Fahmi Ben Abdelkader HEC, Paris Fall Student version 10/18/2012 5:40 PM 1

Transcription

1 Financial Economics Optimal Portolio Choice and the CAPM Fahmi Ben Abdelkader HEC, Paris Fall 2012 Student version 10/18/2012 5:40 PM 1

2 Introduction Diversiication with an Equally Weighted Portolio o Many Stocks Diversiiable risk Systematic risk Source : Berk J. and DeMarzo P. (2011), Corporate Finance, Second Edition. Pearson Education. (Figure 11.2 p.339) The beneit o diversiication declines as the number o stocks in the portolio grows The decrease in volatility when going rom one to two stocks is much larger than the decrease when going rom 100 to 101 stocks To what extent should we keep adding stocks? How an investor can reach an optimal diversiication? 2

3 Learning Objectives Deine an eicient portolio and a market portolio Explain how an individual investor will choose rom the set o eicient portolios. Describe what is meant by a short sale, and illustrate how short selling extends the set o possible portolios. Explain the eect o combining a risk-ree asset with a portolio o risky assets, and compute the expected return and volatility or that combination. Illustrate why the risk-return combinations o the risk-ree investment and a risky portolio lie on a straight line. Understand the relation between systematic risk and the market portolio 3

4 Chapter Outline Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM 4

5 Choosing an Eicient Portolio Eicient portolios with two stocks Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Example Consider a portolio o Intel and Coca-Cola. Suppose an investor believes these stocks are uncorrelated and will perorm as ollows: Expected Return Volatility Intel 26% 50% Coca-Cola 6% 25% How should the investor choose a portolio o these two stocks? Are some portolios preerable to others? Let s compute the expected return and volatility or dierent combinations o the stocks 0% 100% 6% 25.0% 20% 80% 10% 22.3% 40% 60% 14% 25.0% 60% 40% 18% 31.6% 80% 20% 22% 40.3% 100% 0% 26% 50.0% 5

6 Choosing an Eicient Portolio Eicient portolios with two stocks Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Volatility Versus Expected Return or Portolios o Intel and Coca-Cola Stock C B The choice between B and C depends on investor s preerences or return versus risk A Coca-Cola The portolio A is beaten by B (B has a higher expected return and lower volatility) Example: Your riend has invested 100% o its money in Coca-Cola stock and is seeking investment advice. He would like to earn the highest expected return possible without increasing the risk (volatility). Which portolio would you recommend? 6

7 Choosing an Eicient Portolio Eicient Portolio: Main Characteristics Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Volatility Versus Expected Return or Portolios o Intel and Coca-Cola Stock Eicient Frontier In an eicient portolio there is no way to reduce the volatility o the portolio without lowering its expected return Minimum Variance Portolio P 2 P 1 Ineicient Portolios Coca-Cola In an ineicient portolio, it is possible to ind another portolio that is better in terms o both expected return and volatility. Eicient Portolio: or a given level o risk, it oers the highest possible expected return 7

8 The Eect o Correlation Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Eect on Volatility and Expected Return o Changing the Correlation between Intel and Coca-Cola Stock Recall :Correlation has no eect on the expected return o a portolio The lower the correlation, the lower the volatility we can obtain 8

9 Short Sales: principle Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Short sale transaction In a short sale, you sell a stock that you do not own, with the obligation to buy it back in the uture. We can include a short position as part o a portolio by assigning that stock a negative portolio weight Short Position in a portolio A negative portolio weight o that security Long Position A positive portolio weight o that security What is the rationale behind short selling? Short selling is an advantageous strategy i you expect a stock price to decline in the uture. 9

