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Return and Risk: The Capital Asset Pricing Model (Text reference: Chapter 10) Topics general notation single security statistics covariance and correlation return and risk for a portfolio diversification efficient set with two assets diversification with many assets efficient set with many assets the capital market line the capital asset pricing model AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 1 General Notation E(R j ) = expected return on a security/portfolio j σ 2j = variance of some security/portfolio j σAB = covariance between two variables A and B ρAB = correlation between two variables A and B Ω = the number of possible future states of the economy ωi = a possible future state of the economy, i ∈ {1, . . . , Ω} pi = probability of occurrence of state ωi , i ∈ {1, . . . , Ω} Ri, j = return of security j in state ωi , i ∈ {1, . . . , Ω} Rt, j = return of security j in period t, t ∈ {1, . . . , T } β j = beta of some security/portfolio j AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 2 Single Security Statistics expected return historical sample: R̄ j = 1 T T Rt, j ∑t=1 population: E(R j ) = ∑Ω i=1 pi × Ri, j example: ωi pi RA - auto stock RB - gold stock recession 0.25 20% normal 0.50 −8% 5% 3% boom 0.25 18% −20% calculate the expected return for each stock: AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 3 Cont’d variance and standard deviation T 1 (Rt, j − R¯ j )2 historical sample: σ 2j = T −1 ∑t=1 2 population: σ 2j = ∑Ω i=1 pi × (Ri, j − E(R j )) standard deviation: s.d.(R j ) = σ j a measure of risk: a probability weighted average of squared deviations of a security’s return from its expected return calculate the variances and standard deviations for A and B AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 4 Covariance and Correlation variance (s.d.) measures variability of a single variable (e.g. stock) correlation and covariance measure the statistical relationship between two variables (e.g. 2 stocks) covariance (σAB ) - direction of the relationship correlation (ρAB ) - strength and direction of relationship (−1 ≤ ρAB ≤ 1) examples: negative covariance: interest rates and bond prices, housing starts and interest rates, exchange rates and exports, etc. positive covariance: profits and stock prices, exchange rates and imports, dividends and stock prices, etc. AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 5 Cont’d positive correlation/covariance two variables X and Y tend to move together when X is above (below) its mean, Y tends to be above (below) its mean Perfect Positive Correlation 120 Perfect Positive Correlation 90 110 85 100 80 90 75 Y Value X Y 80 70 70 65 60 0 0.1 0.2 0.3 0.4 0.5 Time 0.6 0.7 0.8 0.9 60 90 1 Correlation = 0.85 120 95 100 105 X 110 115 120 110 115 120 Correlation = 0.85 90 85 100 80 90 75 Y Value X Y 110 80 70 70 65 60 0 0.1 0.2 0.3 0.4 0.5 Time 0.6 AFM 271 - Return and Risk: The Capital Asset Pricing Model 0.7 0.8 0.9 1 60 90 95 100 105 X Slide 6 Cont’d negative correlation/covariance two variables X and Y tend to move in opposite directions when X is above (below) its mean, Y tends to be below (above) its mean Perfect Negative Correlation 120 Perfect Negative Correlation 90 110 85 100 80 90 75 Y Value X Y 80 70 70 65 60 0 0.1 0.2 0.3 0.4 0.5 Time 0.6 0.7 0.8 0.9 60 90 1 Correlation = −0.85 120 95 100 105 X 110 115 120 110 115 120 Correlation = −0.85 90 85 100 80 90 75 Y Value X Y 110 80 70 70 65 60 0 0.1 0.2 0.3 0.4 0.5 Time 0.6 0.7 0.8 0.9 1 60 90 95 100 105 X AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 7 Cont’d zero correlation/covariance two variables X and Y are not (linearly) related Zero Correlation 120 Zero Correlation 90 110 85 100 80 90 75 Y Value X Y 80 70 70 65 60 0 0.1 0.2 0.3 0.4 0.5 Time 0.6 AFM 271 - Return and Risk: The Capital Asset Pricing Model 0.7 0.8 0.9 1 60 90 95 100 105 X 110 115 120 Slide 8 Cont’d covariance formulas historical sample: 1 T σAB = cov(RA , RB ) = ∑ [(Rt,A − R̄A ) × (Rt,B − R̄B)] T − 1 t=1 population: Ω σAB = cov(RA , RB ) = ∑ [pi × (Ri,A − E(RA )) × (Ri,B − E(RB ))] i=1 calculate the covariance between stock A and stock B: AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 9 Cont’d correlation formula historical sample/population: ρAB = corr(RA , RB ) = σAB cov(RA , RB ) = σA × σ B σA × σ B calculate the correlation between stock A and stock B: graphically: RA Ri, j 20% 20% 10% 10% 0% R N B state −10% −20%−10% −10% −20% −20% AFM 271 - Return and Risk: The Capital Asset Pricing Model 10% 20% RB Slide 10 Cont’d correlation and covariance - further observations σAB = σBA ρAB = ρBA even if two variables are actually uncorrelated, a historical sample will not yield a zero covariance due to measurement errors, sampling error, etc. covariance σAB can take any value, and is in squared deviation units indicates direction of (linear) relationship correlation coefficient ρAB can take on values between -1 and 1 (inclusive) indicates direction and strength of (linear) relationship AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 11 Return and Risk for a Portfolio let Y, Z be random variables, and a, b be real numbers, recall E(aY + bZ) = aE(Y ) + bE(Z) var(aY + bZ) = a2 var(Y ) + b2 var(Z) + 2abcov(Y, Z) the expected return of a portfolio P is a weighted average of the expected returns of the individual securities: E(RP ) = N ∑ X j E(R j ) j=1 where N is the number of securities in P, X j is the percentage of funds invested in security j (∑Nj=1 X j = 1), and E(R j ) is the expected return of security j e.g. find the expected return on a portfolio consisting of 75% in stock A and 25% in stock B: AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 12 Cont’d to see why the E(RP ) formula holds, consider the N = 2 case and let ni be the number of shares held of stock i (i = 1, 2) letting Si (t) be the price of stock i at time t, the portfolio value at time t = 0 is V (0) = n1 S1 (0) + n2 S2 (0) note that X1 = n1 S1 (0)/V (0), X2 = n2 S2 (0)/V (0) since Si (1) = Si (0)(1 + Ri ), we have V (1) = n1 S1 (1) + n2 S2 (1) = n1 S1 (0)(1 + R1 ) + n2 S2 (0)(1 + R2 ) ⇒ V (1) n1 S1 (0)(1 + R1 ) + n2 S2 (0)(1 + R2 ) − 1 = Rp = −1 V (0) n1 S1 (0) + n2 S2 (0) = X1 (1 + R1 ) + X2 (1 + R2 ) − 1 = X1 R1 + X2 R2 ⇒ E(R p ) = X1 E(R1 ) + X2 E(R2 ) AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 13 Cont’d variance of a portfolio P: σP2 = ∑Nj=1 ∑Nk=1 X j Xk σ jk where j and k each represent a security, σ jk = cov(R j , Rk ), and σ j j = σ 2j = var(R j ) for N = 2 σP2 2 = ∑ ∑ X j Xk σ jk j=1 k=1 2 = 2 ∑ j=1 X j X1 σ j1 + X j X2 σ j2 = X1 X1 σ11 + X1 X2 σ12 + X2 X1 σ21 + X2 X2 σ22 = X12 σ12 + X22 σ22 + 2X1 X2 σ12 = X12 σ12 + X22 σ22 + 2X1 X2 σ1 σ2 ρ12 AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 14 Cont’d s.d. of a portfolio P: σP = √ varP e.g. find the variance and standard deviation of a portfolio 75% in stock A and 25% in stock B observations: AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 15 Diversification combining stocks into a portfolio reduces risk: how does this happen? claim: as long as ρAB < 1, we get diversification (i.e. the s.d. of the portfolio of 2 securities is less than the weighted average of the s.d.’s of the individual securities). Proof: σP2 = X12 σ12 + X22 σ22 + 2X1 X2 σ1 σ2 ρ12 < X12 σ12 + X22 σ22 + 2X1 X2 σ1 σ2 (if ρ12 < 1) = (X1 σ1 + X2 σ2 )2 ⇒ σP < X1 σ1 + X2 σ2 thus portfolio risk is not a weighted average of the risks of the stocks in the portfolio; we get diversification whenever correlation is less than +1 AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 16 Efficient Set With Two Assets suppose E(RA ) = 0.20, E(RB ) = 0.15, σA = 0.3098, σB = 0.0775, ρAB = −0.5 some possible risk/return combinations: XA 0.0 0.2 0.4 0.6 0.8 1.0 E(RP ) 0.15 0.16 0.17 0.18 0.19 0.20 σP 0.0775 0.0620 0.1084 0.1725 0.2405 0.3098 Feasible set with −0.5 correlation 0.22 0.21 0.2 E(Rp) 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0 0.05 0.1 0.15 s.d. 0.2 0.25 0.3 0.35 AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 17 Cont’d opportunity set: (a.k.a. feasible set) set of all attainable portfolios (attained via various combinations of two stocks) domination: P1 dominates P2 if E(RP1 ) = E(RP2 ) and σP1 < σP2 , or σP1 = σP2 and E(RP1 ) > E(RP2 ) efficient set: set of attainable portfolios which result in maximum expected return for a given s.d. (or alternatively, minimum s.d. for a given expected return) note that the efficient set does not contain any portfolios which are dominated minimum variance portfolio: portfolio having the lowest s.d. of all the portfolios in the feasible set AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 18 Cont’d finding the minimum variance portfolio: σP2 = XA2 σA2 + (1 − XA )2 σB2 + 2XA (1 − XA )σAB this is a differentiable concave function, so