Investment Analysis and Portfolio Management 23 November 2018 Investment Analysis and Portfolio Management The Capital Asset Pricing Model (CAPM) Review of Portfolio Theory The Capital Asset Pricing Model Implementing the CAPM F-F 3 Factor Model 23.11.2018 Investments and Portfolio Mgmt, ao

Capital Asset Pricing Model (CAPM) Equilibrium model that underlies all modern financial theory The model predicts the relationship between the risk and expected returns on risky assets in a state of equilibrium. Derived using principles of diversification, but with other simplifying assumptions Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development This was Nobel Prize winning work first proposed by William Sharpe. This model captures all of the risk/return tradeoff we have been discussing up until now. Derived with very strong limiting assumptions (discuss on next slides)

Underlying Assumptions Individual investors are price takers Single-period investment horizon Investments are limited to publicly traded financial assets No taxes and no transaction costs Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations Think of a world where individuals are all very similar except their initial wealth and their level of risk aversion. ( I was born poor and chicken) “Price Takers” means that individuals do not effect prices. (Big assumption! An individual’s behavior does not effect price.)

CAPM, equilibrium state CAPM states that in an equilibrium state (and given the previous assumptions) all investors should hold the same risky portfolio M, This tangency portfolio M should be the market portfolio, and Hence, the market portfolio (the tangency portfolio) has the highest Sharpe Ratio The result will be that all investors will find that the Market portfolio is on the efficient frontier. In fact, it will be the Tangent portfolio. So all investors will choose to hold a portion of their wealth in the risk free and a portion in the risky Market portfolio.

Equilibrium Conditions In the equilibrium state demand is equal to supply, all publicly traded assets are included in the optimal risky portfolio, In the market portfolio, the proportion of each security is its market value as a percentage of total market value Weight of asset ‘i’ in the market portfolio: Pi = price of one share of risky asset i ni = number of shares outstanding for risky security i

Market Portfolio 23.11.2018

The Capital Market Line (CML) From the above analysis, the CAPM predicts that the Market Portfolio is the optimal risky portfolio. The CAL that connects the riskless security and the market portfolio is called the Capital Market Line. Then, investors should decide how to allocate their wealth between: The market portfolio, and A riskless security 23.11.2018

Capital Market Line – combines rf and ‘M’ M = The value weighted “Market” Portfolio of all risky assets. E(r) CML The Capital Market Line should look very familiar. It is basically the Capital allocation line with the Market portfolio, M, as the tangent portfolio. All investors should now choose to invest somewhere along the Capital Market Line. Remember, any other portfolio on the efficient frontier will be dominated by the tangent portfolio or any other point on the Capital market line. C E(rM) M C Efficient Frontier rf s sm

Expected return of an efficient portfolio Now, an investor’s ultimate portfolio is a combination of the market portfolio and T-Bills Expected return on such an efficient portfolio is: It describes the Trade-off between risk and expected return Multiplier is the ratio of portfolio risk to market risk What about other (non-efficient) assets or portfolios? 23.11.2018

The Capital Asset Pricing Model For any asset, define its market beta as: Then the Sharpe-Lintner CAPM implies that: Describes the relationship between the expected return and risk of any given asset. Risk/reward relation is linear. E(ri) = rf + bi[E(rM) - rf] 23.11.2018

Beta Beta measures sensitivity of stock volatility to market volatility. The bi measures the risk of an asset that we cannot diversify away by combining it with other assets in the market portfolio. The higher the Beta is for any asset, the higher must be its return. 23.11.2018

Security Market Line 23.11.2018

Security Market Line 23.11.2018

Sample Calculations for SML Equation of the SML E(ri) = rf + bi[E(rM) - rf] Given E(rm) - rf = 0.08 and rf = 0.03 and bx = 1.25 E(rx) = 0.03 + 1.25(.08) = .13 or 13% If by = .6 E(ry) = 0.03 + 0.6(0.08) = 0.078 or 7.8% If b = 1 E(r) = 0.03 + 1 (0.08) = 0.11 If b = 0? If b < 0? Any Beta above one should result in a return above the market return. What is the return of a stock with a beta of 1?

