset involve borrowing or lending byinvestors at the risk free rate

Since each borrower’s position is offset by acorresponding lender (creditor)

Implies net borrowing and lending across allinvestors must be zero

Therefore the average position in the risky assetmust be 100%

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Therefore we have y-bar = 1

Setting y = 1 in our optimal position in the risky portfoliogives us our result

2

m

f m

0.01A σ

r)E(r y*

And setting y=1

0.01*σ A r)E(r m2

f m

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Expected Return on Individual Securities

CAPM is built on the insight that the appropriate risk premium willbe determined by the assets contribution to the risk of theinvestor’s overall portfolio

Portfolio risk is what governs the risk premiums that investorsdemand

Assume there is a single stock GM and we want to gauge theportfolio risk of GM

We measure GM’s contribution to the overall risk of the portfolioby its covariance with the market portfolio

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Show that GM’s contribution to the risk of the marketportfolio by its covariance with that portfolio GM’s contribution = WGMCov(rGM,rM)

If the covariance with the rest of the market isnegative Provides returns that move inversely with the market –

stabilises the return on the overall portfolio

If the covariance with the rest of the market ispositive GM’s returns amplify swings in the rest of the portfolio

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The contribution of GM to the risk premium of themarket portfolio is given by WGM[E(rGM)-rf]

Therefore the reward-to-risk ratio for investmentsin GM is given by: GM’s contribution to risk premium / GM’s contribution to

variance

)r,Cov(r)r[E(r

)r,Cov(rW)r[E(rW

mGM

f GM

mGMGM

f GMGM

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The market portfolio is the tangency portfolio

Its reward to risk ratio is given by:

Market Risk Premium/Market Variance

M

2

f M

σ

r)E(r

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This ratio is often called the market price of risk

Extra return that investors demand to bear portfolio risk

For components of the market portfolio e.g. GM, riskis the contribution to portfolio variance

For the portfolio itself, risk is the variance of the

market

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General principle of equilibrium in a market

All investments MUST offer the same reward to risk ratio

Forces of demand and supply

If GM’s ratio was better – increase proportion of GM stock anddecrease proportion of other stocks

Security prices would change such that the ratios were equalised

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Therefore the reward to risk ratio of GMshould be equal to the reward to risk ratio ofthe market portfolio

M

2

f M

σ

r)E(r

)r,Cov(r

)r[E(r

mGM

f GM

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To determine the risk premium of GM stock werearrange as follows:

]f r

M[E(r

2σ

)mr,GMCov(r)

f r

GM[E(r

M

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The ratio Cov/ σ2m measures the contribution of GM stock to

the variance of the market portfolio as a fraction of the totalvariance of the market portfolio

We call this ratio the beta of the stock-β

We can restate the relationship as follows:

]

f

r

M

[E(r

GM

β

f

r)

GM

E(r

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The assumptions we established earlier have animportant role to play

Each investor uses the same input list and henceall hold some proportion of the market portfolio

Therefore the beta of an asset with the market

portfolio will equal the beta of the asset with theindividuals own risky portfolio

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But we don´t all hold the market portfolio!

Does the CAPM actually have relevance?

Yes!

A well-diversified portfolio can shed itsunsystematic risk and we are only left withexposure to the market

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A well diversified portfolio will be so highly correlatedwith the market that the assets covarying with themarket will still be a useful measure

It can be shown that the expected return – betarelationship can hold for any combination of assets

Beta of a portfolio is simply a weighted average

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This relationship also applies to the market portfolio:

implying βm is one.

]r[E(rβr)E(r f MMf M

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This establishes 1 as the weighted averagevalue of beta across all assets

Hence betas greater than 1 are consideredaggressive Investments in high beta stocks entail above

average sensitivity to market swings

Betas less than 1 are considered defensivestocks

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Slope and Market Risk Premium

M = Market portfoliorf = Risk free rate

E(rM) - rf = Market risk premium

E(rM) - rf = Market price of risk

= Slope of the CML M

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Return and Risk For Individual Securities

The risk premium on individual securities is afunction of the individual security’s

contribution to the risk of the market portfolio.

An individual security’s risk premium is a

function of the covariance of returns with the

assets that make up the market portfolio.

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Security Market Line

The beta of a security is the appropriate measure of risk becausebeta is a measure of the risk that the security contributes to theoptimal risky portfolio

In our CAPM simplisiic world, we expect the risk premium on

individual assets to depend on the contribution of the asset to therisk of the portfolio

Hence we expect the required risk premium to be a function ofbeta

The CAPM confirms this intuition by stating that the risk premiumis proportional to both the beta and market risk premium

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Expected return-beta relationship can be depictedgraphically by the Security Market Line (SML)

Because the beta of the market is one, the slope ofthe SML is equal to the market risk premium

Use the E(rm)-beta of the market as the first point

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E(r)

E(rM)

rf

SML

bbM= 1.0

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SML Relationships

b = [COV(ri,rm)] / m2

Slope SML = E(rm) - rf

= market risk premium

SML = rf + b[E(rm) - rf]

Betam

= [Cov (ri,r

m)] /

m

2

= m2 / m

2 = 1

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Sample Calculations for SML

E(rm) - rf = .08 rf = .03

bx = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

by

= .6

e(ry) = .03 + .6(.08) = .078 or 7.8%

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Graph of Sample Calculations

E(r)

Rx=13%

SML

b1.0

Rm=11%

Ry=7.8%

3%

1.25

bx

.6

by

.08

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SML vs CML

CML plots the risk premiums of efficient portfolios as

a function of portfolio std deviation

This is appropriate as std dev is an appropriatemeasure of risk for efficiently diversified portfolios

that are candidates for the overall portfolio

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In contrast, the SML graphs INDIVIDUAL asset riskpremiums as a function of asset risk

The relevant measure of risk for assets held as partof an efficiently diversified portfolio IS NOT theasset’s standard deviation or variance

It is the contribution of the asset to the portfolio’svariance – beta!

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The SML provides a benchmark return for theevaluation of investment performance

Given an assets beta, we can read off itsrequired rate of return necessary tocompensate the investor for both risk as well

as the time value of money

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As a result, we can conclude that only fairly pricedassets will plot on the SML

Their expected returns are commensurate with their

risk

If a stock is perceived to be underpriced it will plotabove the SML Given their betas, their E(r) are greater than that dictated

by CAPM

Overpriced stocks plot below the SML

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The difference between the fair and actualexpected returns is called the stocks alpha

Security analysis is about the search foralpha

Investor would increase the weights ofsecurities with positive alphas

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Disequilibrium Example

E(r)

15%

SML

b1.0

Rm=11%

rf =3%

1.25

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Disequilibrium Example

Suppose a security with a b of 1.25 is offeringexpected return of 15%.

According to SML, it should be 13%.

Under-priced: offering too high of a rate of

return for its level of risk.

05+-+CAPM

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