10 Choosing an Eicient Portolio The Mechanics o a Short Sale Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Example: Short sale transaction Suppose you expect a decrease o the AXA s stock price. You decide to short sell 1000 shares o AXA and to close your position as soon as the AXA s stock price drop to 10. AXA price= 15.1/sh AXA Div= 1.1/sh AXA Price= 10/sh Today 30 May 30 June 15 July Short seller cash lows +P 0 = + 15,100 - Div t = - 1,100 - P 1 = - 10,000 Short seller net cash lows = +P 0 - Div t - P 1 =+ 4,000 Broker Broker borrows 1000 shares and sells them Dividend paid Broker buys 1000 shares and returns them Jean-Mouloud: owner o AXA shares Jeanmouloud s account Jeanmouloud s account Jeanmouloud s account 1,100 Jeanmouloud s account 1, AXA 1000 AXA 1000 AXA 1000 AXA 1000 AXA 1000 AXA 1000 AXA 1000 AXA 10

11 Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Short Selling: the case o Volkswagen s Stock in October 2008 VW Shares Plunge, a Day Ater Surge: the mystery o a 200% increase Short sellers yesterday took a pounding as shares in Volkswagen, Europe's biggest carmaker, soared as much as 200% on the back o Porsche's move to seize ull control The Guardian, Tuesday 28 October

12 Choosing an Eicient Portolio The Mechanics o a Short Sale Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Problem: Expected Return and Volatility with a Short Sale Suppose you have 20,000 in cash to invest. You decide to short sell 10,000 worth o Coca-Cola stock and invest the proceeds rom your short sale, plus your 20,000, in Intel. What is the expected return and volatility o your portolio? Expected Return Volatility Cov (Intel, Coca) Intel 26% 50% 0 Coca-Cola 6% 25% In this case, Short selling.. the expected return o your portolio, but also its volatility, above those o the individual stocks 12

13 Choosing an Eicient Portolio Eicient Portolios Allowing or Short Sales Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Portolios o Intel and Coca-Cola Allowing or Short Sales Short selling Coca-Cola to invest in Intel is eicient and might be attractive to an aggressive investor who is seeking high expected return Short selling Intel to invest in Coca- Cola is ineicient Short selling extents investment possibilities or investors 13

14 Choosing an Eicient Portolio Eicient Portolios with Many Stocks Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Consider adding Bore Industries to the two stock portolio: Bore is uncorrelated with Intel and Coca-Cola, but is expected to have a very low return o 2%, and the same volatility as Coca-Cola. Should you add Bore in the portolio Intel-Coca? By adding Bore, we introduce new investment possibilities We can also do better than two-stock portolios: B+I+C > I+C 14

15 Choosing an Eicient Portolio Eicient Frontier with Many Stocks Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio The Volatility and Expected Return or All Portolios o Intel, Coca-Cola, and Bore Stock Source : Berk J. and DeMarzo P. (2011), Corporate Finance, Second Edition. Pearson Education. (Figure 11.7 p.348) In this case none o the stocks, on its own, is on the eicient rontier, so it would not be eicient to put all our money in a single stock. 15

16 Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Eicient Frontier and the Eect o Diversiication Eicient Frontier with Ten Stocks Versus Three Stocks Source : Berk J. and DeMarzo P. (2011), Corporate Finance, Second Edition. Pearson Education. (Figure 11.8 p.349) In general, adding new investment opportunities allows or greater diversiication and improves the eicient rontier 16

17 Choosing an Eicient Portolio Investing in Risk-Free Securities Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Is it proitable to include non-risky investments in our risky-portolio? Consider two types o investments: Return rate in one year Security Sunny weather Rainy weather Expected Return E[R] Volatility Risk-ree bond +3% +3% 3% 0% Umbrella SA -10% +20% 10% 20% You are planning to invest 100,000 in Umbrella s securities. A riend suggests you to invest a raction o your money in risk-ree bonds. What is the eect on return and risk o investing 50,000 in Umbrella s Securities, while leaving the remaining 50,000 in risk-ree bonds? Security Cash lows in one year Average expected Expected Return Amount Sunny Rainy cash lows E[R] invested weather weather Volatility Risk-ree bond 50,000 51,500 51,500 51,500 3% 0% Umbrella SA 50,000 45,000 65,000 55,000 10% 20% Total 100,000 95, , , % 10% Risk can be reduced by investing a portion o a portolio in a risk-ree investment, like T-Bills. However, doing so will likely reduce the expected return 17