Graph of Sample Calculations If the CAPM is correct, only β risk matters in determining the risk premium for a given slope of the SML. E(r) SML Rx=13% ßx 1.25 .08 RM=11% Here we can look at the returns with respect to the asset betas in a graphical format. Ry=7.8% ßy .6 3% ß 1.0 ßM

Positive Alpha, Underpriced Stock Suppose a security with a b of ____ is offering an expected return of ____ According to the SML, the E(r) should be 13% 1.25 13% 15% E(r) = 0.03 + 1.25(.08) = 13% a= Exp Actual return – CAPM derived return a= 15% - 13% = 2% Positive alpha gives the buyer a +2% abnormal return.

Negative Alpha, Overpriced stock Suppose a security with a b of ____ is offering an expected return of ____ According to the SML, the E(r) should be 13% 1.25 13% 11% E(r) = 0.03 + 1.25(.08) = 13% a= Exp Actual return – CAPM derived return a= 11% - 13% = -2% Negative alpha gives the buyer a -2% abnormal return.

The SML can be used to measure performance Suppose three mutual funds have the same average return of 15%, and the same volatility of 20%. Are all three managers equally talented? E(r) SML B R=15% ßC 1.25 A C Here we can look at the returns with respect to the asset betas in a graphical format. .6 3% ß 1.0

Disequilibrium in Return-Beta Space E(r) Any portfolio constructed by combining A and B will stay on the line, for ex., C= %50A+%50B Assume a portfolio ‘D’ exists with the following: E(r)=13% and Beta=1.2 How D compares to C? Can such an investment exist for long? Why? % 13.2 11.6 10 D B C A 1.0 1.2 1.4 Beta Investment Cash Invested (t=0) Exp Cash (t=1) Beta Port C (short) +$100 -$111.6 -1.2 Port D -$100 +$113 +1.2 Total $0 $1.4 0.0 Zero risk, zero net investment and a positive exp return. Riskless arbitrage. 23.11.2018

Does a Portfolio have Beta? If you put half your money in a stock with a beta of 1.4 and half of your money in a stock with a beta of 1.0, what is the portfolio beta? βP = 0.50(1.4) + 0.50(1.0) = 1.2 Wi βi Get Beta of portfolio by the sum of each weight times the beta of the asset. Get the risk premium of the portfolio by the sum of each weight times the risk premium of the asset. Or, by the portfolio Beta times the Market risk premium. All portfolio beta & expected return combinations should also fall on the SML. All (E(ri) – rf) / βi should be the same for all stocks. 20

Measuring Beta Concept: Method: We need to estimate the relationship between the returns of a given security and the “Market” portfolio. Can calculate the Security Characteristic Line, SCL, using historical time series excess returns of the security, and a proxy for the Market portfolio.

Implementing the CAPM CAPM relies on theoritical market portfolio and deals with expected returns To implement the CAPM, we cast it in the form of an index model and use past returns: E(rit) – rft = ai + Bi[E(rmt) – rft] rit – rft = ai + Bi[rmt – rft] + eit CAPM expects ai to be equal to zero.

CAPM and positive alpha What happens if ai is significantly different than zero? Interpretation: The index used is a bad Proxy for the market portfolio, or Theory is not useful Positive alphas persistantly observed in empirical research. 23.11.2018

Implementing the CAPM Use of Market Model: rit = rft + Bi[rmt – rft] + eit rit = rft + Birmt – Birft + eit rit = rft – Birft + Birmt + eit rit = ai + Birmt + eit where ai = rft (1 – Bi) 23.11.2018

Security Characteristic Line (SCL) Excess Returns (i) Dispersion of the points around the line measures ______________. The statistic is called e SCL . . . . Slope =  . . . unsystematic risk . . . . . . . . . . . . . . . The security characteristic line is basically the historical values of the excess returns of an individual asset versus those of the market portfolio. The CAPM model suggests that alpha should equal zero since all returns are based on the constant rf, the stock beta and the market risk. . . . . . Excess returns on market index . . . . . . . . . . =  What should  equal? . . . . . . . . . . . . . Ri = a i + ßiRM + ei

