Return and Risk The Capital Asset Pricing Model (CAPM)

2 Important Security Characteristics μ, which ____ σ 2 and σ, which _____ Covariance and Correlation, which are _____

3 Important Security Characteristics μ, which is What we expect to make σ 2 and σ, which _____ Covariance and Correlation, which are _____

4 Important Security Characteristics μ, which is What we expect to make σ 2 and σ, which measure the risk Covariance and Correlation, which are _____

5 Important Security Characteristics μ, which is What we expect to make σ 2 and σ, which measure the risk Covariance and Correlation, which are what we are learning about now

6 Statistics Review: Covariance Covariance: measures how two variables move in relation to one another  Positive: The two variables move up together or down together Ex: Height and Weight  Negative: When one moves up the other moves down Ex: Sleep and Coffee consumption

7 Calculation Cov (X,Y) = σ XY = Σ p i * (X i – μ x ) * (Y i – μ Y )  σ XY = p 1 *(X 1 – μ X )*(Y 1 – μ Y )+p 2 *(X 2 – μ X )*(Y 2 – μ Y )+….+p N *(X N – μ X )*(Y N – μ Y ) Note that σ XX = Σ p i * (X i – μ x ) * (X i – μ X )  Which implies?  The covariance of an asset with itself is variance

8 Covariance Strength If the covariance of asset 1 and 2 is -1,000, and between asset 1 and 3 is 500. Which asset is more closely related to 1? We don’t know Covariance gives the direction, BUT NOT the strength of the relation.

9 Correlation Coefficient Correlation Coefficient: Measures the strength of the covariance relationship  The correlation coefficient is a standardization of covariance ρ 12 = σ 12 / (σ 1 * σ 2 )

10 Correlation Coefficient Formula ρ 12 = σ 12 / (σ 1 * σ 2 ) ρ 12 – Correlation Coefficient σ 12 - σ 1 - σ 2 -

11 Correlation Coefficient Formula ρ 12 = σ 12 / (σ 1 * σ 2 ) ρ 12 – Correlation Coefficient σ 12 – Covariance between 1 and 2 σ 1 - σ 2 -

12 Correlation Coefficient Formula ρ 12 = σ 12 / (σ 1 * σ 2 ) ρ 12 – Correlation Coefficient σ 12 – Covariance between 1 and 2 σ 1 - Std Dev of asset 1 σ 2 -

13 Correlation Coefficient Formula ρ 12 = σ 12 / (σ 1 * σ 2 ) ρ 12 – Correlation Coefficient σ 12 – Covariance between 1 and 2 σ 1 – Std Dev of asset 1 σ 2 – Std Dev of asset 2

14 Possible Correlation Coefficients ρ 12 has to be between -1 and 1  +1 implies that the assets are perfectly positively correlated One goes up 10% the other goes up 10%  0 implies that the assets are not related One goes up 10% the other does nothing  -1 implies that the assets are perfectly negatively correlated One goes up 10% the other goes down 10%

15 Comparing Strength When determining the strength of a correlation all we care about is the absolute value of the correlation coefficient If ρ 13 is -0.8, and ρ 23 is 0.5, which asset is more correlated with 3? Series 1

16 Correlation Coefficient Example σ 13 is -1,000; σ 23 is 500 σ 1 is 10; σ 2 is 1,000; σ 3 is 250 Is 1 or 2 more strongly correlated with 3? ρ 13 = σ 13 / (σ 1 * σ 3 ) = -1000/(10*250) = -0.4 ρ 23 = σ 23 / (σ 2 * σ 3 ) = 500/(1000*250) = Since |ρ 13 | is greater than |ρ 23 |, the relationship between 1 and 3 is stronger

17 Portfolio Risk So far we’ve been examining the risk of individual assets, but what happens when we combine individual assets into a portfolio? DIVERSIFICATION

18 Portfolio Illustration

19 Diversification Reduces risk by combining assets that are unlikely to all move in the same direction, without sacrificing expected return Intuition: “Don’t put all your eggs in one basket”  Stocks don’t move in exactly the same way. On a given day, while Boeing may yield positive 1% return, Microsoft may have gone down 1%. So, if you had invested a dollar each in Boeing and Microsoft, you would not have lost any money.