18 Choosing an Eicient Portolio Investing in Risk-Free Securities Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Let s generalize Consider an arbitrary risky portolio (with returns R p ) and the eect on risk and return o putting a raction x o the money in the portolio, while leaving the remaining raction in risk-ree Treasury bills with a yield o r. What is the expected return and the volatility o the total portolio? Expected Return [ R ] = x. E[ R ] x E[ R ] E P [ R ] = ( 1 x). r + x E[ R ] E. xp E [ R ] = r + x. E[ R ] xp 2 P ( r ) P 1 Volatility 2 2 Var RP ) = x Var( R ) + x Var( R ) + 2x x Cov( R, ) ( R2 2 2 ( 1 x ) Var ( r ) + x Var ( R ) + 2(1 x ) x. Cov ( r, R ) Var ( RxP ) = P P 0 0 Var 2 ( RxP) = x Var( R P ) SD R ) = x. SD( R ) ( xp P E [ R ] xp = r + ( E[ R ] r ) P SD( R P ). SD( R 18 xp )

19 Choosing an Eicient Portolio The Capital Allocation Line (CAL) Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio The Capital Allocation Line The Risk Return Combinations rom Combining a Risk-Free Investment and a Risky Portolio CAL Portolio combining exclusively risky investments E [ R ] E xp ( r ) [ R ] = r + x. E[ R ] xp = r + ( E[ R ] r ) P SD( R P ) P. SD( R xp ) As we increase the raction x invested in P, we increase both our risk and our risk premium proportionally 19

20 Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Buying Stocks Using Leverage: A Levered Portolio Is it proitable to borrow money at the risk-ree interest rate and invest it in stocks? Problem: Suppose you have 10,000 in cash, and you decide to borrow another 10,000 at 5% interest rate in order to invest 20,000 in Umbrella s securities (P U ). What is the expected return and risk o your investment? What is your realized return in both sunny or rainy weather? Return rate in one year Security Sunny weather Rainy weather Expected Return E[R] Volatility Umbrella SA -10% +30% 10% 20% E [ R xp ]= SD( RxP) = Security Cash lows in one year Average Expected Return Amount Sunny weather Rainy weather expected cash E[R] invested lows Volatility Loan - 10,000-10,500-10,500-10,500-5% - Umbrella SA 20,000 18,000 26,000 22,000 10% 20% Total 10,000 7,500 (-25%) 15,500 (+55%) 11,500 15% 40% Using leverage provided higher expected returns than investing in Umbrella s stocks using only the unds we have available. However, it has doubled the risk o the portolio 20

21 Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Levered Investment is a Risky Investment Strategy! The Capital Allocation Line The Risk Return Combinations rom Combining a Risk-Free Investment and a Risky Portolio CAL E [ R ] E xp ( r ) [ R ] = r + x. E[ R ] xp = r + ( E[ R ] r ) P SD( R P ) P. SD( R xp ) Borrowing money to invest in P Is this portolio eicient? The levered portolio can provide higher expected returns than investing in P using only the unds we have available. However, it has much higher risk 21

22 Choosing an Eicient Portolio How to Identiy The Optimal Risky Portolio? Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio To earn the highest possible expected return or any level o volatility we must ind the portolio that generates the steepest possible line when combined with the risk-ree investment. CAL E [ R ] xp = r + ( E [ R ] r ) P SD( R P ). SD( R xp ) The slope o this line is oten reerred to as the Sharpe ratio o the portolio: Sharpe Ratio = Portolio Excess Return Portolio Volatility [ ] E RP r Sharpe Ratio = SD( R ) P Sharpe Ratio measures the ratio o reward-to-volatility provided by a portolio 22

