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Topic 5: Capital Asset Pricing Model Purpose: Review the highlights of Asset Pricing Theory, focusing on the CAPM Apply these concepts to portfolio performance evaluation -1- Asset pricing models in general: • Creating surrogate portfolios • Using surrogate portfolios for arbitrage – Goal: buy something cheap and sell something just like it for a higher price – Goal: hedge against market movements as protection while trading on scarce information -2- Betting Without Hedge • Suppose you sell GM short, based on scarce information GM Factor Uncertain Future Take Position -3- Betting Without Hedge • Then market rises, carrying GM with it GM Factor Uncertain Future Take Position -4- Betting Without Hedge • Then market rises, carrying GM with it • GM takes hit, but you still lose GM takes hit GM Uncertain Future Factor Take Position -5- A formal look at the assumptions necessary to partition risk n Markowitz formula Security 1 Security 1 Security 2 Security 3 n n s 2p = Â Â wi w j s i s j r ij i=1 j=1 Security 2 Security 3 w12 s 12 2w1w2 s 1 s 2 r12 w22 s 22 2w1w3 s 1s 3r 13 2w2w3 s 2 s 3r 23 w32 s 32 -6- Complex analysis becomes simple n Sharpe’ Sharpe’s approximation n s 2p = Â wi2 b i2 s m2 i=1 Security 1 Security 1 Security 2 Security 3 w12 b 12 s m2 Security 2 Security 3 w22 b 22 s m2 w32 b 32 s m2 -7- Measuring systematic risk with a singleindex model (the CAPM) • Regression analysis: tool for measuring interaction between two variables related by a linear function y = a + bx + e R j - R f = a j + b j ( Rm - Rf ) + e j -8- Coin Tossing Game Heads: Up 20% Tails: Down 10% Start at 100 300.00 250.00 200.00 150.00 100.00 50.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -9- Another Coin Tossing Game 2 stocks, start at 100, tend to follow market except for epsilon Stock A: Beta 1 Stock B: Beta 2 Market Stock A Stock B 500.00 450.00 400.00 350.00 300.00 250.00 200.00 150.00 100.00 50.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 - 10 - Example of company-specific news • DELL COMPUTER said it will meet quarterly forecasts, bucking a gloomy industry trend. This was a surprise to many analysts, who had predicted a shortfall. Shares rose 8.1%. – Source: Wall Street Journal, Oct. 4, 2001 - 11 - Partitioning total risk into systematic and unsystematic elements: The Intuition • Assignment: design a video game to mimic stock market • Step 1: Random move applies equally to all stocks • Step 2: Assign response coefficient to each stock – Average of all coefficients is one • Step 3: Add unique random move for each stock – Average of all these is zero • Question: What happens if a player diversifies across all the stocks in this game? - 12 - Criticism of the Model • The model is very simple • Two kinds of news – Company-specific – Global • Question for discussion: Does the model need additional factors, such as industry effects or regional effects? - 13 - How does a single-index model work? • For the moment, let us suspend the criticism and consider how we could analyze this simple game - 14 - Discovering beta • The characteristic line • The slope is the b Ri - Rf Rm - Rf - 15 - Estimating beta from historical data • The meaning of r Ri - Rf Rm - Rf - 16 - Estimating beta from historical data • The meaning of r Ri - Rf Rm - Rf - 17 - Estimating beta from historical data r2 is the proportion of uncertainty that is explained when we consider the influence of the market (this is the proportion of risk that is systematic) after before - 18 - Estimating beta from historical data • The meaning of a Market pressure Ri - Rf A B Rm - Rf Market pressure - 19 - Which is riskier? Ri - Rf Ri - Rf Rm - Rf Rm - Rf - 20 - Which is realistic? Ri - Rf Ri - Rf Rm - Rf Rm - Rf - 21 - Security Market Line – Market portfolio – T-bill L SM Return • Linear relationship between reward and risk – Risk premium proportional to beta • Two things are on the line: M kt Rf 1 Beta - 22 - Practice See problems 19 and 20 in the problem set • Calculate the beta for a portfolio made from equal proportions of securities with individual betas of .75, 1.0, and 1.25 • Calculate the beta for