Historical Non-Diversifiers People on record for saying: “Put all your eggs in one basket and watch it closely”  Mark Twain: Died penniless  Andrew Carnegie: known as “The Richest Man in the World” 20

21 Two Types of Risk Diversifiable/Unsystematic/Unique risk  Diversifiable risk affects individual or small groups of firms (industries) Ex: Lawsuits, Strikes Non-Diversifiable/Systematic/Market risk  Affects all firms, economy wide risks Ex: Business Cycle, Inflations Shocks Which does σ measure?  Total Risk: Unsystematic plus Systematic

How Diversification Works Reduces/eliminates unsystematic risk. Why can’t we diversify away systematic risk?  It affects everything  Nowhere to hide 22

23 The Wonders of Diversifying

24 Portfolio Variances Formulas

25 Portfolio Covariance Matrix Stock 1 Stock 2 Stock N Stock 3 Stock 2 Stock 3 Stock N Var (1,1) Var (2,2) Var (3,3) Var (N,N) Cov (1,2) Cov (1,3) Cov (1,N)Cov (2,N)Cov (3,N) Cov (N,3) Cov (N,2) Cov (N,1) Cov (2,3 ) Cov (2,1)Cov (3,1) Cov (3,2)

26 Portfolio Variance Example Two stocks A and B have expected returns of 10% and 20%. In the past, A and B have had std dev of 15% and 25%, respectively, with a correlation coefficient of 0.2. You decide to invest 30% in A and the rest in B. Calculate the portfolio return and portfolio risk. Has diversification been of any use? Explain.

27 Calculations w A = 30%w B =  A = 10%  B = 20%  A = 15%  B = 25%  AB = 0.2 Portfolio Return = Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

28 Calculations w A = 30%w B = 70%  A = 10%  B = 20%  A = 15%  B = 25%  AB = 0.2 Portfolio Return = Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

29 Calculations w A = 30%w B = 70%  A = 10%  B = 20%  A = 15%  B = 25%  AB = 0.2 Portfolio Return = 0.3* *0.20 = 17% Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

30 Calculations w A = 30%w B = 70%  A = 10%  B = 20%  A = 15%  B = 25%  AB = 0.2 Portfolio Return = 17% Portfolio Variance = * * *0.3*0.7*0.2*0.15*0.25= 358 % 2 Portfolio Standard Deviation = Weighted Average Standard Deviation

31 Calculations w A = 30%w B = 70%  A = 10%  B = 20%  A = 15%  B = 25%  AB = 0.2 Portfolio Return = 17% Portfolio Variance =358% 2 Portfolio Standard Deviation = 18.9% Weighted Average Standard Deviation

32 Calculations w A = 30%w B = 70%  A = 10%  B = 20%  A = 15%  B = 25%  AB = 0.2 Portfolio Return = 17% Portfolio Variance =358% 2 Portfolio Standard Deviation = 18.9% Weighted Average Standard Deviation 0.3* *0.25= 22%

33 Remarks on Diversification Diversification reduces the  p from 22% to 18.9% What happens if  AB = 1?  There is no diversification benefit What happens as  AB approaches -1?  Diversification increases

34 Which Stock do you Prefer? Stock A :   = 10%;  = 2% Stock B :   = 10%;  = 3% Stock C :   = 12%;  = 2% Alone C, but if we have other investments we need correlation

35 Fundamental Premise of Portfolio Theory Rational investors prefer the highest expected return at the lowest possible risk How can investors lower risk without sacrificing return? Diversification

36 Possible two asset portfolios What are the possible portfolios we can create using only stocks and bonds  The correlation coefficient is  Stock: std dev = 14.3% expected return = 11.0%  Bond: std dev = 8.2% expected return = 7.0%

37 Possible Portfolios Some portfolios are better: why? which ones? They offer a higher return for the same risk 100% stocks 100% bonds Efficient Portfolios

38 Efficient Portfolio Efficient Portfolios  Offers the highest return for a given level of risk  Offers the lowest risk for a given level of return Can these same risk and return pairing be achieved with a single stock?  NO

39 Including More Assets In the real world there are more than two assets The efficient frontier is the outer most envelope of possible portfolios, given the universe of available assets  Form every possible portfolio of the various assets and the efficient frontier are the portfolios on the edge

40 return PP Possible Portfolios Including More Assets Possible Efficient Portfolios Efficient Portfolios

41 Including a Risk Free Asset How will the ability to lend and borrow at the risk free rate affect our possible risk-return combinations?