23 Choosing an Eicient Portolio How to Identiy The Optimal Risky Portolio? Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio To earn the highest possible expected return or any level o volatility we must ind the portolio that generates the steepest possible line when combined with the risk-ree investment. CAL E [ R ] xp = r + ( E [ R ] r ) P SD( R P ). SD( R xp ) P 2 P 1 The slope o this line is oten reerred to as the Sharpe ratio o the portolio: SD( R ) P E [ RP ] r Portolio Excess Return Sharpe Ratio = Portolio Volatility Sharpe Ratio = E [ ] R r P SD( R P ) Sharpe Ratio measures the ratio o reward-to-volatility provided by a portolio 23

24 Choosing an Eicient Portolio How to Identiy The Optimal Risky Portolio? Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio To earn the highest possible expected return or any level o volatility we must ind the portolio that generates the steepest possible line when combined with the risk-ree investment. CAL E [ R ] xt = r + ( E[ R ] r ) T SD( R T ). SD( R xt ) CAL E [ R ] xp = r + ( E[ R ] r ) P SD( R P ). SD( R xp ) The portolio with the highest Sharpe ratio is the portolio where the line with the risk-ree investment is tangent to the eicient rontier o risky investments. The portolio that generates this tangent line is known as the Optimal Risky Portolio or Tangent Portolio. 24

25 The Bottom Line Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Combinations o the risk-ree asset and the tangent portolio provide the best risk and return tradeo available to an investor The tangent portolio is eicient and all eicient portolios are combinations o the risk-ree investment and the tangent portolio. Every investor should invest in the tangent portolio independent o his or her taste or risk An investor s preerences will determine only how much to invest in the tangent portolio versus the risk-ree investment. Conservative investors will invest a small amount in the tangent portolio. Aggressive investors will invest more in the tangent portolio. Both types o investors will choose to hold the same portolio o risky assets, the tangent portolio, which is the optimal risky portolio 25

26 Optimal Portolio Choice Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Problem Your uncle asks or investment advice. Currently, he has 100,000 invested in portolio P (see ig. below), which has an expected return o 10.5% and a volatility o 8%. Suppose the risk-ree rate is 5%, and the tangent portolio has an expected return o 18.5% and a volatility o 13%. 1. To Maximize his expected return without increasing his volatility, which portolio would you recommend? 2. I your uncle preers to keep his expected return the same but minimize his risk, which portolio would you recommend? 3. Your uncle asked you also to give him the expected return o an eicient portolio that has a standard deviation o return o 12%. CAL E [ R ] xt = r + ( E[ R ] r ) T SD( R T ). SD( R xt ) 26

27 Optimal Portolio Choice Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Problem Your uncle asks or investment advice. Currently, he has 100,000 invested in portolio P (see ig. below), which has an expected return o 10.5% and a volatility o 8%. Suppose the risk-ree rate is 5%, and the tangent portolio has an expected return o 18.5% and a volatility o 13%. 1. To Maximize his expected return without increasing his volatility, which portolio would you recommend? 2. I your uncle preers to keep his expected return the same but minimize his risk, which portolio would you recommend? 3. Your uncle asked you also to give him the expected return o an eicient portolio that has a standard deviation o return o 12%. Solution Q.1 Solution Q.2 27

28 Optimal Portolio Choice Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Problem Your uncle asks or investment advice. Currently, he has 100,000 invested in portolio P (see ig. below), which has an expected return o 10.5% and a volatility o 8%. Suppose the risk-ree rate is 5%, and the tangent portolio has an expected return o 18.5% and a volatility o 13%. 1. To Maximize his expected return without increasing his volatility, which portolio would you recommend? 2. I your uncle preers to keep his expected return the same but minimize his risk, which portolio would you recommend? 3. Your uncle asked you also to give him the expected return o an eicient portolio that has a standard deviation o return o 12%. Solution Q.3 28

