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# PPT 7 - 7 Capital Asset Pricing MODEL Reading Luenberger...

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Unformatted text preview: 7. Capital Asset Pricing MODEL Reading: Luenberger Chapter 7 10/19/2011 1 Sharpe (1964), Lintner (1965) Nobel Prize: Sharpe (1990)   Based on Mean Variance Portfolio Optimization ‒  Rules for how to invest (normative economics)   Main Result Under Equilibrium ‒  How assets are priced (positive economics) ‒  Asset expected return is related to a measure of risk called Beta   Requires Strong Assumptions ‒  Unrealistic but necessary to obtain a very simple model 2 7.1 Market Equilibrium Assumptions   All investors use mean-variance optimal portfolios with ‒  Same single period investment horizon ‒  Same values of returns means and covariances   Unlimited buy or selling of a common risk-free asset   Security markets are perfect ‒  Assets are infinitely divisible ‒  zero taxes ‒  No transaction costs 3 Under the previous assumptions: The risky asset holdings of all investors: Portfolio T µP . T The capital market line rf σP The only difference between investors is their mix between risky assets and the risk-free asset, and this is determined by their differing risk aversions. 4 Under Equilibrium T is the Market Portfolio Claim: If all market participants hold the tangency portfolio T (or any fixed portfolio) of risky assets then under equilibrium they all hold the market portfolio of risky assets. It seems obvious: If every investor holds the same fund in the proportional weights determined by T (or any other optimization scheme as long as it is acknowledged by all to be optimal), then since the totality of all investments in these assets must be the market capitalization, the market portfolio must have the same weights as T. What is more subtle is the claim that such weights are capitalization weights: the larger the market cap a stock has, the higher the weight given to that stock in T. 5 N = number of securities in market V j = market capitalization of asset j, j = 1, L , N V = ∑ j =1V j N total market capitalization Market weights based on capitalization: wM , j = Vj V Investor k s weights: wk , j Want to show: wM , j = wk , j under the assumption wk , j ≡ w j , j = 1, L , N common investor portfolio weights 6 Wk = total investment of k -th investor , k = 1, L , K V j = ∑ k =1 wk , jWk total of all investments in j -th asset K = w j ⋅ ∑ k =1Wk K = w j ⋅W , W = total investment of all investors Under equilibrium and so W =V V j w j ! " k =1W k = = =wj . V W K wM , j ! 7 Capital Market Line The risky asset holdings of all investors: Portfolio T µP . T The capital market line rf σP On the capital market line, assets obey: µ M − rf µ − rf = σ σM 8 Capital Market Line Example 7.2 (An oil venture): The price of a share of this oil venture is \$875. It is expected to yield \$1000 after 1 year, with standard deviation of 40%. The risk-free rate is 10%. The expected market return is 17% with standard deviation of 12%. Expected return of this venture is: µ = (1000 − 875) / 875 = 14% If it were on the capital market line it should have returned: 0.17 − 0.10 µ = 0.10 + Not efficient. 