Capital Asset Pricing Model pptx

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Capital Asset Pricing ModelBy:Amit Shankar Choudhary


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Introduction

Asset Pricing how assets arepriced?

Equilibrium concept Portfolio Theory ANY individual

investors optimal selection of portfolio ( partial equilibrium )

CAPM equilibrium of ALL individualinvestors (and asset suppliers)(general equilibrium )


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Our expectation

Risky asset i: Its price is such that:

E(return) = Risk-free rate of return + Risk premium specific toasset i

= R f + (Market price of risk)x(quantity of risk of asset i)

CAPM tells us 1) what is the price of risk?

2) what is the risk of asset i?


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An example to motivateExpected Return Standard Deviation

Asset i 10.9% 4.45%

Asset j 5.4% 7.25%

E(return) = Risk-free rate of return + Risk premium specific to asset i= R f + (Market price of risk)x(quantity of risk of asset i)

Question: According to the above equation, given that asset j has higher riskrelative to asset i, why wouldnt asset j has higher expected return as well?

Possible Answers: (1) the equation, as intuitive as it is, is completelywrong.

(2) the equation is right. But market price of risk isdifferent for different assets.

(3) the equation is right. But quantity of risk of anyrisky asset is not equal to the standard deviation of its

return.


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CAPMs AnswersE(return) = Risk-free rate of return + Risk premium specific to

asset i= R f + (Market price of risk)x(quantity of risk of asset

i)

The intuitive equation is right. The equilibrium price of risk is the

same across all marketable assets In the equation, the quantity of risk

of any asset, however, is only PART

of the total risk (s.d) of the asset.


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CAPMs Answers

Specifically: Total risk = systematic risk + unsystematic risk

CAPM says:(1)Unsystematic risk can be diversified away. Since

there is no free lunch, if there is something youbear but can be avoided by diversifying at NO cost, the market will not reward the holder of

unsystematic risk at all.(2)Systematic risk cannot be diversified awaywithout cost. In other words, investors need to becompensated by a certain risk premium forbearing systematic risk.


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CAPM resultsE(return) = Risk-free rate of return + Risk premium specific to asset i

= R f + (Market price of risk)x(quantity of risk of asset i)

Precisely:[1] Expected Return on asset i = E(R i )[2] Equilibrium Risk-free rate of return = R

f [3] Quantity of risk of asset i = COV(R i , R M )/Var(R M )[4] Market Price of risk = [E(R M )-R f ]

Thus, the equation known as the Capital Asset Pricing Model:

E(R i ) = R f + [E(R M )-R f ] x [COV(R i , R M )/Var(R M )]

Where [COV(R i, R M )/Var(R M )] is also known as BETA of asset I

Or

E(R i ) = R f + [E(R M )-R f ] x i


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Pictorial Result of CAPM

E(Ri )

E(R M)

R f

SecurityMarket

Line

=[COV(R i , R M )/Var(R M )] = 1.0

slope = [E(R M ) - R f ] = Eqm. Price of risk


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CAPM in Details:What is an equilibrium?

CONDITION 1: Individual investors equilibrium: Max U Assume: [1] Market is frictionless

=> borrowing rate = lending rate=> linear efficient set in the return-risk space

[2] Anyone can borrow or lend unlimited amount at risk-free rate

[3] All investors have homogenous beliefs=> they perceive identical distribution of expected

returns on ALL assets=> thus, they all perceive the SAME linear efficient set

(we called the line: CAPITAL MARKET LINE=> the tangency point is the MARKET PORTFOLIO


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CONDITION 1: Individual investors equilibrium: Max U

R f

A

Market Portfolio

Q

B

Capital Market Line

p

E(R p)

E(R M)

M


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CONDITION 2: Demand = Supply for ALL risky assets Remember expected return is a function of price. Market price of any asset is such that its expected return is

just enough to compensate its investors to rationally holdit.

CONDITION 3: Equilibrium weight of any risky assets The Market portfolio consists of all risky assets. Market value of any asset i (V i ) = P i xQ i Market portfolio has a value of iV i Market portfolio has N risky assets, each with a weight of w i

Such that

w i = V i / iV i for all i


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CONDITION 4: Aggregate borrowing = Aggregatelending

Risk-free rate is not exogenously given, but is determined

by equating aggregate borrowing and aggregate lending.


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Two-Fund Separation:

Given the assumptions of frictionless market, unlimitedlending and borrowing, homogenous beliefs, and if theabove 4 equilibrium conditions are satisfied, we then have

the 2-fund separation.

