Capm

Capital Asset Pricing ModelPart 1: The Theory

Introduction

• Asset Pricing – how assets are priced?

• Equilibrium concept

• Portfolio Theory – ANY individual investor’s optimal selection of portfolio (partial equilibrium)

• CAPM – equilibrium of ALL individual investors (and asset suppliers) (general equilibrium)

Our expectation

• Risky asset i:• Its price is such that:

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

= Rf + (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?

An example to motivate

Expected 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 = Rf + (Market price of risk)x(quantity of risk of asset i)

Question: According to the above equation, given that asset j has higher risk relative to asset i, why wouldn’t asset j has higher expected return as well?

Possible Answers: (1) the equation, as intuitive as it is, is completely wrong.(2) the equation is right. But market price of risk is different for different assets.(3) the equation is right. But quantity of risk of any risky asset is not equal to the standard deviation of its return.

CAPM’s Answers

E(return) = Risk-free rate of return + Risk premium specific to asset i = Rf + (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.

CAPM’s 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 you bear 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 away without cost. In other words, investors need to be compensated by a certain risk premium for bearing systematic risk.

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

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

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

[3] Quantity of risk of asset i = COV(Ri, RM)/Var(RM)[4] Market Price of risk = [E(RM)-Rf]

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

E(Ri) = Rf + [E(RM)-Rf] x [COV(Ri, RM)/Var(RM)]

Where [COV(Ri, RM)/Var(RM)] is also known as BETA of asset I

Or

E(Ri) = Rf + [E(RM)-Rf] x βi

Pictorial Result of CAPM

E(Ri)

E(RM)

Rf

SecurityMarketLine

[COV(Ri, RM)/Var(RM)]= 1.0

slope = [E(RM) - Rf] = Eqm. Price of risk

CAPM in Details: What is an equilibrium?

CONDITION 1: Individual investor’s 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


CONDITION 1: Individual investor’s equilibrium: Max U

Rf

A

Market Portfolio

Q

B

Capital Market Line

σp

E(Rp)

E(RM)

σM


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 hold it.

CONDITION 3: Equilibrium weight of any risky assets

• The Market portfolio consists of all risky assets.• Market value of any asset i (Vi) = PixQi

• Market portfolio has a value of ∑iVi

• Market portfolio has N risky assets, each with a weight of wi

Such that

wi = Vi / ∑iVi for all i


CONDITION 4: Aggregate borrowing = Aggregate lending

• Risk-free rate is not exogenously given, but is determined by equating aggregate borrowing and aggregate lending.


Two-Fund Separation:

Given the assumptions of frictionless market, unlimited lending and borrowing, homogenous beliefs, and if the above 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 Capital market line tangent to the investor’s efficient set of risky assets

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


Two-fund separation

Rf

A

Market Portfolio

Q

B

Capital Market Line

σp

E(Rp)

E(RM)

σM

The Role of Capital Market

Efficient set

U’’ U’

P

Endowment Point

E(rp)

σp

The Role of Capital Market

Efficient set

U’’’ U’’ U’

P

Endowment Point

E(rp)

σp

M

U-Max PointCapital Market Line

Rf

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

Derivation of CAPM

• Using equilibrium condition 3wi = Vi / ∑iVi for all i

market value of individual assets (asset i)wi = ------------------------------------------------

market value of all assets (market portfolio) • Consider the following portfolio:

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

Derivation of CAPM

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

E(Rp) = aE(Ri) + (1-a)E(Rm)(Rp) = [ a2i

2 + (1-a)2m

2 + 2a (1-a) im ]

1/2

• Since the market portfolio already contains asset i and, most importantly, the equilibrium value weight is wi

• therefore, the percent a in the above equations represent excess demands for a risky asset

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

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

Derivation of CAPME(Rp) = aE(Ri) + (1-a)E(Rm)

(Rp) = [ a2i2

+ (1-a)2m2

+ 2a (1-a) im ] 1/2

• Consider the change in the mean and standard deviation with respect to the percentage change in the portfolio invested in asset i

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

)RE( - )RE( = a

)RE( mi

p

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

1=

a

)R( imim

2m

2m

2i

1/2-im

2m

22i

2p

)RE( - )RE( = a

)RE( mi

p

m

2mimp -

= a

)R(

(evaluated at a = 0)


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

market equilibrium is

• but we know that the slope of the opportunity set at point M must also equal 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

- )RE( - )RE(

= a)/R(

a)/RE(

m

fm R - )RE(

m

fm

m2mim

mi R - )RE( =

/ ) - (

)RE(-)RE(

]R - )R[E( + R = )RE( fm2m

imfi


Derivation of CAPM• From previous page

• Rearranging

• Where

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

CAPM Equation

)RVAR(

)R,RCOV( = =

m

mi

2m

imi

ifmfi ]R - )R[E( + R = )RE(

]R - )R[E( + R = )RE( fm2m

imfi

Pictorial Result of CAPM

E(Ri)

E(RM)

Rf

Security MarketLine

[COV(Ri, RM)/Var(RM)]= 1.0

slope = [E(RM) - Rf] = Eqm. Price of risk

Properties of CAPM• In equilibrium, every asset must be priced so that its risk-

adjusted required rate of return falls exactly on the security 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 to asset i, (can be diversified away)

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

• CAPM quantifies the systematic risk of any asset as its β

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

• Assets with a higher β require a higher risk-adjusted rate of return. In other words, in market equilibrium, investors are only rewarded for bearing the market risk.

Use of CAPM

• For valuation of risky assets

• For estimating required rate of return of risky projects

Empirical Tests on CAPM

• In the next lecture, we’ll go over some of the empirical 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?

Capm

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