GM Excess Returns May 00 to April 05 “True”  is between 0.81 and 1.74! If rf = 5% and rm – rf = 6%, then we would predict GM’s return (rGM) to be 5% + 1.276(6%) = 12.66% Note I used FF Rm-Rf from the same time period. Tell them this is not the same as the Excess Returns in the text. (Note there is a conceptual error here, because I used book’s ER for GM and FF data for RM-Rf, however it works for illustrative purposes) -0.0143 1.276 0.01108 0.2318 0.5858 (Adjusted) = 33.18% 8.57% 7-26

Predicting Betas - Adjusted Betas Calculated betas are adjusted to account for the empirical finding that betas different from 1 tend to move toward 1 over time. A firm with a beta >1 will tend to have a ___________________ in the future. A firm with a beta <1 will tend to have a ____________________ in the future. lower beta (closer to 1) higher beta (closer to 1) The adjusted beta forecast is adjusted to account for the empirical finding that betas tend to have a regression toward the mean. A firm with a high beta (Beta >1) will tend to have a beta closer to 1 in the future, and vice versa. Adjusted β = = 2/3 (Calculated β) + 1/3 (1) 2/3 (1.276) + 1/3 (1) 1.184 27

Predictions and applications CAPM: In market equilibrium, investors are only rewarded for bearing the market risk, Applications: Evaluating security returns Corporate manager: capital budgeting decisions, company valuations Regulatory commissions: cost of capital for regulated firms (eg., utilities) Professional portfolio managers: evaluating security returns and fund performance. 23.11.2018

Evaluating security returns CAPM gives the required (or expected) return for a stock. Example: Given GE’s GE = 1.25, [E(rm) - rf ] = .08 , rf = .03, E(rGE) = .03 + 1.25(.08) = .13 or 13% If GE’s actually expected return is indeed 14.5%, then the extra 1.5% is considered reward without risk (abnormal return). In this case: ai = Rait – Rft - bi(RMt – Rft) aGE = 14.5%– 3% - 1.25(8%)= 1.5% 23.11.2018

Applying the CAPM to Valuation Recall the one-period rate of return: Solve for P0: 23.11.2018

Example: Stock Valuation using CAPM E[D1] = 5, g = 0.10, rf = 0.03 Beta = 1.5, E[rm – rf] = 0.075 E[r] = rf + B[E(rm) – rf) = 0.03+1.5(0.075)= 0.1425 Constant growth: 23.11.2018

In utility rate-making The rate of return a utility should be allowed to earn on its investment can be computed via CAPM. Ex: Utility equity investment is 100 mio $. Beta= 0.6; rf= 6%; MRP= 8% A fair annual profit for the utility would be: R= 6% + 0.6 ( 8%) = 10.8% That is 100mio x 0.108 = 10.8 mio $ profit Regulators will allow the utility to set its prices at a level to generate these profits. 23.11.2018

Other Applications of CAPM Market timing: If Betas are correctly estimated and likely to stay constant in the (near?) future then you can use betas for market timing. How? Used in portfolio choice: If you want to hold a diversified portfolio but one that has a specific beta that you want, you can use the concept of portfolio beta. 23.11.2018

Fama-French (FF) 3-Factor Model Fama and French noted that stocks of smaller firms and stocks of firms with a high book to market ratio have had higher stock returns than predicted by single factor models. Problem: Empirical model without a theory Will the variables continue to have predictive power?

Fama-French (FF) 3-factor Model FF proposed a 3-factor model of stock returns as follows: rM – rf = Market index excess return Ratio of ______________________________________ measured with a variable called : HML: High minus low or difference in returns between firms with a high versus a low book to market ratio. _______________ measured by the SMB variable SMB: Small minus big or the difference in returns between small and large firms. book value of equity to market value of equity Firm size variable

Fama-French (FF) 3 factor Model rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM

Fama-French (FF) 3 factor Model rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM If rf = 5%, rm – rf = 6%, & return on HML portfolio will be 5%, then we would predict GM’s return (rGM) to be 5% + -2.62% + 1.2029(6%) + 0.6923(5%) = 13.06% Note with negative alpha you don’t want to recommend this stock. -0.0262* 1.2029* 0.6923* 0.3646 0.0116 0.2411 0.2749 0.3327 0.6454 (Adjusted) = 38.52% 8.22% Compared to single factor model: Better Adjusted R2; lower βGM higher E(r), but negative alpha.