TWO-FUND SEPARATION:Each investor will have a utility-maximizing portfolio

that is a combination of the risk-free asset and a portfolio(or fund) of risky assets that is determined by the Capitalmarket line tangent to the investors efficient set of riskyassets

Analogy of Two-fund separationFisher Separation Theorem in a world of certainty


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Two-fund separation

R f

A

Market Portfolio

Q

B

Capital Market Line

p

E(R p)

E(R M)

M


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The Role of Capital Market

Efficient set

U U

P

Endowment Point

E(r p)

p


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The Role of Capital Market

Efficient set

U U U

P

Endowment Point

E(r p)

p

M

U-Max PointCapital Market Line

R f


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CAPM

2 sets of Assumptions:[1] Perfect market:

Frictionless, and perfect information No imperfections like tax, regulations,

restrictions to short selling All assets are publicly traded and perfectly

divisible Perfect competition everyone is a price-taker

[2] Investors: Same one-period horizon Rational, and maximize expected utility over a

mean-variance space Homogenous beliefs


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Derivation of CAPM

Using equilibrium condition 3w i = V i / iV i for all i

market value of individual assets (asset

i) w i =------------------------------------------------

market value of all assets (marketportfolio)

Consider the following portfolio:hold a% in asset iand (1-a%) in the market portfolio


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Derivation of CAPM

The expected return and standard deviation of such a portfolio can be written as:

E(R p ) = aE(R i) + (1-a)E(R m )

(R p ) = [ a 2 i2 + (1-a) 2 m 2 + 2a (1-a)im ] 1/2

Since the market portfolio already contains asset i and, most importantly, the equilibrium valueweight is w i

therefore, the percent a in the above equationsrepresent excess demands for a risky asset

We know from equilibrium condition 2 that inequilibrium, Demand = Supply for all asset.

Therefore, a = 0 has to be true in equilibrium.


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Derivation of CAPME(R p ) = aE(R i) + (1-a)E(R m )

(Rp ) = [ a 2 i2 + (1-a) 2 m 2 + 2a (1-a) im ] 1/2

Consider the change in the mean and standarddeviation with respect to the percentage change inthe portfolio invested in asset i

Since a = 0 is an equilibrium for D = S, we mustevaluate these partial derivatives at a = 0

) R E( - ) R E( =a

) R E( mi p

] 4a-2+2a+2-2a [ * ]a)-2a(1+ )a-(1+a [ 21

=a

) R( imim

2m

2m

2i

1/2-im

2m

22i

2 p

) R E( - ) R E( = a

) R E( mi

p

m

2mim p - =

a ) R(

(evaluated at a = 0)



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Derivation of CAPM the slope of the risk return trade-off evaluated at point M in marketequilibrium is

but we know that the slope of the opportunity set at point M must alsoequal the slope of the capital market line. The slope of the capital

market line is

Therefore, setting the slope of the opportunity set equal to the slope of the capital market line

rearranging,

m

2mim

mi

p

p

- ) R E( - ) R E( =

a )/ R( a )/ R E(

m f m R- ) R E(

m f m

m2mim

mi R- ) R E( = / )-(

) R E( - ) R E(

] R- ) R[E( + R= ) R E( f m2m

im f i



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Derivation of CAPM

From previous page Rearranging Where

E(return) = Risk-free rate of return + Risk premium specific toasset i

E(R i ) = R f + (Market price of risk)x(quantity of risk of

asset i)

CAPM Equation

) RVAR( ) R , RCOV( ==

m

mi2m

imi

i f m f i ] R- ) R[E( + R= ) R E(

] R- ) R[E( + R= ) R E( f m2m

im f i


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Pictorial Result of CAPM

E(Ri )

E(R M)

R f

Security Market

Line

=[COV(R i , R M )/Var(R M )] = 1.0

slope = [E(R M ) - R f ] = Eqm. Price of risk


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Properties of CAPM

In equilibrium, every asset must be priced so that its risk-adjusted required rate of return falls exactly on thesecurity market line.

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk a measure of how the asset co-varies

with the entire economy (cannot be diversified away)e.g., interest rate, business cycle

Unsystematic Risk idiosyncratic shocks specific toasset i, (can be diversified away)

e.g., loss of key contract, death of CEO

CAPM quantifies the systematic risk of any asset as its

Expected return of any risky asset depends linearly on itsexposure to the market (systematic) risk, measured by.

Assets with a higher require a higher risk-adjusted rate of return. In other words, in market equilibrium, investors

are only rewarded for bearing the market risk.


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Use of CAPM

For valuation of risky assets For estimating required rate of return

of risky projects


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Empirical Tests on CAPM

In the next lecture, well go over some of theempirical tests of CAPM.

Think about the following questions:

[1] What are the predictions of the CAPM?[2] Are they testable?[3] What is a regression?

[4] How to test hypothesis? What is t-test?

Capital Asset Pricing Model pptx

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