29 Portolio Theory Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio «Portolio Selection». The Journal o Finance, Vol. 7, No. 1. (Mar., 1952), pp Fundamentals o Portolio Theory Developed techniques o mean-variance portolio optimization, which allow an investor to ind the portolio with the highest expected return or any level o variance Harry Markowitz Nobel Prize in 1990 Markowitz s Approach has evolved into one o the main methods o portolio optimization used on Wall Street 29

30 Chapter Outline Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM 30

31 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Recall: Standard Deviation is the Measure o The Total Risk o a Stock Total Risk o a Stock = Diversiiable Risk + Systematic Risk How to measure Systematic Risk? The measure o total risk is the Standard Deviation o stock returns What is the measure o systematic risk? 31

32 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Recall: the Systematic Risk is Non Diversiiable In a competitive Market investors could eliminate the irm-speciic risk or ree by diversiying their portolios They will not require a reward or risk premium or holding it However, diversiication does not reduce systematic risk The risk premium o a security is determined by its systematic risk and does not depend on its diversiiable risk 32

33 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Measuring Systematic Risk via The Market Portolio To measure the systematic risk o a stock we must determine how much o the variability o its return is due to systematic risk how sensitive a stock is to systematic shocks that aect the economy as a whole? look at the average change in the return or each 1% change in the return o a portolio that luctuates solely due to systematic risk. How can we identiy such a portolio that contains only systematic risk? A ully diversiied portolio A large portolio containing many dierent stocks The Market Portolio: which contains the largest number o shares and securities traded in the capital market In practice, we use the CAC40 or the S&P500 portolio as an approximation or the market portolio 33

34 Measuring Systematic Risk: the Beta How to Invest in The Market Portolio? Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Paris Stock Market The weights o main CAC40 securities CAC40 40 SBF % SBF % Number o irms traded in the Paris Stock Exchange % o thetotal Market Value o Paris Stock Exchange A Market Index reports the value o a particular portolio o securities. Example: CAC40, S&P 500, etc. The CAC40 is an index that represents a value-weighted portolio o 40 o the largest French stocks Investing in a Market Index: invest in index unds or ETF (exchange-traded und) that invest in market portolios. Ex. Lyxor ETF CAC40, SPDR (S&P Depository Receipts, nick-named Spiders ), etc. 34

35 Measuring Systematic Risk: the Beta How to Invest in The Market Portolio? Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Constructing the market portolio Market Capitalization The total market value o a irm s outstanding shares Example : Market Capitalisation o EDF (Number o shares = 1.8 billion ; Stock price=17 ) MV(EDF) = Value-Weighted Portolio A portolio in which each security is held in proportion to its market capitalization 35

36 Measuring Systematic Risk: the Beta How to Invest in The Market Portolio? Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Investing or constructing a market portolio 36

37 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta and the Real World The Market Portolio is a Good Proxy or Systematic Risk I we assume that changes in the value o the market portolio represent systematic shocks to the economy we can measure the systematic risk o a security by calculating the sensitivity o the security s return to the return o the market portolio Monthly excess returns or Apple stock and or S&P500, ( ) 37

38 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta and the Real World The Beta I we assume that changes in the value o the market portolio represent systematic shocks to the economy we can measure the systematic risk o a security by calculating the sensitivity o the security s return to the return o the market portolio Sensitivity to Systematic Risk: Beta (β) The expected percent change in the excess return o a security or a 1% change in the excess return o the market portolio. Quick Chek Question The Alstom s stock has aβ = How to interpret this data with respect to CAC40 returns? Each 1% change in the return o the market is likely to lead, on average, to a 2.43% change in the return or Alstom The stock o Alstom is exposed to two times more systematic risk than the CAC40 38

39 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta and the Real World Real-Firm Betas Betas with respect to the CAC40 or individual stocks (based on monthly data or ) Each 1% change in the return o the CAC40 is likely to lead, on average, to a 1.99% change in the return or Air France, but only a 0.54% change in the return or Total 39

