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# Chapter 7

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### Chapter 7

1. 1. Capital Asset Pricing and Arbitrage Pricing Theory Chapter 7
2. 2. The Capital Asset Pricing Model Equating the CML with the efficient frontier at M and solving for E(Ri) yields: RFR)-(RRFR)E(R Mi i For a complete derivation of the CAPM pls see: http://e.viaminvest.com/A2MonitorSystems/AppA2MonitorSystems/A ppBtoA2CAP_model/CAP_Model.asp#_ftn1
3. 3. Assumptions  Individual investors can borrow or lend money at the risk-free rate of return  Investors are rational mean-variance optimizers  Homogeneous expectations  Single-period investment horizon  Investments are limited to traded financial assets  Information is costless and available to all investors  No taxes, and transaction costs
4. 4. Security Market Line )E(Ri )Beta(Cov 2 Mim/0.1 mR SML 0 Negative Beta RFR
5. 5. Plot of Estimated Returns on SML Graph )E(R i Beta0.1 mR SML 0 .20 .40 .60 .80 1.20 1.40 1.60 1.80-.40 -.20 .22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02 A B C D E
6. 6. Inputs to the CAPM (a) the current risk-free rate (b) the expected return on the market index and (c) the beta of the asset being analyzed
7. 7. Riskfree Rate & Time Horizon  For an investment to be riskfree, i.e., to have an actual return be equal to the expected return, two conditions have to be met –  There has to be no default risk, which generally implies that the security has to be issued by the government. Note, however, that not all governments can be viewed as default free.  There can be no uncertainty about reinvestment rates, which implies that it is a zero coupon security with the same maturity as the cash flow being analyzed.
8. 8. http://www.ustreas.gov/offices/domestic-finance/debt-management/interest-rate/yield.shtml
9. 9. Measurement of the risk premium  The risk premium is the premium that investors demand for investing in an average risk investment, relative to the riskfree rate.  As a general proposition, this premium should be  greater than zero  increase with the risk aversion of the investors in that market  increase with the riskiness of the “average” risk investment
10. 10. Estimating Risk Premiums in Practice  Survey investors on their desired risk premiums and use the average premium from these surveys.  Assume that the actual premium delivered over long time periods is equal to the expected premium - i.e., use historical data  Estimate the implied premium in today’s asset prices.
11. 11. The Survey Approach  Surveying all investors in a market place is impractical. However, you can survey a few investors (especially the larger investors) and use these results. In practice, this translates into surveys of money managers’ expectations of expected returns on stocks over the next year.  The limitations of this approach are:  there are no constraints on reasonability (the survey could produce negative risk premiums or risk premiums of 50%)  they are extremely volatile  they tend to be short term; even the longest surveys do not go beyond one year
12. 12. The Historical Premium Approach  This is the default approach used by most to arrive at the premium to use in the model  In most cases, this approach does the following  it defines a time period for the estimation (1926-Present, 1962- Present....)  it calculates average returns on a stock index during the period  it calculates average returns on a riskless security over the period  it calculates the difference between the two and uses it as a premium looking forward  The limitations of this approach are:  it assumes that the risk aversion of investors has not changed in a systematic way across time. (The risk aversion may change from year to year, but it reverts back to historical averages)  it assumes that the riskiness of the “risky” portfolio (stock index) has not changed in a systematic way across time.
13. 13. Historical Average Premiums for the United States Historical period Stocks - T.Bills Stocks - T.Bonds Arith. Geom Arith Geom 1926-2003 7.92% 5.50% 6.54% 4.82% 1962-2003 6.09% 4.85% 4.70% 3.82% 1992-2003 8.43% 6.68% 4.87% 3.57% What is the right premium?