42 Risk Free and Risky Assets Start with our risky assets Add a risk free asset Reform our possible portfolios return PP rfrf

43 Risk Free and Risky Assets Start with our risky assets Add risk free assets, and form portfolios What are the new efficient portfolios? What assets comprise the new efficient portfolios? Risk Free & an Efficient Portfolio return PP rfrf CML Capital Market Line

44 Which Efficient Portfolio? M, the MARKET PORTFOLIO  It offers the best risk return trade off SHARPE RATIO is the “price” of risk: (r i - r f ) /  i  Measure the return per unit of risk  A higher Sharpe Ratio implies that we receive more return per unit of risk

45 Why only two assets? Holding any risky asset other than M offers a lower risk adjusted return  The investor is bearing more risk than necessary  The investor is accepting too low a return R f is how we adjust for our risk tolerance

46 Investing on the CML: In Two Easy Steps 1. Find M 2. Determine your level of risk aversion  Risk aversion determines location on the CML  The more risk averse the investor the safer the portfolio Closer to the risk free

Where on the CML to invest? Why do people move along the CML?  Risk Aversion Risk averse investors hold more?  Risk Free Asset Risk tolerant investors hold more?  Market Portfolio 47 rfrf M CML

48 How do we Move Along the CML? We move along the CML by altering our holdings of the risk free asset Risk Averse investors buy T-Bills  Guaranteed return  This is similar to lending money Less Risk Averse investors short T-Bills  They borrow T-Bills, sell them, and use the proceeds to buy more of the market portfolio They must return the T-Bill plus interest at some future point  This is similar to borrowing money

49 Where are we Buying/Shorting T-Bills rfrf return  M Buy T-Bills Short T-Bills

Risk & Pricing: Aside Investor A is undiversified, while B has a diversified portfolio. Both investors want to buy a share in Facebook. Who gets the share?  Which investor has a higher discount rate? Assume: A’s discount rate is 20% and B’s is 10% & Facebook is expected to pay a constant $20 dividend  Which investor offers a higher price for the share? (Hint: The price offered is the PV of expected cash flows) B = 20/.1 = $200 A = 20/.2 = $ A

51 Risk compensation In a diversified portfolio, risk depends exclusively on the underlying securities exposure to systematic risk Since unique risk can be diversified away the market will not compensate an investor for holding it  Within the market the undiversified investor will always loss to the diversified investor

52 Measuring Systematic Risk BETA (β): is how we measure systematic risk exposure  β – measures the sensitivity of the stock return to the market return (ex S&P 500)

53 β Formulas β i =  i,m /  m 2 β i = ρ im  m  i /  m 2 β i = (ρ im  i )/  m

54 β i = (ρ im  i )/  m Interpretation  i – Measures asset ‘i’ total risk ρ im – Measures the proportion of i’s total risk that is systematic  m – Measures the total market risk  Which is??? ρ im  i – Measures the systematic risk of asset ‘i’ So β i :

55 β i = (ρ im  i )/  m Interpretation  i – Measures asset ‘i’ total risk ρ im – Measures the proportion of i’s total risk that is systematic  m – Measures the total market risk  Which is??? Systematic Risk ρ im  i – Measures the systematic risk of asset ‘i’ So β i :

56 β i = (ρ im  i )/  m Interpretation  i – Measures asset ‘i’ total risk ρ im – Measures the proportion of i’s total risk that is systematic  m – Measures the total market risk  Which is??? Systematic Risk ρ im  i – Measures the systematic risk of asset ‘i’ So β i : is the ratio of the asset’s systematic risk to the systematic risk of the market  An asset’s marginal contribution to the risk of the portfolio

