Capital Asset Pricing Model CAPM I: 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)

Intuition Risky Asset i: Its price is such that: E(Return i ) = Risk-free rate of return + Risk premium specific to Asset i = R f + (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?

An example to motivate Expected ReturnStandard Deviation Asset I10.9%4.45% Asset j5.4%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 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 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.

Answers from CAPM 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) 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.

CAPM’s 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.

CAPM results 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) 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 ] The following equation, a.k.a., 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 referred to as the BETA of asset i Or E(R i ) = R f + [E(R M )-R f ] x β i

Pictorial Result of CAPM E(R i ) E(R M ) RfRf Security Market Line  [COV(R i, R M )/Var(R M )]   = 1.0 slope = [E(R M ) - R f ] = Eqm. Price of risk

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

CAPM in Details: What is an equilibrium? CONDITION 1: Consistent with Individual investor’s Eqm.: 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 NOTE: 2-Fund Separation must hold under the above assumptions.

CAPM in Details: What is an equilibrium? CONDITION 1: Individual investor’s equilibrium: Max U RfRf A Market Portfolio Q B Capital Market Line σpσp E(R p ) E(R M ) σMσM

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 (V i ) = P i xQ i Market portfolio has a value of ∑ i V i Market portfolio has N risky assets, each with a weight of w i Such that w i = V i / ∑ i V i for all i

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.

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 investor’s 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.

CAPM in Details: What is an equilibrium? Two-fund separation RfRf A Market Portfolio Q B Capital Market Line σpσp E(R p ) E(R M ) σMσM

The Role of Capital Market Efficient set U’’ U’ P Endowment Point E(r p ) σpσp

The Role of Capital Market Efficient set U’’’ U’’ U’ P Endowment Point E(r p ) σpσp M U-Max Point Capital Market Line RfRf

Derivation of CAPM Using equilibrium condition 3 w i = V i / ∑ i V i for all i market value of individual assets (asset i) w i = 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

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  i 2 + (1-a) 2  m 2 + 2a (1-a)  im ] 1/2 Since the market portfolio already contains asset i and, most importantly, the equilibrium value weight is w i 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.

Derivation of CAPM E(R p ) = aE(R i ) + (1-a)E(R m )  (R p ) = [ a 2  i 2 + (1-a) 2  m 2 + 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)

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)

Derivation of CAPM From previous page Rearranging Where E(return) = Risk-free rate of return + Risk premium specific to asset i E(R i ) = R f + (Market price of risk)x(quantity of risk of asset i) CAPM Equation

Pictorial Result of CAPM E(R i ) E(R M ) RfRf Security Market Line  [COV(R i, R M )/Var(R M )]   = 1.0 slope = [E(R M ) - R f ] = 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 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 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”s” of 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-spreaded. Think of other uses of CAPM as an exercise for yourself. Do some research on it to help yourself understand more.

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?