14. 14. What about historical premiums for other Markets?  Historical data for markets outside the United States tends to be unreliable.  Ibbotson, for instance, estimates the following premiums for major markets from 1970-1990 Country Period Stocks Bonds Risk Premium Australia 1970-90 9.60% 7.35% 2.25% Canada 1970-90 10.50% 7.41% 3.09% France 1970-90 11.90% 7.68% 4.22% Germany 1970-90 7.40% 6.81% 0.59% Italy 1970-90 9.40% 9.06% 0.34% Japan 1970-90 13.70% 6.96% 6.74% Netherlands 1970-90 11.20% 6.87% 4.33% Switzerland 1970-90 5.30% 4.10% 1.20% UK 1970-90 14.70% 8.45% 6.25%
15. 15. Implied Equity Risk Premiums  If we use a basic discounted cash flow model, we can estimate the implied risk premium from the current level of stock prices.  For instance, if stock prices are determined by the simple Gordon Growth Model:  Value = Expected Dividends next year/ (Required Returns on Stocks - Expected Growth Rate)  Plugging in the current level of the index, the dividends on the index and expected growth rate will yield a “implied” expected return on stocks.  Subtracting out the riskfree rate will yield the implied premium.  The problems with this approach are:  the discounted cash flow model used to value the stock index has to be the right one.  the inputs on dividends and expected growth have to be correct  it implicitly assumes that the market is currently correctly valued
16. 16. S&P 500 Implied Equity Premium  Current level – 1052  5 yr ave dividend yield 5.48%  Dividend growth rate 4.5%  1052 = 57.65(1.045) Ke - .045 Ke = 10.23% Risk premium – 10.23% - 3.5% = 6.73% http://www.wstreet.com/investing/stocks/18204_is_the_standard_and_poors_500_running_out_of_stream.html
17. 17. Implied Premiums in the US
18. 18. Estimating Beta  The standard procedure for estimating betas is to regress stock returns (Rj) against market returns (Rm) -  Rj = a + b Rm  where a is the intercept and b is the slope of the regression.  The slope of the regression corresponds to the beta of the stock, and measures the relative volatility of the stock to the market.
19. 19. Firm Specific and Market Risk  The R squared (R2) of the regression provides an estimate of the proportion of the risk (variance) of a firm that can be attributed to market risk;  The balance (1 - R2) can be attributed to firm specific risk.
20. 20. Setting up for the Estimation  Decide on an estimation period  Services use periods ranging from 2 to 5 years for the regression  Longer estimation period provides more data, but firms change.  Shorter periods can be affected more easily by significant firm- specific event that occurred during the period  Decide on a return interval - daily, weekly, monthly  Shorter intervals yield more observations, but suffer from more noise.  Noise is created by stocks not trading and biases all betas towards one.  Estimate returns (including dividends) on stock  Return = (PriceEnd – PriceBeginning + DividendsPeriod)/ PriceBeginning  Included dividends only in ex-dividend month  Choose a market index, and estimate returns (inclusive of dividends) on the index for each interval for the period.
21. 21. Choosing the Parameters: Disney  Period used: 5 years  Return Interval = Monthly  Market Index: S&P 500 Index.  For instance, to calculate returns on Disney Price for Disney at end of March = \$ 37.87  Price for Disney at end of April = \$ 36.42  Dividends during month = \$0.05 (It was an ex-dividend month)  Return =(\$36.42 - \$ 37.87 + \$ 0.05)/\$ 37.87=-3.69%  To estimate returns on the index in the same month  Index level (including dividends) at end of March = 404.35  Index level (including dividends) at end of April = 415.53  Return =(415.53 - 404.35)/ 404.35 = 2.76%
22. 22. Disney’s Historical Beta
23. 23. The Regression Output ReturnsDisney = -0.01% + 1.40 ReturnsS & P 500 (0.27) (R squared=32.41%) Intercept = -0.01% Slope = 1.40
24. 24. Estimating Disney’s Beta  Slope of the Regression of 1.40 is the beta  Regression parameters are always estimated with noise. The noise is captured in the standard error of the beta estimate, which in the case of Disney is 0.27.  Assume that I asked you what Disney’s true beta is, after this regression.  What is your best point estimate?  What range would you give me, with 67% confidence?  What range would you give me, with 95% confidence?