57 Market β β m =  m,m /  m 2 β m =  m 2 /  m 2 β m = 1 What is the β of the risk free asset?  rf,m = 0 so β rf = 0

58 Notes on β β – tells us how sensitive a stock is to market movements  “Average Stock” has a β of 1  Stocks with β > 1 amplify market movements  Stocks with 0 < β < 1 reduce market movements  Stocks with negative β?  Move opposite the market

59 Portfolio β The weighted average of the component stocks’ β Example: You invested 40% of your money in asset A, β A is 1.5 and the balance in asset B, β B is 0.5. What is the portfolio beta? β Port = 0.4* *0.5 = 0.9

60 CAPM and β CAPM states that expected returns are proportional to an investment’s systematic risk  A stock’s expected risk premium varies in proportion to it’s β E(R i ) = R f + β i (R M - R f ) Security Market Line (SML) is the graphical representation of this relation Stock’s risk premium

CAPM Assumptions Investors all have the same expectations Investors are risk averse and utility-maximizing Investors only care about mean and variance  Expected return and risk Perfect markets  No taxes  No transactions costs  Unlimited borrowing and lending at the risk-free rate 61

62 Security Market Line What is the slope of the SML?

63 Security Market Line What is the slope of the SML? The market risk premium, r m - r f

64 Putting Stocks in the SML In equilibrium, all stocks should lie on the SML. This means that all stocks are correctly priced  A stock under the SML is: Under or Over priced  A stock above the SML is: Under or Over priced

65 Putting Stocks in the SML In equilibrium, all stocks should lie on the SML. This means that all stocks are correctly priced  A stock under the SML is: Under or Over priced  A stock above the SML is: Under or Over priced

66 Putting Stocks in the SML In equilibrium, all stocks should lie on the SML. This means that all stocks are correctly priced  A stock under the SML is: Under or Over priced  A stock above the SML is: Under or Over priced

67 Dis-Equilibrium SML A B CE(r i ) β SML A: is over-priced, people won’t want it. Demand falls so the price falls C: is under-priced, people really want it. Demand rises so the price rises B: is correctly priced

68 CAPM Formula CAPM: E(R i ) = R f + β i (R M - R f ) Market Risk Premium: (R M - R f ) Asset i’s Risk Premium: β i (R M - R f )

69 Example R f = 5% Historical average risk premium is 8.4% β = 1.5, return = β = 1, return = β = 0.5, return = β = 0, return = β = -1, return =

70 Example R f = 5% Historical average risk premium is 8.4% β = 1.5, return = *0.084 = β = 1, return = β = 0.5, return = β = 0, return = β = -1, return =

71 Example R f = 5% Historical average risk premium is 8.4% β = 1.5, return = *0.084 = β = 1, return = *0.084 = β = 0.5, return = β = 0, return = β = -1, return =

72 Example R f = 5% Historical average risk premium is 8.4% β = 1.5, return = *0.084 = β = 1, return = *0.084 = β = 0.5, return = *0.084 = β = 0, return = β = -1, return =

73 Example R f = 5% Historical average risk premium is 8.4% β = 1.5, return = *0.084 = β = 1, return = *0.084 = β = 0.5, return = *0.084 = β = 0, return = *0.084 = β = -1, return =

74 Example R f = 5% Historical average risk premium is 8.4% β = 1.5, return = *0.084 = β = 1, return = *0.084 = β = 0.5, return = *0.084 = β = 0, return = *0.084 = β = -1, return = *0.084 =

75 Example 1 The stock market moves up by 10%. Assume stock A has a beta of 1.5, stock B has a beta of 0.5 and stock C has a beta of Predict A, B, and C’s response to the market? A: 15% B: 5% C: -5%

76 Example 2 β A is 1.5, and  A is 20% β B is 2, and  B is 15%; Which stock has a higher expected return? Stock B

Why We Care Basic investment rule  Big rewards are accompanied by large risks Explains the risk return trade off 77