40 Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta and the Real World The Beta Versus the Standard Deviation Standard Deviation: SD The beta : β Systematic Risk + Diversiiable Risk Systematic Risk Example Michelin and Air France have similar standard deviation or (SD ~ 45%). However, the beta o Air France is higher than the beta o Michelin. How to interpret this data? Security SD β Air France-KLM 45% 1.99 Michelin 45% 0.21 Michelin and Air France have similar volatility, but the exposure o Air France to systematic risk is.. than Michelin 40

41 Measuring Systematic Risk: the Beta The Beta Versus the Standard Deviation Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta and the Real World Standard Deviation: SD The beta : β Systematic Risk + Diversiiable Risk Systematic Risk 41

42 Measuring Systematic Risk: the Beta Estimating Beta rom Historical Returns Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta and the Real World Monthly excess returns or STMicroelectronics and or CAC40, ( ) Représentation chronologique des rentabilités à l aide d histogrammes STM s returns tend to move in the same direction but arther than those o the CAC40 42

43 Measuring Systematic Risk: the Beta Estimating Beta rom Historical Returns Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Monthly excess returns or STMicroelectronics and or CAC40, ( ) 47.13% Example: In octobre 2002, R STM =47.13% and R CAC40 =13.4% 13,4% Chaque point représente un couple de rentabilité (CAC40/STM) un mois donné entre 2002 et

44 Measuring Systematic Risk: the Beta Estimating Beta rom Historical Returns Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Monthly excess returns or STMicroelectronics and or CAC40, ( ) The slope o the linear regression line = 1.8 = Beta On average, a 1% change in the return o the CAC40 during this period led to a 1.8% change in the return or STM 44

45 Measuring Systematic Risk: the Beta Estimating Beta rom Historical Returns Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Average Betas or some european stocks based on historical data 45

46 Measuring Systematic Risk: the Beta Estimating Beta rom Historical Returns Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Average Betas or stocks by industry and the betas o slected company in each industry (based on historical data) 46

47 Measuring Systematic Risk: the Beta Estimating Beta rom Covariance Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice I we assume that the Market Portolio is represented by the CAC40 index, and the beta o a security i is β i =1.5 Each 1% change in the return o the CAC40 is likely to lead, on average, to a 1.5% change in the return or the security i β i 1.5% = 1 % = Variation o R i ralative to variation o R Mkt Variation o R Mkt = Cov ( R i, R Var ( R Mkt Mkt ) ) β i = Cov ( R i, [ SD ( R )] 2 Mkt R Mkt ) Volatility o i that is common with the market β = i SD ( R i ). Corr SD ( R ( R ) Mkt i, R Mkt ) 47

48 Interpreting Beta Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice A security s beta is related to how sensitive its underlying revenues and cash lows are to general economic conditions I β i =1 The security i tend to move in the same direction than the market I β i >1 The security i is likely to be more sensitive to systematic risk than the market Shocks in the economy have an ampliied impact (negatively or positively) on this stock I β i <1 The security i is likely to be more stable and less sensitive to systematic risk than the market Shocks in the economy on this stock are dampened I β i <0 When the market as a whole stumbles, the return o i tends to rise 48

49 Interpreting Beta Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Quick Check Question Is Apple s Beta > or < to 1? Monthly excess returns or Apple stock and or S&P500, ( ) Apple s returns tend to move in the same direction than those o the market, but its movements are larger 49

50 Measuring Systematic Risk: the Beta To what extent does Beta estimations reliable? Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Time Horizon: trade-o regarding which time horizon to use to measure returns The Market Proxy: which index should we use? Beta variation and extrapolation: Adjusted betas 2 Adjusted Beta o Security i = βi + 3 ( 1.0) Estimation Methodologies Used by Selected Data Providers 1 3 How to deal with outliers (returns o unusually large magnitude)? 50