a minimum exists. To find it, differentiate w.r.t. XA and set the result to zero: so d σP2 = 2XA σA2 + 2(1 − XA )(−1)σB2 + 2(1 − 2XA )σAB dXA = 2 XA σA2 − (1 − XA )σB2 + (1 − 2XA )σAB = 2 XA (σA2 + σB2 − 2σAB ) + (σAB − σB2 ) σ 2 − σAB d σP2 = 0 ⇒ XA∗ = 2 B 2 dXA σA + σB − 2σAB AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 19 Cont’d what happens for different values of ρAB ? Portfolio Standard Deviation σP XA E(RP ) ρAB = 1.0 ρAB = 0.5 ρAB = 0.0 0.0 15% 0.0775 0.0775 0.0775 0.2 16% 0.1240 0.1074 0.4 17% 0.1704 0.6 18% 0.8 19% 1.0 20% ρAB = −0.5 ρAB = −1.0 0.0877 0.0620 0.0000 0.1526 0.1324 0.1084 0.0774 0.2169 0.2032 0.1884 0.1725 0.1549 0.2633 0.2559 0.2483 0.2405 0.2323 0.3098 0.3098 0.3098 0.3098 0.3098 0.0775 0.0775 Feasible set with varying correlations 0.21 0.2 0.19 E(Rp) 0.18 0.17 0.16 0.15 0 AFM 271 - Return and Risk: The Capital Asset Pricing Model 0.05 0.1 0.15 s.d. 0.2 0.25 0.3 Slide 20 Cont’d observations: lower ρAB : more bend in the curve lower s.d. (see table) greater diversification if ρAB = −1, we can find a risk free portfolio (i.e. having σP = 0) recall that this is extremely unlikely with two stocks, but can happen in a portfolio consisting of a stock and a derivative (e.g. stock and a put on that stock) over short periods in reality, only one curve is possible, since ρAB is unique; other curves in our graph are hypothetical AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 21 Diversification With Many Assets the risk an asset adds to a portfolio must be measured with reference to its relationship to other securities in the portfolio the variance of the return on a portfolio with many securities is more dependent on the covariances between the individual securities than on the variances of the individual securities proof: AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 22 Cont’d proof cont’d: AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 23 Cont’d σP2 = total risk = diversifiable risk + undiversifiable risk σP2 N about 30 stocks required for optimal diversification diversifiable risk: risk relating to uncorrelated events that gets eliminated when we hold many securities undiversifiable risk: risk relating to correlated events that is not eliminated by owning many securities AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 24 Efficient Set With Many Assets feasible set becomes an area efficient frontier is still a curve E(RP ) σP how do we find the efficient frontier? AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 25 Summarizing to Here investors are risk-averse; try to increase expected return and reduce risk of their portfolios investors can reduce risk by choosing stocks which are not perfectly correlated the risk added by a security to a portfolio has to be measured with reference to its relationship to other securities in the portfolio (variances and covariances determine risk) a rational risk-averse investor chooses investments that are efficient the optimal investment lies on the efficient frontier investors are not compensated for bearing diversifiable risk, and only undiversifiable risk is priced in the market AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 26 The Capital Market Line introduce risk-free borrowing/lending asset f with return R f : σ f = 0 by definition σA f = 0 by definition for any risky asset A portfolio combinations of risky portfolio B and risk free asset f : E(RP ) = (1 − XB )R f + XB E(RB ) 1/2 = XB σB σP = (1 − XB )2 σ 2f + XB2 σB2 + 2(1 − XB )XB σ f σB ρB f σP ⇒ XB = σ B σP σP R f + E(RB ) ⇒ E(RP ) = 1 − σB σB E(RB ) − R f = Rf + · σP σB AFM 271 - Return and Risk: The Capital Asset Pricing Model E(RP ) Slide 27 Cont’d σP choose point of tangency A to get capital market line (CML) separation principle: 1. determine point A, independent of personal preferences; and 2. based on degree of personal risk aversion, pick a point on CML as your investment point AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 28 The Capital Asset Pricing Model so far we have considered decisions of one investor now consider many investors: homogeneous expectations assumption: all individuals have same expectations regarding returns, variances, and covariances under homogeneous expectations, all investors choose the same portfolio of risky assets represented by A in equilibrium, A must be the market portfolio - a value-weighted portfolio of all existing