0.12 0.40 = 33% 9 7.2 The CAPM Formula: Relation of expected return to risk for individual stocks ri rM return for the i -th asset, µi = E ( ri ) return for the market, µ M = E ( rM ) µi − rf = β i ⋅ ( µM − rf ) where cov(ri , rM ) βi = var(rM ) In terms of excess returns: µi ,e = β i ⋅ µM ,e 10 CAPM in vector notation: µ M ,e µ e = cov(r, rM ) ⋅ 2 = β ⋅ µM ,e σM where β= cov(r, rM ) σ 2 M = Ωw M 2 σM 11 Capital Asset Pricing Model:   The expected excess return of an asset is proportional to the excess return of the market portfolio, and the proportionality constant is beta.   The expected excess return of an asset is proportional to its covariance with the market. 12 Proof of the CAPM A proof is provided in Section 7.3 of Luenberger. Consider a portfolio consisting of a portion α invested in asset i, and (1-α) invested in the market portfolio M. The expected return is µ = !µi + (1 ! ! )µ M The standard deviation is 2 ! = [" 2! i2 + 2" (1 ! " )! i, M + (1 ! " )2 ! M ]1/2 13 Proof of the CAPM 2 2 dµ d" !" i + (1 ! 2! )" i, M + (! ! 1)" M = µi ! µ M = d! d! "M d µ d µ / d! (µi ! µ M )" M = = 2 d" d" / d! !" i2 + (1 ! 2! )" i, M + (! ! 1)" M dµ d" (µi ! µ M )" M µ M ! rf = = 2 " i, M ! " M "M ! =0 since ! = 0 corresponds to the slope of the capital market line. Solving for µi : µ M ! rf µi = rf + ( )" i, M = rf + #i (µ M ! rf ) 2 "M 14 Alternative Proof of the CAPM An alternative proof follows from the derivation of the efficient frontier with a cash mix in Lecture 6. Recall that the Lagrangian for that derivation is: 1 L(w ) = w′Σw − γ ⎡(1 − w′1) rf + w ′µ − µ P ⎤ ⎣ ⎦ 2 and setting the derivative equal to zero gives Σw − γ ⎡µ − 1rf ⎤ = Σw − γ µ e = 0 ⎣ ⎦ w = γΣ −1µe 15 This gave the following optimal weight vector on the linear efficient frontier w = γ ⋅ Σ −1µ e At the tangency portfolio w ' 1 → 1 or 1 ' w → 1 γ = (1′Σ µ e ) −1 −1 Under the assumptions for the CAPM the tangency portfolio is the market portfolio so we have the market weights vector Σ −1µ e wT = = wM −1 1′Σ µ e 16 Thus µ e = (1′Σ −1µ e ) ⋅ Σw M Noting that ( Σw M )i = ∑ j =1 cov(ri , rj ) ⋅ wM , j N = cov(ri , ∑ j =1 wM , j rj ) N = cov(ri , rM ) ′Σ −1µ e ) ⋅ cov(ri , rM ) µe,i = (1 Calculate (1′Σ −1µ e ) next. 17 The market portfolio has mean excess return and variance: µ M ,e 2 σM µ′ Σ −1µ e = E (rM − rf ) = E (re′w M ) = µ′ w M = e −1 e 1′Σ µ e µ′ Σ −1ΣΣ −1µ e µ′ Σ −1µ e = var(re′w M ) = e −1 2 = e −1 2 (1′Σ µ e ) (1′Σ µ e ) This give µ M ,e 1′Σ µ e = 2 σM −1 µ M ,e and so µe ,i = cov( ri , rM ) ⋅ 2 σM = cov(ri , rM ) σ 2 M ⋅ µ M ,e = β i ⋅ µ M ,e (CAPM) 18 Two Ways to View CAPM View 1: E (ri ) − rf as a (straight line) function of E (rM ) − rf with slope βi . This is the usual simple linear regression model view View 2: E (ri ) − rf as a (straight line) function of βi with slope The resulting straight-line plot is called the security E (rM ) − rf market line. 19 . Market risk premium (assume positive) µi = rf + β i ⋅ ( µM − rf ) βi = cov( ri , rM ) σ 2 M Risk premium Example: β i = 1.2, µM = 6%, rf = 3% (annual) µi = 3% + 1.2 ⋅ ( 6% − 3% ) = 6.6% β i = 1 ⇒ µi = µM β i < 1 ⇒ µi < µM β i > 1 ⇒ µi > µM β i = ρi , M σi σM This is good but it comes at the price of increased risk! 20 Portfolio Beta n E (rP ) = ∑ wi ⋅ E (ri ) Portfolio expected returns: i =1 Use the CAPM for each asset to get ( E(ri ) − rf = βi ⋅ E(rM ) − rf E(rP ) − rf = β P ( E(rM ) − rf ) where ) n β P = ∑ wi ⋅ βi i =1 Note linearity: The beta of a sum of stocks is the same as the weighted sum of the betas of individual stocks. n β P = ∑ wi ⋅ i =1 cov(ri , rM ) σ 2 M = ⎛n ⎞ cov ⎜ ∑ wi ⋅ ri , rM ⎟ ⎝ i =1 ⎠ σ 2 M = cov(rP , rM ) 2 σM 21 CAPM as a Pricing Model   Suppose an asset is purchased at price P and later (one year later) sold at price Q. P is known and Q is random.   (Annual) Return: r = Q − P P   Expected return: µ = E[r ] = E[Q] − P = rf + β ( µ M − rf ) P   Price according to CAPM (efficient): E[Q] P= 1 + rf + β ( µ M − rf ) 22 Same example for the oil venture   The oil venture has an expected payoff of \$1000 a share but with a high standard deviation of 40% because of the uncertainty associated with whether or not there is oil at the site, and with the future price of oil. The beta is relatively low at 60% because the uncertainty related to exploration is not correlated with the market. CAPM price: \$1000 P= = \$876 1.1 + 0.6(0.17 − 0.10) 23 Certainty equivalent pricing formula:   Price according to CAPM (efficient): E[Q] P= 1 + rf + β ( µ M − rf ) It can be rearranged into the form (LU p189): cov(Q, rM )(rM − rf ) 1 P= {E[Q] − }. 2 1 + rf σM {E[Q] − cov(Q, rM )(rM − rf ) σ 2 M } is the certainty equivalent of Q 24 7.3 General Portfolio Betas and Risk For any portfolio B, not necessarily the market portfolio, define the beta of the i-th stock with respect to the portfolio B as βi = In vector notation: cov(ri , rB ) β= σ 2 B , i = 1, 2,L , n cov(r, rB ) 2 σB MAIN APPLICATION: Active Portfolio Management where B will be a benchmark portfolio. 25 Covariance Reflects Asset Risk Contribution Just as in the case of the market portfolio we have for any portfolio B , with cov(r, rB ) = Σw 2 σ B = w′Σw = ∑ wi cov(ri , rB ) = w′ cov(r, rB ) i =σ 2 B ∑ wi ⋅ i !i cov(ri , rB ) 2 σB 2 = σ B ∑ wi ⋅ βi i measures the contribution of asset i to the overall 2 σB portfolio s variance Also: ∑ wi ⋅ βi = w′β = 1 if beta is defined with respect to B. i 26 Systematic vs nonsystematic risk Let the random rate of return of asset i be ( ε is whatever it needs to be to make the formula true): ri = rf + !i (rM ! rf ) + "i Expected return: µi = rf + !i (µ M ! rf ) + E["i ] E[!i ] = 0 CAPM says: (It assumes only that the market M is efficient) 27 Systematic vs nonsystematic risk Variance: 2 ! i2 = var("i (rM ! rf ) + #i ) = "i2! M + var(#i ). if !i is uncorrelated with the market. Risk= systematic risk + nonsystematic risk 2 Systematic risk= !i2" M The smallest risk is the asset on the capital market line with variance=systematic risk Nonsystematic risk can be reduced by diversification 28 References Book Treatments of CAPM Connor, G., Goldberg, L. and Korajczyk, R. (2010). Portfolio Risk Management, Oxford University Press. Grinold, R. C. and Kahn, R. N. (2000), Active Portfolio Management, Chapter 2. Luenberger, D. G. (1987). Investment Science, Oxford University Press. Pennachi, G. (2008). The Theory of Asset Pricing, Pearson Addison-Wesley. CAPM Papers W. F. Sharpe (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk , Jour. of finance, pp. 425-442. L. Lintner (1965). Security Prices, risk and Maximal gains from Diversification , Jour. of Finance, pp. 587-615. E. F. Fama (1968). Risk, Return and Equilibrium: Some Clarifying Comments , Jour. of Finance, pp. 29-40. 29 ...
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