51 Measuring Systematic Risk: the Beta To what extent does Beta estimations reliable? Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice Beta Estimation with and without Outliers or Genentech Using Monthly Returns or

52 Chapter Outline Choosing an Eicient Portolio Eicient Portolios Short Sales Eicient Frontier and diversiication Identiying The Optimal Risky Portolio Measuring Systematic Risk: the Beta Identiying Systematic Risk: The Market Portolio The Beta: A measure o Systematic Risk Calculating and Interpreting Betas The Beta in practice The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM 52

53 The Capital Asset Pricing Model (CAPM) The CAPM : the purpose The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Estimating the Equity Cost o Capital o an investment using its Beta «Capital Asset Prices. A Theory o Market Equilibrium under Conditions o Risk». The Journal o Finance, Vol. 19, No. 3, (1964), pp William F. Sharpe Nobel prize

54 The Capital Asset Pricing Model (CAPM) The CAPM Assumptions The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Three Main Assumptions Assumption 1 Investors can buy and sell all securities at competitive market prices (without incurring taxes or transactions costs) and can borrow and lend at the risk-ree interest rate. Assumption 2 Investors hold only eicient portolios o traded securities portolios that yield the maximum expected return or a given level o volatility. Assumption 3 Investors have homogeneous expectations regarding the volatilities, correlations, and expected returns o securities. Main implication According to the CAPM: The eicient portolio is the market portolio P e = P Mkt 54

55 The Capital Asset Pricing Model (CAPM) The Capital Market Line The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM According to the CAPM: The eicient portolio is the market portolio P e = P Mkt the CML represents the highest expected return available or any level o volatility 55

56 Reminder The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Risk Premium Additional return (compared to risk-ree interest rate) that investors expect to earn to compensate them or the security s risk Expected Risk Premium Expected return o a risky investment The riskree interest rate E [ R ] = r ( Risk Premium o s ) s + [ R ]: Expected Return o a risky investment s E s r : The risk - ree interest rate 56

57 Market risk and Beta The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM E [ R ] r ( Risk Premium o s ) = 1 s + When the CAPM assumptions hold: The risk premium o a security is determined by its Market risk (systematic risk) The risk premium o a security is proportional to the risk premium o the market portolio Risk Premium o s = β. Risk Premium o P Mkt E [ Rs ] = r + β. Risk Premium o PMkt E [ R ] = r + ( E [R ] r ) s β. PMkt Risk Premium o s The CAPM Equation 57

58 Market risk and Beta The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Problem Assume the risk-ree return is 4% and market inormation below. What is Renault s beta with the market? Under the CAPM assumptions, what is its expected return? Expected Return SD Corr (RNO, Pm) Renault (RNO)? 35% 27% P Mkt 30% 20% - 58

59 Market risk and Beta The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Problem: A Negative Beta Stock Assume the risk-ree return is 4% and the stock o AXA has a negative beta o Under the CAPM assumptions, how does its expected return compare to the risk-ree rate? Does this result make sense? E [ R ] = r + ( E [R ] r ) s Expected Return AXA? P Mkt 30% 1. PMkt β E[ ] = 4 % 0.1* ( 30% 4% ) = 1.4% R AXA β This result seems odd: why would investors be willing to accept a 1.4% expected return on this stock when they can invest in a sae investment and earn 4%? A savvy investor will AXA alone; instead, he will hold it in combination with other securities as part o a well-diversiied portolio Because AXA will tend to when the market and most other securities all, AXA provides.or the portolio, and investors pay or this insurance by accepting an expected return below the risk-ree rate. 59

60 The Capital Asset Pricing Model (CAPM) The Security Market Line (SML) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM The SML shows the expected return or each security as a unction o its beta with the market. E [ R ] = r + ( E [R ] r ) s β. PMkt Security Market Line (SML) Market Portolio A B β<1 β>1 Expected return P Mkt 30%? Risk-ree Investment 4%? β C β<0 According to the CAPM, the expected return o a security depends exclusively on its beta : its common risk 60