a portfolio with $200 invested in stock A, $300 in stock B, and $500 in stock C. Betas for the individual stocks are .75 for stock A, 1.0 for stock B, and 1.25 for stock C Beta = (.2*.75) + (.3*1) + (.5*1.25) = 1.075 - 23 - Practice See problems 21 and 22 in the problem set • When the T-Bill rate is 9% and the expected return for the market portfolio is 12%, what is the opportunity cost of capital for an investment with beta of 1.5? • Assume the T-Bill rate is 10% and the expected return on the market portfolio is 18%. An investment is twice as risky as the market portfolio, in terms of its systematic risk. What is the opportunity cost of capital for this investment? - 24 - Estimating beta from subjective inputs • Estimate s from confidence interval • Estimate r from coefficient of determination: bbii=(s =(sii/s /smm )r )rim im r2 = proportion of risk arising from systematic factors - 25 - Evaluating an investment from subjective inputs Assume the following: • 95% confidence interval for stock ranges from -30% to +70% • 95% confidence interval for market portfolio: -10% to +30% • 75% of risk is diversifiable • T-bill rate is 5% bbii=(s =(sii/s /smm )r )rim im bbi= (.25/ .10 )* .5 i= (.25/ .10 )* .5 == 1.25 1.25 OCC OCC =5% =5%++1.25(10% 1.25(10%-5%) -5%) ==11.25% 11.25% E(R) E(R)==20%, 20%,so soaccept accept - 26 - Practice See problem 23 in the problem set • An initial public offering (IPO) under analysis by Ajax Pension Fund is in a relatively new technology, and 75% of its risk is considered to be unsystematic. The 95% confidence interval for its rate of return ranges from 4% to 40%, and the probability distribution is symmetric normal. The expected return for the market portfolio is being quoted at 15% with standard deviation of 5%. Estimate the beta for the IPO, and then recommend whether to accept or reject. E(R) = 22% Beta = (9%/5%) * .5 = 0.9 Conclusion: This IPO has above-average reward with below-average risk, so is attractive OCC = 5% + 0.9(15% - 5%) = 14% - 27 - Practice See problem 24 in the problem set • Another IPO under analysis by Ajax Pension Fund is in an established technology, and only 19% of its risk is considered to be unsystematic. The 95% confidence interval for its rate of return ranges from -5% to 35%, and the probability distribution is symmetric normal. The expected return for the market portfolio is being quoted at 15% with standard deviation of 5%. Estimate the beta for this IPO, and then recommend whether to accept or reject. E(R) = 15% Beta = (10%/5%) * .9 = 1.8 Conclusion: This IPO has average reward with above-average risk, so is not attractive OCC = 5% + 1.8(15% - 5%) = 23% - 28 - Practice See problem 25 in the problem set • Large State Pension Fund recently installed an evaluation system centered on the capital asset pricing model. Data for actual outcomes of several managers and accompanying market conditions during the evaluation period are given below. See if you can identify any possible problem areas in the fund managers’ performance. Manager 1 1 2 2 3 3 Fund A B C D E F bj 0.4 1.1 0.8 1.7 0.5 1.3 Rj 12% 18% 13% 15% 13% 25% RTbill 9% 10% 10% 9% 9% 10% Rmkt 14% 15% 15% 13% 14% 15% - 29 - Practice See problems 26 through 28 in the problem set • If you were certain that the market was going to rise, what sort of beta would you prefer for your portfolio? • If you were certain that the market was going to drop, what sort of beta would you prefer for your portfolio? • If you thought the market might be turning up, but still recognize the possibility for a downturn, what sort of beta would you prefer for your portfolio? - 30 - What’s wrong with WACC? L SM Return • Project A is above WACC, but below SML • Project B is below WACC, but above SML • WACC would wrongly reject B and accept A B A WACC Rf Beta WACC Ignores Risk - 31 - • Empirical SML has higher intercept than theoretical SML • Empirical SML has a more gradual slope than the theoretical SML (there is less additional reward for added risk, in practice compared with theory) • Evidence motivates search for improved Arbitrage Pricing Theories Return Empirical evidence concerning the CAPM Theory Actual Beta - 32 -