25. 25. Standard Errors for US Stocks
26. 26. Breaking Down Disney’s Risk  R Squared = 32%  This implies that  32% of the risk at Disney comes from market sources  68%, therefore, comes from firm-specific sources  The firm-specific risk is diversifiable and will not be rewarded
27. 27. The Relevance of R Squared You are a diversified investor trying to decide whether you should invest in Disney or Amgen. They both have betas of 1.35, but Disney has an R Squared of 32% while Amgen’s R squared of only 15%. Which one would you invest in:  Amgen, because it has the lower R squared  Disney, because it has the higher R squared  You would be indifferent
28. 28. The Relevance of R Squared…  I would be indifferent, because they have the same beta  I am well diversified, and thus can eliminate all firm-specific risk. If I were not, I would have picked Disney, since it has less firm-specific risk.
29. 29. Estimating Expected Returns  Disney’s Beta = 1.40  Riskfree Rate = 3.50% (Long term Government Bond rate)  Risk Premium = 7% (Implied)  Expected Return = 3.5% + 1.40 (7%) = 13.3%
30. 30. Use to a Potential Investor in Disney  As a potential investor in Disney, what does this expected return of 13.30% tell you?  This is the return that I can expect to make in the long term on Disney, if the stock is correctly priced and the CAPM is the right model for risk,  This is the return that I need to make on Disney in the long term to break even on my investment in the stock  Both  Assume now that you are an active investor and that your research suggests that an investment in Disney will yield 25% a year for the next 5 years. Based upon the expected return of 14.70%, you would  Buy the stock  Sell the stock
31. 31. Arbitrage Pricing Theory •Developed as an alternative to the CAPM •Reasonably intuitive •Required limited assumptions •Allowed for multiple dimensions of investment risk
32. 32. Arbitrage Pricing Theory  Based on the Law of One Price  Since two otherwise identical assets cannot sell at different prices, equilibrium prices adjust to eliminate all arbitrage opportunities  Arbitrage opportunity  arises if an investor can construct a zero investment portfolio with no risk, but with a positive profit  Since no investment is required, an investor can create large positions in long and short to secure large levels of profits  In an efficient market, profitable arbitrage opportunities will quickly disappear
33. 33. APT Model  APT assumes returns generated by a factor model  Factor Characteristics  Each risk factor must have a pervasive influence on stock returns  Risk factors must have nonzero prices  Risk factors must be unpredictable to the market  The expected return-risk relationship for the APT: E(Ri) = RF + bi1 (risk premium for factor 1) + bi2 (risk premium for factor 2) + ... + bin (risk premium for factor n)
34. 34. APT and CAPM Compared  APT applies to well diversified portfolios, and not necessarily to individual stocks  With APT, it is possible for some individual stocks to be mispriced - not lie on the SML  APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio  Unlike CAPM, APT does not assume mean-variance decisions, riskless borrowing or lending, and existence of a market portfolio  APT can be extended to multifactor models
35. 35. Arbitrage Pricing Theory (APT) where: = the expected return on an asset with zero systematic risk where ikkiii bbbRE   ...)( 22110 0 1 = the risk premium related to the common jth factor bij = the pricing relationship between the risk premium and asset - that is how responsive asset i is to jth common factor
36. 36. Using the APT = unanticipated changes in the rate of inflation. The risk premium related to this factor is 2 percent for every 1 percent change in the rate 1 )02.( 1  = percent growth in real GDP. The average risk premium related to this factor is 3 percent for every 1 percent change in the rate = the rate of return on a zero-systematic-risk asset (zero beta) is 4 percent 2 )03.( 2  )04.( 0  0
