Capital Asset Pricing Model CAPM I: The Theory

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Capital Asset Pricing ModelCAPM I The Theory
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Introduction

• Asset Pricing how assets are priced?
• Market Equilibrium concept
• Portfolio Theory ANY individual investors
  optimal selection of portfolio (partial
  equilibrium)
• CAPM equilibrium of ALL individual investors
  (and asset suppliers) (general equilibrium)

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Intuition

• Risky Asset i
• Its price is such that
• E(Returni) 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) The price of risk
• 2) The risk of Asset i?

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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 wouldnt asset j has higher expected
return as well? Possible Answers (1) The
equation, as intuitive as it is, is wrong. (2)
The equation is right, but the market prices of
risk are different for different
assets. (3) The equation is right, but the
quantity of risk of any risky asset is not
equal to the standard deviation of its
return.
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Answers from CAPM

• 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 market price of risk in equilibrium should be
  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 costlessly
  diversified away. No free lunch implies the
  market will NOT reward you for bearing
  unsystematic risk if there is NO cost of
  eliminating it through well diversification.
• (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.

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CAPM results
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)
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 The following equation, a.k.a., the
Capital Asset Pricing Model E(Ri) Rf
E(RM)-Rf x COV(Ri, RM)/Var(RM) Where
COV(Ri, RM)/Var(RM) is referred to as the BETA
of asset i Or E(Ri) Rf E(RM)-Rf x ßi
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Pictorial Result of CAPM
E(Ri)
Security Market Line
E(RM)
slope E(RM) - Rf Eqm. Price of risk
Rf
b COV(Ri, RM)/Var(RM)
bM 1.0
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CAPM

• 2 Sets of Assumptions
• 1 Perfect market
• Frictionless, and Perfect information
• No imperfections like tax, regulations,
  restrictions on 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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CAPM in Details What is an equilibrium?

• CONDITION 1 Consistent with Individual
  investors Eqm. Max U
• Assume
• 1 Market is frictionless
• gt borrowing rate lending rate
• gt 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
• gt they perceive identical distribution of
  expected returns on ALL assets
• gt thus, they all perceive the SAME linear
  efficient set (we called the line CAPITAL
  MARKET LINE
• gt the tangency point is the MARKET PORTFOLIO
• NOTE 2-Fund Separation must hold under the above
  assumptions.

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CAPM in Details What is an equilibrium?

• CONDITION 1 Individual investors equilibrium
  Max U

Capital Market Line
E(Rp)
B
Q
E(RM)
Market Portfolio
A
Rf
sM
sp
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CAPM in Details What is an equilibrium?

• 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 its outstanding
  shares.
• 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

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CAPM in Details What is an equilibrium?

• CONDITION 4 Aggregate borrowing Aggregate
  lending
• Risk-free rate is not exogenously given, but is
  determined by equating aggregate borrowing and
  aggregate lending.

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CAPM in Details What is an equilibrium?

• 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 investors efficient set of risky
  assets
• Analogy of Two-fund separation
• Fisher Separation Theorem in a world of
  certainty. Related the two separation theorems to
  help your understanding.

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CAPM in Details What is an equilibrium?

• Two-fund separation

Capital Market Line
E(Rp)
B
Q
E(RM)
Market Portfolio
A
Rf
sM
sp
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The Role of Capital Market
U
U
E(rp)
Efficient set
P
Endowment Point
sp
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The Role of Capital Market
U
U
U
E(rp)
Capital Market Line
U-Max Point
Efficient set
P
M
Endowment Point
Rf
sp
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Derivation of CAPM

• Using equilibrium condition 3
• wi 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 ) in asset i
• and (1-a) (in ) 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(Rp) aE(Ri) (1-a)E(Rm)
• ?(Rp) a2?i2 (1-a)2?m2 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 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.

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Derivation of CAPM

• E(Rp) aE(Ri) (1-a)E(Rm)
• ?(Rp) a2?i2 (1-a)2?m2 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

(evaluated at a 0)
(evaluated at a 0)
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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 to the slope of the
  capital market line, which is given as
• Therefore, setting the slope of the opportunity
  set equal to the slope of the capital market line
• rearranging,

(evaluated at a 0)
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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
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Pictorial Result of CAPM
E(Ri)
Security Market Line
E(RM)
slope E(RM) - Rf Eqm. Price of risk
Rf
b COV(Ri, RM)/Var(RM)
bM 1.0
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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 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 costlessly)
• e.g., interest rate, business cycle
•  
• Unsystematic Risk idiosyncratic shocks
  specific to asset i, (CAN be diversified away
  costlessly)
•   e.g., loss of key contract, death of CEO
• CAPM quantifies the systematic risk of any asset
  as its ß
• Expected return of any risky asset depends
  linearly on its exposure to the market
  (systematic) risk, measured by ß.
• Assets with a higher ß 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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How to use CAPM?

• For valuation of risky assets
• For estimating required rate of return of risky
  projects
• As you can see from stocks quotations, beta is a
  prominent measure everywhere. The usage of CAPM
  is wide-spread. Think of other uses of CAPM as an
  exercise for yourself. Do some research on it to
  help yourself understand more.

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Empirical Tests on CAPM

• In the next lecture, well 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?