61 The Capital Asset Pricing Model (CAPM) The Security Market Line (SML) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM The SML shows the expected return or each security as a unction o its beta with the market. E [ R ] = r + ( E [R ] r ) s β. PMkt According to the CAPM, the market portolio is eicient, so all stocks and portolios should lie on the SML. 61

62 The Capital Asset Pricing Model (CAPM) Recall : The Capital Market Line (CML) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM The CML depicts portolios combining the risk-ree investment and the eicient portolio, and shows the highest expected return that we can attain or each level o volatility. According to the CAPM, the market portolio is on the CML and all other stocks and portolios contain diversiiable risk and lie to the right o the CML, as illustrated or Exxon Mobil (XOM). 62

63 The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM The Capital Market Line Versus The Security Market Line The Capital Market Line and the Security Market Line E [ R] = unction o its volatility ( SD) E [ R] = unction o its β No risky security on the Capital Market Line There is no relatioship between an individual stock s volatility and its expected return 63

64 The Beta o a portolio The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Beta o a Portolio The beta o a portolio is the weighted average beta o the securities in the portolio. I = i β = x. β P i i 64

65 The Beta o a portolio The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Problem Assume the risk-ree return is 4% and the market inormation below. What is the expected return o an equally weighted portolio o Danone and Inograe, according to the CAPM? Rentabilité espérée Action Danone (BN)? 0.5 Action Inogrames (IFG)? 1.25 P Mkt 10% β = [ R ] x E[ R ] E E[ R ] = r + β ( E [R ] r ) P i i i s. PMkt β = x. β P i i i 1 st way 2 nd way 65

66 The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Putting It All Together: The Capital Asset Pricing Model According to the CAPM, the risk premium o any security is equal to the market risk premium multiplied by the beta o the security. This relationship is called the Security Market Line (SML), and it determines the expected or required return or an investment : its equity cost o capital E The CAPM Equation [ R ] = r + ( E [R ] r ) s β. PMkt Risk Premium o s Volatility o i that is common with the market With β = i Cov ( R i, R [ SD ( R )] 2 Mkt Mkt ) β = i SD ( R i ).Corr SD ( R ( R ) Mkt i, R Mkt ) 66

67 The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Putting It All Together: The Capital Asset Pricing Model Recall: the discount rate to compute the NPV is the OCC Opportunity Cost o Capital: The best available expected return oered in the market on an investment o comparable risk and term to the cash low being discounted (Also reerred to as Cost o Capital) According to the CAPM, the expected return on a risky investment only depends on its beta The OCC used to compute the NPV should be the expected return on an asset that has the same beta o the project 67

68 The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Putting It All Together: The Capital Asset Pricing Model Example : The Equity Cost o Capital Stocks Volatility Beta Arcelor Mittal 25% 1.9 Vallourec 35% 1.15 Risk-ree investment Expected Return o P Mkt 4% 10% Which stock carries more total risk? Which has more market risk? What is the cost o capital o an investment that has equivalent Market risk than Vallourec? What is the cost o capital o an investment that has equivalent Market risk than Arcelor-Mittal? Which investment has a higher cost o capital? Mittal has..cost o capital than Vallourec, even though it is less Because market risk cannot be, it is.. risk that determines the cost o capital 68

69 The Capital Asset Pricing Model (CAPM) The CAPM Assumptions and the Market Portolio Measuring the Cost o Capital: the CAPM Equation The Capital Market Line Versus The Security Market Line Summary o the CAPM Putting It All Together: The Capital Asset Pricing Model The Bottom line Under the CAPM assumptions the tangency portolio is equal to the market portolio An eicient portolio is a combination o the risk ree asset and the market portolio. The Beta o an asset represents the risk contribution o this asset to the risk o the market. The risk premium required by investors to hold a risky asset in their portolio is proportional to the asset s Beta. 69