securities (often proxied by indexes (e.g. S&P/TSX)) one implication: index investing (pick a broad stock index such as S&P/TSX, then divide your investment between this index and the risk free asset) AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 29 Cont’d β measures the sensitivity of the change in return of an individual security to the change in return of the market portfolio M: cov(R j , RM ) βj = 2 σM sign of β depends on sign of covariance/correlation example: βC = 2 E(RC ) E(RM ) AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 30 Cont’d β can be estimated by linear regression: Rt, j = α j + β j × Rt,M + εt, j , where α j = intercept, β j = security β , and εt, j = regression error term characteristics of β βM = 1 βR f = 0 β j and E(R j ) have a positive, linear relationship β j > 1 ⇒ unusually sensitive to market movements β j < 1 ⇒ unusually insensitive to market movements AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 31 Aside: Linear Regression linear regression is a widely used tool in statistical analysis consider a simple case with a dependent variable y and a single independent variable x: y x AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 32 Cont’d the simple linear regression model is yt = α + β xt + εt basic idea is to minimize the variance of the error term the “best fit” line is given by α̂ = ȳ − β̂ x̄ β̂ = cov(x, y) var(x) simple e.g.: let x be monthly returns on TSE 300 and y be monthly returns on RBC (last half of 2002): y 2.53 4.77 -5.89 4.00 7.61 -1.20 x -7.45 0.22 -6.29 1.21 5.28 0.91 AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 33 Cont’d we can calculate: x̄ = −1.02 cov(x, y) = 15.05 ȳ = 1.97 var(x) = 23.81 α̂ = 2.61 β̂ = 0.63 y 2.53 4.77 -5.89 4.00 7.61 -1.20 x -7.45 0.22 -6.29 1.21 5.28 0.91 ŷ -2.09 2.75 -1.36 3.38 5.95 3.19 ε 4.62 2.02 -4.53 0.62 1.66 -4.39 since var(y) = 23.18 = β̂ 2 var(x) + var(ε ) = 9.51 + 13.67, the regression “explains” 9.51/23.18 = 41% of the variation in y the implication here is that of RBC’s total risk, 41% is market risk and 59% is unique risk AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 34 Cont’d RRBC 8 6 4 2 -8 -6 -4 -2 2 4 6 8 RM -2 -4 -6 -8 AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 35 Back to CAPM there is positive linear relationship between risk and expected return E(R j ) = R f + β j × E(RM ) − R f basic idea: investors expect a reward for waiting (R f ) and worrying (risk premium) CAPM conclusions: expected return on a security depends on security’s risk relative to the risk of the market portfolio, i.e. undiversifiable risk β is the only reason that expected returns differ between securities AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 36 Cont’d CAPM applies to portfolios as well as to individual securities for portfolios: E(RP ) = R f + βP × E(RM ) − R f βP = N ∑ Xj × βj j=1 for well-diversified portfolios: 2 var(RP ) = βP2 × σM s.d.(RP ) = σP = βP × σM AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 37 Cont’d the security market line (SML): E(R j ) E(RM ) Rf 0 1 βj there is a positive linear relationship between E(R j ) and β j : AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 38 Cont’d in equilibrium, all securities must line on the SML, e.g.: comparison of capital market line and security market line: CML - traces efficient set of portfolios formed from risky assets and riskless asset SML - describes the return-β relationship - relates return to total risk - relates expected return to systematic/undiversifiable risk - holds only for efficient portfolios - holds for all individual securities and all possible portfolios AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 39 Cont’d overall summary: the stock market is risky and investors want a reward for risk a measure of risk for a single security is σ or σ 2 in a portfolio, do not look at the risk of a security in isolation risk consists of diversifiable and undiversifiable risk only undiversifiable/systematic risk is rewarded a security’s contribution to the total risk of a portfolio is measured by β , which represents sensitivity of the security to market changes (i.e. the systematic risk) CAPM/SML - positive linear relationship between expected returns and systematic risk in equilibrium, all stocks must lie on the SML CAPM is the best known model of risk and return AFM 271 - Return and Risk: The Capital Asset Pricing Model Slide 40