37. 37. Using the APT = the response of asset X to changes in the inflation factor is 0.50 1xb )50.( 1 xb = the response of asset Y to changes in the inflation factor is 2.00 )00.2( 1 yb 1yb = the response of asset X to changes in the GDP factor is 1.50 = the response of asset Y to changes in the GDP factor is 1.75 2xb 2yb )50.1( 2 xb )75.1( 2 yb
38. 38. Using the APT = .04 + (.02)bi1 + (.03)bi2 E(Rx) = .04 + (.02)(0.50) + (.03)(1.50) = .095 = 9.5% E(Ry) = .04 + (.02)(2.00) + (.03)(1.75) = .1325 = 13.25% 22110)( iii bbRE  
39. 39. Security Valuation with the APT: An Example Three stocks (A, B, C) and two common systematic risk factors have the following relationship( )0 0  21 21 21 )5.0()8.1()( )3.1()2.0()( )9.0()8.0()(       C B A RE RE RE %7.9)( %7.5)( %7.7)(    C B A RE RE RE %5and%4 21  if 40.38\$%)7.91(35\$)( 00.37\$%)7.51(35\$)( 70.37\$%)7.71(35\$)(    C B A PE PE PE Currently priced at \$35 each and will not pay dividend Expected prices a year from now Your estimates a year from now (you are sure!) E(PA) = \$37.20 E(PB) = \$37.80 E(PC) = \$38.50
40. 40. Security Valuation with the APT: An Example  Riskless arbitrage  Requires no net wealth invested initially  Will bear no systematic or unsystematic risk but  Still earns a profit  Condition must be satisfied as follow:  1.  2  3 0i iw 0 iji ibw 0 ii i Rw i.e. actual portfolio return is positive For all K factors [i.e. no systematic risk] and w is small for all I [ unsystematic risk is fully diversified] i.e. no net wealth invested Wi the percentage investment in security i
41. 41. Security Valuation with the APT: An Example  Example:  Stock A is overvalued; Stock B and C are two undervalued securities  Consider the following investment proportions  WA=-1.0  WB=+0.5  WC=+0.5 These investment weight imply the creation of a portfolio that is short two shares of Stock A for each share of Stock B and one share of Stock C held long
42. 42. Security Valuation with the APT: An Example Net Initial Investment: Short 2 shares of A: +70 Purchase 1 share of B: -35 Purchase 1 share of C: -35 Net investment: 0 Net Exposure to Risk Factors: Factor 1 Factor 2 Weighted exposure from Stock A: (-1.0)(0.8) (-1.0)(0.9) Weighted exposure from Stock B: (0.5)(-0.2) (0.5)(1.3) Weighted exposure from Stock C: (0.5)(1.8) (0.5)(0.5) Net risk exposure 0 0 Net Profit: Stock A Stock B Stock C [2(35)-2(37.20)]+[37.80-35]+[38.50-35] =\$1.90
43. 43. Security Valuation with the APT: An Example  The price of stock A will be bid down while the prices of stock B and C will be bid up until arbitrage trading in the current market is no long profitable. 10.35097.1/40.38\$ 76.35\$057.1/00.37\$ 54.34\$077.1/20.37\$    C B A P P P
44. 44. Microeconomic-Based Risk Factor Models (Fama-French)  The Fama-French Three Factor Model ittititmtiitit eHMLbSMBbRFRRbaRFRR  321 )()( SMB (i.e. small minus big) is the return to a portfolio of small capitalization stocks less the return to a portfolio of large capitalization stocks HML (i.e. high minus low) is the return to a portfolio of stocks with high ratios of book-to-market values less the return to a portfolio of low book-to-market value stocks
45. 45. Summary  APT model has fewer assumptions than the CAPM and does not specifically require the designation of a market portfolio.  The APT posits that expected security returns are related in a linear fashion to multiple common risk factors.  Unfortunately, the theory does not offer guidance as to how many factors exist or what their identifies might be
46. 46. Summary  APT is difficult to put into practice in a theoretically rigorous fashion. Multifactor models of risk and return attempt to bridge the gap between the practice and theory by specifying a set of variables.  Macroeconomic variables have been successfully applied  An equally successful second approach to identifying the risk exposures in a multifactor model has focused on the characteristics of securities themselves. (Microeconomic approach)